A-Level Mathematics Exam Questions

38 Question 12 continued _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________

39 Turn over Question 14 continued _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________

page 07 MARKS 15. (a) Solve the equation sin cos + = x x 2 ° 6 ° 0 for ≤ < x 0 360. (b) Hence solve sin cos + = x x 4 ° 6 2 ° 0 for ≤ < x 0 360. 16. The point P has coordinates (4,k). C is the centre of the circle with equation ( ) ( ) − + + = x y 2 2 1 2 25. (a) Show that the distance between the points P and C is given by + + k k 2 4 13. (b) Hence, or otherwise, find the range of values of k such that P lies outside the circle. 17. (a) Express ( ) sin cos − x x 2 in the form sin + p q rx where p, q and r are integers. (b) Hence, find ( ) sin cos . − ∫ x x dx 2 [END OF QUESTION PAPER] 4 1 2 4 3 2 page 08 [BLANK PAGE] DO NOT WRITE ON THIS PAGE Write your answers clearly in the spaces provided in the answer booklet. The size of the space provided for an answer is not an indication of how much to write. You do not need to use all the space. Additional space for answers is provided at the end of the answer booklet. If you use this space you must clearly identify the question number you are attempting. Use blue or black ink. Before leaving the examination room you must give your answer booklet to the Invigilator; if you do not, you may lose all the marks for this paper. FOR OFFICIAL USE X847/76/01 Mathematics Paper 1 (Non-calculator) Answer booklet Fill in these boxes and read what is printed below. Full name of centre Town Forename(s) Surname Number of seat Day Month Year Scottish candidate number © Mark Date of birth National Qualications 2019 THURSDAY, 2 MAY 9:00 AM – 10:30 AM A/PB H page 02 DO NOT WRITE IN THIS MARGIN QUESTION NUMBER 1. 2. page 03 DO NOT WRITE IN THIS MARGIN QUESTION NUMBER 3. 4.(a) 4.(b) page 04 DO NOT WRITE IN THIS MARGIN QUESTION NUMBER 5.(a) 6. 5.(b) page 05 DO NOT WRITE IN THIS MARGIN QUESTION NUMBER 7. 8.(b) 8.(a) page 06 DO NOT WRITE IN THIS MARGIN QUESTION NUMBER 9.(a) (i) 9.(a) (ii) 9.(b) page 07 DO NOT WRITE IN THIS MARGIN QUESTION NUMBER 10.(a) 10.(b) 11. page 08 DO NOT WRITE IN THIS MARGIN QUESTION NUMBER 12.(a) 12.(b) page 09 DO NOT WRITE IN THIS MARGIN QUESTION NUMBER 13.(a) (i) 13.(a) (ii) 13.(b) page 10 DO NOT WRITE IN THIS MARGIN QUESTION NUMBER 14.(a) 14.(b) page 11 DO NOT WRITE IN THIS MARGIN QUESTION NUMBER 15.(a) 15.(b) page 12 DO NOT WRITE IN THIS MARGIN QUESTION NUMBER 16.(a) 16.(b) page 13 17.(a) 17.(b) DO NOT WRITE IN THIS MARGIN QUESTION NUMBER page 14 DO NOT WRITE IN THIS MARGIN QUESTION NUMBER ADDITIONAL SPACE FOR ANSWERS page 15 ADDITIONAL SPACE FOR ANSWERS DO NOT WRITE IN THIS MARGIN QUESTION NUMBER page 16 For marker’s use Question no Marks/Grades © National Qualications 2019 H Total marks — 80 Attempt ALL questions. You may use a calculator. To earn full marks you must show your working in your answers. State the units for your answer where appropriate. You will not earn marks for answers obtained by readings from scale drawings. Write your answers clearly in the spaces provided in the answer booklet. The size of the space provided for an answer is not an indication of how much to write. You do not need to use all the space. Additional space for answers is provided at the end of the answer booklet. If you use this space you must clearly identify the question number you are attempting. Use blue or black ink. Before leaving the examination room you must give your answer booklet to the Invigilator; if you do not, you may lose all the marks for this paper. X847/76/12 Mathematics Paper 2 THURSDAY, 2 MAY 11:00 AM – 12:45 PM A/PB page 02 FORMULAE LIST Circle The equation 2 2 2 2 0 x y gx fy c + + + + = represents a circle centre (−g, −f ) and radius 2 2 g f c + − . The equation ( ) ( ) 2 2 2 x a y b r − + − = represents a circle centre (a, b) and radius r. Scalar product cos , where is the angle between and θ θ = a.b a b a b or 1 1 1 1 2 2 3 3 2 2 3 3 where and . a b a b a b a b a b a b ⎛ ⎞ ⎛ ⎞ ⎜ ⎟ ⎜ ⎟ = + + = = ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎝ ⎠ ⎝ ⎠ a.b a b Trigonometric formulae sin (A B) sin A cos B cos A sin B ± = ± cos (A B) cos A cos B sin A sin B ± = ∓ sin A sin AcosA 2 2 = cos A cos A sin A 2 2 2 = − cos A 2 2 1 = − sin A 2 1 2 = − Table of standard derivatives ( ) f x ( ) f x ′ sin ax cosax cos a ax sin a ax − Table of standard integrals ( ) f x ( ) f x dx ∫ sin ax cosax cos 1 ax c a − + sin 1 ax c a +

40 Question 14 continued _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ (Total for Question 14 is 9 marks) TOTAL FOR PAPER IS 100 MARKS

Turn over 35 Question 14 continued _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________

page 03 MARKS Attempt ALL questions Total marks — 80 1. Triangle ABC has vertices A(-5,-12), B(11,-8) and C(-3,6). E D O A B C y x (a) Find the equation of the median BD. (b) Find the equation of the altitude AE. (c) Find the coordinates of the point of intersection of BD and AE. 2. Find ( ) 3 6 4 5 x x dx − − + ∫ . [Turn over 3 3 2 4

12 4. (a) Use de Moivre’s theorem to prove that sin 7θ = 7 sin θ − 56 sin3 θ + 112 sin5 θ − 64 sin7 θ (5) (b) Hence find the distinct roots of the equation 1 + 7x − 56x3 + 112x5 − 64x7 = 0 (5) giving your answer to 3 decimal places where appropriate. _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________

13 Turn over Question 4 continued _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________

14 Question 4 continued _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________

15 Turn over Question 4 continued _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ (Total for Question 4 is 10 marks)

14 5. The ellipse E has equation x y 2 2 36 16 1 + = The points S and S ′ are the foci of E. (a) Find the coordinates of S and S ′ (3) (b) Show that for any point P on E, the triangle PSS ′ has constant perimeter and determine its value. (4) _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________

15 Turn over Question 5 continued _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ (Total for Question 5 is 7 marks)

9 P62678A Turn over 7. A maximisation linear programming problem in x, y and z is to be solved using the two-stage simplex method. The partially completed initial tableau is shown below. Basic variable x y z s1 s2 s3 a1 a2 Value s1 1 2 3 1 0 0 0 0 45 a1 3 2 0 0 –1 0 1 0 9 a2 –1 0 4 0 0 –1 0 1 4 P –2 –1 –3 0 0 0 0 0 0 A (a) Using the information in the above tableau, formulate the linear programming problem. State the objective and list the constraints as inequalities. (4) (b) Complete the bottom row of Table 1 in the answer book. You should make your method and working clear. (2) The following tableau is obtained after two iterations of the first stage of the two-stage simplex method. Basic variable x y z s1 s2 s3 a1 a2 Value s1 0 5 6 0 1 7 12 3 4 −7 12 −3 4 147 4 x 1 2 3 0 0 −1 3 0 1 3 0 3 z 0 1 6 1 0 −1 12 −1 4 1 12 1 4 7 4 P 0 5 6 0 0 −11 12 −3 4 11 12 3 4 45 4 A 0 0 0 0 0 0 1 1 0 (c) (i) Explain how the above tableau shows that a basic feasible solution has been found for the original linear programming problem. (ii) Write down the basic feasible solution for the second stage. (3) (d) Taking the most negative number in the profit row to indicate the pivot column, perform one complete iteration of the second stage of the two-stage simplex method, to obtain a new tableau, T. Make your method clear by stating the row operations you use. (5) 10 P62678A (e) (i) Explain, using T, whether or not an optimal solution to the original linear programming problem has been found. (ii) Write down the value of the objective function. (iii) State the values of the basic variables. (3) (Total for Question 7 is 17 marks) TOTAL FOR PAPER IS 75 MARKS 11 P62678A BLANK PAGE 12 P62678A BLANK PAGE Please check the examination details below before entering your candidate information Candidate surname Other names Total Marks Centre Number Candidate Number Pearson Edexcel Level 3 GCE P62678A ©2020 Pearson Education Ltd. 1/1/1/1/1/1/1/ Turn over Answer book Do not return the question paper with the answer book. Paper Reference 9FM0/3D Afternoon (Time: 1 hour 30 minutes) Wednesday 17 June 2020 Turn over Further Mathematics Advanced Paper 3D: Decision Mathematics 1 2 DO NOT WRITE IN THIS AREA DO NOT WRITE IN THIS AREA DO NOT WRITE IN THIS AREA 1. A B C D E F G A – 24 – 22 35 – – B 24 – 25 27 – – – C – 25 – 33 31 36 26 D 22 27 33 – – 42 – E 35 – 31 – – 37 29 F – – 36 42 37 – 40 G – – 26 – 29 40 – C G E D B A F 24 25 35 29 26 31 40 37 36 Diagram 1 _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ 3 Turn over DO NOT WRITE IN THIS AREA DO NOT WRITE IN THIS AREA DO NOT WRITE IN THIS AREA Question 1 continued _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ (Total for Question 1 is 6 marks) 4 DO NOT WRITE IN THIS AREA DO NOT WRITE IN THIS AREA DO NOT WRITE IN THIS AREA 2. (a) _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ (b) Activity Immediately preceding activities Activity Immediately preceding activities Activity Immediately preceding activities A E I B F J C G K D H L Key: Early event time Late event time A(6) B(7) E(5) I(4) J(6) G(4) F(3) H(3) D(4) C(5) K(5) L(6) Diagram 1 5 Turn over DO NOT WRITE IN THIS AREA DO NOT WRITE IN THIS AREA DO NOT WRITE IN THIS AREA Question 2 continued _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ time (hours) O 6 5 4 3 2 1 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 number of workers Grid 1 _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ (Total for Question 2 is 15 marks) 6 DO NOT WRITE IN THIS AREA DO NOT WRITE IN THIS AREA DO NOT WRITE IN THIS AREA 3. (a) Initial time matrix Initial route matrix A B C D E A B C D E A B C D E A B C D E (b) Time matrix Route matrix A B C D E A B C D E A B C D E A B C D E Time matrix Route matrix A B C D E A B C D E A B C D E A B C D E If you make an error there are spare copies of these matrices on Page 9. 7 Turn over DO NOT WRITE IN THIS AREA DO NOT WRITE IN THIS AREA DO NOT WRITE IN THIS AREA Question 3 continued _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ 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_____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ 9 Turn over DO NOT WRITE IN THIS AREA DO NOT WRITE IN THIS AREA DO NOT WRITE IN THIS AREA Question 3 continued Only use these matrices if you have made an error in your working. (a) Spare copy of initial time matrix Spare copy of initial route matrix A B C D E A B C D E A B C D E A B C D E (b) Spare copy of time matrix Spare copy of route matrix A B C D E A B C D E A B C D E A B C D E Spare copy of time matrix Spare copy of route matrix A B C D E A B C D E A B C D E A B C D E (Total for Question 3 is 9 marks) 10 DO NOT WRITE IN THIS AREA DO NOT WRITE IN THIS AREA DO NOT WRITE IN THIS AREA 4. y O 7 – 6 – 5 – 4 – 3 – 2 – 1 – 1 2 3 4 5 x 2y = 5x 6x + 5y = 30 R y = x + 1 – – – – – – – Figure 3 _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ 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_____________________________________________________________________________________ 11 Turn over DO NOT WRITE IN THIS AREA DO NOT WRITE IN THIS AREA DO NOT WRITE IN THIS AREA Question 4 continued _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ 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_____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ 12 DO NOT WRITE IN THIS AREA DO NOT WRITE IN THIS AREA DO NOT WRITE IN THIS AREA Question 4 continued _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ 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_____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ 13 Turn over DO NOT WRITE IN THIS AREA DO NOT WRITE IN THIS AREA DO NOT WRITE IN THIS AREA Question 4 continued _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ 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_____________________________________________________________________________________ _____________________________________________________________________________________ 15 Turn over DO NOT WRITE IN THIS AREA DO NOT WRITE IN THIS AREA DO NOT WRITE IN THIS AREA Question 5 continued _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ 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_____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ (Total for Question 5 is 7 marks) 16 DO NOT WRITE IN THIS AREA DO NOT WRITE IN THIS AREA DO NOT WRITE IN THIS AREA 6. (a) ________________________________________________________________________________________________________________________ ________________________________________________________________________________________________________________________ ________________________________________________________________________________________________________________________ Vertex Order of labelling Final value Working values Key: D F E B C A G 24 y 42 15 65 21 34 31 24 x 19 45 17 Turn over DO NOT WRITE IN THIS AREA DO NOT WRITE IN THIS AREA DO NOT WRITE IN THIS AREA Question 6 continued _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ 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_____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ Basic variable x y z s1 s2 s3 a1 a2 Value s1 1 2 3 1 0 0 0 0 45 a1 3 2 0 0 –1 0 1 0 9 a2 –1 0 4 0 0 –1 0 1 4 P –2 –1 –3 0 0 0 0 0 0 A Table 1 _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ 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_____________________________________________________________________________________ b.v. x y z s1 s2 s3 Value s1 0 5 6 0 1 7 12 3 4 147 4 x 1 2 3 0 0 −1 3 0 3 z 0 1 6 1 0 −1 12 −1 4 7 4 P 0 5 6 0 0 −11 12 −3 4 45 4 b.v. x y z s1 s2 s3 Value Row Ops P Spare copy b.v. x y z s1 s2 s3 Value Row Ops P _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ 20 DO NOT WRITE IN THIS AREA DO NOT WRITE IN THIS AREA DO NOT WRITE IN THIS AREA Question 7 continued _____________________________________________________________________________________ _____________________________________________________________________________________ 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Turn over 11 Question 5 continued _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ (Total for Question 5 is 4 marks)

5 Turn over Question 2 continued _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ (Total for Question 2 is 8 marks) 6 3. y O x R Figure 1 Figure 1 shows a circle with radius r and centre at the origin. The region R, shown shaded in Figure 1, is bounded by the x-axis and the part of the circle for which y > 0 The region R is rotated through 360° about the x-axis to create a sphere with volume V Use integration to show that 3 4 3 V πr = (5) _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________

6 Question 2 continued _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________

27 Turn over Question 10 continued _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________

28 Question 10 continued _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ (Total for Question 10 is 7 marks) TOTAL FOR CORE PURE MATHEMATICS IS 80 MARKS

4 2. By taking logarithms of both sides, solve the equation 43 p −1 = 5210 giving the value of p to one decimal place. (3) _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________

2 1. An estate agent in Tornep believes that houses further from the railway station are more expensive than those that are closer. She took a random sample of 22 three‑bedroom houses in Tornep and calculated the product moment correlation coefficient between the house price and the distance from the station to be 0.3892 Stating your hypotheses clearly, use a 5% level of significance to test the estate agent’s belief. State the critical region used in your test. (3) _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________

3 Turn over Question 1 continued _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ (Total for Question 1 is 3 marks)

Turn over 5 Question 2 continued _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ (Total for Question 2 is 3 marks)

Turn over 7 Question 2 continued _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ (Total for Question 2 is 7 marks)

36 Question 14 continued _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________

24 8. (a) Prove by induction that, for all positive integers n, r n =∑ 1 r(r + 1)(2r + 1) = 1 2 n(n + 1)2(n + 2) (6) (b) Hence, show that, for all positive integers n, r n n =∑ 2 r(r + 1)(2r + 1) = 1 2 n(n + 1)(an + b)(cn + d) where a, b, c and d are integers to be determined. (3) _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________

6 3. Relative to a fixed origin O ● point A has position vector 2i + 5j − 6k ● point B has position vector 3i − 3j − 4k ● point C has position vector 2i − 16j + 4k (a) Find AB → (2) (b) Show that quadrilateral OABC is a trapezium, giving reasons for your answer. (2) _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________

4 2. [In this question the unit vectors i and j are due east and due north respectively.] A coastguard station O monitors the movements of a small boat. At 10:00 the boat is at the point (4i – 2j) km relative to O. At 12:45 the boat is at the point (–3i – 5j) km relative to O. The motion of the boat is modelled as that of a particle moving in a straight line at constant speed. (a) Calculate the bearing on which the boat is moving, giving your answer in degrees to one decimal place. (3) (b) Calculate the speed of the boat, giving your answer in km h–1 (3) _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________

5 Turn over Question 2 continued _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ (Total for Question 2 is 6 marks) 6 3. In this question you must show all stages of your working. Solutions relying on calculator technology are not acceptable. (i) Solve the equation x 2 18 − = x writing the answer as a surd in simplest form. (3) (ii) Solve the equation 4 3x – 2 = 1 2 2 (3) _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________

7 Turn over Question 3 continued _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ (Total for Question 3 is 6 marks)

8 4. In 1997 the average CO2 emissions of new cars in the UK was 190 g/km. In 2005 the average CO2 emissions of new cars in the UK had fallen to 169 g/km. Given A g/km is the average CO2 emissions of new cars in the UK n years after 1997 and using a linear model, (a) form an equation linking A with n. (3) In 2016 the average CO2 emissions of new cars in the UK was 120 g/km. (b) Comment on the suitability of your model in light of this information. (3) _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________

Turn over 7 Question 3 continued _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ (Total for Question 3 is 4 marks)

Turn over 25 Question 8 continued _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________

8 4. The function f is defined by f (x) = 3 7 2 x x − − x  , x ≠ 2 (a) Find f −1(7) (2) (b) Show that ff (x) = ax b x + −3 where a and b are integers to be found. (3) _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________

Turn over 9 Question 4 continued _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ (Total for Question 4 is 5 marks)

26 Question 8 continued _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________

Turn over 13 Question 5 continued _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________

10 5. A car has six forward gears. The fastest speed of the car ● in 1st gear is 28 km h–1 ● in 6th gear is 115 km h–1 Given that the fastest speed of the car in successive gears is modelled by an arithmetic sequence, (a) find the fastest speed of the car in 3rd gear. (3) Given that the fastest speed of the car in successive gears is modelled by a geometric sequence, (b) find the fastest speed of the car in 5th gear. (3) _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________

Turn over 11 Question 5 continued _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ (Total for Question 5 is 6 marks)

2 1. A small sports club has 12 adult members and 14 junior members. The club needs to enter a team of 8 players for a particular competition. Determine the number of ways in which the team can be selected if (i) there are no restrictions on the team, (1) (ii) the team must contain 4 adults and 4 juniors, (2) (iii) more than half the team must be adults. (3) _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________

3 Turn over Question 1 continued _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ (Total for Question 1 is 6 marks)

Turn over 7 Question 3 continued _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ (Total for Question 3 is 5 marks)

14 Question 5 continued _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________

Page 04 MARKS 6. A function, h, is defined by ( ) 3 7 h x x = + , where x∈\. Determine an expression for ( ) 1 h x − . 7. A (−3, 5), B (7, 9) and C (2, 11) are the vertices of a triangle. Find the equation of the median through C. 8. Calculate the rate of change of ( ) 1 2 d t t = , t ≠ 0, when t = 5. 9. A sequence is generated by the recurrence relation 1 6 n n u mu + = + where m is a constant. (a) Given u1 = 28 and u2 = 13, find the value of m. (b) (i) Explain why this sequence approaches a limit as n →∞. (ii) Calculate this limit. 3 3 3 2 1 2

4 2. Solve the recurrence system u1 = 1 u2 = 4 9un + 2 – 12un + 1 + 4un = 3n (9) _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________

5 Turn over Question 2 continued _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________

6 Question 2 continued _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________

Turn over 15 Question 5 continued _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ (Total for Question 5 is 8 marks)

32 Question 8 continued _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ (Total for Question 8 is 12 marks) TOTAL FOR PAPER IS 75 MARKS

5 P62682RA Turn over 4. The complementary function for the second order recurrence relation un + 2 + αun + 1 + βun = 20 (−3) n    n  0 is given by un = A (2) n + B (−1) n where A and B are arbitrary non-zero constants. (a) Find the value of α and the value of β. (2) Given that 2u0 = u1 and u4 = 164 (b) find the solution of this second order recurrence relation to obtain an expression for un in terms of n. (6) (Total for Question 4 is 8 marks) 6 P62682RA 5. C1 C1 A D C B E F G H J (5, 8) (5, 7) (6, 10) (1, 2) (0, 2) (4, 6) (4, 8) (1, 6) (1, 1) (0, 4) (2, 3) (3, 6) (4, 6) (3, 7) (4, 5) (2, 5) (5, 7) Figure 2 Figure 2 shows a capacitated, directed network. The network represents a system of pipes through which fluid can flow. The weights on the arcs show the lower capacities and upper capacities for the corresponding pipes, in litres per second. (a) State the source node. (1) (b) Explain why the sink node must be G. (1) (c) Calculate the capacity of the cut C1 (1) (d) Assuming that a feasible flow exists, (i) explain why arc JH must be at its upper capacity, (ii) explain why arcs AD and CD must be at their lower capacities. (2) (e) Use Diagram 1 in the answer book to show a flow of 18 litres per second through the system. (2) (f) Prove that the answer to (e) is the maximum flow through the system. (3) (Total for Question 5 is 10 marks) 7 P62682RA Turn over 6. Player B Option X Option Y Option Z Player A Option Q 1 5 3 Option R 4 −3 1 Option S 2 −4 −2 Option T 3 −2 0 A two person zero-sum game is represented by the pay-off matrix for player A, shown above. (a) Explain, with justification, why this matrix may be reduced to a 3 × 3 matrix by removing option S from player A’s choices. (2) (b) Verify that there is no stable solution to the reduced game. (3) Player A intends to make a random choice between options Q, R and T, choosing option Q with probability p1, option R with probability p2 and option T with probability p3 Player A wants to find the optimal values of p1, p2 and p3 using the Simplex algorithm. Player A formulates the following linear programme, writing the constraints as inequalities. Maximise P = V, where V = the value of original game + 3 subject to V  4p1 + 7p2 + 6p3 V  8p1 + p3 V  6p1 + 4p2 + 3p3 p1 + p2 + p3  1 p1  0, p2  0, p3  0, V  0 (c) Explain why V cannot exceed any of the following expressions 4p1 + 7p2 + 6p3            8p1 + p3            6p1 + 4p2 + 3p3 (1) (d) Explain why it is necessary to use the constraint p1 + p2 + p3  1 (1) The Simplex algorithm is used to solve the linear programming problem. Given that the optimal value of p1 = 7 11 and the optimal value of p3 = 0 (e) calculate the value of the game to player A. (3) Player B intends to make a random choice between options X, Y and Z, choosing option X with probability q1, option Y with probability q2 and option Z with probability q3 (f) Determine the optimal strategy for player B, making your working clear. (4) (Total for Question 6 is 14 marks)

8 P62682RA 7. A manufacturer can export five batches of footwear each year. Each exported batch contains just one type of footwear. The types of footwear are trainers, sandals or high heels. The table below shows the profit, in £1000s, for the number of batches of each type of footwear. Number of batches 0 1 2 3 4 5 Trainers 0 50 90 170 225 295 Sandals 0 70 110 165 245 300 High heels 0 75 115 x 235 305 The total annual profit is to be maximised. (a) Use dynamic programming to determine the two possible values of the maximum total annual profit, giving one of these values in terms of x. (10) Given that the maximum total annual profit is £320 000 (b) advise the manufacturer on the possible ways in which the five batches should be allocated. (2) (Total for Question 7 is 12 marks) TOTAL FOR PAPER IS 75 MARKS Please check the examination details below before entering your candidate information Candidate surname Other names Total Marks Centre Number Candidate Number Pearson Edexcel Level 3 GCE Answer Book Do not return the question paper with the answer book Further Mathematics Advanced Paper 4D: Decision Mathematics 2 Paper Reference 9FM0/4D Tuesday 23 June 2020 Afternoon (Time: 1 hour 30 minutes) P62682RA ©2020 Pearson Education Ltd. 1/1/1/1/1/1/ Turn over 2 DO NOT WRITE IN THIS AREA DO NOT WRITE IN THIS AREA DO NOT WRITE IN THIS AREA 1. 1 2 3 4 A 29 20 – 23 B 32 30 28 – C 35 32 34 25 D 29 31 27 30 1 2 3 4 A B C D 1 2 3 4 A B C D 1 2 3 4 A B C D 1 2 3 4 A B C D ____________________________________________________________________________ ____________________________________________________________________________ ____________________________________________________________________________ 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B C D 1 2 3 4 A B C D (Total for Question 1 is 8 marks) ____________________________________________________________________________ ____________________________________________________________________________ ____________________________________________________________________________ ____________________________________________________________________________ ____________________________________________________________________________ ____________________________________________________________________________ ____________________________________________________________________________ ____________________________________________________________________________ ____________________________________________________________________________ ____________________________________________________________________________ ____________________________________________________________________________ ____________________________________________________________________________ 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A C B 0.6 0.4 0.75 0.25 0.8 0.2 350 −140 260 −190 220 −230 Figure 1 _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ 5 Turn over DO NOT WRITE IN THIS AREA DO NOT WRITE IN THIS AREA DO NOT WRITE IN THIS AREA Question 2 continued A C B 0.6 0.4 0.75 0.25 0.8 0.2 350 −140 260 −190 220 −230 Figure 1 _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ (Total for Question 2 is 7 marks) 6 DO NOT WRITE IN THIS AREA DO NOT WRITE IN THIS AREA DO NOT WRITE IN THIS AREA 3. P Q R Supply A 25 24 17 42 B 7 12 14 68 C 13 11 20 25 D 16 15 13 40 Demand 59 72 44 Table 1 P Q R A B C D P Q R A B C D P Q R A B C D P Q R A B C D __________________________________________________________________________ __________________________________________________________________________ __________________________________________________________________________ __________________________________________________________________________ __________________________________________________________________________ __________________________________________________________________________ __________________________________________________________________________ __________________________________________________________________________ __________________________________________________________________________ __________________________________________________________________________ __________________________________________________________________________ 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__________________________________________________________________________ __________________________________________________________________________ __________________________________________________________________________ 7 Turn over DO NOT WRITE IN THIS AREA DO NOT WRITE IN THIS AREA DO NOT WRITE IN THIS AREA Question 3 continued P Q R A B C D P Q R A B C D P Q R A B C D P Q R A B C D P Q R A B C D __________________________________________________________________________ __________________________________________________________________________ __________________________________________________________________________ __________________________________________________________________________ __________________________________________________________________________ __________________________________________________________________________ __________________________________________________________________________ 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_____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ (Total for Question 6 is 14 marks) 18 DO NOT WRITE IN THIS AREA DO NOT WRITE IN THIS AREA DO NOT WRITE IN THIS AREA 7. Stage State Action Destination Value Trainers 0 0 0 0 19 Turn over DO NOT WRITE IN THIS AREA DO NOT WRITE IN THIS AREA DO NOT WRITE IN THIS AREA Question 7 continued Stage State Action Destination Value _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ 20 DO NOT WRITE IN THIS AREA DO NOT WRITE IN THIS AREA DO NOT WRITE IN THIS AREA Question 7 continued _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ (Total for Question 7 is 12 marks) TOTAL FOR PAPER IS 75 MARKS

9 Turn over Question 4 continued _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ (Total for Question 4 is 6 marks) 10 5. Not to scale A B C D 27° Figure 1 Figure 1 shows the design for a structure used to support a roof. The structure consists of four steel beams, AB, BD, BC and AD. Given AB = 12 m, BC = BD = 7m and angle BAC = 27° (a) find, to one decimal place, the size of angle ACB. (3) The steel beams can only be bought in whole metre lengths. (b) Find the minimum length of steel that needs to be bought to make the complete structure. 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11 Turn over Question 5 continued _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ (Total for Question 5 is 6 marks)

20 Question 5 continued _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ (Total for Question 5 is 10 marks) TOTAL FOR FURTHER PURE MATHEMATICS 1 IS 40 MARKS

MARKS Page three Attempt ALL questions Total marks – 60 1. Vectors u = 8i + 2j − k and v = −3i + tj − 6k are perpendicular. Determine the value of t. 2. Find the equation of the tangent to the curve y = 2x3 + 3 at the point where x = −2. 3. Show that (x + 3) is a factor of x3 − 3x2 − 10x + 24 and hence factorise x3 − 3x2 – 10x + 24 fully. 4. The diagram shows part of the graph of the function y = p cos qx + r. y 4 x −2 π 2 Write down the values of p, q and r. 5. A function g is defined on , the set of real numbers, by g(x) = 6 − 2x. (a) Determine an expression for g−1(x). (b) Write down an expression for g(g−1(x)). 6. Evaluate log612 + 1 3 log627. 7. A function f is defined on a suitable domain by f (x) = 2 3 – x x x x ⎛ ⎞⎟ ⎜ ⎟ ⎜ ⎟⎟ ⎜⎝ ⎠ . Find f ′(4). [Turn over 2 4 4 3 2 1 3 4 O

12 6. (a) Find the first 4 terms, in ascending powers of x, in the binomial expansion of (1 + kx)10 where k is a non-zero constant. Write each coefficient as simply as possible. (3) Given that in the expansion of (1 + kx)10 the coefficient x3 is 3 times the coefficient of x, (b) find the possible values of k. (3) _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________

MARKS Page four 8. ABCD is a rectangle with sides of lengths x centimetres and (x − 2) centimetres, as shown. If the area of ABCD is less than 15 cm2, determine the range of possible values of x. 9. A, B and C are points such that AB is parallel to the line with equation y + 3 = 0 x and BC makes an angle of 150º with the positive direction of the x-axis. Are the points A, B and C collinear? 10. Given that 3 π tan2 = , 0 < < 4 4 x x , find the exact value of (a) cos 2x (b) cos x. 11. T(−2, −5) lies on the circumference of the circle with equation (x + 8)2 + (y + 2)2 = 45. (a) Find the equation of the tangent to the circle passing through T. (b) This tangent is also a tangent to a parabola with equation y = −2x2 + px + 1 − p, where p > 3. Determine the value of p. 4 3 1 2 4 6 x cm A D C B (x – 2) cm

MARKS Page five 12. The diagram shows part of the graph of y = a cos bx. The shaded area is 1 2 unit2. What is the value of ∫ (acosbx)dx? 13. The function f (x) = 2x + 3 is defined on ,the set of real numbers. The graph with equation y = f (x) passes through the point P(1, b) and cuts the y-axis at Q as shown in the diagram. f (x) = 2x + 3 P(1, b) Q y x (a) What is the value of b? (b) (i) Copy the above diagram. On the same diagram, sketch the graph with equation y = f  –1(x). (ii) Write down the coordinates of the images of P and Q. (c) R (3,11) also lies on the graph with equation y = f (x). Find the coordinates of the image of R on the graph with equation y = 4 − f (x + 1). 2 1 1 3 2 y x π 4 3π 4 O O [Turn over 3π 4 0

MARKS Page six 14. The circle with equation x2 + y2 − 12x − 10y + k = 0 meets the coordinate axes at exactly three points. What is the value of k? 15. The rate of change of the temperature, T ºC of a mug of coffee is given by 1 25 = – dT t k dt ,  0 ≤ t ≤ 50 • t is the elapsed time, in minutes, after the coffee is poured into the mug • k is a constant • initially, the temperature of the coffee is 100 ºC • 10 minutes later the temperature has fallen to 82 ºC. Express T in terms of t. [END OF QUESTION PAPER] 2 6 [BLANK PAGE] DO NOT WRITE ON THIS PAGE Page seven [BLANK PAGE] DO NOT WRITE ON THIS PAGE Page eight H © National Qualications 2015 Total marks — 70 Attempt ALL questions. You may use a calculator Full credit will be given only to solutions which contain appropriate working. State the units for your answer where appropriate. Answers obtained by readings from scale drawings will not receive any credit. Write your answers clearly in the spaces in the answer booklet provided. Additional space for answers is provided at the end of the answer booklet. If you use this space you must clearly identify the question number you are attempting. Use blue or black ink. Before leaving the examination room you must give your answer booklet to the Invigilator; if you do not, you may lose all the marks for this paper. X747/76/12 Mathematics Paper 2 WEDNESDAY, 20 MAY 10:30 AM – 12:00 NOON HTP Page two FORMULAE LIST Circle: The equation x2 + y2 + 2gx + 2fy + c = 0 represents a circle centre (−g, −f ) and radius The equation (x − a)2 + (y − b)2 = r2 represents a circle centre (a, b) and radius r. Scalar Product: a.b = |a||b| cos θ, where θ is the angle between a and b or a.b = a1b1 + a2b2 + a3b3 where a = 1 1 2 2 3 3 = and a b a b a b b ⎛ ⎞ ⎛ ⎞ ⎟ ⎟ ⎜ ⎜ ⎟ ⎟ ⎜ ⎜ ⎟ ⎟ ⎜ ⎜ ⎟ ⎟ ⎜ ⎜ ⎟ ⎟ ⎜ ⎜ ⎟ ⎟ ⎜ ⎜ ⎟ ⎟ ⎟ ⎟ ⎜ ⎜ ⎝ ⎠ ⎝ ⎠ . Trigonometric formulae: sin (A ± B) = sin A cos B ± cos A sin B cos (A ± B) = cos A cos B ± sin A sin B sin 2A = 2 sin A cos A cos 2A = cos2 A − sin2 A = 2 cos2 A − 1 = 1 − 2 sin2 A Table of standard derivatives: Table of standard integrals: f (x) f ′(x) sin ax cos ax a cos ax –a sin ax f (x) ∫ f (x)dx sin ax cos ax cos ax + c sin ax + c 1 – a 1 a 2 2 + – . g f c

MARKS Page three Attempt ALL questions Total marks – 70 1. The vertices of triangle ABC are A(−5, 7), B(−1, −5) and C(13, 3) as shown in the diagram. The broken line represents the altitude from C. (a) Show that the equation of the altitude from C is x – 3y = 4. (b) Find the equation of the median from B. (c) Find the coordinates of the point of intersection of the altitude from C and the median from B. 2. Functions f and g are defined on suitable domains by f(x) = 10 + x and g(x) = (1 + x) (3 – x) + 2. (a) Find an expression for f ( g (x)). (b) Express f (g(x)) in the form p(x + q)2 + r. (c) Another function h is given by ( ) ( ) ( ) 1 = h x f g x . What values of x cannot be in the domain of h ? [Turn over 4 3 2 2 3 2 y x A C B

MARKS Page four 3. A version of the following problem first appeared in print in the 16th Century. A frog and a toad fall to the bottom of a well that is 50 feet deep. Each day, the frog climbs 32 feet and then rests overnight. During the night, it slides down 2 3 of its height above the floor of the well. The toad climbs 13 feet each day before resting. Overnight, it slides down 1 4 of its height above the floor of the well. Their progress can be modelled by the recurrence relations: • +1 1 = +32 3 , n n f f f1 = 32 • +1 3 = +13 4 , n n t t t1 = 13 where fn and tn are the heights reached by the frog and the toad at the end of the nth day after falling in. (a) Calculate t2, the height of the toad at the end of the second day. (b) Determine whether or not either of them will eventually escape from the well. 1 5

MARKS Page five 4. A wall plaque is to be made to commemorate the 150th anniversary of the publication of “Alice’s Adventures in Wonderland”. The edges of the wall plaque can be modelled by parts of the graphs of four quadratic functions as shown in the sketch. ( ) ( ) ( ) ( ) 2 2 2 2 1 1 = – +3 4 2 1 3 = – +5 4 2 3 9 = – +3 8 4 3 3 = – 8 4 f x x x g x x x h x x x k x x x i i i i (a) Find the x-coordinate of the point of intersection of the graphs with equations y = f (x) and y = g(x). The graphs of the functions f (x) and h(x) intersect on the y-axis. The plaque has a vertical line of symmetry. (b) Calculate the area of the wall plaque. [Turn over 2 7 y = f (x) y = h(x) y = g(x) y = k(x) y x O

MARKS Page six 5. Circle C1 has equation x2 + y2 + 6x + 10y + 9 = 0. The centre of circle C2 is (9, 11). Circles C1 and C2 touch externally. (a) Determine the radius of C2. A third circle, C3, is drawn such that: • both C1 and C2 touch C3 internally • the centres of C1, C2 and C3 are collinear. (b) Determine the equation of C3. 4 4 C1 C2

MARKS Page seven 6. Vectors p, q and r are represented on the diagram as shown. • BCDE is a parallelogram • ABE is an equilateral triangle • |p| = 3 • Angle ABC = 90º (a) Evaluate p . (q+r). (b) Express EC in terms of p, q and r. (c) Given that AE.EC = 9 9 3 – 2 , find |r|. [Turn over 3 1 3 D r E B C q p A

MARKS Page eight 7. (a) Find 3 2 1 cos x dx + ( ) ∫ . (b) Show that 3cos 2x + 1 = 4cos2x − 2sin2x. (c) Hence, or otherwise, find sin cos 2 2 2 x x dx − ( ) ∫ . 8. A crocodile is stalking prey located 20 metres further upstream on the opposite bank of a river. Crocodiles travel at different speeds on land and in water. The time taken for the crocodile to reach its prey can be minimised if it swims to a particular point, P, x metres upstream on the other side of the river as shown in the diagram. 20 metres x metres P The time taken, T, measured in tenths of a second, is given by ( ) ( ) 2 = 5 36 + + 4 20 – T x x x (a) (i) Calculate the time taken if the crocodile does not travel on land. (ii) Calculate the time taken if the crocodile swims the shortest distance possible. (b) Between these two extremes there is one value of x which minimises the time taken. Find this value of x and hence calculate the minimum possible time. 2 2 2 1 1 8

MARKS Page nine 9. The blades of a wind turbine are turning at a steady rate. The height, h metres, of the tip of one of the blades above the ground at time, t seconds, is given by the formula h = 36sin(1·5t) – 15cos(1·5t) + 65. Express 36sin(1·5t) – 15cos(1·5t) in the form ksin(1·5t – a), where k > 0 and 0 < a < 2 π , and hence find the two values of t for which the tip of this blade is at a height of 100 metres above the ground during the first turn. [END OF QUESTION PAPER] 8 [BLANK PAGE] do not write on this page Page ten [BLANK PAGE] do not write on this page Page eleven [BLANK PAGE] do not write on this page Page twelve

Page 04 MARKS Total marks — 100 Attempt ALL questions 1. (a) Differentiate tan . 12 y x x − = (b) Given ( ) 2 2 1 1 4 x f x x − = + , find f ′(x), simplifying your answer. (c) A curve is given by the parametric equations x = 6t  and  y = 1 − cos t. Find dy dx in terms of t. 2. A geometric sequence has second and fifth terms 108 and 4 respectively. (a) Calculate the value of the common ratio. (b) State why the associated geometric series has a sum to infinity. (c) Find the value of this sum to infinity. 3. Write down and simplify the general term in the binomial expansion of 13 3 2x x ⎛ ⎞ − ⎜ ⎟ ⎝ ⎠ . Hence, or otherwise, find the term in x9. 4. Below is a system of equations: x y z x y z x y λz + + = − + = − + = 2 3 3 2 4 5 3 2 2 Use Gaussian elimination to find the value of λ which leads to redundancy. 5. Prove by induction that ( ) ( ) 2 1 3 1 1 n r r r n n = − = + ∑ , n ∀∈` . 3 3 2 3 1 2 5 4 4

Page 05 MARKS 6. Find Maclaurin expansions for sin 3x  and  e4x up to and including the term in x3. Hence obtain an expansion for e4x sin 3x up to and including the term in x3. 7. A is the matrix 2 0 1 λ ⎛ ⎞ ⎜ ⎟ − ⎝ ⎠ . (a) Find the determinant of matrix A. (b) Show that A2 can be expressed in the form pA + qI , stating the values of p and q. (c) Obtain a similar expression for A4. 8. Let 3 z i = −. (a) Plot z on an Argand diagram. (b) Let w = az where a > 0 , a ∈ . Express w in polar form. (c) Express w8 in the form ( ) n ka x i y + where k, x, y ∈. 9. Obtain ( ) ln 2 7x x dx ∫ . 10. For each of the following statements, decide whether it is true or false. If true, give a proof; if false, give a counterexample. A. If a positive integer p is prime, then so is 2p + 1. B. If a positive integer n has remainder 1 when divided by 3, then n3 also has remainder 1 when divided by 3. 11. The height of a cube is increasing at the rate of 5 cm s-1. Find the rate of increase of the volume when the height of the cube is 3 cm. [Turn over 6 1 3 2 1 2 3 6 4 4

Page 06 MARKS 12. Below is a diagram showing the graph of a linear function, ( ) y f x = . y x c −c O On separate diagrams show: (a) ( ) y f x c = − (b) ( ) y f x = 2 13. Express ( )( ) 3 32 4 6 x x x + + − in partial fractions and hence evaluate ( )( ) x dx x x + + − ⌠ ⎮⌡ 4 3 3 32 4 6 . Give your answer in the form ln p q ⎛ ⎞ ⎜ ⎟ ⎝ ⎠ . 2 2 9

Page 07 MARKS 14. Two lines L1 and L2 are given by the equations: , , L x λ y λ z λ x y z L = + = + = − − − + = = − 1 2 4 3 2 4 7 3 8 1 1 3 2 : : (a) Show that the lines L1 and L2 intersect and find the point of intersection. (b) Calculate the obtuse angle between the lines L1 and L2. 15. Solve the differential equation 2 2 2 5 6 12 2 5 d y dy y x x dx dx + + = + − given y = -6 and 3 dy dx = , when x = 0. 16. A beaker of liquid was placed in a fridge. The rate of cooling is given by ( ), 0, F dT k T T k dt = − − > where TF is the constant temperature in the fridge and T is the temperature of the liquid at time t. • The constant temperature in the fridge is 4 °C. • When first placed in the fridge, the temperature of the liquid was 25 °C. • At 12 noon, the temperature of the liquid was 9·8 °C. • At 12:15 pm, the temperature of the liquid had dropped to 6·5 °C. At what time, to the nearest minute, was the liquid placed in the fridge? [END OF QUESTION PAPER] 5 4 10 9 Page 08 [BLANK PAGE] DO NOT WRITE ON THIS PAGE

Page 03 MARKS Attempt ALL questions Total marks – 60 1. Find the equation of the line passing through the point ( ) 2, 3 − which is parallel to the line with equation 4 7 + = y x . 2. Given that 3 12 8 = + y x x, where 0 > x , find dy dx . 3. A sequence is defined by the recurrence relation 1 10 3 + = + n n u u 1 with 3 6 = u . (a) Find the value of u4. (b) Explain why this sequence approaches a limit as →∞ n . (c) Calculate this limit. 4. A and B are the points −7, 3 ( ) and ( ) 1, 5 . AB is a diameter of a circle. y x B A O Find the equation of this circle. 2 3 1 1 2 3 [Turn over

Page 04 MARKS 5. Find ∫ ( ) 8cos 4 1 + x dx. 6. Functions f and g are defined on , the set of real numbers. The inverse functions f −1 and g−1 both exist. (a) Given ( ) 3 5 = + f x x , find ( ) 1 − f x . (b) If g ( ) 2 = 7, write down the value of g−1 ( ) 7 . 7. Three vectors can be expressed as follows: FG = −2i −6j + 3k GH = 3i + 9j −7k EH = 2i + 3j + k (a) Find FH. (b) Hence, or otherwise, find FE. 8. Show that the line with equation = 3 5 − y x is a tangent to the circle with equation 2 2 2 4 5 0 + + − − = x y x y and find the coordinates of the point of contact. 2 3 1 2 2 5

Page 05 MARKS 9. (a) Find the x-coordinates of the stationary points on the graph with equation ( ) = y f x , where 3 2 ( ) 3 24 = + − f x x x x. (b) Hence determine the range of values of x for which the function f is strictly increasing. 10. The diagram below shows the graph of the function ( ) 4 log = f x x, where 0 > x . y x O (1, 0) (4, 1) f x( )=log4 x The inverse function, f −1, exists. On the diagram in your answer booklet, sketch the graph of the inverse function. 11. (a) A and C are the points ( ) 1, 3, 2 − and ( ) 4, , 4 −3 respectively. Point B divides AC in the ratio 1 : 2. Find the coordinates of B. (b) kAC is a vector of magnitude 1, where 0 > k . Determine the value of k. 4 2 2 2 3 [Turn over

Page 06 MARKS 12. The functions f and g are defined on , the set of real numbers by ( ) 2 2 4 5 = − + f x x x and ( ) 3 = − g x x. (a) Given ( ) ( ) ( ) = h x f g x , show that ( ) 2 2 8 11 = − + h x x x . (b) Express ( ) h x in the form p x q r + ( ) + 2 . 13. Triangle ABD is right-angled at B with angles BAC = p and BAD = q and lengths as shown in the diagram below. D C B 2 1 4 A q p Show that the exact value of ( ) cos − q p is 19 17 85 . 2 3 5

Page 07 MARKS 14. (a) Evaluate log5 25. (b) Hence solve ( ) 4 4 5 log log 6 log 25 x x + − = , where 6 > x . 15. The diagram below shows the graph with equation ( ) = y f x , where ( ) ( )( ) 2 = − − f x k x a x b . y (1, 9) −5 O 4 x (a) Find the values of a, b and k. (b) For the function ( ) ( ) = − g x f x d , where d is positive, determine the range of values of d for which ( ) g x has exactly one real root. [END OF QUESTION PAPER] 1 5 3 1 Page 08 [BLANK PAGE] do not write on this page Write your answers clearly in the spaces provided in this booklet. The size of the space provided for an answer should not be taken as an indication of how much to write. It is not necessary to use all the space. Additional space for answers is provided at the end of this booklet. If you use this space you must clearly identify the question number you are attempting. Use blue or black ink. Before leaving the examination room you must give this booklet to the Invigilator; if you do not you may lose all the marks for this paper. FOR OFFICIAL USE X747/76/01 Mathematics Paper 1 (Non-Calculator) Answer Booklet Fill in these boxes and read what is printed below. Full name of centre Town Forename(s) Surname Number of seat Day Month Year Scottish candidate number © Mark Date of birth H National Qualications 2016 THURSDAY, 12 MAY 9:00 AM – 10:10 AM A/PB Page 02 DO NOT WRITE IN THIS MARGIN QUESTION NUMBER 2. 1. Page 03 DO NOT WRITE IN THIS MARGIN QUESTION NUMBER 3.(a) 3.(b) 3.(c) Page 04 DO NOT WRITE IN THIS MARGIN QUESTION NUMBER 4. Page 05 DO NOT WRITE IN THIS MARGIN QUESTION NUMBER 5. 6.(a) 6.(b) Page 06 DO NOT WRITE IN THIS MARGIN QUESTION NUMBER 7.(b) 7.(a) Page 07 DO NOT WRITE IN THIS MARGIN QUESTION NUMBER 8. Page 08 DO NOT WRITE IN THIS MARGIN QUESTION NUMBER 9.(b) 9.(a) Page 09 DO NOT WRITE IN THIS MARGIN QUESTION NUMBER 10. y (1, 0) O (4, 1) x f(x) = log4 x Page 10 DO NOT WRITE IN THIS MARGIN QUESTION NUMBER 11.(a) Page 11 DO NOT WRITE IN THIS MARGIN QUESTION NUMBER 11.(b) Page 12 DO NOT WRITE IN THIS MARGIN QUESTION NUMBER 12.(a) Page 13 DO NOT WRITE IN THIS MARGIN QUESTION NUMBER 12.(b) Page 14 DO NOT WRITE IN THIS MARGIN QUESTION NUMBER 13. Page 15 DO NOT WRITE IN THIS MARGIN QUESTION NUMBER 14.(a) 14.(b) Page 16 DO NOT WRITE IN THIS MARGIN QUESTION NUMBER 15.(a) 15.(b) Page 17 DO NOT WRITE IN THIS MARGIN QUESTION NUMBER ADDITIONAL SPACE FOR ANSWERS Page 18 DO NOT WRITE IN THIS MARGIN QUESTION NUMBER ADDITIONAL SPACE FOR ANSWERS Page 19 ADDITIONAL SPACE FOR ANSWERS DO NOT WRITE IN THIS MARGIN QUESTION NUMBER Page 20 For Marker’s Use Question No Marks/Grades © National Qualications 2016 H Total marks — 70 Attempt ALL questions. You may use a calculator. Full credit will be given only to solutions which contain appropriate working. State the units for your answer where appropriate. Answers obtained by readings from scale drawings will not receive any credit. Write your answers clearly in the spaces provided in the answer booklet. The size of the space provided for an answer should not be taken as an indication of how much to write. It is not necessary to use all the space. Additional space for answers is provided at the end of the answer booklet. If you use this space you must clearly identify the question number you are attempting. Use blue or black ink. Before leaving the examination room you must give your answer booklet to the Invigilator; if you do not, you may lose all the marks for this paper. X747/76/12 Mathematics Paper 2 THURSDAY, 12 MAY 10:30 AM – 12:00 NOON A/PB Page 02 FORMULAE LIST Circle: The equation x2 + y2 + 2gx + 2fy + c = 0 represents a circle centre (−g, −f ) and radius The equation (x − a)2 + (y − b)2 = r2 represents a circle centre (a, b) and radius r. Scalar Product: a.b = |a||b| cos θ, where θ is the angle between a and b or a.b = a1b1 + a2b2 + a3b3 where a = 1 1 2 2 3 3 = and a b a b a b b ⎛ ⎞ ⎛ ⎞ ⎟ ⎟ ⎜ ⎜ ⎟ ⎟ ⎜ ⎜ ⎟ ⎟ ⎜ ⎜ ⎟ ⎟ ⎜ ⎜ ⎟ ⎟ ⎜ ⎜ ⎟ ⎟ ⎜ ⎜ ⎟ ⎟ ⎟ ⎟ ⎜ ⎜ ⎝ ⎠ ⎝ ⎠ . Trigonometric formulae: sin (A ± B) = sin A cos B ± cos A sin B cos (A ± B) = cos A cos B ± sin A sin B sin 2A = 2 sin A cos A cos 2A = cos2 A − sin2 A = 2 cos2 A − 1 = 1 − 2 sin2 A Table of standard derivatives: Table of standard integrals: f (x) f ′(x) sin ax cos ax a cos ax – a sin ax f (x) ∫ f (x)dx sin ax cos ax cos ax + c sin ax + c 1 – a 1 a 2 2 + – g f c .

Page 03 MARKS Attempt ALL questions Total marks — 70 1. PQR is a triangle with vertices P( ) 0, 4 − , Q( ) 6,2 − and R( ) 10,6 . y x Q O P R M (a) (i) State the coordinates of M, the midpoint of QR. (ii) Hence find the equation of PM, the median through P. (b) Find the equation of the line, L, passing through M and perpendicular to PR. (c) Show that line L passes through the midpoint of PR. 2. Find the range of values for p such that x2 − 2x + 3 − p = 0 has no real roots. [Turn over 1 2 3 3 3

Page 04 MARKS 3. (a) (i) Show that ( )1 x + is a factor of  2x3 − 9x2 + 3x + 14. (ii) Hence solve the equation  2x3 − 9x2 + 3x + 14 = 0. (b) The diagram below shows the graph with equation y = 2x3 − 9x2 + 3x + 14. The curve cuts the x-axis at A, B and C. y = 2x3 − 9x2 + 3x + 14 y x A B C O (i) Write down the coordinates of the points A and B. (ii) Hence calculate the shaded area in the diagram. 4. Circles C1 and C2 have equations ( ) ( ) 2 2 5 6 9 x y + + − = and x2 + y2 − 6x −16 = 0 respectively. (a) Write down the centres and radii of C1 and C2. (b) Show that C1 and C2 do not intersect. 2 3 1 4 4 3

Page 05 MARKS 5. The picture shows a model of a water molecule. H H O Relative to suitable coordinate axes, the oxygen atom is positioned at point ( ) 2 2 5 − A , , . The two hydrogen atoms are positioned at points ( ) 10 18 7 − B , , and ( ) 4 6 21 − − C , , as shown in the diagram below. ( ) 10 18 7 − B , , ( ) 2 2 5 − A , , ( ) 4 6 21 − − C , , (a) Express AB and AC in component form. (b) Hence, or otherwise, find the size of angle BAC. 6. Scientists are studying the growth of a strain of bacteria. The number of bacteria present is given by the formula ( ) 0 107 200 t B t e =  , where t represents the number of hours since the study began. (a) State the number of bacteria present at the start of the study. (b) Calculate the time taken for the number of bacteria to double. [Turn over 2 4 1 4

Page 06 MARKS 7. A council is setting aside an area of land to create six fenced plots where local residents can grow their own food. Each plot will be a rectangle measuring x metres by y metres as shown in the diagram. y x (a) The area of land being set aside is 108 m2. Show that the total length of fencing, L metres, is given by ( ) 144 9 L x x x = + . (b) Find the value of x that minimises the length of fencing required. 3 6

Page 07 MARKS 8. (a) Express 5cos x − 2sin x in the form k cos (x + a), where k > 0 and 0 < a < 2π. (b) The diagram shows a sketch of part of the graph of y = 10 + 5cos x − 2sin x and the line with equation 12 y = . The line cuts the curve at the points P and Q. y x P Q O y = 10 + 5cos x − 2sin x y = 12 Find the x-coordinates of P and Q. 9. For a function f , defined on a suitable domain, it is known that: • ( ) 2 1 x f x x + = ′ • ( ) 9 40 f = Express ( ) f x in terms of x. [Turn over for next question 4 4 4

Page 08 MARKS 10. (a) Given that ( ) 1 2 2 7 y x = + , find dy dx . (b) Hence find 2 4 7 x dx x + ⌠⎮⌡ . 11. (a) Show that  sin 2x tan x = 1 − cos 2x,  where 3 2 2 x π π < < . (b) Given that ( ) sin 2 tan f x x x = ,  find ( ) f x ′ . [END OF QUESTION PAPER] 2 1 4 2 Write your answers clearly in the spaces provided in this booklet. The size of the space provided for an answer should not be taken as an indication of how much to write. It is not necessary to use all the space. Additional space for answers is provided at the end of this booklet. If you use this space you must clearly identify the question number you are attempting. Use blue or black ink. Before leaving the examination room you must give this booklet to the Invigilator; if you do not you may lose all the marks for this paper. FOR OFFICIAL USE X747/76/02 Mathematics Paper 2 Answer Booklet Fill in these boxes and read what is printed below. Full name of centre Town Forename(s) Surname Number of seat Day Month Year Scottish candidate number © Mark Date of birth H National Qualications 2016 THURSDAY, 12 MAY 10:30 AM – 12:00 NOON A/PB Page 02 1.(a) (i) 1.(b) 1.(a) (ii) DO NOT WRITE IN THIS MARGIN QUESTION NUMBER

13 Turn over Question 6 continued _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ (Total for Question 6 is 6 marks)

8 4. In the binomial expansion of (a + 2x) 7        where a is a constant the coefficient of x4 is 15 120 Find the value of a. (3) _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________

Page 03 1. (c) 2. DO NOT WRITE IN THIS MARGIN QUESTION NUMBER Page 04 3.(a) (i) DO NOT WRITE IN THIS MARGIN QUESTION NUMBER 3.(a) (ii) Page 05 3.(b) (i) DO NOT WRITE IN THIS MARGIN QUESTION NUMBER 3.(b) (ii) Page 06 4.(a) DO NOT WRITE IN THIS MARGIN QUESTION NUMBER Page 07 4.(b) DO NOT WRITE IN THIS MARGIN QUESTION NUMBER Page 08 DO NOT WRITE IN THIS MARGIN QUESTION NUMBER 5.(b) 5.(a) Page 09 DO NOT WRITE IN THIS MARGIN QUESTION NUMBER 6.(a) 6.(b) Page 10 7.(a) DO NOT WRITE IN THIS MARGIN QUESTION NUMBER Page 11 7.(b) DO NOT WRITE IN THIS MARGIN QUESTION NUMBER Page 12 8.(a) DO NOT WRITE IN THIS MARGIN QUESTION NUMBER Page 13 8.(b) DO NOT WRITE IN THIS MARGIN QUESTION NUMBER Page 14 9. DO NOT WRITE IN THIS MARGIN QUESTION NUMBER Page 15 10.(a) DO NOT WRITE IN THIS MARGIN QUESTION NUMBER 10.(b) Page 16 11.(a) DO NOT WRITE IN THIS MARGIN QUESTION NUMBER Page 17 DO NOT WRITE IN THIS MARGIN 11.(b) QUESTION NUMBER Page 18 DO NOT WRITE IN THIS MARGIN ENTER NUMBER OF QUESTION ADDITIONAL SPACE FOR ANSWERS Page 19 DO NOT WRITE IN THIS MARGIN ENTER NUMBER OF QUESTION ADDITIONAL SPACE FOR ANSWERS Page 20 For Marker’s Use Question No Marks/Grades

Page 04 MARKS Total marks – 100 Attempt ALL questions 1. Write down the binomial expansion of 3 2 2 5y y ⎛ ⎞ − ⎜ ⎟ ⎝ ⎠ and simplify your answer. 2. Express ( )( ) 2 2 6 20 1 2 x x x x − + + − in partial fractions. 3. On a suitable domain, a function is defined by ( ) 2 1 2 1 xe f x x − = − . Find ( ) f x ′ , simplifying your answer. 4. The fifth term of an arithmetic sequence is -6 and the twelfth term is -34. (a) Determine the values of the first term and the common difference. (b) Obtain algebraically the value of n for which 144 n S = − . 5. (a) (i) Use Gaussian elimination on the system of equations below to give an expression for z in terms of λ. 2 3 4 2 3 11 3 2 8 x y z x y z x y λz + − = − − + = + + = (ii) For what value of λ is this system of equations inconsistent? (b) Determine the solution of this system when λ = −2·5. 6. Use the substitution 2 5 u x = to find the exact value of 1 10 4 1 25 0 x dx x − ⌠ ⎮⌡ . 4 4 3 2 3 4 1 1 6

Page 05 MARKS 7. Matrices P and Q are defined by 2 5 1 x P ⎛ ⎞ = ⎜ ⎟ − − ⎝ ⎠ and 2 3 4 Q y − ⎛ ⎞ = ⎜ ⎟ ⎝ ⎠ , where ,x y ∈\. (a) Given the determinant of P is 2, obtain: (i) The value of x. (ii) 1 P−. (iii) 1 P−Q′, where Q′ is the transpose of Q. (b) The matrix R is defined by , 5 2 where 6 R z z − ⎛ ⎞ = ∈ ⎜ ⎟ − ⎝ ⎠ \. Determine the value of z such that R is singular. 8. Use the Euclidean algorithm to find integers a and b such that 1595a + 1218b = 29. 9. Solve ( ) 2 2 1 x dy e y dx = + given that when x = 0, y = 1. Express y in terms of x. 10. n S is defined by 2 1 1 3 n r r r = ⎛ ⎞ + ⎜ ⎟ ⎝ ⎠ ∑ . (a) Find an expression for n S , fully factorising your answer. (b) Hence find an expression for 2 2 10 1 3 p r r r = ⎛ ⎞ + ⎜ ⎟ ⎝ ⎠ ∑ where p > 5. [Turn over 1 1 2 2 4 5 2 2

Page 06 MARKS 11. Given 3 2 1 x y x + = , use logarithmic differentiation to find dy dx . Write your answer in terms of x. 12. In the diagram below part of the graph of ( ) y f x = has been omitted. The point (−1, −2) lies on the graph and the line 1 3 2 y x = − is an asymptote. y 0 −3 (−1, −2) x Given that ( ) f x is an odd function: (a) Copy and complete the diagram, including any asymptotes and any points you know to be on the graph. (b) ( ) ( ) g x f x = . On a separate diagram, sketch ( ) g x . Include known asymptotes and points. (c) State the range of values of ( ) f x ′ given that ( ) 0 2 f ′ = . 13. Let n be an integer. Using proof by contrapositive, show that if 2 n is even, then n is even. 5 2 2 1 4

Page 07 MARKS 14. Find the particular solution of the differential equation 2 2 6 d y dy dx dx − + 9 y = 8 sin x + 19 cos x given that y = 7 and 1 2 dy dx = when x = 0. 15. (a) A beam of light passes through the points B(7, 8, 1) and T(-3, -22, 6). Obtain parametric equations of the line representing the beam of light. (b) A sheet of metal is represented by a plane containing the points P(2, 1, 9), Q(1, 2, 7) and R(-3, 7, 1). Find the Cartesian equation of the plane. (c) The beam of light passes through a hole in the metal at point H. Find the coordinates of H. 16. On a suitable domain, a curve is defined by the equation 2 2 4 9 36 x y + = . A section of the curve in the first quadrant, illustrated in the diagram below, is rotated 360° about the y-axis. y x 0 Calculate the exact value of the volume generated. [Turn over for next question 10 2 4 3 5

Page 08 MARKS 17. The complex number z = 2 + i is a root of the polynomial equation 4 3 2 6 16 22 0 z z z z q − + − + = , where q∈]. (a) State a second root of the equation. (b) Find the value of q and the remaining roots. (c) Show the solutions to 4 3 2 6 16 22 0 z z z z q − + − + = on an Argand diagram. 18. The position of a particle at time t is given by the parametric equations x = t cos t,    y = t sin t,    t ≥ 0. (a) Find an expression for the instantaneous speed of the particle. The diagram below shows the path that the particle takes. y x A 0 (b) Calculate the instantaneous speed of the particle at point A. [END OF QUESTION PAPER] 1 6 1 5 2

Page 03 MARKS ATTEMPT ALL QUESTIONS Total marks — 60 1. Functions f and g are defined on suitable domains by ( ) 5 f x x = and ( ) cos 2 g x x = . (a) Evaluate ( ) ( ) 0 f g . (b) Find an expression for ( ) ( ) g f x . 2. The point P (−2, 1) lies on the circle 2 2 8 6 15 0 x y x y + − − − = . Find the equation of the tangent to the circle at P. 3. Given ( ) 12 4 1 y x = − , find dy dx . 4. Find the value of k for which the equation ( ) 2 4 5 0 x x k + + − = has equal roots. 5. Vectors u and v are 5 1 1 ⎛ ⎞ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ − ⎝ ⎠ and 3 8 6 ⎛ ⎞ ⎜ ⎟ − ⎜ ⎟ ⎜ ⎟ ⎝ ⎠ respectively. (a) Evaluate u .v. (b) u w 3 π Vector w makes an angle of 3 π with u and 3 = w . Calculate u .w. 1 2 4 2 3 1 3 [Turn over

Page 05 MARKS 10. Two curves with equations y = x3 − 4x2 + 3x + 1 and y = x2 − 3x + 1 intersect as shown in the diagram. y x O 2 y = x3 − 4x2 + 3x + 1 y = x2 − 3x + 1 (a) Calculate the shaded area. The line passing through the points of intersection of the curves has equation y = 1 − x. y = x3 − 4x2 + 3x + 1 y = x2 − 3x + 1 y = 1 − x y x O 2 (b) Determine the fraction of the shaded area which lies below the line y = 1 − x. [Turn over 5 4

Page 06 MARKS 11. A and B are the points (−7, 2) and (5, a). AB is parallel to the line with equation 3y − 2x = 4. Determine the value of a. 12. Given that log log 1 36 4 2 a a − = , find the value of a. 13. Find ( ) 1 2 1 5 4 dx x − ⌠ ⎮ ⌡ , 5 4 x < . 14. (a) Express sin cos 3 ° ° x x − in the form ( ) sin ° k x a − , where k > 0 and 0 < a < 360. (b) Hence, or otherwise, sketch the graph with equation sin cos 3 ° ° y x x = − , 0 ≤ x ≤ 360. Use the diagram provided in the answer booklet. 3 3 4 4 3

Page 07 MARKS 15. A quadratic function, f, is defined on , the set of real numbers. Diagram 1 shows part of the graph with equation ( ) y f x = . The turning point is (2, 3). Diagram 2 shows part of the graph with equation ( ) y h x = . The turning point is (7, 6). Diagram 2 (7, 6) y y x x O O 3 6 8 1 (2, 3) Diagram 1 ( ) y f x = ( ) y h x = (a) Given that ( ) ( ) h x f x a b = + + . Write down the values of a and b. (b) It is known that ( ) 3 1 4 f x dx = ∫ . Determine the value of ( ) 8 6 h x dx ∫ . (c) Given ( ) 1 6 f ′ = , state the value of ( ) 8 h′ . [END OF QUESTION PAPER] 2 1 1 Page 08 [BLANK PAGE] DO NOT WRITE ON THIS PAGE Write your answers clearly in the spaces provided in this booklet. The size of the space provided for an answer should not be taken as an indication of how much to write. It is not necessary to use all the space. Additional space for answers is provided at the end of this booklet. If you use this space you must clearly identify the question number you are attempting. Use blue or black ink. Before leaving the examination room you must give this booklet to the Invigilator; if you do not you may lose all the marks for this paper. FOR OFFICIAL USE X747/76/01 Mathematics Paper 1 (Non-Calculator) Answer Booklet Fill in these boxes and read what is printed below. Full name of centre Town Forename(s) Surname Number of seat Day Month Year Scottish candidate number © Mark Date of birth H National Qualications 2017 FRIDAY, 5 MAY 9:00 AM – 10:10 AM A/PB Page 02 DO NOT WRITE IN THIS MARGIN QUESTION NUMBER 1.(a) 1.(b) 2. Page 03 DO NOT WRITE IN THIS MARGIN QUESTION NUMBER 3. 4. Page 04 DO NOT WRITE IN THIS MARGIN QUESTION NUMBER 5.(a) 5.(b) 6. Page 05 DO NOT WRITE IN THIS MARGIN QUESTION NUMBER 7. 8. Page 06 DO NOT WRITE IN THIS MARGIN QUESTION NUMBER 9.(a) 9.(b) (i) 9.(b) (ii) Page 07 DO NOT WRITE IN THIS MARGIN QUESTION NUMBER 10.(a) Page 08 DO NOT WRITE IN THIS MARGIN QUESTION NUMBER 10.(b) Page 09 DO NOT WRITE IN THIS MARGIN QUESTION NUMBER 11. 12. Page 10 DO NOT WRITE IN THIS MARGIN QUESTION NUMBER 13. 14.(a) Page 11 DO NOT WRITE IN THIS MARGIN QUESTION NUMBER 14.(b) An additional diagram, if required, can be found on Page 13. 30 x y 4 3 2 1 0 −1 −2 −3 −4 60 90 120 150 180 210 240 270 300 330 360 Page 12 DO NOT WRITE IN THIS MARGIN QUESTION NUMBER 15.(a) 15.(b) 15.(c) Page 13 DO NOT WRITE IN THIS MARGIN QUESTION NUMBER ADDITIONAL SPACE FOR ANSWERS 30 x y 4 3 2 1 0 −1 −2 −3 −4 60 90 120 150 180 210 240 270 300 330 360 Additional diagram for Question 14(b). Page 14 DO NOT WRITE IN THIS MARGIN QUESTION NUMBER ADDITIONAL SPACE FOR ANSWERS Page 15 ADDITIONAL SPACE FOR ANSWERS DO NOT WRITE IN THIS MARGIN QUESTION NUMBER Page 16 For Marker’s Use Question No Marks/Grades © National Qualications 2017 H Total marks — 70 Attempt ALL questions. You may use a calculator. Full credit will be given only to solutions which contain appropriate working. State the units for your answer where appropriate. Answers obtained by readings from scale drawings will not receive any credit. Write your answers clearly in the spaces provided in the answer booklet. The size of the space provided for an answer should not be taken as an indication of how much to write. It is not necessary to use all the space. Additional space for answers is provided at the end of the answer booklet. If you use this space you must clearly identify the question number you are attempting. Use blue or black ink. Before leaving the examination room you must give your answer booklet to the Invigilator; if you do not, you may lose all the marks for this paper. X747/76/12 Mathematics Paper 2 FRIDAY, 5 MAY 10:30 AM – 12:00 NOON A/PB Page 02 FORMULAE LIST Circle: The equation 2 2 2 2 0 x y gx fy c + + + + = represents a circle centre (−g, −f ) and radius 2 2 g f c + − . The equation ( ) ( ) 2 2 2 x a y b r − + − = represents a circle centre (a, b) and radius r. Scalar Product: cos , where is the angle between and θ θ = a.b a b a b or 1 1 1 1 2 2 3 3 2 2 3 3 where and . a b a b a b a b a b a b ⎛ ⎞ ⎛ ⎞ ⎜ ⎟ ⎜ ⎟ = + + = = ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎝ ⎠ ⎝ ⎠ a.b a b Trigonometric formulae: sin (A B) sin A cos B cos A sin B ± = ± cos (A B) cos A cos B sin A sin B ± = ∓ sin A sin AcosA 2 2 = cos A cos A sin A 2 2 2 = − cos A 2 2 1 = − sin A 2 1 2 = − Table of standard derivatives: ( ) f x ( ) f x ′ sin ax cosax cos a ax sin a ax − Table of standard integrals: ( ) f x ( ) f x dx ∫ sin ax cosax cos 1 ax c a − + sin 1 ax c a +

Page 03 MARKS Attempt ALL questions Total marks — 70 1. Triangle ABC is shown in the diagram below. The coordinates of B are (3,0) and the coordinates of C are (9,−2). The broken line is the perpendicular bisector of BC. y x A B C O (a) Find the equation of the perpendicular bisector of BC. (b) The line AB makes an angle of 45° with the positive direction of the x-axis. Find the equation of AB. (c) Find the coordinates of the point of intersection of AB and the perpendicular bisector of BC. 2. (a) Show that ( ) 1 x − is a factor of ( ) 3 2 2 5 2 f x x x x = − + + . (b) Hence, or otherwise, solve ( ) 0 f x = . [Turn over 4 2 2 2 3

Page 04 MARKS 3. The line y = 3x intersects the circle with equation ( ) ( ) 2 2 2 1 25 x y − + − = . y O x Find the coordinates of the points of intersection. 4. (a) Express 2 3 24 50 x x + + in the form ( ) 2 a x b c + + . (b) Given that ( ) 3 2 12 50 11 f x x x x = + + − , find ( ) f x ′ . (c) Hence, or otherwise, explain why the curve with equation ( ) y f x = is strictly increasing for all values of x. 5 3 2 2

Page 05 MARKS 5. In the diagram, PR 9 5 2 = + + i j k and RQ 12 9 3 = − − + i j k. Q S P R (a) Express PQ in terms of i, j and k. The point S divides QR in the ratio 1:2. (b) Show that PS 4 = −+ i j k. (c) Hence, find the size of angle QPS. 6. Solve sin cos 5 4 2 2 x x − = for 0 2 x ≤ < π. 7. (a) Find the x-coordinate of the stationary point on the curve with equation 3 6 2 y x x = − . (b) Hence, determine the greatest and least values of y in the interval 1 9 x ≤ ≤. [Turn over 2 2 5 5 4 3

Page 06 MARKS 8. Sequences may be generated by recurrence relations of the form 1 20 n n u k u + = − , 0 5 u = where k ∈R. (a) Show that 2 2 5 20 20 u k k = − − . (b) Determine the range of values of k for which 2 0 u u < . 9. Two variables, x and y, are connected by the equation n y kx = . The graph of log2 y against log2 x is a straight line as shown. O 3 −12 log2 y log2 x Find the values of k and n. 2 4 5

Page 07 MARKS 10. (a) Show that the points A(−7, −2), B(2, 1) and C(17, 6) are collinear. Three circles with centres A, B and C are drawn inside a circle with centre D as shown. A B C D The circles with centres A, B and C have radii rA, rB and rC respectively. • A 10 r = • B A 2 r r = • C A B r r r = + (b) Determine the equation of the circle with centre D. 11. (a) Show that sin sin cos sin cos 2 3 2 2 x x x x x − = , where 0 2 x π < < . (b) Hence, differentiate sin sin cos cos 2 2 2 x x x x − , where 0 2 x π < < . [END OF QUESTION PAPER] 3 4 3 3 Page 08 [BLANK PAGE] DO NOT WRITE ON THIS PAGE Write your answers clearly in the spaces provided in this booklet. The size of the space provided for an answer should not be taken as an indication of how much to write. It is not necessary to use all the space. Additional space for answers is provided at the end of this booklet. If you use this space you must clearly identify the question number you are attempting. Use blue or black ink. Before leaving the examination room you must give this booklet to the Invigilator; if you do not you may lose all the marks for this paper. FOR OFFICIAL USE X747/76/02 Mathematics Paper 2 Answer Booklet Fill in these boxes and read what is printed below. Full name of centre Town Forename(s) Surname Number of seat Day Month Year Scottish candidate number © Mark Date of birth H National Qualications 2017 FRIDAY, 5 MAY 10:30 AM – 12:00 NOON A/PB Page 02 DO NOT WRITE IN THIS MARGIN QUESTION NUMBER 1.(a) 1.(b) 1.(c) Page 03 DO NOT WRITE IN THIS MARGIN QUESTION NUMBER 2.(a) 2.(b) Page 04 3. DO NOT WRITE IN THIS MARGIN QUESTION NUMBER Page 05 DO NOT WRITE IN THIS MARGIN QUESTION NUMBER 4.(a) 4.(b) 4.(c) Page 06 DO NOT WRITE IN THIS MARGIN QUESTION NUMBER 5.(a) 5.(b) Page 07 DO NOT WRITE IN THIS MARGIN QUESTION NUMBER 5.(c) Page 08 DO NOT WRITE IN THIS MARGIN QUESTION NUMBER 6. Page 09 DO NOT WRITE IN THIS MARGIN QUESTION NUMBER 7.(a) 7.(b) Page 10 DO NOT WRITE IN THIS MARGIN QUESTION NUMBER 8.(a) 8.(b) Page 11 DO NOT WRITE IN THIS MARGIN QUESTION NUMBER 9. Page 12 DO NOT WRITE IN THIS MARGIN QUESTION NUMBER 10.(a) 10.(b) Page 13 DO NOT WRITE IN THIS MARGIN QUESTION NUMBER 11.(a) 11.(b) Page 14 DO NOT WRITE IN THIS MARGIN ENTER NUMBER OF QUESTION ADDITIONAL SPACE FOR ANSWERS Page 15 DO NOT WRITE IN THIS MARGIN ENTER NUMBER OF QUESTION ADDITIONAL SPACE FOR ANSWERS Page 16 For Marker’s Use Question No Marks/Grades

page 09 MARKS 16. Planes 1π , 2 π and 3 π have equations: : : : 1 2 3 2 4 3 5 2 1 7 11 11 π x y z π x y z π x y az − + = − − − = − + + = − where a∈\. (a) Use Gaussian elimination to find the value of a such that the intersection of the planes 1π , 2 π and 3 π is a line. (b) Find the equation of the line of intersection of the planes when a takes this value. The plane 4 π has equation x y z − + + = 9 15 6 20. (c) Find the acute angle between 1π and 4 π . (d) Describe the geometrical relationship between 2 π and 4 π . Justify your answer. 17. (a) Given ( ) x f x e = 2 , obtain the Maclaurin expansion for ( ) f x up to, and including, the term in x3. (b) On a suitable domain, let ( ) tan g x x = . (i) Show that the third derivative of ( ) g x is given by ( ) sec tan sec g x x x x = + ′′′ 4 2 2 2 4 . (ii) Hence obtain the Maclaurin expansion for ( ) g x up to and including the term in x3. (c) Hence, or otherwise, obtain the Maclaurin expansion for tan x e x 2 up to, and including, the term in x3. (d) Write down the first three non-zero terms in the Maclaurin expansion for tan sec x x e x e x + 2 2 2 2 . [END OF QUESTION PAPER] 4 2 3 1 2 3 2 2 1 page 10 [BLANK PAGE] DO NOT WRITE ON THIS PAGE page 11 [BLANK PAGE] DO NOT WRITE ON THIS PAGE page 12 [BLANK PAGE] DO NOT WRITE ON THIS PAGE

6 (B420U10-1) Examiner only 3. When colliding two protons at very high energy the following interaction has been observed at the large hadron collider in CERN. p + p → H + e+ + x (a) Use the conservation of lepton number and conservation of charge to identify the particle x. [3] (b) State which force is responsible for this interaction. Give your reasoning. [2] (c) State which of the above particles can be classed as: [2] (i) a baryon; (ii) an antiparticle. (d) At present, countries in Europe contribute to funding research at CERN. Evaluate whether or not the money could be better spent on humanitarian aid. [2] © WJEC CBAC Ltd. 2 1 9 (B420U10-1) Turn over. B 420 U101 07 7 Examiner only 4. In this question the Sun and the star Vega can be considered to be black bodies. (a) (i) Define a black body. [1] (ii) The surface temperature of the Sun is approximately 6 000 K and that of Vega approximately 10 000 K. Calculate the wavelength of peak spectral intensity for each star and name the region of the electromagnetic spectrum within which they lie. [3] (iii) Sketch a black body spectrum for each star on the axis provided. [3] © WJEC CBAC Ltd. Spectral intensity Wavelength 8 (B420U10-1) Examiner only (b) The radius of Vega is approximately 2.71 times that of the Sun. Determine the ratio: [4] © WJEC CBAC Ltd. total power output of Vega total power output of Sun 11 BLANK PAGE 9 © WJEC CBAC Ltd. (B420U10-1) Turn over.

10 (B420U10-1) Examiner only 5. George attempts to determine the acceleration due to gravity using the following apparatus. The ball bearing starts from rest just above the top beam of light. When the ball bearing cuts the top beam it starts the timer and when it breaks the bottom beam it stops the timer. © WJEC CBAC Ltd. (a) (i) The vertical distance, h, between the two light beams was measured using a metre ruler of resolution 1 cm and was found to be 1.25 m. Determine the percentage uncertainty in this result. [1] 0:00 Light sensor 1 Light sensor 2 Timer Height, h Ball bearing Laser 2 Laser 1 (ii) The time, t, taken to fall through the distance, h, was measured three times and the following results were obtained. Time 1 / s Time 2 / s Time 3 / s 0.51 0.53 0.50 Determine the mean time taken for the ball bearing to fall along with its percentage uncertainty. [2] (B420U10-1) Turn over. 11 Examiner only © WJEC CBAC Ltd. (iii) Use the equation: gt2 2 to determine a value for the acceleration due to gravity along with its absolute uncertainty. [4] (iv) State why the value you obtained in (a)(iii) is less than 9.81 m s–2. [1] (b) (i) Annabel suggests that the experiment would be more accurate if times were obtained for the ball bearing to fall different vertical distances and a graph was drawn. Do you believe Annabel is correct? Justify your answer. [2] (ii) Explain which graph should be drawn to determine g and how it can be used to find a value for g. [2] 12 h = 12 (B420U10-1) Examiner only 6. (a) Explain, with the aid of a diagram, what is meant by the moment of a force about a point. [2] (b) © WJEC CBAC Ltd. pivot uniform metre ruler newtonmeter 95 cm 10 cm (i) A uniform metre ruler is pivoted at the 10 cm mark and the other end is supported by a newtonmeter attached to the 95 cm mark. The ruler balances horizontally when the newtonmeter reads 0.8 N. Determine the mass of the ruler to an appropriate number of significant figures. [3] (B420U10-1) Turn over. 13 Examiner only (ii) A mass of 0.5 kg is now added at the 70 cm mark on the ruler and the ruler is adjusted to be horizontal once more. I. How could you check the ruler is horizontal? [1] II. What is the new reading on the newtonmeter when the ruler is horizontal? [2] (c) If a uniform metre ruler of greater mass than the one in part (b) were used, describe how you could alter the apparatus so the ruler is horizontal once more. Explain your reasoning. [2] © WJEC CBAC Ltd. 10 14 (B420U10-1) Examiner only 7. Louise and Ryan stand a few metres apart on a stationary boat. Louise throws a heavy ball to Ryan who catches it. Describe and explain the motion of the boat from the moment Louise starts to throw the ball until just after the ball is caught by Ryan. Ignore all resistive forces. [6 QER] © WJEC CBAC Ltd. END OF PAPER 6 (B420U10-1) 15 © WJEC CBAC Ltd. BLANK PAGE

4 (B420U20-1) 3. Abigail investigates how the power dissipated in a variable resistor varies as its resistance is altered. The diagram shows the circuit that Abigail uses (meters not shown). The variable resistor is connected to a battery of emf, E, and internal resistance, r. © WJEC CBAC Ltd. Abigail obtains the following data as the resistance is varied from 0.5 Ω to 6.0 Ω. Resistance, R / Ω Power dissipated in R / W 0.5 2.5 1.0 3.3 2.0 3.8 3.0 3.8 4.0 3.7 5.0 3.6 6.0 3.5 E r R (B420U20-1) Turn over. B 420 U201 0 5 5 Examiner only (a) Plot a graph of power dissipated in R (on the y-axis) against resistance (on the x-axis) and draw a smooth curve through the data. [3] © WJEC CBAC Ltd. 6 (B420U20-1) Examiner only (b) The emf of the battery is 6.0 V and the resistance, R, is now set at 4.5 Ω. (i) State what is meant by an emf of 6.0 V. [2] (ii) Calculate the current through the battery using data from your graph. [3] (iii) Calculate the internal resistance, r, of the battery. [3] (c) Abigail repeats the experiment but with a battery of the same emf but smaller internal resistance. Explain how the graph would change. [2] © WJEC CBAC Ltd. 13 (B420U20-1) Turn over. 7 Examiner only 4. (a) Calculate the mean drift velocity of the free electrons in a wire, which has a diameter of 0.50 mm and carries a current of 2.8 A. Assume each aluminium atom contributes 3 free electrons, and there are 6.0 × 1028 atoms per m3 of aluminium. [4] (b) The wire is thinner in a small section as shown below. © WJEC CBAC Ltd. Paula claims that within the thinner section the mean drift velocity of the free electrons will be greater. Evaluate whether she is correct. [3] (c) (i) State what is meant by a superconductor. [1] (ii) State and explain one advantage and one disadvantage of using superconductors to carry large currents. [4] Advantage:................................................................................................................................................................... Disadvantage:............................................................................................................................................................ 12 8 (B420U20-1) Examiner only 5. The apparatus shown below is used to produce a visible interference pattern on the screen. © WJEC CBAC Ltd. Slits B and C act as coherent sources. (a) (i) Explain what is meant by this statement. [1] (ii) Explain why destructive interference is observed at certain points on the screen. [3] (b) (i) The separation of the slits, B and C is 0.090 mm and the perpendicular distance between the slits and the screen is 3.60 m. The slits are illuminated with light of wavelength 4.4 × 10-7 m. A point X on the screen is 52.8 mm away from the central bright fringe at O. Brian states that a bright fringe will be formed at X. Evaluate whether or not he is correct. [4] Laser Screen B X O C (B420U20-1) Turn over. 9 Examiner only (ii) Determine the distance from X to the next dark fringe formed. [2] (c) Explain the historical significance of this experiment. [2] (d) A beam of electrons is fired at a thin sheet of aluminium. A pattern of light and dark circular fringes is observed as shown below. © WJEC CBAC Ltd. (i) Explain how this pattern is formed. [2] (ii) The thin sheet of aluminium is replaced with a foil of copper. The distance between the copper atoms is smaller than in aluminium. Describe how you expect the pattern to change. [2] 16 10 (B420U20-1) Examiner only 6. (a) Describe an experiment using a range of LEDs to determine the Planck constant. [6 QER] (b) A university research group uses a new experimental method to determine a value for the Planck constant that is higher than the accepted value of 6.63 × 10–34J s. Explain how other scientists would attempt to confirm this result. [2] © WJEC CBAC Ltd. 8 (B420U20-1) Turn over. 11 Examiner only 7. When sunlight shines on the atmosphere of Mars, carbon dioxide molecules at a height of 75 km behave like the amplifying medium in a laser. The energy levels involved are shown in the following diagram. A population inversion occurs between energy levels E1 and E2. © WJEC CBAC Ltd. Pumped level E2 = 0.289 eV E1 = 0.165 eV E0 = 0 (a) (i) State what is meant by a population inversion. [1] (ii) Explain why a population inversion is needed for laser action. [2] (b) Calculate the output wavelength of this naturally occurring laser and determine the region of the electromagnetic spectrum in which it lies. [3] TURN OVER FOR THE LAST PART OF THE QUESTION 12 (B420U20-1) Examiner only (c) The Llanesco crater is on the surface of Mars and has a cross-sectional area of 2 290 km2. Carbon dioxide molecules in the Mars atmosphere collide with the surface and give rise to a surface pressure of 600 Pa. The mass of a carbon dioxide molecule is 7.3 × 10–26 kg and the perpendicular component of the speed of the molecules to the surface is 550 m s–1 before and after they collide with the surface. Calculate the number of carbon dioxide molecules that collide with the Llanesco crater in 1 s. [5] END OF PAPER © WJEC CBAC Ltd. 11

Figure 1 (A420U10-1) SECTION B 7. Read through the following article carefully. Cool Physics for Smart Phones by Justino Luis Moreno Some meters can be expensive to buy but if you’ve got a smart phone it’s surprising just how many meters it can be employed as, if you get hold of the right apps. Right now on my iphone I’ve got a seismometer, magnetometer, accelerometer, tiltometer, protractor, light meter, ruler, decibel meter, oscilloscope and frequency meter. In the hands of a physicist your smart phone is transformed into a whole lot of experimental fun. Experiment 1 - Measuring the speed of sound All you need is a hollow cardboard tube of length between 1 and 3 metres. You’ll also need a frequency meter app (mine’s part of a free guitar tuner) and somebody who can make didgeridoo noises into the cardboard tube. Anyone can make these noises after a bit of practice and you should even be able to play the two lowest notes. The cardboard tube didgeridoo can be classed as a pipe that is closed at one end. There is a node of air displacement at the closed end, and an antinode at the open end. The second lowest note is called the third harmonic because it has a frequency of 3 times the lowest (or ‘fundamental’ or ‘first harmonic’). When carrying out this experiment with a hollow tube of length 1.800 m, the fundamental note had a frequency of 47 Hz. This gave a value of the speed of sound of around 340 m s–1 which is not bad with a piece of throw-away cardboard and a free guitar tuning app. Experiment 2 - Investigating polarisation of light Unfortunately, light meter software tends to use a unit called ‘lux,’ but as long as we don’t change light source mid-experiment the lux is proportional to the intensity. We shall concentrate on an experiment to do with polarisation - the variation of transmitted intensity through a polaroid with angle of the polaroid relative to the polarised light. The set-up is shown below. 16 © WJEC CBAC Ltd. The polariser provides the polarised light, the analyser is rotated in steps through 180° and the intensity of light is recorded using the smart phone. You can measure the angle (θ ) with a plastic protractor but there is a smarter way. You can tape the analyser over the smart phone camera and have 2 apps running simultaneously – a light meter to measure the light intensity and a tiltometer to measure the angle of the analyser. Unpolarised light Polariser Polarised light Transmission axis Analyser smart phone θ E0 Figure 2 1 2 3 4 Paragraph Turn over. 17 (A420U10-1) These were the results obtained: © WJEC CBAC Ltd. 0 0 100 200 40 80 120 160 20 60 100 140 180 0 300 40 80 120 160 20 60 100 140 180 These results are in excellent agreement with Malus’s Law – that the intensity, I, of light is given by: I = I0 cos2 θ where θ is the angle of the analyser relative to the polariser. Experiment 3 - Measuring the coefficient of friction There is a simple rule of friction that states: F = μR Light intensity / arbitrary units Figure 3 Figure 4 θ / ° where F is the maximum frictional force, R is the normal contact force and μ is the coefficient of friction. Basically, μ is a dimensionless constant that tells you how slippery the contact is between two smooth surfaces. The greater the value of μ the greater the frictional force (for a given value of R). One of the easiest methods of obtaining values of the coefficient of friction is to place a block on a slope and tilt the slope until the block slips. When the block slips it’s easy to show that the coefficient of friction is given by: μ = tanθ R F 5 6 7 Paragraph Figure 5 36.5° θ (A420U10-1) 18 © WJEC CBAC Ltd. Table 1 The angle can be measured easily with an uncertainty of less than 1° using a smart phone with a tiltometer placed on the slope. Here are some results obtained for rubber on tarmac. Surface Mean slide angle (degrees) Mean coefficient of friction smooth rubber on tarmac 44.4 0.98 rubber with treads on tarmac 40.4 0.85 Experiment 4 - Investigating magnetic field strengths The phone was used as a magnetometer to take readings of the magnetic field strength, B, at two distances, r, from the centre of a small bar magnet (with the magnet end-on to the magnetometer). The results are shown in Table 2. r / cm B / mT 8.0 9.8 12.0 2.9 These results suggest that B depends on r according to an inverse cube law, that is a law of the form: where K is a constant. B K r3 = 8 9 10 11 Table 2 Experiment 5 - Investigating the acceleration of a car All sorts of investigations can be done with a mobile phone used as an accelerometer. Oscillations particularly lend themselves to this method. As an even simpler case, though, here is a (tidied up) acceleration-time graph, based on accelerometer readings for a car starting from rest at time t = 0 and going along a straight road. 0 0.2 0.4 0 0.1 0.3 0.5 0.6 10 20 30 40 50 a / m s–2 Figure 6 Paragraph t / s Turn over. 19 (A420U10-1) Examiner only © WJEC CBAC Ltd. Answer all questions. Answer the following questions in your own words. Direct quotes from the original article will not be awarded marks. (a) Experiment 1 - Draw diagrams showing the stationary wave patterns for the lowest frequency (first harmonic) and the next lowest frequency (third harmonic) in the hollow cardboard tube (Paragraph 2). [2] (b) Experiment 1 - Calculate the frequency and wavelength for the third harmonic in the 1.800 m tube (Paragraph 2). [2] (c) Experiment 2 - Explain qualitatively why the variation in light intensity detected by the smart phone is as shown in Figure 3 (see Paragraphs 4 & 5 and Figure 2 also). Do not refer to Malus’s law or the cos2 θ  dependency. [4] (A420U10-1) 20 Examiner only © WJEC CBAC Ltd. (d) Experiment 2 - In paragraph 5 it is claimed that the results plotted in Figure 3 are in excellent agreement with the law I = I0 cos2θ . (i) Show clearly that for these results, using the arbitrary units of the graph scale, I0 = 340. [1] (ii) Check whether or not the point plotted in Figure 3 for θ = 140° is in agreement with the law I = I0 cos2θ. Show all steps in your working. [2] (e) Experiment 3 - Some textbooks claim that “rubber tyres have patterns of grooves or tread: these patterns increase the roughness and give better grip.” Evaluate whether the data in Table 1 (Paragraph 8) disproves this theory or whether this result is due to experimental uncertainty. [3] (f) Experiment 4 - According to the writer, the results in Table 2 (Paragraph 9) suggest that B depends on r according to an inverse cube law, that is a law of the form: (Paragraph 10) Justify this statement. [2] B K r3 = (A420U10-1) 21 Examiner only (g) Experiment 5 (i) For the car, starting from rest, whose acceleration-time graph is given in Figure 6, calculate the velocity at t = 25 s. [1] (ii) Sketch a velocity-time graph for the car, on the blank grid below. Put a scale on the vertical axis. (A copy of Figure 6 is provided so that you can refer to it easily.) [3] © WJEC CBAC Ltd. 0 0.2 0.4 0 0.1 0.3 0.5 0.6 10 20 30 40 50 0 0 10 20 30 40 50 a / m s–2 v / m s–1 t / s t / s END OF PAPER 20 Turn over. BLANK PAGE (A420U10-1) 22 © WJEC CBAC Ltd. 23 Examiner only © WJEC CBAC Ltd. For continuation only. (A420U10-1) BLANK PAGE (A420U10-1) 24 © WJEC CBAC Ltd.

PEARSON EDEXCEL GCE MATHEMATICS General Instructions for Marking 1. The total number of marks for the paper is 60. 2. These mark schemes use the following types of marks: • M marks: Method marks are awarded for ‘knowing a method and attempting to apply it’, unless otherwise indicated. • A marks: Accuracy marks can only be awarded if the relevant method (M) marks have been earned. • B marks are unconditional accuracy marks (independent of M marks) • Marks should not be subdivided. 3. Abbreviations These are some of the traditional marking abbreviations that will appear in the mark schemes. • bod – benefit of doubt • ft – follow through • the symbol will be used for correct ft • cao – correct answer only • cso - correct solution only. There must be no errors in this part of the question to obtain this mark • isw – ignore subsequent working • awrt – answers which round to • SC: special case • o.e. – or equivalent (and appropriate) • d or dep – dependent • indep – independent • dp decimal places • sf significant figures

• The answer is printed on the paper or ag- answer given 4. All M marks are follow through. A marks are ‘correct answer only’ (cao.), unless shown, for example, as A1 ft to indicate that previous wrong working is to be followed through. After a misread however, the subsequent A marks affected are treated as A ft, but answers that don’t logically make sense e.g. if an answer given for a probability is >1 or <0, should never be awarded A marks. be awarded A marks. 5. For misreading which does not alter the character of a question or materially simplify it, deduct two from any A or B marks gained, in that part of the question affected. 6. Where a candidate has made multiple responses and indicates which response they wish to submit, examiners should mark this response. If there are several attempts at a question which have not been crossed out, examiners should mark the final answer which is the answer that is the most complete. 7. Ignore wrong working or incorrect statements following a correct answer. 8. Mark schemes will firstly show the solution judged to be the most common response expected from candidates. Where appropriate, alternatives answers are provided in the notes. If examiners are not sure if an answer is acceptable, they will check the mark scheme to see if an alternative answer is given for the method used. If no such alternative answer is provided but the response is deemed to be valid, examiners must escalate the response for a senior examiner to review. General Principles for Mechanics Marking (But note that specific mark schemes may sometimes override these general principles) • Rules for M marks: correct no. of terms; dimensionally correct; all terms that need resolving (i.e. multiplied by cos or sin) are resolved. • Omission or extra g in a resolution is an accuracy error not method error. • Omission of mass from a resolution is a method error. • Omission of a length from a moments equation is a method error. • Omission of units or incorrect units is not (usually) counted as an accuracy error. • DM indicates a dependent method mark i.e. one that can only be awarded if a previous specified method mark has been awarded. • Any numerical answer which comes from use of g = 9.8 should be given to 2 or 3 SF. • Use of g = 9.81 should be penalised once per (complete) question. N.B. Over-accuracy or under-accuracy of correct answers should only be penalised once per complete question. However, premature approximation should be penalised every time it occurs. • Marks must be entered in the same order as they appear on the mark scheme. • In all cases, if the candidate clearly labels their working under a particular part of a question i.e. (a) or (b) or (c),……then that working can only score marks for that part of the question. • Accept column vectors in all cases. • Misreads – if a misread does not alter the character of a question or materially simplify it, deduct two from any A or B marks gained, bearing in mind that after a misread, the subsequent A marks affected are treated as A ft • Mechanics Abbreviations M(A) Taking moments about A. N2L Newton’s Second Law (Equation of Motion) NEL Newton’s Experimental Law (Newton’s Law of Impact) HL Hooke’s Law SHM Simple harmonic motion PCLM Principle of conservation of linear momentum RHS, LHS Right hand side, left hand side. 8MA0 22: Mechanics AS 1906 Mark Scheme Question Scheme Marks AOs Notes 1 (a) V = 30 ( 1 m s) B1 3.4 cao (1) (b) 30 shape figs 0 3 5 T B1 1.1b Overall shape of the graph, starting at the origin. Dotted vertical line at end is OK but solid vertical line is B0 B1ft 1.1b 3, 5 and T marked on the t-axis, and ft on their 30 marked on the speed axis. 3 must be where graph reaches a peak. Allow delineators: 3, 2 and T – 5 or a mixture (2) (c) Using total area = 550 to set up an equation in one unknown, Or they may use suvat on one or more of the sections (but must still be considering all sections) M0 if they use one suvat equation for the whole motion M1 2.1 Need all sections to be included, with correct structure for each section. e.g. triangle + trapezium + rectangle oe = 550 to give an equation in one unknown (may not be T) 1 (30 6) 3 30 2 6( 5) 550 2 2 T       OR: 1 1 3 30 2 24 6( 3) 550 2 2 T       OR: 1 1 3 30 2 24 (2 6) 6( 5) 550 2 2 T         A2 ft 1.1b ft on their answer to (a). -1 each error. N.B. If ‘6’ is incorrect, treat as one error, unless it is correct ft from their 30. Solve for T M1 1.1b Attempt to solve for T provided they have tried to find the area using at least 3 sections. (M0 if they only solve for their unknown and never try to find T) T = 83 (nearest whole number) A1 1.1b 83 is the only answer (5) (d) New value of T would be bigger (ignore their reasons whether correct or not) B1 3.5a Clear statement about the value of T Allow ‘it would increase, get larger etc’ B0 for ‘Takes longer’ or ‘the value of T would be longer’ (1) (e) e.g. effect of wind; allow for dimensions of parachutist; use a more accurate value for g; parachutist does not fall vertically after chute opens; smooth changes in v; time for parachute to open; deceleration not constant (but B0 if they say acceleration not constant); smooth changes in a; B0 for: moves horizontally; mass/weight of parachutist; upthrust; air pressure;air resistance; terminal velocity B1 3.5c Any appropriate refinement of the model. B0 if incorrect (or irrelevant) extras (1) (10 marks) N.B. Omission or extra g in a resolution is an accuracy error not a method error In 2(a), use the mass which appears in the ‘ma’ term of an equation of motion, to identify which particle that equation of motion applies to. Question Scheme Marks AOs Notes 2(a) Equation of motion for Q M1 3.3 Equation of motion for Q with correct no. of terms, condone sign errors. 0.6 0.6 g T a   A1 1.1b A correct equation Equation of motion for P M1 3.3 Equation of motion for Q with correct no. of terms, condone sign errors. 0.8 T a  A1 1.1b A correct equation 2 4.2 (m s ) a   * A1* 2.2a Given acceleration obtained correctly. You must see an equation in a only before reaching a = 4.2 (5) N.B. if they just use the whole system equation: 0.6 1.4 g a  , can only score max M1A1M0A0A0 N.B. Use of g = 9.81 or 10 loses final A mark only. N.B. Complete verification, using both equations, can score full marks. (b) 2 1 1 0.4= 4.2 2 t   or e.g. they may find v first and then use v = 4.2 1t M1 2.1 Complete method (they may use more than one suvat equation) to find time for Q to hit the floor (M0 if 0.4 not used as distance moved and/or if 4.2 is not used as acceleration and this applies to finding v as well if they use v to find 1t ) 1 0.436 t  (4357…..) Allow 0.43, 0.44, 0.436, or better, or any surd form e.g. 2 21 A1 1.1b See alternatives 1 4.2 or 2 4.2 0.4 v t v      or 0.4 = 1 (0 ) 2 v t   (v = 1.8330…) M1 3.4 Complete method to find speed of Q as it hits the floor (M0 if 0.4 not used as distance moved and/or if 4.2 is not used as acceleration and this applies to finding 1t as well if they use 1t to find v) 2 1.5 0.4 t v   M1 1.1b Uses distance/speed to find time for P to hit the pulley after Q has hit the floor. N.B. This is independent of previous M mark. Complete strategy to solve the problem by finding the sum of the two times 1 2 t t  DM1 3.1b Complete method to solve the problem by finding and adding the two required times, dependent on previous three M marks 1.0 (s) or 1.04 (s) A1 1.1b (6) (c) e.g. rope being light; rope being inextensible; pulley being smooth; pulley being small; balls being particles B1 3.5b Clear statement. Allow negatives of these i.e. the rope may not be light, the rope may not be inextensible etc Must be a limitation of the model stated in the question Penalise incorrect or irrelevant extras (1) B0 for: Air resistance, table being smooth (12 marks) Question Scheme Marks AOs Notes 3(a) 2 12 4 v t t    = 0 and solving M1 3.1a Equating v to 0 and solving the quadratic If no evidence of solving, and at least one answer wrong, M0 t = 6 (or -2) A1 1.1b 6 but allow -2 as well at this stage Differentiate v wrt t M1 1.1a For differentiation (both powers decreasing by 1) d ( ) 4 2 d v a t t    A1 1.1b Cao; only need RHS When t = 6, a = - 8; Magnitude is 8 ( 2 m s) A1 1.1b Substitute in t = 6 and get 8 (m s-2) as the answer . Must be positive. (A0 if two answers given) (5) (b) Integrate v wrt t M1 3.1a For integration (at least two powers increasing by 1) 2 3 1 ( )12 2 ( ) 3 s t t t C     A1 1.1b Correct expression (ignore C) only need RHS Must be used in part (b) t = 3 => distance = 45 (m) A1 1.1b Correct distance. Ignore units (3) (8 marks) Pearson Education Limited. Registered company number 872828 with its registered office at 80 Strand, London, WC2R 0RL, United Kingdom

Mark scheme notes Underlying principle The mark scheme will clearly indicate the concept that is being rewarded, backed up by examples. It is not a set of model answers. 1. Mark scheme format 1.1 You will not see ‘wtte’ (words to that effect). Alternative correct wording should be credited in every answer unless the MS has specified specific words that must be present. Such words will be indicated by underlining e.g. ‘resonance’ 1.2 Bold lower case will be used for emphasis e.g. ‘and’ when two pieces of information are needed for 1 mark. 1.3 Round brackets ( ) indicate words that are not essential e.g. “(hence) distance is increased”. 1.4 Square brackets [ ] indicate advice to examiners or examples e.g. [Do not accept gravity] [ecf]. 2. Unit error penalties 2.1 A separate mark is not usually given for a unit but a missing or incorrect unit will normally mean that the final calculation mark will not be awarded. 2.2 This does not apply in ‘show that’ questions or in any other question where the units to be used have been given, for example in a spreadsheet. 2.3 The mark will not be awarded for the same missing or incorrect unit only once within one clip in epen. 2.4 Occasionally, it may be decided not to insist on a unit e.g the candidate may be calculating the gradient of a graph, resulting in a unit that is not one that should be known and is complex. 2.5 The mark scheme will indicate if no unit error is to be applied by means of [no ue]. 3. Significant figures 3.1 Use of too many significant figures in the theory questions will not be prevent a mark being awarded if the answer given rounds to the answer in the MS. 3.2 Too few significant figures will mean that the final mark cannot be awarded in ‘show that’ questions where one more significant figure than the value in the question is needed for the candidate to demonstrate the validity of the given answer. 3.3 The use of one significant figure might be inappropriate in the context of the question e.g. reading a value off a graph. If this is the case, there will be a clear indication in the MS. 3.4 The use of g = 10 m s-2 or 10 N kg-1 instead of 9.81 m s-2 or 9.81 N kg-1 will mean that one mark will not be awarded. (but not more than once per clip). Accept 9.8 m s-2 or 9.8 N kg-1 3.5 In questions assessing practical skills, a specific number of significant figures will be required e.g. determining a constant from the gradient of a graph or in uncertainty calculations. The MS will clearly identify the number of significant figures required. 4. Calculations 4.1 Bald (i.e. no working shown) correct answers score full marks unless in a ‘show that’ question. 4.2 If a ‘show that’ question is worth 2 marks. then both marks will be available for a reverse working; if it is worth 3 marks then only 2 will be available. 4.3 use of the formula means that the candidate demonstrates substitution of physically correct values, although there may be conversion errors e.g. power of 10 error. 4.4 recall of the correct formula will be awarded when the formula is seen or implied by substitution. 4.5 The mark scheme will show a correctly worked answer for illustration only. 1. Quality of Written Communication 1.1 Indicated by QoWC in mark scheme. QWC – Work must be clear and organised in a logical manner using technical wording where appropriate. 1.2 Usually it is part of a max mark, the final mark not being awarded unless the QoWC condition has been satisfied. 2. Graphs 2.1 A mark given for axes requires both axes to be labelled with quantities and units, and drawn the correct way round. 2.2 Sometimes a separate mark will be given for units or for each axis if the units are complex. This will be indicated on the mark scheme. 2.3 A mark given for choosing a scale requires that the chosen scale allows all points to be plotted, spreads plotted points over more than half of each axis and is not an awkward scale e.g. multiples of 3, 7 etc. 2.4 Points should be plotted to within 1 mm.  Check the two points furthest from the best line. If both OK award mark.  If either is 2 mm out do not award mark.  If both are 1 mm out do not award mark.  If either is 1 mm out then check another two and award mark if both of these OK, otherwise no mark. For a line mark there must be a thin continuous line which is the best-fit line for the candidate’s results. Question Number Acceptable answers Additional guidance Mark 1 The only correct answer is D A is not correct because as it is 2A + 1A B is not correct because 2A +2A C is not correct because 2A + 3A 6 A 1 2 The only correct answer is A B is not correct because V decreases as I decreases C is not correct because I decreases as R increases D is not correct because I decreases as R increases decreases decreases 1 3 The only correct answer is C A is not correct because acceleration is 0 as v constant B is not correct because there is a resistive force opposing weight D is not correct because there is a weight There is a resultant force acting on the object. 1 4 The only correct answer is B A is not correct because R decreases as more conduction electrons C is not correct because lattice vibrations not affected D is not correct because lattice vibrations not affected It increases because there is an increase in the number of conduction electrons. 1 5 The only correct answer is C A is not correct because it shows constant resistance B is not correct because it shows decreasing resistance D is not correct because it shows an I independent of V 1 6 The only correct answer is A B is not correct because these are base units of force C is not correct because these are not base units D is not correct because these are not base units kg m s-1 1 V I 7 The only correct answer is B A is not correct because this is a uniform field so F constant C is not correct because this is a uniform field so F constant D is not correct because this is a uniform field so F constant F 1 8 The only correct answer is B A is not correct because it is 3 divided by 2 C is not correct because it is 3 x root 2 D is not correct because it is 32 2.1 A 1 9 The only correct answer is B A is not correct because a cyclotron uses a magnetic field C is not correct because a LINAC uses an electric field D is not correct because a LINAC does not use a magnetic field Cyclotron accelerated 1 10 The only correct answer is C A is not correct because lepton number is not conserved B is not correct because charge conservation is not obeyed D is not correct because charge conservation is not obeyed p → n + β 1 (Total for Multiple Choice Questions = 10 marks) Question Number Acceptable answers Additional Guidance Mark 11a  Use of A l R    Use of area formula with correct value of radius  R = 8.9  (1) (1) (1) Use of: any dimensionally correct substitutions eg using a diameter squared Accept R=8.62 as due to rounding Area Example of calculation: R = 1.12 x 10-6 m x 1m / (0.2x10-3)2 m2 3 11bi  As resistance increases with length of wire  potential (difference) proportional to length of wire (1) (1) Alt to MP1: Current same through whole length of wire and V=IR 2 11bii  Use of ratio of lengths = ratio of potentials  Potential at P = 1.13 V (1) (1) Alternative method uses ratio of resistances. Example of calculation: V 125 .1 50 .1 100 0. 75    2 11(c) Either  Calculates current correctly using I=V/R  R = 1.10   Or  Use of ratios of lengths = ratios of resistances  R = 1.10   Or  Use ratio of resistances = ratio of p.d.s  R = 1.10  (1) (1) (1) (1) (1) (1) Show that value gives 1.2  Example of calculation:      1.1 34 .0 375 .0 34 .0 30 .3 125 .1 R A I Or 75 25 30 . 3  R 2 (Total for Question 11 = 9 marks) Question Number Acceptable Answers Additional Guidance Mark *12 This question assesses a student’s ability to show a coherent and logically structured answer with linkages and fully-sustained reasoning. Marks are awarded for indicative content and for how the answer is structured and shows lines of reasoning. The following table shows how the marks should be awarded for indicative content. Indicative content:  (Maximum/Initial) current is equal to battery emf divided by R Or current as switch closed Or current as complete circuit Or current due to battery  Coil rotates  (movement of) coil “cuts/changes” (magnetic) flux (linkage) / field  Which induces an emf (according to Faraday’s law)  Opposes original emf/current according to Lenz’s law Or current reduced as effect opposes change  The faster the coil rotates the larger this (back) emf/effect the smaller the current IC points IC mark Max linkage mark available Max final mark 6 4 2 6 5 3 2 5 4 3 1 4 3 2 1 3 2 2 0 2 1 1 0 1 0 0 0 0 ic3 needs a link to coil moving ic4 depends on ic3 6 (Total for question 12 = 6 Marks) Question Number Acceptable answers Additional guidance Mark 13 (a)  Take a correct moment about pivot P (1) eg T.6.cos20 or sin70 4  Converts the mass to weight of beam ie ×9.81 seen  Appreciates centre of mass 0.5 m from P  T = 25 kN (1) (1) (1) If cos20’s are absent from both sides of equation then can still credit 4 marks Example of Calculation: kN 9. 24 20 cos ) m ( 5.0 ) ms ( 81 .9 ) kg ( 10 05 .3 20 cos ) m ( 6 -2 4         T T 13 (b)  Use of Egrav= mgh  convert for a unit time, e.g. day or second ie W  Calculation of energy input provided by coal in unit time  Use of Efficiency = energy output/energy input Or in terms of power  Efficiency = 1.5 % so not correct (1) (1) (1) (1) (1) (allow reverse argument starting with 10% efficiency for full credit) Example of calculation: day per MJ 424 24 60 kJ 294 minute per kJ 294 12 81 .9 2500             grav grav grav grav E E E E energy input = 1250  22.3 MJ Efficiency = 424 MJ/ 27900 MJ = 1.5% 5 (Total for Question 13 = 9 marks) Question Number Acceptable Answers Additional Guidance Mark 14 (a)  Attempts to find area under graph  Height = 19 m (1) (1) Range for base of triangle between 1.8 and 2s to recognise area Example of calculation: Area = 1 ms 20 s 9.1 2 1    2 Alternative:  Use equations of motion to Height = 19 m (2) = 19 m 14 (b)  Positive value of acceleration for first 0.2 s  Constant at 100 m s-2  Negative value of acceleration up to 4 s  Initially at -12 m s-2 (value between -10 and -15 on graph)  Negative value reducing in magnitude in last 1.0 s to 2.0 s (can be a step change) (1) (1) (1) (1) (1) 5 (Total for Question 14 = 7 marks) Question Number Acceptable Answers Additional guidance Mark 15a  fundamental – quarks and leptons  Baryons made of 3 q  Mesons made of quark and antiquark (1) (1) (1) MP2 and 3 could be given for a named particle and its quark composition 5  6 quark Or 6 leptons  Each particle has an antiparticle (1) (1) Can be inferred if either set named 15b  Use of ΔE = Δmc2  Conversion of J to eV  mass = 120 GeV/c2 (1) (1) (1) Example of calculation: eV 10 124 JeV 10 1.6 J 10 1.98 J 10 98 .1 ) (ms ) 10 3.0 ( kg 10 2.2 9 1- 19 - 8 - 8 2 -1 2 8 25               E E E E 3 15c(i)  Energy (of protons) converted to mass (of Higgs) Or Energy is required to overcome electrostatic repulsion between protons  Reference to E = mc2 (can be written in any form)  Because c2 is very large (E must be large) Or Higgs particle is massive so needs a lot of energy to create it (1) (1) (1) Alternative based on numerical values: Observation that Higgs mass is 120 GeV/c2 This requires an energy of at least 120 GeV Each beam of protons would need an energy of at least 60 GeV 3 15c(ii)  Use of circumference = r  2  Use of Bqr p  p = 5.7  10-15 Ns (1) (1) (1) Example of calculation: Ns 10 7.5 m 4300 C 10 1.6 8.3T m 4300 π 2 27000 15 19 -           p p r r 3 Question Number Acceptable Answers Additional guidance Mark 15ciii 0 (1) zero 1 15d  High speeds Or relativistic  Mass (of proton) increases  Or this equation is only valid at non-relativistic speeds (1) (1) Alt: speeds close to speed of light 2 (Total for Question 15 = 17 marks) Question number Acceptable Answers Additional guidance Mark 16a  At least 4 radial lines  arrow pointing outwards  straight, symmetrical and equally distributed (1) (1) (1) Ignore dotted lines 3 16b  tangent at correct point  triangle with base at least 0.4 m  5.3  106 (Vm-1) (range 4.9  106 to 6.1  106) (1) (1) (1) Example of calculation: Gradient = 3200000 / 0.6 E = 5.3  106 V m-1  So would ionise as value greater than 3  106 Alternative:  Correct value of V at 30 cm  Use of ܧൌ݇ ொ ௥మ and ܸൌ݇ ொ ௥  5.3  106 (Vm-1)  So would ionise as value greater than 3  106 (1) (1) (1) (1) (1) MP4 to be consistent with calculated value V = 1.6 106 V m-1 4 16c Maximum 3 marks  There cannot be a p.d. across his body  Electric field strength inside cage is zero  As no potential gradient  Current/electrons/charge would conduct through suit Or the current would not pass through body (1) (1) (1) (1) Accept reference to Faraday cage for MP2 3max (Total for Question 16 = 10 marks) Question mark Acceptable Answers Additional Guidance Mark 17a  vector velocities at two positions as part of a triangle and third side identified as v  Acceleration a = v/t (i)  Use of trigonometry: v/v  sin   for small angles (ii)  Use of v = rt (iii)  Combine i, ii, iii to final equation OR  Diagram shows components of v with angle turned through  Acceleration = 2vsint  Use of trigonometry: v/v  sin   for small angles  t = rv and 2s cancel  Simplify to final equation (1) (1) (1) (1) (1) (1) (1) (1) (1) (1) Example of diagram Ignore arrow directions Combine (i) and (ii) a = vt Substitute for using (iii) a = r vt t v  then “t”s cancel Allow other fully correct methods 5 17b(i)  Correct conversion to angle in radians  = 5.2 (rads-1 ) (1) (1) Example of calculation = 50  2/ 60 s = 5.24 rads-1 2 17b(ii)  Reference to F = mr2  appreciation that r is large Or (the equipment) has a high (linear) velocity (1) (1) Alt: mass (of equipment) could be large 2 17b(iii)  use of r   a = 25g and appropriate comment (1) (1) Show that value gives 22.5g Allow reverse argument starting with  25g to 5.28 rads-1 Example of calculation a = 8.8 (m) x 5.242 (rads-1)2 a = 238 (ms-2) 9.81 (ms-2) = 24.6 g 2 (Total for Question 17 = 11 marks) -v v  v Question Marks Acceptable Answers Additional guidance Mark 18a  Magnet accelerates ball Or magnet increases ball’s KE  Momentum is conserved in the collision(s)  (Since collisions are elastic) KE conserved so third ball moves off with the same velocity/KE as incoming ball hit magnet with (1) (1) (1) Marks can be gained by discussing either set of balls 3 18bi  Use of 2 2 1 CV W   W = 45 J (1) (1) Example of calculation 2) V 5.1( μF 40 2 1   W W = 45 J Alt: Use Q = CV then E = QV/2 for MP1 2 18bii  Use of RC t oe V V /    Time = 0.14 (s) (1) (1) Example of calculation s 14 .0 2.0 / 5.0 ln 5.0 6 10 40 5000 /         t t e t 2 18biii  Use of speed = d/t  Speed = 3.6 ms-1 Allow ecf from ii (1) (1) Show that value gives 5.0 ms-1 Example of calculation s 14 .0 / m 5.0  v = 3.6 ms-1 2 18biv  use of 2 2 at s   s = 9 cm + comment that foil is not broken at its centre (comment consistent with calculation) Allow ecf from ii (1) (1) Show that value gives 0.049 m Example of calculation ݏൌ ଽ.଼ଵ ௠௦షమ ൈ଴.ଵସమ ௦ ଶ = 0.094 m 2 (Total for Question 18 = 11 marks) Pearson Education Limited. Registered company number 872828 with its registered office at 80 Strand, London, WC2R 0RL, United Kingdom

Mark scheme notes Underlying principle The mark scheme will clearly indicate the concept that is being rewarded, backed up by examples. It is not a set of model answers. 1. Mark scheme format 1.1 You will not see ‘wtte’ (words to that effect). Alternative correct wording should be credited in every answer unless the MS has specified specific words that must be present. Such words will be indicated by underlining e.g. ‘resonance’ 1.2 Bold lower case will be used for emphasis e.g. ‘and’ when two pieces of information are needed for 1 mark. 1.3 Round brackets ( ) indicate words that are not essential e.g. “(hence) distance is increased”. 1.4 Square brackets [ ] indicate advice to examiners or examples e.g. [Do not accept gravity] [ecf]. 2. Unit error penalties 2.1 A separate mark is not usually given for a unit but a missing or incorrect unit will normally mean that the final calculation mark will not be awarded. 2.2 This does not apply in ‘show that’ questions or in any other question where the units to be used have been given, for example in a spreadsheet. 2.3 The mark will not be awarded for the same missing or incorrect unit only once within one clip in epen. 2.4 Occasionally, it may be decided not to insist on a unit e.g the candidate may be calculating the gradient of a graph, resulting in a unit that is not one that should be known and is complex. 2.5 The mark scheme will indicate if no unit error is to be applied by means of [no ue]. 3. Significant figures 3.1 Use of too many significant figures in the theory questions will not be prevent a mark being awarded if the answer given rounds to the answer in the MS. 3.2 Too few significant figures will mean that the final mark cannot be awarded in ‘show that’ questions where one more significant figure than the value in the question is needed for the candidate to demonstrate the validity of the given answer. 3.3 The use of one significant figure might be inappropriate in the context of the question e.g. reading a value off a graph. If this is the case, there will be a clear indication in the MS. 3.4 The use of g = 10 m s-2 or 10 N kg-1 instead of 9.81 m s-2 or 9.81 N kg- 1 will mean that one mark will not be awarded. (but not more than once per clip). Accept 9.8 m s-2 or 9.8 N kg-1 3.5 In questions assessing practical skills, a specific number of significant figures will be required e.g. determining a constant from the

gradient of a graph or in uncertainty calculations. The MS will clearly identify the number of significant figures required. 4. Calculations 4.1 Bald (i.e. no working shown) correct answers score full marks unless in a ‘show that’ question. 4.2 If a ‘show that’ question is worth 2 marks. then both marks will be available for a reverse working; if it is worth 3 marks then only 2 will be available. 4.3 use of the formula means that the candidate demonstrates substitution of physically correct values, although there may be conversion errors e.g. power of 10 error. 4.4 recall of the correct formula will be awarded when the formula is seen or implied by substitution. 4.5 The mark scheme will show a correctly worked answer for illustration only. 5. Quality of Written Communication 5.1 Indicated by QoWC in mark scheme. QWC – Work must be clear and organised in a logical manner using technical wording where appropriate. 5.2 Usually it is part of a max mark, the final mark not being awarded unless the QoWC condition has been satisfied. 6. Graphs 6.1 A mark given for axes requires both axes to be labelled with quantities and units, and drawn the correct way round. 6.2 Sometimes a separate mark will be given for units or for each axis if the units are complex. This will be indicated on the mark scheme. 6.3 A mark given for choosing a scale requires that the chosen scale allows all points to be plotted, spreads plotted points over more than half of each axis and is not an awkward scale e.g. multiples of 3, 7 etc. 6.4 Points should be plotted to within 1 mm.  Check the two points furthest from the best line. If both OK award mark.  If either is 2 mm out do not award mark.  If both are 1 mm out do not award mark.  If either is 1 mm out then check another two and award mark if both of these OK, otherwise no mark. For a line mark there must be a thin continuous line which is the best-fit line for the candidate’s results. Question Number Acceptable answers Additional guidance Mark 1 The only correct answer is B because the power of a diverging lens is negative, so the total power = 9.4 D – 4.2 D = 5.2 D A is not correct because the total power should be obtained from (9.4 D – 4.2 D), but this is (9.4 D + 4.2 D) C is not correct because this is (4.2 D – 9.4 D) using negative power for a converging lens and positive for a diverging lens where it should be the opposite so that (9.4 D – 4.2 D) is used D is not correct because – 13.6 D = − 9.4 D – 4.2 D, as if both lenses are diverging, which is not the case 1 2 The only correct answer is D because pressure is proportional to absolute temperature and inversely proportional to volume, so the effect of the volume change is to increase the pressure by 3/2 and the effect of the temperature change is to increase the pressure by 6/5, and 18/10 = 9/5 A is not correct because a pressure of 5/9 p would depend on pressure being proportional to volume and inversely proportional to absolute temperature rather than being proportional to absolute temperature and inversely proportional to volume B is not correct because this assumes that pressure is proportional to both volume and absolute temperature, giving an answer of 4/5 p, instead of assuming that pressure is proportional to absolute temperature and inversely proportional to volume C is not correct because this assumes that pressure is inversely proportional to both volume and absolute temperature, giving an answer of 5/4 p, instead of assuming that pressure is proportional to absolute temperature and inversely proportional to volume 1 3 The only correct answer is B because the gradient of this graph is change in length ÷ change in force and the change in length is the same as the change in extension, so the gradient is equal to stiffness A is not correct because a graph of extension against force will have a gradient of 1/k C is not correct because a graph of stress against strain will have a gradient equal to the Young modulus for the sample D is not correct because a graph of strain versus length is equivalent to a graph of extension versus (length)2, so it does not have a gradient equal to k 1 4 The only correct answer is B because at angles less than or equal to the critical angle not all of the light is reflected internally such that angle of incidence is equal to the angle of reflection A is not correct because total internal reflection occurs at angles greater than the critical angle but at the critical angle the angle of refraction is 90 degrees, so the reflection is not total C is not correct because internal reflection is not total at angles less than the critical angle D is not correct because internal reflection is not total at angles less than the critical angle 1 5 The only correct answer is B because alpha is the most ionising and gamma is the most penetrating A is not correct because although alpha is the most ionising, gamma and not alpha is the most penetrating C is not correct because alpha, not gamma, is the most ionising and gamma, not alpha is the most penetrating D is not correct because alpha, not gamma is the most ionising, even though gamma is the most penetrating 1 6 The only correct answer is A because, using Einstein’s photoelectric equation, hf = φ + ½ mv2 max, since the work function is constant, an increase in frequency results in an increase in the maximum kinetic energy of the photoelectrons B is not correct because, using Einstein’s photoelectric equation, hf = φ + ½ mv2 max, intensity has no effect on the maximum kinetic energy of the photoelectrons, just the rate at which they are emitted C is not correct because , using Einstein’s photoelectric equation, hf = φ + ½ mv2 max, and since the work function is equal to (the Planck constant × threshold frequency), a higher threshold frequency will lead to a lower maximum kinetic energy of the photoelectrons D is not correct because, using Einstein’s photoelectric equation, hf = φ + ½ mv2 max, a higher work function will lead to a lower maximum kinetic energy of the photoelectrons 1 7 The only correct answer is C emitting an alpha particle decreases the nucleon number by 4 and decreases the proton number by 2 A is not correct because emitting an alpha particle decreases the nucleon number by 4 and decreases the proton number by 2, but here the nucleon number has been increased by 2 and the proton number has been increased by 4 B is not correct because emitting an alpha particle decreases the nucleon number by 4 and decreases the proton number by 2, but here the nucleon number has been increased by 4 and the proton number has been increased by 2 D is not correct because emitting an alpha particle decreases the nucleon number by 4 and decreases the proton number by 2, but here the nucleon number has been decreased by 2 and the proton number has been decreased by 4 1 8 The only correct answer is D because velocity is equal to the gradient of the displacement-time graph A is not correct because velocity is equal to the gradient of the displacement-time graph, but here velocity is shown as proportional to −1 times the displacement B is not correct because velocity is equal to the gradient of the displacement-time graph, but here velocity is shown as −1 times the gradient C is not correct because velocity is equal to the gradient of the displacement-time graph, but here velocity is shown as proportional to the displacement 1 9 The only correct answer is D because magnification is numerically equal to image distance divided by object distance A is not correct because magnification is numerically equal to image distance divided by object distance, but this is focal length divided by object distance B is not correct because magnification is numerically equal to image distance divided by object distance, but this is object distance divided by image distance C is not correct because magnification is numerically equal to image distance divided by object distance, but this is object distance divided by focal length 1 10 The only correct answer is C because the maximum order reached corresponds to the highest integer value less than or equal to line spacing divided by wavelength, which is 4, and there are that many orders either side of the maximum plus a central order A is not correct because the maximum order reached corresponds to the highest integer value less than or equal to line spacing divided by wavelength, which is 4, and there are that many orders either side of the maximum plus a central order, but this answer only gives the number of orders on one side of the central order B is not correct because the maximum order reached corresponds to the highest integer value less than or equal to line spacing divided by wavelength, which is 4, but this order rounds 4.7 to 5 and doesn’t consider the central maximum or that there are orders on either side D is not correct because the maximum order reached corresponds to the highest integer value less than or equal to line spacing divided by wavelength, which is 4, but this order rounds 4.7 to 5 and then adds the orders on the other side and the central maximum 1 (Total for Multiple Choice Questions = 10 marks) (Total for Question 11 = 5 marks) Question Number Acceptable answers Additional guidance Mark 11(a)  Equates right hand sides  Final Ek formula (½ m <c2> = 3/2 kT) and k is constant (1) (1) 2 11( b)  Use of ½ m <c2> = 3/2 kT  with T in Kelvin  √<c2> = 497 m s−1 (1) (1) (1) Example of calculation ½ m <c2> = 3/2 kT ½ × 5.0 × 10−26 kg × <c2> = 3/2 × 1.38 × 10−23 J K−1 × 298 K <c2> = 247 000 m2 s−2 √<c2> = 497 m s−1 3 (Total for Question 12 = 6 marks) Question Number Acceptable answers Additional guidance Mark 12(a)  Use of λmaxT = 2.898 × 10−3 m K  … with sensible temperature expressed in Kelvin  E.g. λmax = 9.89 × 10−6 m for 293 K  Value is greater than max wavelength of red light so is in IR region Or (if starting from 700 nm)  Use of λmaxT = 2.898 × 10−3 m K  T = 4100 K  Comparison with stated sensible temperature (C or K)  Temperature is too high so wavelength greater than max wavelength of red light so is in IR region (1) (1) (1) (1) Example of calculation λmaxT = 2.898 × 10−3 m K λmax = 2.898 × 10−3 m K / 293 K λmax = 9.89 × 10−6 m MP4 consistent with calculation 4 12 (b)  Glass reflects (these wavelengths of) IR radiation  Glass does not transmit (these wavelengths of) IR radiation (1) (1) Accept IR does not pass through glass 2 (Total for Question 13 = 7 marks) Question Number Acceptable answers Additional guidance Mark 13(a)  Weight and drag force are equal for terminal velocity stated or implied  Quotes F = 6πηrv and mg = 4(πr3)ρg/3 and suitable working to obtain ݒൌ ଶ௚ఘ௥మ ଽఎ (1) (1) 2 13(b)  Use of ݒ ൌ ଶ௚ఘ௥మ ଽఎ  v = 760 (m s−1) (1) (1) Example of calculation v = 2 × 9.81 N kg−1 × 1.0 × 103 kg m−3 × (2.5 × 10−3 m)2 / 9 × 1.8 × 10−5 Pa s v = 757 m s−1 2 13(c)  Measured value much less than calculated value Max 2 from  The raindrop is moving very fast so Stokes’ law does not apply  Flow is not laminar so Stokes’ law does not apply  Raindrops not small so Stokes’ law does not apply  Raindrops not spherical so Stokes’ law does not apply  Argument based on increased upward force if upthrust taken into account so it doesn’t apply (1) (1) (1) (1) (1) (1) 3 (Total for Question 14 = 6 marks) Question Number Acceptable answers Additional guidance Mark 14  Use of EK = p2 / 2m  Use of λ = h/p  λ =5.0 × 10−11 (m) calculated from EK Or EK = 9.7 × 10−17 (J) calculated from λ =5.0 × 10−11 m Or p = 1.3 × 10−23 (kg m s−1) calculated from EK and p = 1.3 × 10−23 (kg m s−1) calculated from λ =5.0 × 10−11 m  path difference at X is λ/2 Or path difference at Y is λ  (electron) waves at X are in antiphase Or (electron) waves at Y are in phase  at X destructive interference/superposition takes place Or at Y constructive interference/superposition takes place (1) (1) (1) (1) (1) (1) MP1 accept use of p = mv and Use of Ek = ½ mv2 MP4 accept (n + ½) λ or n λ respectively Example of calculation p = √(2 × 9.11 × 10−31 kg × 9.6 × 10−17 J) p = 1.32 × 10−23 kg m s−1 λ =6.63 × 10−34 Js / 1.32 × 10−23 kg m s−1 λ =5.0 × 10−11 m 6 Question Number Acceptable answers Additional guidance Mark 15(a)(i)  use of F = Gm1m2/r2 and use of F = mr2 Or use of F = Gm1m2/r2 and use of F = mv2/r  use of T= 2π/ Or use of T = 2πr/v  T = 12 hours Or F = 120 N by gravitational approach and centripetal force approach Or  = 1.45 × 10-4 radians s-1 by gravitational approach and circular motion approach Or height of orbit = 7700 km  Comparative statement consistent with their value(s) (1) (1) (1) (1) MP3 and 4 - for force and angular velocity, both approaches required Example of calculation T2 = 4π2r3 / G m1 T2 = 4π2 × (2 430 000 m + 7 690 000 m)3 / 6.67 × 10-11 N m2 kg-2 × 3.30 × 1023 kg T = 43115 s = 11.98 hours 4 15(a)(ii) Max 2  Allows satellite to get (much) closer to surface  So more detailed photographs/scans possible OR  Allows satellite to spend time further from the surface  So prevents exposure to prolonged heat from planet damaging probe OR  Satellite varies distance from surface  So it can take wide-angle and close-up pictures of the planet (1) (1) (1) (1) (1) (1) For each, the second marking point is dependent on the first. Award second marking point for other sensible advantages 2 (Total for Question 15 = 10 marks) Question Number Acceptable Answers Additional Guidance Mark 15(b)  Use of Vgrav = - GM/r Or Use of EP = - GMm/r with an assumed mass  Subtraction of potential at surface from potential at orbital height Or Subtraction of potential energy at surface from potential energy at orbital height  Use of difference in potential = EK/m = ½ v2 Or Use of difference in potential energy = EK = ½ mv2  v = 3948 m s−1 (1) (1) (1) (1) Example of calculation ΔV = 6.67 × 10−11 N m2 kg−2 × 3.30 × 1023 kg (1/2.43 × 106 m – 1/1.743 × 107 m) = 7.795 × 106 J kg−1 ½ mv2 = 7.795 × 106 J kg−1 × m Assume 1 kg v = 3948 m s−1 4 Question Number Acceptable Answers Additional Guidance Mark *16(a) This question assesses a student’s ability to show a coherent and logically structured answer with linkages and fully-sustained reasoning. Marks are awarded for indicative content and for how the answer is structured and shows lines of reasoning. The following table shows how the marks should be awarded for indicative content. Number of indicative marking points seen in answer Number of marks awarded for indicative marking points Max linkage mark available Max final mark 6 4 2 6 5 3 2 5 4 3 1 4 3 2 1 3 2 2 0 2 1 1 0 1 0 0 0 0 The following table shows how the marks should be awarded for structure and lines of reasoning. Guidance on how the mark scheme should be applied: The mark for indicative content should be added to the mark for lines of reasoning. For example, an answer with five indicative marking points which is partially structured with some linkages and lines of reasoning scores 4 marks (3 marks for indicative content and 1 mark for partial structure and some linkages and lines of reasoning). If there are no linkages between points, the same five indicative marking points would yield an overall score of 3 marks (3 marks for indicative content and no marks for linkages). 3 Indicati ve content  he pendulums have the same length, so they have the same time period/freq uency  he first pendulum causes forced oscillations of the second pendulum  The driving frequency equals the natural frequency  Resonance occurs, so there is maximum transfer of energy so the amplitude increases until all energy is transferred  The second pendulum then acts as a driver for the first pendulum Or the process repeats with energy transfer from B to A  When the lengths differ the driving frequency is not the natural frequency of the second pendulum so little energy transfer occurs Number of marks awarded for structure of answer and sustained line of reasoning Answer shows a coherent and logical structure with linkages and fully sustained lines of reasoning demonstrated throughout 2 Answer is partially structured with some linkages and lines of reasoning 1 Answer has no linkages between points and is unstructured 0 16(b)(i)  Pendulum A is π/2 ahead of pendulum B (1) 1 16(b)(ii)  T = 1.2 s from graph  Use of T = 2π√(l/g)  l = 0.36 m (1) (1) (1) T = 3.0 s / 2.5 oscillations 1.2 s = 2π√(l/9.81 N kg−1) l = 0.36 m 3 (Total for Question 16 = 10 marks Question Number Acceptable answers Additional guidance Mark 17(a)  Two rays correctly drawn including extrapolation  Completes diagram with image at 10.6 cm (range 9.0 cm to 12.0 cm)  Magnification = 3.8 (3.5 to 4.0)  Conclusion consistent with values for distance and M (1) (1) (1) (1) Acceptable rays:  from arrowhead on object through the optical centre of the lens  from arrowhead on object parallel to the axis up to the lens and then through the principal focus on the other side  from the principal focus on the same side and through the arrowhead on the object to the lens and then parallel to the axis Example of calculation M = image size / object size (accept use of distances) = 8.0 cm / 2.0 cm = 4.0 4 (Total for Question 17 = 11 marks) Question Number Acceptable Answers Additional Guidance Mark 17 (b)(i)  use of n1sin i1 = n2sin i2  with angle of incidence in plastic = 28°  angle of deviation = angle of refraction – angle of incidence  angle of deviation = 16° (1) (1) (1) (1) Example of calculation n1sin i1 = n2sin i2 1.47 sin (90° − 62°) = 1.00 sin i2 i2 = 43.6° angle of deviation = 44° − 28° = 16° 4 17 (b)(ii)  Going from the centre of the lens towards the edge the angle of incidence in the plastic increases  The angle of deviation increases  (So) all rays cross (the axis) at the principal focus (1) (1) (1) Accept focal point for principal focus 3 (Total for Question 18 = 11 marks) Question Number Acceptable answers Additional guidance Mark 18(a)(i)  Use of λt½ = ln 2  Use of dN/dt = −λN  N = 9.5 × 1010 (1) (1) (1) λ× (1600 × 365 × 24 × 60 × 60) s = ln 2 λ = 1.37 × 10−11 s−1 1.3 Bq = 1.37 × 10−11 s−1 × N N = 9.46 × 1010 3 18(a)(ii)  Use of A = A0 e –λt  Use of lnA = lnA0 –λt or equivalent  t = 8.58 × 1010 s = 2700 years (1) (1) (1) ecf λ calculated in (i) ln 0.4 = ln1.3 –1.37 × 10−11 s−1 × t t = 8.58 × 1010 s = 2721 years 3 18(b)  attempt to determine mass difference between radium and radon-plus-alpha  conversion to kg  Use of ΔE = c2Δm  Use of 1.6 × 10−19 factor  Answer = 4.87 (MeV) (1) (1) (1) (1) (1) Δm = 225.97713u – (221.97040u + 4.00151 u) = 5.22 × 10−3 u = 5.22 × 10−3 × 1.66 × 10−27 kg = 8.67 × 10−30 kg ΔE = c2Δm = (3 × 108 m s−1)2 × 8.67 × 10−30 kg = 7.80 × 10−13 J ΔE in MeV = 7.80 × 10−13 J ÷ 1.6 × 10−19 C = 4.87 MeV 5 Question Number Acceptable Answers Additional Guidance Mark 19(a)(i)  use of I = L / 4πd2  L = 6.53 × 1023 W  = 0.17% of Sun (1) (1) (1) Example of calculation 3.25 × 10−11 W m−2 = L / 4π(4.00 × 1016 m)2 L = 6.53 × 1023 W 6.53 × 1023 W / 3.85 × 1026 W = 0.17% 3 19(a)(ii)  use of L = σAT4  T = 3124 (K)  Statement relating calculated values of T and L to main sequence on H-R diagram (1) (1) (1) Example of calculation 6.53 × 1023 W = 5.67 × 10−8 W m−2 K−4 × 4π (9.81 × 107 m)2 × T4 T = 3124 K 3 19(b)  atoms/electrons have fixed/discrete/specific energy levels  electrons get excited by absorbing photons  energy of photon absorbed = difference in energy levels  only certain transitions possible, so only certain photon energies absorbed so only certain frequencies missing  the set of frequencies absorbed depends on the element (1) (1) (1) (1) (1) Answers in terms of emission spectrum can be awarded MP1, 4 and 5 5 19(c)  The star is viewed from two positions at 6 month intervals Or The star is viewed from opposite ends of its orbit diameter about the Sun  The change in angle/position of the star against backdrop of fixed stars is measured (1) (1) Marks may be obtained from suitably annotated diagrams e.g MP1 and MP2: 3  Trigonometry is used to calculate the distance to the star [Do not accept Pythagoras] Or The diameter/radius of the Earth’s orbit about the Sun must be known (1) (Total for Question 19 = 14 marks) Pearson Education Limited. 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Mark scheme notes Underlying principle The mark scheme will clearly indicate the concept that is being rewarded, backed up by examples. It is not a set of model answers. 1. Mark scheme format 1.1 You will not see ‘wtte’ (words to that effect). Alternative correct wording should be credited in every answer unless the MS has specified specific words that must be present. Such words will be indicated by underlining e.g. ‘resonance’ 1.2 Bold lower case will be used for emphasis e.g. ‘and’ when two pieces of information are needed for 1 mark. 1.3 Round brackets ( ) indicate words that are not essential e.g. “(hence) distance is increased”. 1.4 Square brackets [ ] indicate advice to examiners or examples e.g. [Do not accept gravity] [ecf]. 2. Unit error penalties 2.1 A separate mark is not usually given for a unit but a missing or incorrect unit will normally mean that the final calculation mark will not be awarded. 2.2 This does not apply in ‘show that’ questions or in any other question where the units to be used have been given, for example in a spreadsheet. 2.3 The mark will not be awarded for the same missing or incorrect unit only once within one clip in epen. 2.4 Occasionally, it may be decided not to insist on a unit e.g the candidate may be calculating the gradient of a graph, resulting in a unit that is not one that should be known and is complex. 2.5 The mark scheme will indicate if no unit error is to be applied by means of [no ue]. 3. Significant figures 3.1 Use of too many significant figures in the theory questions will not be prevent a mark being awarded if the answer given rounds to the answer in the MS. 3.2 Too few significant figures will mean that the final mark cannot be awarded in ‘show that’ questions where one more significant figure than the value in the question is needed for the candidate to demonstrate the validity of the given answer. 3.3 The use of one significant figure might be inappropriate in the context of the question e.g. reading a value off a graph. If this is the case, there will be a clear indication in the MS. 3.4 The use of g = 10 m s-2 or 10 N kg-1 instead of 9.81 m s-2 or 9.81 N kg-1 will mean that one mark will not be awarded. (but not more than once per clip). Accept 9.8 m s-2 or 9.8 N kg-1 3.5 In questions assessing practical skills, a specific number of significant figures will be required e.g. determining a constant from the gradient of a graph or in uncertainty calculations. The MS will clearly identify the number of significant figures required. 4. Calculations 4.1 Bald (i.e. no working shown) correct answers score full marks unless in a ‘show that’ question. 4.2 If a ‘show that’ question is worth 2 marks. then both marks will be available for a reverse working; if it is worth 3 marks then only 2 will be available. 4.3 use of the formula means that the candidate demonstrates substitution of physically correct values, although there may be conversion errors e.g. power of 10 error. 4.4 recall of the correct formula will be awarded when the formula is seen or implied by substitution. 4.5 The mark scheme will show a correctly worked answer for illustration only. 5. Quality of Written Communication 5.1 Indicated by QoWC in mark scheme. QWC – Work must be clear and organised in a logical manner using technical wording where appropriate. 5.2 Usually it is part of a max mark, the final mark not being awarded unless the QoWC condition has been satisfied. 6. Graphs 6.1 A mark given for axes requires both axes to be labelled with quantities and units, and drawn the correct way round. 6.2 Sometimes a separate mark will be given for units or for each axis if the units are complex. This will be indicated on the mark scheme. 6.3 A mark given for choosing a scale requires that the chosen scale allows all points to be plotted, spreads plotted points over more than half of each axis and is not an awkward scale e.g. multiples of 3, 7 etc. 6.4 Points should be plotted to within 1 mm.  Check the two points furthest from the best line. If both OK award mark.  If either is 2 mm out do not award mark.  If both are 1 mm out do not award mark.  If either is 1 mm out then check another two and award mark if both of these OK, otherwise no mark. For a line mark there must be a thin continuous line which is the best-fit line for the candidate’s results. (Total for Question 1 = 4 marks) Question Number Acceptable Answer Additional Guidance Mark 1(a)  Accuracy is (a measure of) how close a measured/calculated value is to the true value  Precision is (a measure of) the consistency of values obtained by repeated measurements (1) (1) 2 1(b)  Repeat readings can give very similar measurements so value precise  Value is not accurate because of a systematic error (1) (1) In MP2 accept zero error or calibration or parallax error for systematic error 2 Question Number Acceptable Answer Additional Guidance Mark 2(a)  The data has to be collected over a long period of time (1) 1 2(b)(i) An explanation that makes reference to the following:  The (terminal) p.d. (of the cell) decreases.  I = ௏ ோ, so the current decreases as the (external) resistance in the circuit stays constant (1) (1) In MP1 do not accept voltage for p.d. MP2: Accept I  V because (external) resistance in the circuit is constant, so the current decreases 2 2(b)(ii)  Use of I = ௏ ோ with R = 220 Ω or (220 + r)  (Sum of e.m.f. = sum of p.d. leading to) use of r = ሺߝ ି ௏ሻ ூ with I from MP1  r = 30  (allow answers in range 28  – 32 ) (1) (1) (1) Answer in range with no working shown scores MP3 only Example of calculation ܫൌ3.52 V 220 Ω ൌ1.60ൈ10‐2A ݎൌߝെܸ ܫ ൌሺ4.0‐3.52ሻ V 1.60ൈ10‐2A ൌ30.0 Ω 3 (Total for Question 2 = 6 marks) Question Number Acceptable Answer Additional Guidance Mark 3(a) Micrometer (screw gauge) (1) Accept digital calipers 1 3(b) Use of ܸൌπݎଶݐ Use of ߩൌ ௠ ௏ to find m Use of 0.5% to find total mass needed Number of discs = 10 (1) (1) (1) (1) Example of calculation ܸൌπݎଶݐൌπ ቆ1.3ൈ10‐2 m 2 ቇ 2 ൈ 2 ൈ10‐3m ∴ܸൌ2.65ൈ10‐7m3 ݉ൌߩܸൌ7900 kg m‐3ൈ2.65ൈ10‐7m3 ∴݉ൌ2.10ൈ10‐3kg 0.1 g ܯ ൌ0.5% ∴ܯൌ 0.1 g 0.5 100 ൗ ൌ20 g ∴number of discsൌ 20 g 2.10 g ൌ9.5 4 3(c) % uncertainty calculated for d or t % uncertainty calculated for cross sectional area Uncertainty in volume = (±) 4% [accept 4.0%] (1) (1) (1) Example of calculation uncertainty in ݀ = 0.1 mm 13 mm ×100% = 0.77% ∴uncertainty in ܣൌ2 ൈ0.77% ൌ1.5% uncertainty in ݐ= 0.05 mm 2 mm ×100% = 2.5% ∴ uncertainty in ܸ= 1.5% + 2.5% ൌ4.0 % 3 (Total for Question 3 = 8 marks) Question Number Acceptable Answer Additional Guidance Mark 4(a)  all I values should be recorded to the same number of decimal places Or all I values should be recorded to the same resolution (1) Do not accept same number of significant figures 1 4(b)  Ohm’s law requires current to be (directly) proportional to the (applied) p.d  Hence the line should pass through the origin  (There is scatter around the line drawn by the student so) the correct line may be a curve  Conclusion that this graph does not meet the conditions for Ohm’s law so the student’s statement is invalid (1) (1) (1) (1) For MP1 accept Ohm’s law requires ܫ∝ܸ In MP2, credit students who check values from graph to see if I doubles when V doubles In MP3, credit students who draw a curve onto the graph MP4 dependent upon MP2 OR MP3 4 4(c)  potential divider circuit diagram drawn as shown  correctly incorporated into the given circuit (1) (1) Switch not essential, accept potential divider with arrow drawn through resistor 2 (Total for Question 4 = 7 marks) Question Number Acceptable Answer Additional Guidance Mark 5(a)(i)  discards value for l3  ݈௠=85.7 ሺcmሻ (1) (1) MP2: answer to 1 d.p. only Example of calculation ݈௠ൌ85.5൅86.0൅85.5 3 ൌ85.7 cm 2 5(a)(ii)  Use of ܶൌ2ߨට ℓ ௚  ܶൌ1.86 s (1) (1) ECF from (i) MP2: accept T = 1.9 s Example of calculation ܶൌ2ߨඨℓ ݃ൌ2π ൈඨ0.857 m 9.81 m s‐2 ൌ1.86 s 2 5(b) Max 4 from 2 out of 3 pairs  The student should let the pendulum swing back and to before starting the stopwatch.  The first swing may be affected by the student pushing the bob as they release it  The student should use a (fiducial) marker at O  Easier to determine when it passes O  Time more oscillations  A longer time reduces (%) uncertainty (in T) (1) (1) (1) (1) (1) (1) For each pair, the second marking point is dependent on the first marking point MP4: Accept the pendulum travelling fastest when it passes O 4 (Total for Question 5 = 8 marks) Question Number Acceptable Answer Additional Guidance Mark 6(a)  See ܫൌܫଵ൅ܫଶ  Use of ܫൌ ௏ ோ with the same V for every term  Algebra to show ଵ ோ౛౜౜ൌ ଵ ோభ൅ ଵ ோమ (1) (1) (1) If student assumes that ܴଵൌܴଶൌܴ→ܴeff ൌ ோ ଶ , then MP1 and MP2 only For MP3, can assume algebra if equation seen Example of derivation ܫൌܫଵ൅ܫଶ ܫൌ ௏ ோ ∴ܸ ܴୣ୤୤ ൌܸ ܴଵ ൅ܸ ܴଶ 3 6(b)  If two springs are added in parallel the stretching force is shared between the springs  Hence the extension for a given force is half of what it would be for a single spring  So parallel combination has twice the stiffness of a single spring  For two identical resistors in parallel ଵ ோ౛౜౜ൌ ଵ ோ൅ ଵ ோ  So, adding two equal resistors in parallel halves the effective resistance of the combination  This is in contrast to the springs and so the student’s suggestion is invalid (dependent upon MP3 and MP5) (1) (1) (1) (1) (1) (1) MP3: Allow parallel combination has a greater stiffness than a single spring MP5: Allow adding two resistors in parallel decreases the effective resistance of the combination Equivalent points for MP4 – MP6  For two identical resistors in series, Reff = R + R  So adding two equal resistors in series doubles/increases the effective resistance  This is equivalent to parallel springs, so the student’s statement is invalid (dependent upon MP3 and MP5) 6 (Total for Question 6 = 9 marks) Question Number Acceptable Answer Additional Guidance Mark 7 (a)  Use of ∆ܧൌܮ∆݉  Use of ∆ܧൌ݉ܿ∆ߠ  L = 2.1 × 106 J kg–1 (1) (1) (1) In MP1 ∆m must be an attempt at a mass difference In MP2 allow any m, ∆θ from the question data Example of calculation heat transfer as steam condenses ൌ ሺ258.3‐255.0ሻൈ10‐3kg ൈܮ heat transfer as steam cools ൌ 3.3ൈ10‐3kgൈ4190 J kg‐1 K‐1ൈሺ100‐26ሻK heat transfer as water is heated ൌ 0.255 kgൈ4190 J kg‐1 K‐1ൈሺ26‐18.5ሻK ∴3.3ൈ10‐3kg ൈ L ൅1023 J ൌ 8013 J ∴ܮ ൌ8013 J‐1023 J 3.3ൈ10‐3 kg ൌ2.12 ൈ 106 J kg‐1 3 7(b)(i) An explanation that makes reference to the following:  To bring tubing up to temperature (of steam)  So steam only condenses in the cup Or steam doesn’t condense in the tubing (1) (1) 2 7(b)(ii)  Thermal energy will be transferred from the steam/tubing to the surroundings  Lagging/insulating/shortening the tubing (1) (1) Accept:  Thermal energy is transferred to the cup/ probe  These should have a small a heat capacity 2 (Total for Question 7 = 7 marks) Question Number Acceptable Answer Additional Guidance Mark 8(a)  Use of ݉ൌ ୧୫ୟ୥ୣ ୦ୣ୧୥୦୲ ୭ୠ୨ୣୡ୲ ୦ୣ୧୥୦୲  Use of ଵ ௨൅ ଵ ௩ൌ ଵ ௙  ∴ݒൌ28 m (1) (1) (1) For MP2 allow (u ≈ f, so) v = f × magnification Example of calculation ݒ ݑൌ݄i ݄o ൌ 0.75 m 4.0ൈ10‐3m ൌ187.5 1 ݑ൅1 ݒൌ1 ݂ ∴ݒ ݑ൅1 ൌݒ ݂ ∴ݒൌሺ187.5൅1ሻൈ15.0ൈ10‐2mൌ28.3 m 3 8(b) Either  (Inverse square law states that) the intensity is inversely proportional to the square of the distance Or ܫ∝ ଵ ௫మ with symbols defined  So ܫ→ ூ ସ and statement is incorrect Or intensity/brightness falls to one quarter and statement is incorrect OR  The power is spread over 4 times the area  So intensity/brightness falls to one quarter and statement is incorrect (1) (1) (1) (1) 2 (Total for Question 8 = 5 marks) Question Number Acceptable Answer Additional Guidance Mark 9(a) The student’s answer should  Include the idea that ‘threshold’ refers to a (minimum) frequency  state that photons have an energy given by hf  recognise that the energy used to release electrons is called the work function  include the idea that one photon is absorbed by one electron (1) (1) (1) (1) For MP1, accept that wavelength has to be below a certain ‘threshold’ Max 3 if the response is not a discussion of the student’s answer 4 9(b)  Photoelectric equation stated in words Or ݄݂ൌ ൅ ଵ ଶ݉ݒmax ଶwith  defined  Hence ܸ݁௦ൌ݄݂െ Or Ek max = ݄݂െ and Ek max = eVs  Compare with ݕൌ݉ݔ൅ܿ  So plot a graph of Vs against f Or plot a graph of eVs against f  Gradient ൌ ௛ ௘ Or gradient = h (1) (1) (1) (1) (1) MP1: Accept hf0 for  [with f0 defined], and Ek max for ଵ ଶ݉ݒmax ଶ MP2: ܸ݁௦ does not have to be the subject of the equation MP5 is dependent upon MP4 5 9(c)  Light consists of (particles called) photons  These particles: are discrete packets of energy Or are quanta of energy Or have momentum (1) (1) 2 (Total for Question 9 = 11 marks) Question Number Acceptable Answer Additional Guidance Mark 10(a)(i)  Use of ݒൌݑ൅ܽݐ  a = 0.81 m s–2 (1) (1) Example of calculation ܽൌݒെݑ ݐ ൌ97 m s‐1 120 s ൌ0.808 m s‐2 2 10(a)(ii)  Use of ݏൌݑݐ൅ ଵ ଶܽݐଶ Or v2 = u2 + 2as Or s = ቀ ሺ௩ା௨ሻ ଶ ቁݐ  Use of ݒavൌ s t with t = 320 s  vav = 75 m s–1 (1) (1) (1) Ecf acceleration from (a)(i) Example of calculation ݏൌݑݐ൅1 2 ܽݐଶൌ0.5ൈ0.808 m s‐2 ሺ120 sሻ2ൌ5820 m s2 = 29900 m – 5820 m = 24 080 m t = 440s – 120 s = 320 s ݒavൌ 24 080 m 320 s ൌ75.3 m s-1 3 10(b) An explanation that makes reference to the following:  (Magnetic levitation) reduces frictional forces (acting on the train as it moves)  This reduces the work done against friction Or this makes it more efficient Or there is less time to reach maximum speed Or there is a higher maximum speed (1) (1) MP1: accept removes/no frictional forces MP2: accept there is less fuel used Or less energy is wasted 2 *10(c) This question assesses a student’s ability to show a coherent and logically structured answer with linkages and fully sustained reasoning. Marks are awarded for indicative content and for how the answer is structured and shows lines of reasoning. The table shows how the marks should be awarded for indicative content and structure and lines of reasoning. IC Points IC Mark Max linkage mark avail. Max final mark 6 4 2 6 5 3 2 5 4 3 1 4 3 2 1 3 2 2 0 2 1 1 0 1 0 0 0 0 IC1: accept references to flux cutting Alternative indicative content for IC4 – IC5 4. The current is in the magnetic field produced by the magnet 5. The current experiences a magnetic force 6 Indicative content: 1. Removing the magnet from the ring changes the magnetic flux (linked with the ring) 2. This induces an e.m.f. (in the ring) 3. E.m.f. causes a current in the ring 4. Which produces a magnetic field 5. The magnetic fields interact/combine 6. This opposes the change, causing an attractive force to act (Total for Question 10 = 13 marks) Question Number Acceptable Answer Additional Guidance Mark 11(a) An explanation that makes reference to the following:  The astronauts are in free fall in the space craft Or The acceleration of the astronauts and the space station are the same  So there is no contact/reaction force acting on them (1) (1) For MP2, accept a statement that the gravitational force (weight) is the only force acting upon them Or all of their weight is used to provide the centripetal force /acceleration 2 11(b) The acceleration (of the platform) is:  (directly) proportional to displacement from equilibrium position  (always) acting towards the equilibrium position Or the idea that acceleration is in the opposite direction to displacement [An equation with symbols defined correctly is a valid response for both marks. e.g or ] (1) (1) Accept towards undisplaced point/fixed point/central point for equilibrium position. Accept a definition in terms of the force i.e. the (resultant) force (on the platform) is:  (directly) proportional to displacement from equilibrium position  (always) acting towards the equilibrium position Or the idea that force is in the opposite direction to displacement 2 11(c)(i)  (Use the stopwatch to) time a large number of oscillations to determine the time period  Calculate the frequency using ݂ൌ ଵ ் (1) (1) MP1: At least 5 oscillations required 2 11(c)(ii)  (Use the metre rule to) measure the max displacement of the platform (from the equilibrium position)  Calculate the maximum speed using ݒmax ൌ2ߨ݂ݔmax (1) (1) MP2: allow use of ݒmax ൌ ଶ஠ ்ݔmax 2 11(d) An explanation that makes reference to the following:  ܧ௞ൌ ଵ ଶ݉ݒଶ ܉ܖ܌ ݒmax ൌ2ߨ݂ݔmax  Max kinetic energy is quadrupled (1) (1) 2 (Total for Question 11 = 10 marks) x a   x F   *12(a) This question assesses a student’s ability to show a coherent and logically structured answer with linkages and fully sustained reasoning. Marks are awarded for indicative content and for how the answer is structured and shows lines of reasoning. The table shows how the marks should be awarded for indicative content and structure and lines of reasoning. IC Points IC Mark Max linkage mark avail. Max final mark 6 4 2 6 5 3 2 5 4 3 1 4 3 2 1 3 2 2 0 2 1 1 0 1 0 0 0 0 IC3 and IC4 must include a mention of the walls/container 6 Indicative content: 1. As the temperature of the gas increases the (average) speed/Ek of the atoms increases 2. Greater speed/Ek so the momentum of the atoms increases 3. The rate/frequency of collision of atoms with the container walls increases Or the time between collisions with the walls decreases 4. The rate of change of momentum at the walls increases 5. Rate of change of momentum is equal to the force 6. Pressure is ୤୭୰ୡୣ ୟ୰ୣୟ and the force (on the walls) is greater 12(b)(i)  Conversion of MeV to J  See Q1 = 79 × 1.6 × 10−19 and Q2 = 2 × 1.6 × 10−19  Use of ܸൌ ொ ସగఌబ௥ and ܹൌܸܳ  ݎൌ 4.1ൈ10‐14 m (1) (1) (1) (1) Example of calculation ܧαൌ5.5ൈ106eVൈ1.6ൈ10‐19J eV‐1ൌ8.8ൈ10‐13J 8.8ൈ10‐13Jൌ79ൈ1.6ൈ10‐19Cൈ2ൈ1.6ൈ10‐19C 4πൈ8.85ൈ10‐12 F m‐1 ݎ ݎൌ3.64ൈ10‐26 N m2 8.8ൈ10‐13 J ൌ4.1ൈ10‐14 m 4 12(b)(ii)  Electrons are behaving like waves  wavelengthൌ ௛ momentum  Electron wavelength must be similar to the atomic spacing in the foil (1) (1) (1) MP3: Accept electron wavelength must be similar to the distance between (adjacent) nuclei 3 12(c)  Electrons are excited to higher energy states / levels (by incident electrons)  An electron returns to the lower energy state / level resulting in the emission of a photon  The energy of the photon is equal to the difference of the energy states / levels  Large difference in energy states / levels so as E = hf, radiation is high frequency (1) (1) (1) (1) For MP1 and MP2 allow  Electrons knock electrons out of low energy levels  Electrons cascade down to fill up the levels 4 (Total for Question 12 = 17 marks) Question Number Acceptable Answer Additional Guidance Mark 13(a)  Log values calculated in table  Axes correctly labelled  Suitable scales  All points plotted correctly  Line of best fit drawn  Determine gradient using large triangle  Intercept read from graph  p = 5.0 – 5.2 (mm) Or p = (5.0 – 5.2) × 10−3 (m) q = 0.5 (1) (1) (1) (1) (1) (1) (1) (1) Accept log10 or ln (loge) log10 (D/mm) log10 N ln (D/mm) ln N 0.710 0.000 1.64 0.000 0.850 0.301 1.96 0.693 0.940 0.477 2.16 1.10 1.01 0.602 2.33 1.39 1.06 0.699 2.44 1.61 8 y = 0.5017x + 0.7053 0.70 0.75 0.80 0.85 0.90 0.95 1.00 1.05 1.10 0.00 0.10 0.20 0.30 0.40 0.50 0.60 0.70 0.80 log10 (D/mm) log10 N 13(b)(i)  absolute uncertainty in position × 2  % uncertainty = 0.2 % (1) (1) Example of calculation Absolute uncertainty = 2 × 0.005 mm = 0.01 mm ∴% uncertainty= 0.01 mm 5.13 mm ×100% = 0.2 % 2 13(b)(ii)  The edges of the dark circle are not clearly defined (1) Accept: the rings are not perfect circles 1 13(c)  Coherent waves have a constant phase relationship  Coherent waves have the same frequency  However, for each frequency present the (two) reflected waves are coherent  Hence with a non-monochromatic source, a set of dark rings for each frequency would be produced (1) (1) (1) (1) MP2-4: accept wavelength for frequency MP4: hence with a white light source you would see a set of coloured rings 4 (Total for Question 13 = 15 marks) Pearson Education Limited. Registered company number 872828 with its registered office at 80 Strand, London, WC2R 0RL, United Kingdom

Turn over 9 Question 4 continued _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ (Total for Question 4 is 3 marks)

Mark scheme notes Underlying principle The mark scheme will clearly indicate the concept that is being rewarded, backed up by examples. It is not a set of model answers. 1. Mark scheme format 1.1 You will not see ‘wtte’ (words to that effect). Alternative correct wording should be credited in every answer unless the MS has specified specific words that must be present. Such words will be indicated by underlining e.g. ‘resonance’ 1.2 Bold lower case will be used for emphasis e.g. ‘and’ when two pieces of information are needed for 1 mark. 1.3 Round brackets ( ) indicate words that are not essential e.g. “(hence) distance is increased”. 1.4 Square brackets [ ] indicate advice to examiners or examples e.g. [Do not accept gravity] [ecf]. 2. Unit error penalties 2.1 A separate mark is not usually given for a unit but a missing or incorrect unit will normally mean that the final calculation mark will not be awarded. 2.2 This does not apply in ‘show that’ questions or in any other question where the units to be used have been given, for example in a spreadsheet. 2.3 The mark will not be awarded for the same missing or incorrect unit only once within one clip in epen. 2.4 Occasionally, it may be decided not to insist on a unit e.g the candidate may be calculating the gradient of a graph, resulting in a unit that is not one that should be known and is complex. 2.5 The mark scheme will indicate if no unit error is to be applied by means of [no ue]. 3. Significant figures 3.1 Use of too many significant figures in the theory questions will not be prevent a mark being awarded if the answer given rounds to the answer in the MS. 3.2 Too few significant figures will mean that the final mark cannot be awarded in ‘show that’ questions where one more significant figure than the value in the question is needed for the candidate to demonstrate the validity of the given answer. 3.3 The use of one significant figure might be inappropriate in the context of the question e.g. reading a value off a graph. If this is the case, there will be a clear indication in the MS. 3.4 The use of g = 10 m s-2 or 10 N kg-1 instead of 9.81 m s-2 or 9.81 N kg- 1 will mean that one mark will not be awarded. (but not more than once per clip). Accept 9.8 m s-2 or 9.8 N kg-1 3.5 In questions assessing practical skills, a specific number of significant figures will be required e.g. determining a constant from the

gradient of a graph or in uncertainty calculations. The MS will clearly identify the number of significant figures required. 4. Calculations 4.1 Bald (i.e. no working shown) correct answers score full marks unless in a ‘show that’ question. 4.2 If a ‘show that’ question is worth 2 marks. then both marks will be available for a reverse working; if it is worth 3 marks then only 2 will be available. 4.3 use of the formula means that the candidate demonstrates substitution of physically correct values, although there may be conversion errors e.g. power of 10 error. 4.4 recall of the correct formula will be awarded when the formula is seen or implied by substitution. 4.5 The mark scheme will show a correctly worked answer for illustration only. 5. Quality of Written Communication 5.1 Indicated by QoWC in mark scheme. QWC – Work must be clear and organised in a logical manner using technical wording where appropriate. 5.2 Usually it is part of a max mark, the final mark not being awarded unless the QoWC condition has been satisfied. 6. Graphs 6.1 A mark given for axes requires both axes to be labelled with quantities and units, and drawn the correct way round. 6.2 Sometimes a separate mark will be given for units or for each axis if the units are complex. This will be indicated on the mark scheme. 6.3 A mark given for choosing a scale requires that the chosen scale allows all points to be plotted, spreads plotted points over more than half of each axis and is not an awkward scale e.g. multiples of 3, 7 etc. 6.4 Points should be plotted to within 1 mm.  Check the two points furthest from the best line. If both OK award mark.  If either is 2 mm out do not award mark.  If both are 1 mm out do not award mark.  If either is 1 mm out then check another two and award mark if both of these OK, otherwise no mark. For a line mark there must be a thin continuous line which is the best-fit line for the candidate’s results. Question Number Answer Mark 1 D power kg m2 s−3 1 Incorrect Answers: A – Coulombs is not an SI base unit B – Incorrect, as the unit for charge in SI base units is A s C – J s1 is not in SI base units 2 B 1 Incorrect Answers: A – this is the trajectory for a ball kicked at an angle greater than , at a greater speed C – this is the trajectory for a ball kicked at an angle greater than , at a greater speed D – this is the trajectory for a ball kicked at an angle lower than , at a lower speed 3 A amplitude of the vibrations of the lattice ions 1 Incorrect Answers: B – the distance travelled by charge carriers between collisions will decrease if the temperature increases C – the drift velocity of conduction electrons will decrease if the temperature increases D – the number of conduction electrons per unit volume will remain constant if the temperature increases 4 B filament bulb 1 Incorrect Answers: A – this is not the graph for a diode C – this is not the graph for an ohmic resistor D – this is not the graph for a thermistor 5 C ± 0.1 % 1 Incorrect Answers: A – the calculation has not been multiplied by 100 to give the % uncertainty i.e. ଴.ଵ ଽଷ = 0.001 B – the uncertainty in mm has not been converted to cm and the calculation has not been multiplied by 100 i.e. ଵ ଽଷ = 0.01 D – the uncertainty in mm has not been converted to cm i.e. ଵ ଽଷൈ100 = 1 6 C 190 kJ 1 Incorrect Answers: A – The velocity was not squared when using the formula Ek = ½ mv2 e.g. ½ (1.2  103)(18) = 11 kJ B – The velocity was not squared and the ½ was omitted when using the formula Ek = ½ mv2 e.g. (1.2  103)(18) = 22 kJ D – The ½ was omitted when using the formula Ek = ½ mv2 e.g. (1.2  103)(18)2 = 390 kJ 7 A 1 Incorrect Answers: B – the ammeter would measure the current in the cell, but the voltmeter would not be measuring the p.d. across the cell C – the voltmeter would measure the p.d. across the cell but the ammeter would not be measuring the current in the cell D – the voltmeter would measure the p.d. across the cell but the ammeter would not be measuring the current in the cell 8 C 42 m 1 Incorrect Answers: A – 141 m is ¾ of the internal circumference of the track (¾  2    30 = 141 m) B – 141 m is ¼ of the internal circumference of the track (¼  2    30 = 47 m) D – 30 m (the radius) is the displacement travelled in one direction (downwards from the start position) (Total for Multiple Choice Questions = 8 marks) (Total for Question 9 = 6 marks) Question Number Acceptable answers Additional guidance Mark 9(a)  A statement applying the conservation of energy to the circuit  Use of Ohm’s law for each term /individual pd leading to the cancelling of currents in the equation  RT = R1 + R2 + R3 (1) (1) (1) Example of calculation ε = V1 + V2 +V3 ε = IRT = IR1 + IR2 + IR3 RT = R1 + R2 + R3 3 9(b) Either  Use of equation(s) to determine the total resistance of the voltmeter and 40 Ω resistor in parallel (34.3 Ω) i.e. potential divider formula or Ohm’s law  Use of ଵ ோ೅ൌ ଵ ோభ൅ ଵ ோమ  Rv = 240 (Ω) Or  Use of Ohm’s law to determine the current through voltmeter (0.0075 A) i.e. current in 40 Ω resistor calculated (0.045 Ω) and subtracted from current in 80 Ω resistor (0.0525 Ω)  Use of Ohm’s law with 1.8 V to calculate the resistance of the voltmeter  Rv = 240 (Ω) (1) (1) (1) (1) (1) (1) 1 34.29 Ω ൌ 1 40 Ω ൅1 ܴ௏ Example of calculation ቀ ோ ଼଴ ஐାோቁ 6 V = 1.8 V R = 34.29 Ω RV = 240.2 Ω 3 (Total for Question 10 = 7 marks) Question Number Acceptable answers Additional guidance Mark 10(a)  Weight of the picture is equal to the vertical component of tension  mg/2 = Tcosθ where θ is the angle between the wire and the vertical  As the angle (to the vertical) is the smaller in arrangement 1, the cosine of the angle will be larger  Arrangement 1 as the tension in the wire is lower than in arrangement 2 (1) (1) (1) (1) MP1: may be implied in an equation, accept mg = Tcosθ MP1/2: mg/2 = Tcosθ with θ defined scores 2 marks MP2/3: the angle used must be defined in words or on the diagram. Answer can be in terms of sin θ if θ defined as the angle between the wire and the horizontal e.g.MP2: mg = 2Tsinθ MP3: angle larger in 1 so sine of angle is larger MP4 conditional MP2 or MP3 4 10(b)  The weight does not act through the nail/pivot Or the centre of gravity is not in line/below the nail/pivot Or there is a perpendicular distance between the weight and the nail/pivot  There is now a moment of the weight Or the anticlockwise moment is greater than the clockwise moment  The idea that the picture stops moving when the c of g is below the nail (1) (1) (1) Accept centre of mass for centre of gravity (Allow annotations to a diagram with additional explanation for MP1/3) MP3 Accept: the turning moment being 0 Or the clockwise moments equal to the anti-clockwise moments 3 (Total for Question 11 = 8 marks Question Number Acceptable answers Additional guidance Mark 11(a)  Use of v = s/t for the horizontal motion Or see uh = 25 m s−1  Use of s = ut + ½ at2 with s = 0 Or v = u + at with t = 1.0 s Or see uv = 9.81 m s−1  Combining of horizontal velocity and vertical velocity expressions Or see tanθ = ቀ ଽ.଼ଵ ଶହቁ  θ = 21° (1) (1) (1) (1) Example of calculation horizontal motion: uh = ହ଴ ୫ ଶ.଴ ୱ= 25 m s−1 vertical motion: 0 = (uv) (2.0 s) + ½(− 9.81 m s2)(2.0 s)2 uv = 9.81 m s−1 tanθ = ௨౬ ௨౞ = ଽ.଼ଵ ୫ ୱషభ ଶହ ୫ ୱషభ θ = 21.4° 4 11(b)  Construction of vector diagram with 2 N/weight and 9N / catapult force labelled and all three directions shown  Correct scaling of 9 N and 2 N forces  Magnitude = 7.6 N to 8.0 N  Direction = 27° to 31° (1) (1) (1) (1) MP2: Award if MP3 awarded. Otherwise, the ratio of the lengths should lie between 4.3 and 4.8 (if no diagram, only MP3/4 can be awarded if answers obtained by calculation) 4 9.0 N 2.0 N (Total for Question 12 = 7 marks) Question Number Acceptable answers Additional guidance Mark 12(a)  Use of p = mv  Use of principle of conservation of momentum  Magnitude of velocity = 0.2 m s−1 with direction to the left (1) (1) (1) MP1: see 0.3 m, 0.7 m or 2mv MP3: accept ‘in the initial direction of glider 2’ for ‘to the left’ Example of calculation (taking the initial direction of glider 1 as positive) 0.3 m – 0.7 m = 2mv v = − 0.2 m s−1 3 12(b)  Glider 1 exerts this force on glider 2, so according to N3  Glider 2 will exert an (equal and) opposite force on glider 1  There is now a resultant force on glider 1  Glider 1 accelerates according to N1 Or glider 1 now moves to the left according to N1 (1) (1) (1) (1) 4 (Total for Question 13 = 6 marks) Question Number Acceptable answers Additional guidance Mark 13(a)  The (voltmeter) reading will increase  Resistance increases (with length) Or resistance  length (1) (1) MP2: accept idea of a potential divider i.e. the ratio of the of BC to the total length AD will be greater, so the proportion of the total voltage will be greater ( ୆େ ୅ୈ V) MP2: Do not award if there is also a reference to resistivity increasing 2 13(b)  Use of V = IR  Use of R = ρl/A  (Min) resistivity = 160 (Ω m) Or (max) resistivity = 730 (Ω m)  Compacted clay pathways and limestone are present in the soil (1) (1) (1) (1) Example of calculation Rmin = ଵ.଼ ୚ ଽ.ହ ൈଵ଴షయ ୅ = 189.5 Ω Rmax = ଼.଴ ୚ ଽ.ହ ൈଵ଴షయ ୅ = 842.1 Ω ρmin= ோ஺ ௟ = ଵ଼ଽ.ହ  ൈ଴.଺ହ଴ ୫మ ଴.଻ହ ୫ = 164.2 Ω m ρmax= ோ஺ ௟ = ଼ସଶ.ଵ  ൈ଴.଺ହ଴ ୫మ ଴.଻ହ ୫ = 729.8 Ω m conclusion to be consistent with calculated values 4 Question Number Acceptable answers Additional guidance Mark 14(a)  Attempt to find area under the graph  Length from 18 000 m to 20 000 m  Comparison of calculated value to 23 km e.g. The length is long enough (1) (1) (1) MP1: use of triangles or counting squares MP3: conclusion to be consistent with calculated value Example of calculation Area under the graph (counting large squares) = 18.7 × 100 m s1 × 10 s = 18 700 m 3 14 (b)(i)  26 - 28 s (1) Unit required 1 14(b)(ii)  Use of gradient of the graph between 28 and 46 s  Acceleration = 16 - 17 m s−2 (1) (1) Example of calculation Gradient of tangent = ସଽ଴ ୫ ୱషభି ଴ ୫ ୱషభ ହଶୱିଶଶୱ Acceleration = 16.3 m s−2 2 14(b)(iii)  Use of ΣF = ma using a from (ii)  ΣF = (89 + 120) × 103 N − frictional force  Frictional force = 80 kN to 84 kN (full ecf for acceleration) (1) (1) (1) Example of calculation (89 + 120) × 103 N - F = 7790 kg × 16.3 m s−2 F = 82.0 × 103 N 3 (Total for Question 14 = 15 marks) *14(c) This question assesses a student’s ability to show a coherent and logically structured answer with linkages and fully-sustained reasoning. Marks are awarded for indicative content and for how the answer is structured and shows lines of reasoning. The following table shows how the marks should be awarded for indicative content. Number of indicative marking points seen in answer Number of marks awarded for indicative marking points 6 4 5 - 4 3 3 - 2 2 1 1 0 0 Indicative content  At greater speed, the drag force is greater  Resultant force decreases Or acceleration decreases  When the rocket is started the (resultant) force/thrust increases  The mass/weight of the car decreases as fuel is used up  Increasing the acceleration (for a given applied force)  When the brakes are applied, there is a deceleration Or when the brakes are applied the resultant force is in the opposite direction Or deceleration (due to drag forces) decreases due to smaller drag forces acting on the car at lower speeds The following table shows how the marks should be awarded for structure and lines of reasoning Number of marks awarded for structure of answer and sustained line of reasoning Answer shows a coherent and logical structure with linkages and fully sustained lines of reasoning demonstrated throughout 2 Answer is partially structured with some linkages and lines of reasoning 1 Answer has no linkages between points and is unstructured 0 6 Question Number Acceptable answers Additional guidance Mark 15(a)  No repeats Or range too small Or readings should be repeated (and mean calculated) Or separations less than 0.50 m should be used Or separations greater than 1.00 m should be used Or distances should be recorded to the nearest mm Or smaller intervals (for the separation) should be used (1) Do not credit: more readings should be taken references to the number of decimal places 1 15(b)(i)  Axes labelled with quantities and units  Suitable scale  Correct plotting Microphone separation/ m Time interval / ms 1.00 3.2 0.90 2.8 0.80 2.4 0.70 2.1 0.60 1.9 0.50 1.5  Line of best fit (1) (1) (1) (1) Graph may be landscape and axes either way round MP2: Suitable scale – plotted points take up at least half the graph paper in both directions 4 0.0 0.2 0.4 0.6 0.8 1.0 1.2 0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 Microphone separation / m Time interval / ms (Total for Question 15 = 10 marks) 15(b)(ii)  Use of gradient Or use of 1/gradient (if axes swapped)  Speed of sound = 280 - 310 m s−1 (1) (1) MP1: only award for a y/x calculation if the graph passes through the origin Example of calculation Speed of sound = ଴.ଽ଺୫ି଴.଴ସ୫ ሺଷ.଴ି଴.଴ሻൈଵ଴షయ୫ = 307 m s−1 2 15(c)  Value obtained is lower than accepted value Max 2  Measured times are too small  Large percentage uncertainty for time  Parallax when reading from the metre rule  The distance not measured to same position on each microphone (1) (1) (1) (1) (1) MP1: accept answer consistent with calculated value but do not accept different for greater/lower Ignore references to temperature 3 Question Number Acceptable answers Additional guidance Mark 16(a)(i)  Oscillations/vibrations (of the light) are in one plane only  Plane includes the direction of energy transfer Or Plane includes the direction of travel/propagation (1) (1) Accept:  Oscillations/vibrations (of the light) are in one direction only  perpendicular to the direction of propagation/travel Or perpendicular to the direction of energy transfer Allow labelled diagrams for each marking point 2 16(a)(ii)  The (angle of polarisation of the) filters are 90° to one another Either  If plane of polarisation of light is rotated (by 90°) when it passes through the crystal (with no p.d. across it), it can still pass through the upper filter Or  If plane of polarisation of light is not rotated (by 90°) when it passes through the crystal (with a p.d. across it), it cannot pass through the upper filter (1) (1) (1) MP2: it must be clear as to whether the candidate is describing a light screen or a dark screen 2 16(b)(i)  Use of I = P/A  P = 0.014 W (1) (1) Example of calculation P = 7.8 W m−2 × 1.8 × 10−3 m2 P = 0.014 W 2 16(b)(ii)  Use of P = VI  Use of efficiency = ୳ୱୣ୤୳୪ ୮୭୵ୣ୰ ୭୳୲୮୳୲ ୲୭୲ୟ୪ ୮୭୵ୣ୰ ୧୬୮୳୲  Efficiency = 0.19 or 0.20 or 19 % or 20 % (1) (1) (1) Example of calculation Power input into LED = 3.6 V × 20 × 10−3 A = 0.072 W Efficiency = ଴.଴ଵସ ୛ ଴.଴଻ଶ୛ = 0.194 ecf from (b)(i) for the power output of LED 3 (Total for Question 16 = 13 marks) 16(c)  X is brittle at greater stresses/forces  Y will deform plastically at greater stresses/forces  The Young modulus for X is greater than Y  A screen made from material Y would be more suitable as it is more flexible (1) (1) (1) (1) Accept converse for MP3 and MP4 MP4: accept less stiff for flexible. 4 Pearson Education Limited. 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Mark scheme notes Underlying principle The mark scheme will clearly indicate the concept that is being rewarded, backed up by examples. It is not a set of model answers. 1. Mark scheme format 1.1 You will not see ‘wtte’ (words to that effect). Alternative correct wording should be credited in every answer unless the MS has specified specific words that must be present. Such words will be indicated by underlining e.g. ‘resonance’ 1.2 Bold lower case will be used for emphasis e.g. ‘and’ when two pieces of information are needed for 1 mark. 1.3 Round brackets ( ) indicate words that are not essential e.g. “(hence) distance is increased”. 1.4 Square brackets [ ] indicate advice to examiners or examples e.g. [Do not accept gravity] [ecf]. 2. Unit error penalties 2.1 A separate mark is not usually given for a unit but a missing or incorrect unit will normally mean that the final calculation mark will not be awarded. 2.2 This does not apply in ‘show that’ questions or in any other question where the units to be used have been given, for example in a spreadsheet. 2.3 The mark will not be awarded for the same missing or incorrect unit only once within one clip in epen. 2.4 Occasionally, it may be decided not to insist on a unit e.g the candidate may be calculating the gradient of a graph, resulting in a unit that is not one that should be known and is complex. 2.5 The mark scheme will indicate if no unit error is to be applied by means of [no ue]. 3. Significant figures 3.1 Use of too many significant figures in the theory questions will not be prevent a mark being awarded if the answer given rounds to the answer in the MS. 3.2 Too few significant figures will mean that the final mark cannot be awarded in ‘show that’ questions where one more significant figure than the value in the question is needed for the candidate to demonstrate the validity of the given answer. 3.3 The use of one significant figure might be inappropriate in the context of the question e.g. reading a value off a graph. If this is the case, there will be a clear indication in the MS. 3.4 The use of g = 10 m s-2 or 10 N kg-1 instead of 9.81 m s-2 or 9.81 N kg-1 will mean that one mark will not be awarded. (but not more than once per clip). Accept 9.8 m s-2 or 9.8 N kg-1 3.5 In questions assessing practical skills, a specific number of significant figures will be required e.g. determining a constant from the gradient of a graph or in uncertainty calculations. The MS will clearly identify the number of significant figures required. 4. Calculations 4.1 Bald (i.e. no working shown) correct answers score full marks unless in a ‘show that’ question. 4.2 If a ‘show that’ question is worth 2 marks. then both marks will be available for a reverse working; if it is worth 3 marks then only 2 will be available. 4.3 use of the formula means that the candidate demonstrates substitution of physically correct values, although there may be conversion errors e.g. power of 10 error. 4.4 recall of the correct formula will be awarded when the formula is seen or implied by substitution. 4.5 The mark scheme will show a correctly worked answer for illustration only. 5. Quality of Written Communication 5.1 Indicated by QoWC in mark scheme. QWC – Work must be clear and organised in a logical manner using technical wording where appropriate. 5.2 Usually it is part of a max mark, the final mark not being awarded unless the QoWC condition has been satisfied. 6. Graphs 6.1 A mark given for axes requires both axes to be labelled with quantities and units, and drawn the correct way round. 6.2 Sometimes a separate mark will be given for units or for each axis if the units are complex. This will be indicated on the mark scheme. 6.3 A mark given for choosing a scale requires that the chosen scale allows all points to be plotted, spreads plotted points over more than half of each axis and is not an awkward scale e.g. multiples of 3, 7 etc. 6.4 Points should be plotted to within 1 mm.  Check the two points furthest from the best line. If both OK award mark.  If either is 2 mm out do not award mark.  If both are 1 mm out do not award mark.  If either is 1 mm out then check another two and award mark if both of these OK, otherwise no mark. For a line mark there must be a thin continuous line which is the best-fit line for the candidate’s results. Question Number Answer Ma rk 1 C ࡲ࡭ ൗ ∆࢞࢞ ൗ (stress/strain) 1 Incorrect Answers: A incorrect arrangement B incorrect arrangement D incorrect arrangement for strain 2 C ࢚ൌ ૢ૙ൈ૛ ૚૞૙૙ 1 Incorrect Answers: A -no factor of 2 B - no factor of 2 and incorrect equation D - incorrect equation 3 D – Two points ࣅ ૛ apart oscillate with the same amplitude 1 Incorrect Answers: A – Points on a wave do not all oscillate in phase B – A node is formed from destructive interference C – Stationary waves may also be formed from longitudinal waves 4 A – amount of energy of a photon needed to release an electron 1 Incorrect Answers: B- an electron does not release a photon C – reference to frequency incorrect D - reference to frequency incorrect and electron does not release a photon 5 B – decreasing the speed of the electrons in the beam 1 Incorrect Answers: A – the number of electrons does not affect the angle C - the number of electrons does not affect the angle D – increasing the speed would decrease the distance between the rings 6 C - ࣊ ૞ ( ૚ ૚૙ ൈ૛࣊ሻ 1 Incorrect Answers: A – the waves would be in antiphase B – the waves would be a quarter of a wave out of phase D – the waves would be a 20th of a wave out of phase 7 B - ࢜ൌ૛ࢌ࢒ 1 Incorrect Answers: A – wavelength is 2l C - wavelength is 2l D - wavelength is 2l 8 C - µൌ ૚ ܏ܚ܉܌ܑ܍ܖܜ 1 Incorrect Answers: A – incorrect use of ݒൌට ் ஜ B – incorrect use of ݒൌට ் ஜ D - incorrect use of ݒൌට ் ஜ (Total for Multiple Choice Questions = 8 marks) Question Number Acceptable Answers Additional guidance Mark 9 This question assesses a student’s ability to show a coherent and logically structured answer with linkages and fully-sustained reasoning. Marks are awarded for indicative content and for how the answer is structured and shows lines of reasoning. Indicative Content  Band does not obey Hooke’s law Or there is a non-linear relationship between force and extension  (The band is elastic so) the extension returns to zero when the force is removed or size/shape is unchanged  For a given force the extension when loading is less than when unloading or for the same extension more force required when loading  Area under the loading curve is greater than the unloading curve Or Loading increases the elastic strain energy (of the band)  The band absorbs more energy when being loaded than it releases when unloaded Or Unloading: some strain energy transferred by heating  Energy released by heating represented by the area between the lines (1) (1) (1) (1) (1) (1) The following table shows how the marks should be awarded for structure and lines of reasoning Number of marks awarded for structure of answer and sustained line of reasoning Answer shows a coherent and logical structure with linkages and fully sustained lines of reasoning demonstrated throughout 2 Answer is partially structured with some linkages and lines of reasoning 1 Answer has no linkages between points and is unstructured 0 6 Total for question 9 6 Question Number Acceptable Answers Additional guidance Mark 10(a)  Use of ∆ܧ௘௟ൌ ଵ ଶܨ∆ݔ  4.6 J (1) (1) Example of Calculation ∆ܧ௘௟ൌ ଵ ଶൈ29 N ൈ0.32 m ൌ4.6 J 2 10(b)  Use of ܨൌ݇∆ݔ  0.30 m (1) (1) Example of Calculation 29 N 0.32 m ൌ27 N ∆ݔ ∆ݔൌ0.30 m 2 Total for question 10 4 Question Number Acceptable Answers Additional Guidance Mark 11(a) Max 2  Use large angles of incidence or use a wide range of angles  Use narrowest possible ray  Repeat and calculate mean  Work in a dark room.  Use monochromatic light (2) 2 11(b)(i) Either  Resolution of protractor 1°  Recognises that the results are recorded to the same precision as the resolution of the protractor Or  Resolution of protractor 0.5°  Recognises that the results are not recorded to the same precision as the resolution of the protractor (1) (1) (1) (1) MP2 dependent on MP1 2 11(b)(ii)  Use of percentage uncertainty equation with 0.5° as the resolution  percentage uncertainty = 2% (1) (1) Example of calculation 0.5 29 ൈ100% ൌ1.72% 2 11(c)(i)  Appropriate line of best fit (1) 1 11(c)(ii)  Calculates a gradient using at least half the drawn line  ηൌ1.37 to 1.47  leading to a conclusion that glass is silica Or conclusion consistent with their value for η (1) (1) (1) Example of calculation ଴.ଽି଴.଴ହ ଴.ହ଼ = 1.47 silica 3 Total for question 11 10 Question Number Acceptable Answers Additional Guidance Mark 12 (a)  Electrons exist in discrete energy levels  (After the mercury atom is excited) electrons move back to a lower energy level Or (After the mercury atom is excited) electrons move back down from a higher energy level  Photons are emitted  Frequency of emitted radiation / photons depends on the difference in energy between the energy levels Or reference to ∆ܧൌ݄݂  Since there are only certain energy changes possible, only certain frequencies are possible (1) (1) (1) (1) (1) Accept excited state for higher energy level Accept ground state for lower energy state 5 12 (b)  Use of ܧൌ ଵ ଶ݉ݒଶ  Use of ܧൌ݄݂ and ܿൌ݂ߣ  λൌሺ2.4 െ2.5ሻൈ10ି଻ሺm)  leading to a conclusion that radiation is ultraviolet Or conclusion consistent with their value for wavelength (1) (1) (1) (1) ∆ܧ௞ൌଵ ଶൈଽ.ଵଵൈଵ଴షయభ ୩୥ൈሺሺ2.5 ൈ10଺m sିଵሻଶെሺ2.1 ൈ10଺m sିଵሻଶሻ ߣൌ6.63 ൈ10ିଷସJ s ൈ3 ൈ10଼ m sିଵ 8.4 ൈ10ିଵଽ J MP4 dependent on MP2 Example of Calculation ∆ܧ௞ൌ8.4 ൈ10ିଵଽ (J) ߣൌ2.4 ൈ10ି଻ሺm) 4 Total for question 12 9 Question Number Acceptable Answers Additional guidance Mark 13 (a)  Line/surface joining points on a wave that are in phase (1) 1 13(b)(i)  Do not stare into laser (1) 1 13 (b)(ii)  Lack of other evidence (supporting wave theory) Or Was yet to be reproduced Or There was support / evidence for the particle model (1) 1 13(c)(i) An explanation that makes reference to the following:  Path lengths (A-O and B-O) are equal Or Path difference is zero  Will arrive in phase Or phase difference is zero  (Bright line is position of) constructive interference/superposition (1) (1) (1) 3 13(c)(ii)  600 nm Or 600 × 10-9 m Or 6.0 × 10-7 m Or one wavelength Or λ (1) Do not accept (nλ) Accept any correct equivalent value 1 Total for question 13 7 Additional guidance Mark 14(a)(i)  One ray correctly drawn  Second ray correctly drawn  Completes diagram with image at position 3.6 to 3.8 cm and height of 0.7 to 0.8 cm (1) (1) (1) 3 14(a)(ii)  Use of ݉ൌ ௩ ௨ or ݉ൌ ௜௠௔௚௘௛௘௜௚௛௧ ௢௕௝௘௖௧௛௘௜௚௛௧ using values from (a)(i)  Magnification of 0.47 to 0.53 (1) (1) Example of Calculation ݉ൌ ௩ ௨ ൌ ଷ.଻ ଻.ହൌ0.5 2 14(a)(iii)  Real and image on different side of converging lens to object Or rays pass through the image (1) 1 14(b)(i)  Focus image of distant/far object on to a screen  Measure distance from lens to screen Or  Use parallel rays of light  Measure distance from lens to the point where the rays converge (1) (1) (1) (1) MP2 dependent on MP1 2 14(b)(ii)  Greater refraction  To converge (parallel) rays at a point closer to the lens (1) (1) 2 14(b)(iii)  Photograph 2 has a greater magnification  so v is greater  since u is constant  So f is greater  Hence photograph 2 taken with lens of focal length 200 mm (1) (1) (1) (1) (1) MP5 dependent on MP2 and MP4 5 Total for question 14 15 Question Number Acceptable Answers Additional guidance Mark 15(a)  (High intensity of light means) photons strike the photocell at a greater rate  One to one interaction between photons and electrons  So Electrons released at a greater rate  So greater current/charge/power in external circuit (1) (1) (1) (1) 4 15(b)  Use of ܫൌ ௉ ஺  Use of ܲൌ ா ௧  Multiply by 23%  1700 MJ (1) (1) (1) (1) Example of Calculation ܧൌ1300 Wmିଶൈ200 mଶൈሺ8 ൈ60 ൈ60 s ൈ0.23ሻ ܧൌ1700 MJ 4 15(c)  Use of ܧൌܲݐ  430 MJ (1) (1) Example of Calculation ܧൌ4 ൈ7500 W ൈሺ4 ൈ60 ൈ60ሻs ܧൌ432 MJ 2 15(d)  Use of ܧൌ݉݃∆݄  ܧൌ1.1 ൈ10଼ J (1) (1) Example of Calculation ܧൌ݉݃∆݄ൌ1600 ݇݃ൈ9.81݉ݏିଵൈሺ8500 െ1500ሻ݉ ܧൌ1.1 ൈ10଼ J 2 15(e)  Drag increases as speed increases Or Drag increases as density of air increases  Requiring a greater forward force  So motors require more energy/power so plane should fly at a reduced speed (1) (1) (1) Accept air resistance for drag 3 15(f) Max 1  Develops new technologies  Develops alternative energy sources  Raises public awareness  Secures funding for other projects (1) 1 Total for question 15 16 Question Number Acceptable Answers Additional guidance Mark 16(a)  Recognises that F=0 Or Uses D = U – W  Use of W = mg with m = 5.2 g  Use of U = weight of air displaced by balloon Or U = ସ ଷߩ௔ߨݎଷ݃ Or U = ߩ௔ܸ݃ and ܸൌ ସ ଷߨݎଷ Or U = mg and ߩൌ ௠ ௏ and ܸൌ ସ ଷߨݎଷ  0.034 N (1) (1) (1) (1) Example of calculation ܦൌሺర యగൈሺ଴.ଵଶ ୫ሻ య ൈ ଵ.ଶ ୩୥ ୫షయ ൈ 9.81 m sିଶ) – ((4+1.2) kg ൈ 9.81 m sିଶሻ ൌ0.034 N 4 16(b) Viscosity of air increases as temperature increases Or Density of air is lower so upthrust is smaller (1) (1) 1 Total for question 16 5 Pearson Education Limited. 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page 04 MARKS Total marks – 100 Attempt ALL questions 1. (a) Given ( ) sin f x x − = 13 , find ( ) f x ′ . (b) Differentiate x e y x = + 5 7 1 . (c) For cos y x y x + = 2 6 , use implicit differentiation to find dy dx . 2. Use partial fractions to find x dx x x − − − ⌠⎮⌡ 2 3 7 2 15 . 3. (a) Write down and simplify the general term in the binomial expansion of x x ⎛ ⎞ + ⎜ ⎟ ⎝ ⎠ 9 2 5 2 . (b) Hence, or otherwise, find the term independent of x. 4. Given that z i = + 1 2 3 and , , z p i p = − ∈\ 2 6 find: (a) z z 1 2; (b) the value of p such that z z 1 2 is a real number. 5. Use the Euclidean algorithm to find integers a and b such that a b + = 306 119 17. 2 2 4 4 3 2 2 1 4

page 05 MARKS 6. On a suitable domain, a curve is defined parametrically by x t = + 2 1 and ( ) ln y t = + 3 2 . Find the equation of the tangent to the curve where t = −1 3 . 7. Matrices C and D are given by: C − ⎛ ⎞ ⎜ ⎟ = − ⎜ ⎟ ⎜ ⎟ − ⎝ ⎠ 2 1 2 1 1 0 1 0 1 and D k ⎛ ⎞ ⎜ ⎟ = + ⎜ ⎟ ⎜ ⎟ ⎝ ⎠ 1 1 2 3 0 2 1 1 1 , where k ∈\. (a) Obtain C D − ′ 2 where C′ is the transpose of C. (b) (i) Find and simplify an expression for the determinant of D. (ii) State the value of k such that 1 D− does not exist. 8. Using the substitution sin u θ = , or otherwise, evaluate sin cos θ θ dθ π π∫ 2 4 6 2 . 9. Prove directly that: (a) the sum of any three consecutive integers is divisible by 3; (b) any odd integer can be expressed as the sum of two consecutive integers. [Turn over 5 2 2 1 4 2 1

page 06 MARKS 10. Given z x iy = + , sketch the locus in the complex plane given by z z i = − + 2 2 . 11. (a) Obtain the matrix, A, associated with an anticlockwise rotation of π 3 radians about the origin. (b) Find the matrix, B, associated with a reflection in the x-axis. (c) Hence obtain the matrix, P, associated with an anticlockwise rotation of π 3 radians about the origin followed by reflection in the x-axis, expressing your answer using exact values. (d) Explain why matrix P is not associated with rotation about the origin. 12. Prove by induction that, for all positive integers n, ( ) n r n r − = = − ∑ 1 1 1 3 3 1 2 . 3 1 1 2 1 5

page 07 MARKS 13. An engineer has designed a lifting device. The handle turns a screw which shortens the horizontal length and increases the vertical height. The device is modelled by a rhombus, with each side 25 cm. The horizontal length is x cm, and the vertical height is h cm as shown. h x 25 cm (a) Show that h x = − 2 2500 . (b) The horizontal length decreases at a rate of 0·3 cm per second as the handle is turned. Find the rate of change of the vertical height when x = 30. [Turn over 1 5

page 08 MARKS 14. A geometric sequence has first term 80 and common ratio 1 3 . (a) For this sequence, calculate: (i) the 7th term; (ii) the sum to infinity of the associated geometric series. The first term of this geometric sequence is equal to the first term of an arithmetic sequence. The sum of the first five terms of this arithmetic sequence is 240. (b) (i) Find the common difference of this sequence. (ii) Write down and simplify an expression for the nth term. Let Sn represent the sum of the first n terms of this arithmetic sequence. (c) Find the values of n for which n S = 144. 15. (a) Use integration by parts to find sin x x dx ∫ 3 . (b) Hence find the particular solution of sin , dy y x x x dx x − = ≠ 3 2 3 0 given that x = π when y = 0. Express your answer in the form ( ) y f x = . 2 2 2 1 3 3 7

page 03 MARKS Attempt ALL questions Total marks — 60 1. PQR is a triangle with vertices P (−2, 4), Q (4, 0) and R (3, 6). R (3, 6) P (−2, 4) Q (4, 0) Find the equation of the median through R. 2. A function ( ) g x is defined on , the set of real numbers, by ( ) g x x = − 1 4 5 . Find the inverse function, ( ) g x −1 . 3. Given ( ) cos h x x = 3 2 , find the value of π h ⎛ ⎞ ′⎜ ⎟ ⎝ ⎠ 6 . [Turn over 3 3 3

page 04 MARKS 4. The point K (8, −5) lies on the circle with equation x y x y + − − − = 2 2 12 6 23 0. K (8, −5) x y x y + − − − = 2 2 12 6 23 0 Find the equation of the tangent to the circle at K. 5. A (−3, 4, −7), B (5, t, 5) and C (7, 9, 8) are collinear. (a) State the ratio in which B divides AC. (b) State the value of t. 6. Find the value of log log − 5 5 1 250 8 3 . 4 1 1 3

page 05 MARKS 7. The curve with equation y x x x = − + + 3 2 3 2 5 is shown on the diagram. y P Q O x y x x x = − + + 3 2 3 2 5 (a) Write down the coordinates of P, the point where the curve crosses the y-axis . (b) Determine the equation of the tangent to the curve at P. (c) Find the coordinates of Q, the point where this tangent meets the curve again. 8. A line has equation y x − + = 3 5 0. Determine the angle this line makes with the positive direction of the x-axis. [Turn over 1 3 4 2

page 06 MARKS 9. The diagram shows a triangular prism ABC,DEF. AB = t, AC = u and AD = v. B D F C u A t M v E (a) Express BC in terms of u and t. M is the midpoint of BC. (b) Express MD in terms of t, u and v. 10. Given that • dy x x dx = − + 2 6 3 4, and • y = 14 when x = 2, express y in terms of x. 1 2 4

page 07 MARKS 11. The diagram shows the curve with equation log y x = 3 . y x 1 O (3,1) log y x = 3 (a) On the diagram in your answer booklet, sketch the curve with equation log y x = − 3 1 . (b) Determine the exact value of the x-coordinate of the point of intersection of the two curves. 12. Vectors a and b are such that = − + a i j k 4 2 2 and p = − + + b i j k 2 . (a) Express 2a + b in component form. (b) Hence find the values of p for which + = a b 2 7. [Turn over for next question 2 3 1 3

page 08 MARKS 13. The right-angled triangle in the diagram is such that sin x = 2 11 and x π < < 0 4 . 2 x 11 (a) Find the exact value of: (i) sin 2x (ii) cos 2x. (b) By expressing sin 3x as ( ) sin x x + 2 , find the exact value of sin 3x. 14. Evaluate ( ) dx x − + ⌠ ⎮ ⌡ 9 2 3 4 1 2 9 . 15. A cubic function, f, is defined on the set of real numbers. • ( ) x + 4 is a factor of ( ) f x • x = 2 is a repeated root of ( ) f x • ( ) f − ′ = 2 0 • ( ) f x ′ > 0 where the graph with equation ( ) y f x = crosses the y-axis Sketch a possible graph of ( ) y f x = on the diagram in your answer booklet. [END OF QUESTION PAPER] 3 1 3 5 4 Write your answers clearly in the spaces provided in this booklet. The size of the space provided for an answer should not be taken as an indication of how much to write. It is not necessary to use all the space. Additional space for answers is provided at the end of this booklet. If you use this space you must clearly identify the question number you are attempting. Use blue or black ink. Before leaving the examination room you must give this booklet to the Invigilator; if you do not, you may lose all the marks for this paper. FOR OFFICIAL USE X747/76/01 Mathematics Paper 1 (Non-Calculator) Answer Booklet Fill in these boxes and read what is printed below. Full name of centre Town Forename(s) Surname Number of seat Day Month Year Scottish candidate number © Mark Date of birth H National Qualications 2018 THURSDAY, 3 MAY 9:00 AM – 10:10 AM A/PB page 02 DO NOT WRITE IN THIS MARGIN QUESTION NUMBER 1. 2. page 03 DO NOT WRITE IN THIS MARGIN QUESTION NUMBER 3. 4. page 04 DO NOT WRITE IN THIS MARGIN QUESTION NUMBER 5.(a) 6. 5.(b) page 05 DO NOT WRITE IN THIS MARGIN QUESTION NUMBER 7.(a) 7.(c) 7.(b) page 06 DO NOT WRITE IN THIS MARGIN QUESTION NUMBER 8. 9.(a) 9.(b) page 07 DO NOT WRITE IN THIS MARGIN QUESTION NUMBER 10. *X74776010* page 08 DO NOT WRITE IN THIS MARGIN QUESTION NUMBER 11.(a) 11.(b) An additional diagram, if required, can be found on page 13. y x 1 O (3,1) log y x = 3 page 09 DO NOT WRITE IN THIS MARGIN QUESTION NUMBER 12.(a) 12.(b) page 10 DO NOT WRITE IN THIS MARGIN QUESTION NUMBER 13.(a) (i) 13.(a) (ii) 13.(b) page 11 DO NOT WRITE IN THIS MARGIN QUESTION NUMBER 14. page 12 DO NOT WRITE IN THIS MARGIN QUESTION NUMBER 15. –6 –5 –4 –3 –2 –1 O 1 2 3 4 5 6 An additional diagram, if required, can be found on page 14. x y page 13 DO NOT WRITE IN THIS MARGIN QUESTION NUMBER ADDITIONAL SPACE FOR ANSWERS Additional diagram for Question 11(a). y x 1 O (3,1) log y x = 3 page 14 DO NOT WRITE IN THIS MARGIN QUESTION NUMBER ADDITIONAL SPACE FOR ANSWERS Additional diagram for Question 15. –6 –5 –4 –3 –2 –1 O 1 2 3 4 5 6 x y page 15 ADDITIONAL SPACE FOR ANSWERS DO NOT WRITE IN THIS MARGIN QUESTION NUMBER page 16 For Marker’s Use Question No Marks/Grades © National Qualications 2018 H Total marks — 70 Attempt ALL questions. You may use a calculator. Full credit will be given only to solutions which contain appropriate working. State the units for your answer where appropriate. Answers obtained by readings from scale drawings will not receive any credit. Write your answers clearly in the spaces provided in the answer booklet. The size of the space provided for an answer should not be taken as an indication of how much to write. It is not necessary to use all the space. Additional space for answers is provided at the end of the answer booklet. If you use this space you must clearly identify the question number you are attempting. Use blue or black ink. Before leaving the examination room you must give your answer booklet to the Invigilator; if you do not, you may lose all the marks for this paper. X747/76/12 Mathematics Paper 2 THURSDAY, 3 MAY 10:30 AM – 12:00 NOON A/PB page 02 FORMULAE LIST Circle: The equation 2 2 2 2 0 x y gx fy c + + + + = represents a circle centre (−g, −f ) and radius 2 2 g f c + − . The equation ( ) ( ) 2 2 2 x a y b r − + − = represents a circle centre (a, b) and radius r. Scalar Product: cos , where is the angle between and θ θ = a.b a b a b or 1 1 1 1 2 2 3 3 2 2 3 3 where and . a b a b a b a b a b a b ⎛ ⎞ ⎛ ⎞ ⎜ ⎟ ⎜ ⎟ = + + = = ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎝ ⎠ ⎝ ⎠ a.b a b Trigonometric formulae: sin (A B) sin A cos B cos A sin B ± = ± cos (A B) cos A cos B sin A sin B ± = ∓ sin A sin AcosA 2 2 = cos A cos A sin A 2 2 2 = − cos A 2 2 1 = − sin A 2 1 2 = − Table of standard derivatives: ( ) f x ( ) f x ′ sin ax cosax cos a ax sin a ax − Table of standard integrals: ( ) f x ( ) f x dx ∫ sin ax cosax cos 1 ax c a − + sin 1 ax c a +

page 03 MARKS Attempt ALL questions Total marks — 70 1. The diagram shows the curve with equation y x x = + − 2 3 2 . y y = 3 + 2 x − x2 x O 3 −1 Calculate the shaded area. 2. Vectors u and v are defined by − ⎛ ⎞ ⎜ ⎟ = ⎜ ⎟ ⎜ ⎟ − ⎝ ⎠ 1 4 3 u and − ⎛ ⎞ ⎜ ⎟ = ⎜ ⎟ ⎜ ⎟ ⎝ ⎠ 7 8 5 v . (a) Find u.v. (b) Calculate the acute angle between u and v. 3. A function, f , is defined on the set of real numbers by ( ) f x x x = − − 3 7 6 . Determine whether f  is increasing or decreasing when x = 2 . 4. Express 2 3 6 7 x x − − + in the form ( ) a x b c + + 2 . [Turn over 4 1 4 3 3

page 04 MARKS 5. PQR is a triangle with P(3,4) and Q(9,−2). R P (3,4) Q (9,−2) L1 (a) Find the equation of L1 , the perpendicular bisector of PQ. The equation of L2 , the perpendicular bisector of PR is y x + = 3 25. R C P (3,4) Q (9,−2) L1 L2 (b) Calculate the coordinates of C, the point of intersection of L1 and L2. C is the centre of the circle which passes through the vertices of triangle PQR. R C P (3,4) Q (9,−2) L1 L2 (c) Determine the equation of this circle. 3 2 2

page 05 MARKS 6. Functions, f and g, are given by ( ) cos f x x = + 3 and ( ) g x x = 2 , x∈\. (a) Find expressions for (i) ( ) ( ) f g x and (ii) ( ) ( ) g f x . (b) Determine the value(s) of x for which ( ) ( ) ( ) ( ) f g x g f x = where x ≤ < π 0 2 . 7. (a) (i) Show that ( ) x −2 is a factor of x x x − − + 3 2 2 3 3 2. (ii) Hence, factorise x x x − − + 3 2 2 3 3 2 fully. The fifth term, u5, of a sequence is u a = − 5 2 3. The terms of the sequence satisfy the recurrence relation n n u au + = − 1 1. (b) Show that u a a a = − − − 3 2 7 2 3 1. For this sequence, it is known that • u u = 7 5 • a limit exists. (c) (i) Determine the value of a. (ii) Calculate the limit. [Turn over 2 1 6 2 2 1 3 1

page 06 MARKS 8. (a) Express cos sin x x − 2 ° ° in the form ( ) cos k x a − °, k a > < < 0, 0 360. (b) Hence, or otherwise, find (i) the minimum value of cos sin x x − 6 ° 3 ° and (ii) the value of x for which it occurs where x ≤ < 0 360. 9. A sector with a particular fixed area has radius x cm. The perimeter, P cm, of the sector is given by P x x = + 128 2 . Find the minimum value of P. 10. The equation ( ) x m x m + − + = 2 3 0 has two real and distinct roots. Determine the range of values for m. 11. A supermarket has been investigating how long customers have to wait at the checkout. During any half hour period, the percentage, P %, of customers who wait for less than t minutes, can be modelled by ( ) kt P e = − 100 1 , where k is a constant. (a) If 50% of customers wait for less than 3 minutes, determine the value of k. (b) Calculate the percentage of customers who wait for 5 minutes or longer. 4 1 2 6 4 4 2

page 07 MARKS 12. Circle C1 has equation ( ) ( ) x y − + + = 2 2 13 4 100. Circle C2 has equation x y x y c + + − + = 2 2 14 22 0 . P C2 C1 (a) (i) Write down the coordinates of the centre of C1. (ii) The centre of C1 lies on the circumference of C2. Show that c = −455. The line joining the centres of the circles intersects C1 at P. (b) (i) Determine the ratio in which P divides the line joining the centres of the circles. (ii) Hence, or otherwise, determine the coordinates of P. P is the centre of a third circle, C3. C2 touches C3 internally. (c) Determine the equation of C3. [END OF QUESTION PAPER] 1 1 2 2 1 page 08 [BLANK PAGE] DO NOT WRITE ON THIS PAGE Write your answers clearly in the spaces provided in this booklet. The size of the space provided for an answer should not be taken as an indication of how much to write. It is not necessary to use all the space. Additional space for answers is provided at the end of this booklet. If you use this space you must clearly identify the question number you are attempting. Use blue or black ink. Before leaving the examination room you must give this booklet to the Invigilator; if you do not, you may lose all the marks for this paper. FOR OFFICIAL USE X747/76/02 Mathematics Paper 2 Answer Booklet Fill in these boxes and read what is printed below. Full name of centre Town Forename(s) Surname Number of seat Day Month Year Scottish candidate number © Mark Date of birth H National Qualications 2018 THURSDAY, 3 MAY 10:30 AM – 12:00 NOON A/PB page 02 [BLANK PAGE] DO NOT WRITE ON THIS PAGE page 03 DO NOT WRITE IN THIS MARGIN QUESTION NUMBER 1. page 04 DO NOT WRITE IN THIS MARGIN QUESTION NUMBER 2.(a) 2.(b) page 05 3. DO NOT WRITE IN THIS MARGIN QUESTION NUMBER 4. page 06 DO NOT WRITE IN THIS MARGIN QUESTION NUMBER 5.(a) 5.(b) 5.(c) page 07 DO NOT WRITE IN THIS MARGIN QUESTION NUMBER 6.(a) (i) 6.(b) 6.(a) (ii) page 08 DO NOT WRITE IN THIS MARGIN QUESTION NUMBER 7.(a) (i) 7.(a) (ii) 7.(b) page 09 DO NOT WRITE IN THIS MARGIN QUESTION NUMBER 7.(c) (i) 7.(c) (ii) page 10 DO NOT WRITE IN THIS MARGIN QUESTION NUMBER 8.(a) 8.(b) (i) 8.(b) (ii) page 11 DO NOT WRITE IN THIS MARGIN QUESTION NUMBER 9. page 12 DO NOT WRITE IN THIS MARGIN QUESTION NUMBER 10. page 13 DO NOT WRITE IN THIS MARGIN QUESTION NUMBER 11.(a) 11.(b) page 14 DO NOT WRITE IN THIS MARGIN QUESTION NUMBER 12.(a) (i) 12.(b) (i) 12.(a) (ii) page 15 DO NOT WRITE IN THIS MARGIN QUESTION NUMBER 12.(b) (ii) 12.(c) page 16 DO NOT WRITE IN THIS MARGIN ENTER NUMBER OF QUESTION ADDITIONAL SPACE FOR ANSWERS page 17 DO NOT WRITE IN THIS MARGIN ENTER NUMBER OF QUESTION ADDITIONAL SPACE FOR ANSWERS page 18 DO NOT WRITE IN THIS MARGIN ENTER NUMBER OF QUESTION ADDITIONAL SPACE FOR ANSWERS page 19 DO NOT WRITE IN THIS MARGIN ENTER NUMBER OF QUESTION ADDITIONAL SPACE FOR ANSWERS page 20 For Marker’s Use Question No Marks/Grades

14 7. Given that k is a positive constant and 5 2 3 4 1 x x k +     = ∫ d (a) show that 3k + 5 k – 12 = 0 (4) (b) Hence, using algebra, find any values of k such that 5 2 3 4 1 x x k +     = ∫ d (4) _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________

P61639A 2 1. (a) Draw the graph K5 (1) (b) (i) In the context of graph theory explain what is meant by ‘semi-Eulerian’. (ii) Draw two semi-Eulerian subgraphs of K5, each having five vertices but with a different number of edges. (3) (c) Explain why a graph with exactly five vertices with vertex orders 1, 2, 2, 3 and 4 cannot be a tree. (2) (Total for Question 1 is 6 marks) 2. The following algorithm produces a numerical approximation for the integral I = A B∫ x4 dx Step 1 Start Step 2 Input the values of A, B and N Step 3 Let H = (B – A) / N Step 4 Let C = H / 2 Step 5 Let D = 0 Step 6 Let D = D + A4 + B4 Step 7 Let E = A Step 8 Let E = E + H Step 9 If E = B go to Step 12 Step 10 Let D = D + 2 × E4 Step 11 Go to Step 8 Step 12 Let F = C × D Step 13 Output F Step 14 Stop For the case when A = 1, B = 3 and N = 4, (a) (i) complete the table in the answer book to show the results obtained at each step of the algorithm. (ii) State the final output. (4) (b) Calculate, to 3 significant figures, the percentage error between the exact value of I and the value obtained from using the approximation to I in this case. (3) (Total for Question 2 is 7 marks) P61639A 3 Turn over 3. Activity Immediately preceding activities A - B - C A D A E A F B, C G B, C H D I D, E, F, G J D, E, F, G K G (a) Draw the activity network described in the precedence table above, using activity on arc. Your activity network must contain the minimum number of dummies. (5) Every activity shown in the precedence table has the same duration. (b) Explain why activity B cannot be critical. (1) (c) State which other activities are not critical. (1) (Total for Question 3 is 7 marks) P61639A 4 4. B A C D E F G H 10 17 12 16 13 25 3x + y x + y 9 5 9 7 7 3 2 Figure 1 [The total weight of the network is 135 + 4x + 2y] The weights on the arcs in Figure 1 represent distances. The weights on the arcs CE and GH are given in terms of x and y, where x and y are positive constants and 7 x + y 20 There are three paths from A to H that have the same minimum length. (a) Use Dijkstra’s algorithm to find x and y. (7) An inspection route starting at A and finishing at H is found. The route traverses each arc at least once and is of minimum length. (b) State the arcs that are traversed twice. (1) (c) State the number of times that vertex C appears in the inspection route. (1) (d) Determine the length of the inspection route. (1) (Total for Question 4 is 10 marks)

P61639A 5 5. Ben is a wedding planner. He needs to order flowers for the weddings that are taking place next month. The three types of flower he needs to order are roses, hydrangeas and peonies. Based on his experience, Ben forms the following constraints on the number of each type of flower he will need to order. Ɣ At least three-fifths of all the flowers must be roses. Ɣ For every 2 hydrangeas there must be at most 3 peonies. Ɣ The total number of flowers must be exactly 1000 The cost of each rose is £1, the cost of each hydrangea is £5 and the cost of each peony is £4 Ben wants to minimise the cost of the flowers. Let x represent the number of roses, let y represent the number of hydrangeas and let z represent the number of peonies that he will order. (a) Formulate this as a linear programming problem in x and y only, stating the objective function and listing the constraints as simplified inequalities with integer coefficients. (7) Ben decides to order the minimum number of roses that satisfy his constraints. (b) (i) Calculate the number of each type of flower that he will order to minimise the cost of the flowers. (ii) Calculate the corresponding total cost of this order. (3) (Total for Question 5 is 10 marks) TOTAL FOR DECISION MATHEMATICS 1 IS 40 MARKS END P61639A 6 BLANK PAGE P61639A 7 BLANK PAGE P61639A 8 BLANK PAGE Please check the examination details below before entering your candidate information Candidate surname Other names Total Marks Centre Number Candidate Number Answer Book Do not return the question paper with the answer book. Further Mathematics Advanced Subsidiary Further Mathematics options 27: Decision Mathematics 1 (Part of options D, F, H and K) Paper Reference 8FM0-27 Afternoon Thursday 16 May 2019 Pearson Edexcel Level 3 GCE P61639A ©2019 Pearson Education Ltd. 1/1/1/1/1/1/ Turn over 2 DO NOT WRITE IN THIS AREA DO NOT WRITE IN THIS AREA DO NOT WRITE IN THIS AREA 1. _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ 3 Turn over DO NOT WRITE IN THIS AREA DO NOT WRITE IN THIS AREA DO NOT WRITE IN THIS AREA Question 1 continued _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ (Total for Question 1 is 6 marks) 4 DO NOT WRITE IN THIS AREA DO NOT WRITE IN THIS AREA DO NOT WRITE IN THIS AREA 2. 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Vertex Order of labelling Final value Working values Key: C D H G F E B A 9 17 25 10 5 x + y 3x + y 16 3 9 2 13 12 7 7 9 Turn over DO NOT WRITE IN THIS AREA DO NOT WRITE IN THIS AREA DO NOT WRITE IN THIS AREA Question 4 continued _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ 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MATHEMATICS 1 IS 40 MARKS

P61640A 2 1. Three workers, A, B and C, are each to be assigned to one of four tasks, P, Q, R and S. Each worker must be assigned to at most one task, and each task must be done by at most one worker. The amount, in pounds, that each worker will earn while assigned to each task is shown in the table below. P Q R S A 32 40 37 42 B 29 32 35 41 C 37 33 39 40 The Hungarian algorithm is to be used to find the maximum total amount that can be earned by the three workers. (a) Explain how the table should be modified. (2) (b) (i) Reducing rows first, use the Hungarian algorithm to obtain an allocation which maximises the total earnings. (ii) Explain how any initial row and column reductions were made and also how you determined if the table was optimal at each stage. (7) (Total for Question 1 is 9 marks) 2. (a) Find the general solution of the recurrence relation un + 1 = 3un + 2n n . 1 (4) (b) Find the particular solution of this recurrence relation for which u1 = u2 (2) (Total for Question 2 is 6 marks) P61640A 3 Turn over 3. 14 A B C D E F G H J S T 18 4 6 7 24 2 7 2 3 0 11 5 3 0 5 19 14 7 26 0 2 6 5 6 12 0 12 2 6 0 4 5 17 21 4 0 7 Figure 1 Alexa is monitoring a system of pipes through which fluid can flow from the source, S, to the sink, T. Currently, fluid is flowing through the system from S to T. Alexa initialises the labelling procedure for this system, and the excess capacities and potential backflows are shown on the arrows either side of each arc, as shown in Figure 1. (a) State the value of the initial flow. (1) (b) Explain why arcs DF and DG can never both be full to capacity. (1) (c) Obtain the capacity of the cut that passes through the arcs AC, AD, BD, DE, EG and EJ. (1) (d) Use the labelling procedure to find a maximum flow through the network. You must list each flow-augmenting route you use, together with its flow. (3) (e) Use your answers to part (d) to find a maximum flow pattern for this system of pipes and draw it on Diagram 1 in the answer book. (1) (f) Prove that the answer to part (e) is optimal. (3) (Total for Question 3 is 10 marks)

P61640A 4 4. The table below gives the pay-off matrix for a zero-sum game between two players, Aljaz and Brendan. The values in the table show the pay-offs for Aljaz. Brendan Option X Option Y Option Z Aljaz Option P í í 2 Option Q 5 4 í Option R 5 6 3 (a) (i) Show that this game is stable. (ii) State the value of this game to Brendan. (3) Option R is removed from Aljaz’s choices and the reduced game, with option R removed, is no longer stable. (b) Find the best strategy for Aljaz in this reduced game, defining any variable you use. (7) (c) Explain why Brendan should never play option Y (1) Let Brendan play option X with probability q (d) (i) Explain why q satisfies the equation 6q í í q) = 1.6 (ii) Hence find the best strategy for Brendan in this reduced game. (4) (Total for Question 4 is 15 marks) TOTAL FOR DECISION MATHEMATICS 2 IS 40 MARKS END Please check the examination details below before entering your candidate information Candidate surname Other names Total Marks Centre Number Candidate Number Pearson Edexcel Level 3 GCE Answer Book Do not return the question paper with the answer book. Further Mathematics Advanced Subsidiary Further Mathematics options 28: Decision Mathematics 2 (Part of option K only) Paper Reference 8FM0-28 Afternoon Thursday 16 May 2019 P61640A ©2019 Pearson Education Ltd. 1/1/1/1/1/1/ Turn over 2 DO NOT WRITE IN THIS AREA DO NOT WRITE IN THIS AREA DO NOT WRITE IN THIS AREA 1. (a) _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ P Q R S A 32 40 37 42 B 29 32 35 41 C 37 33 39 40 You may not need to use all of these tables You may not need to use all the rows and columns ____________________________________________ ____________________________________________ ____________________________________________ ____________________________________________ ____________________________________________ ____________________________________________ ____________________________________________ ____________________________________________ ____________________________________________ ____________________________________________ ____________________________________________ ____________________________________________ ____________________________________________ ____________________________________________ ____________________________________________ ____________________________________________ ____________________________________________ ____________________________________________ ____________________________________________ ____________________________________________ ____________________________________________ ____________________________________________ ____________________________________________ P Q R S A B C P Q R S A B C P Q R S A B C 3 Turn over DO NOT WRITE IN THIS AREA DO NOT WRITE IN THIS AREA DO NOT WRITE IN THIS AREA Question 1 continued ____________________________________________ ____________________________________________ 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____________________________________________ ____________________________________________ ____________________________________________ ____________________________________________ ____________________________________________ ____________________________________________ ____________________________________________ ____________________________________________ ____________________________________________ ____________________________________________ ____________________________________________ (Total for Question 1 is 9 marks) P Q R S A B C P Q R S A B C P Q R S A B C P Q R S A B C P Q R S A B C 4 DO NOT WRITE IN THIS AREA DO NOT WRITE IN THIS AREA DO NOT WRITE IN THIS AREA 2. _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ 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_____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ (Total for Question 2 is 6 marks) 6 DO NOT WRITE IN THIS AREA DO NOT WRITE IN THIS AREA DO NOT WRITE IN THIS AREA 3. _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ 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_____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ 7 Turn over DO NOT WRITE IN THIS AREA DO NOT WRITE IN THIS AREA DO NOT WRITE IN THIS AREA Question 3 continued Question 3 continues on the next page. 14 A B C D E F G H J S T 18 4 6 7 24 2 7 2 3 0 11 5 3 0 5 19 14 7 26 0 2 6 5 6 12 0 12 2 6 0 4 5 17 21 4 0 7 8 DO NOT WRITE IN THIS AREA DO NOT WRITE IN THIS AREA DO NOT WRITE IN THIS AREA Question 3 continued 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_____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ 9 Turn over DO NOT WRITE IN THIS AREA DO NOT WRITE IN THIS AREA DO NOT WRITE IN THIS AREA Question 3 continued A B C D E F G H J S T Diagram 1 _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ 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_____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ (Total for Question 3 is 10 marks) 10 DO NOT WRITE IN THIS AREA DO NOT WRITE IN THIS AREA DO NOT WRITE IN THIS AREA 4. Brendan Option X Option Y Option Z Aljaz Option P í í 2 Option Q 5 4 í Option R 5 6 3 _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ 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_____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ 12 DO NOT WRITE IN THIS AREA DO NOT WRITE IN THIS AREA DO NOT WRITE IN THIS AREA Question 4 continued _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ 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_____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ (Total for Question 4 is 15 marks) TOTAL FOR DECISION MATHEMATICS 2 IS 40 MARKS

Question 1 continued _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________

Question 1 continued _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________

Question 1 continued _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ (Total for Question 1 is 8 marks) 2. 6 m s–1 A B C Figure 2 Figure 2 represents the plan view of part of a horizontal floor, where AB and BC are fixed vertical walls with AB perpendicular to BC. A small ball is projected along the floor towards AB with speed 6 m s–1 on a path that makes an angle with AB, where tan = . The ball hits AB and then hits BC. Immediately after hitting AB, the ball is moving at an angle to AB, where tan = The coefficient of restitution between the ball and AB is e. The coefficient of restitution between the ball and BC is By modelling the ball as a particle and the floor and walls as being smooth, (a) show that the value of e = (5) (b) find the speed of the ball immediately after it hits BC. (4) (c) Suggest two ways in which the model could be refined to make it more realistic. (2) _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________

Question 2 continued _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________

Question 2 continued _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________

Question 2 continued _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ (Total for Question 2 is 11 marks)

3. A particle P, of mass 0.5 kg, is moving with velocity (4i + 4j) m s–1 when it receives an impulse I of magnitude 2.5 Ns. As a result of the impulse, the direction of motion of P is deflected through an angle of 45 Given that I = ( i + j) Ns, find all the possible pairs of values of and . (9) _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________

Question 3 continued _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________

Question 3 continued _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________

Question 3 continued _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ (Total for Question 3 is 9 marks)

11 Turn over Question 5 continued _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ (Total for Question 5 is 3 marks) 12 6. y O x Figure 4 Figure 4 shows a sketch of the graph of y J ௘[࣠) , where g( ) ( ) x x x x x = − + − ⎧ ⎨ ⎩ 2 1 4 7 2 2 2 - (a) Find the value of gg(0). (2) (b) Find all values of x for which J ௘[࣠) 28 (4) The function h is defined by K ௘[࣠) = (x í 2 + 1 x - 2 (c) Explain why h has an inverse but g does not. (1) (d) Solve the equation h í (x) = −1 2 (3) _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________

4. A car of mass 600 kg pulls a trailer of mass 150 kg along a straight horizontal road. The trailer is connected to the car by a light inextensible towbar, which is parallel to the direction of motion of the car. The resistance to the motion of the trailer is modelled as a constant force of magnitude 200 N. At the instant when the speed of the car is v m s–1, the resistance to the motion of the car is modelled as a force of magnitude (200 + ) N, where is a constant. When the engine of the car is working at a constant rate of 15 kW, the car is moving at a constant speed of 25 m s–1 (a) Show that = 8 (4) Later on, the car is pulling the trailer up a straight road inclined at an angle to the horizontal, where sin = The resistance to the motion of the trailer from non-gravitational forces is modelled as a constant force of magnitude 200 N at all times. At the instant when the speed of the car is v m s–1, the resistance to the motion of the car from non-gravitational forces is modelled as a force of magnitude (200 + 8v) N. The engine of the car is again working at a constant rate of 15 kW. When v = 10, the towbar breaks. The trailer comes to instantaneous rest after moving a distance d metres up the road from the point where the towbar broke. (b) Find the acceleration of the car immediately after the towbar breaks. (4) (c) Use the work-energy principle to find the value of d. (4) _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________

Question 4 continued _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________

Question 4 continued _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________

Question 4 continued _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ (Total for Question 4 is 12 marks)

5. A particle P of mass 3m and a particle Q of mass 2m are moving along the same straight line on a smooth horizontal plane. The particles are moving in opposite directions towards each other and collide directly. Immediately before the collision the speed of P is u and the speed of Q is 2u. Immediately after the collision P and Q are moving in opposite directions. The coefficient of restitution between P and Q is e. (a) Find the range of possible values of e, justifying your answer. (8) Given that Q loses 75% of its kinetic energy as a result of the collision, (b) find the value of e. (3) _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________

Question 5 continued _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________

Question 5 continued _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________

Question 5 continued _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ (Total for Question 5 is 11 marks)

6. [In this question i and j are perpendicular unit vectors in a horizontal plane.] A smooth uniform sphere A has mass 0.2 kg and another smooth uniform sphere B, with the same radius as A, has mass 0.4 kg. The spheres are moving on a smooth horizontal surface when they collide obliquely. Immediately before the collision, the velocity of A is (3i + 2j) m s–1 and the velocity of B is (–4i – j) m s–1 At the instant of collision, the line joining the centres of the spheres is parallel to i The coefficient of restitution between the spheres is (a) Find the velocity of A immediately after the collision. (7) (b) Find the magnitude of the impulse received by A in the collision. (2) (c) Find, to the nearest degree, the size of the angle through which the direction of motion of A is deflected as a result of the collision. (3) _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________

Question 6 continued _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________

Question 6 continued _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________

Question 6 continued _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ (Total for Question 6 is 12 marks)

7. A particle P, of mass m, is attached to one end of a light elastic spring of natural length a and modulus of elasticity kmg. The other end of the spring is attached to a fixed point O on a ceiling. The point A is vertically below O such that OA = 3a The point B is vertically below O such that OB = a The particle is held at rest at A, then released and first comes to instantaneous rest at the point B. (a) Show that k = (3) (b) Find, in terms of g, the acceleration of P immediately after it is released from rest at A. (3) (c) Find, in terms of g and a, the maximum speed attained by P as it moves from A to B. (6) _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________

Question 7 continued _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________

Question 7 continued _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ (Total for Question 7 is 12 marks) TOTAL FOR PAPER IS 75 MARKS

7 P61182A Turn over 6. A linear programming problem in x, y and z is described as follows. Maximise P = 2x + 2y í z subject to 3x + y + 2z - 30 x í y + z . 8 4y + 2z . 15 x, y, z . 0 (a) Explain why the Simplex algorithm cannot be used to solve this linear programming problem. (1) (b) Set up the initial tableau for solving this linear programming problem using the big-M method. (7) After a first iteration of the big-M method, the tableau is b.v. x y z s1 s2 s3 a1 a2 Value s1 3 0 1.5 1 0 0.25 0 í 26.25 a1 1 0 1.5 0 í í 1 0.25 11.75 y 0 1 0.5 0 0 í 0 0.25 3.75 P í M) 0 í M 0 M í M 0 0.5 + 0.75M í M (c) State the value of each variable after the first iteration. (1) (d) Explain why the solution given by the first iteration is not feasible. (1) Taking the most negative entry in the profit row to indicate the pivot column, (e) obtain the most efficient pivot for a second iteration. You must give reasons for your answer. (2) (Total for Question 6 is 12 marks)

8 P61182A 7. A shop sells two types of watch, analogue watches and digital watches. The shop manager knows that, each month, she should order at least 60 watches in total. In addition, at most 80% of the watches she orders must be digital. Let x be the number of analogue watches ordered and let y be the number of digital watches ordered. (a) Write down inequalities, in terms of x and y, to model these constraints. (2) Two further constraints are y + 3x . 140 4y + x . 80 (b) Represent all these constraints on Diagram 1 in the answer book. Hence determine, and label, the feasible region, R. (4) The cost to the shop of ordering an analogue watch is five times the cost of ordering a digital watch. The shop manager wishes to minimise the total cost. (c) Determine the number of each type of watch the shop manager should order. You must make your method clear. (3) Given that the minimum total cost of ordering the watches is £4455 (d) determine the cost of ordering one analogue watch and the cost of ordering one digital watch. You must make your method clear. (3) (Total for Question 7 is 12 marks) TOTAL FOR PAPER IS 75 MARKS Please check the examination details below before entering your candidate information Candidate surname Other names Total Marks Centre Number Candidate Number Pearson Edexcel Level 3 GCE P61182A ©2019 Pearson Education Ltd. 1/1/1/1/1/1/C2/1/ Turn over Answer Book Do not return the question paper with the answer book. Paper Reference 9FM0/3D Morning (Time: 1 hour 30 minutes) Monday 24 June 2019 Turn over Further Mathematics Advanced Paper 3D: Decision Mathematics 1 2 DO NOT WRITE IN THIS AREA DO NOT WRITE IN THIS AREA DO NOT WRITE IN THIS AREA 1. 2.1 1.7 3.0 1.9 3.2 1.2 3.3 1.4 1.5 0.2 _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ 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_____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ 3 Turn over DO NOT WRITE IN THIS AREA DO NOT WRITE IN THIS AREA DO NOT WRITE IN THIS AREA Question 1 continued 2.1 1.7 3.0 1.9 3.2 1.2 3.3 1.4 1.5 0.2 _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ 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_____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ (Total for Question 1 is 8 marks) 4 DO NOT WRITE IN THIS AREA DO NOT WRITE IN THIS AREA DO NOT WRITE IN THIS AREA 2. Vertex Order of labelling Final value Working values Key: B E D J H F A C G 38 42 15 10 17 14 6 23 39 12 7 20 42 75 10 Shortest path from A to D: ________________________________________________________ Length of shortest path from A to D: ________________________________________________ 5 Turn over DO NOT WRITE IN THIS AREA DO NOT WRITE IN THIS AREA DO NOT WRITE IN THIS AREA Question 2 continued H C G E J D B A F 38 75 42 15 10 42 20 12 10 17 14 23 6 39 7 Figure 1 _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ 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_____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ 6 DO NOT WRITE IN THIS AREA DO NOT WRITE IN THIS AREA DO NOT WRITE IN THIS AREA Question 2 continued _____________________________________________________________________________________ _____________________________________________________________________________________ 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THIS AREA DO NOT WRITE IN THIS AREA Question 2 continued _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ 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_____________________________________________________________________________________ _____________________________________________________________________________________ (Total for Question 2 is 14 marks) 8 DO NOT WRITE IN THIS AREA DO NOT WRITE IN THIS AREA DO NOT WRITE IN THIS AREA 3. (a) Initial distance table Initial route table A B C D E A B C D E A A B B C C D D E E (b) 1st iteration Distance table Route table A B C D E A B C D E A A B B C C D D E E 2nd iteration Distance table Route table A B C D E A B C D E A A B B C C D D E E 3rd iteration Distance table Route table A B C D E A B C D E A A B B C C D D E E Turn to page 11 for spare copies of these tables if you need to correct your work. 9 Turn over DO NOT WRITE IN THIS AREA DO NOT WRITE IN THIS AREA DO NOT WRITE IN THIS AREA Question 3 continued _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ 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THIS AREA DO NOT WRITE IN THIS AREA Question 3 continued Only use these tables if you need to correct your work. (a) Initial distance table Initial route table A B C D E A B C D E A A B B C C D D E E (b) 1st iteration Distance table Route table A B C D E A B C D E A A B B C C D D E E 2nd iteration Distance table Route table A B C D E A B C D E A A B B C C D D E E 3rd iteration Distance table Route table A B C D E A B C D E A A B B C C D D E E (Total for Question 3 is 14 marks) 12 DO NOT WRITE IN THIS AREA DO NOT WRITE IN THIS AREA DO NOT WRITE IN THIS AREA 4. Early event time Late event time Key: 25 A(8) D(11) I(6) E(5) K(6) J(4) L(7) F(8) B(6) C(7) G(10) H(x) M(6) Diagram 1 _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ 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_____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ 0 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30 32 34 Grid 1 (Total for Question 4 is 9 marks) 14 DO NOT WRITE IN THIS AREA DO NOT WRITE IN THIS AREA DO NOT WRITE IN THIS AREA 5. 15 Turn over DO NOT WRITE IN THIS AREA DO NOT WRITE IN THIS AREA DO NOT WRITE IN THIS AREA Question 5 continued _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ 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_____________________________________________________________________________________ _____________________________________________________________________________________ (Total for Question 5 is 6 marks) 16 DO NOT WRITE IN THIS AREA DO NOT WRITE IN THIS AREA DO NOT WRITE IN THIS AREA 6. _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ 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_____________________________________________________________________________________ (b) b.v. x y z s1 s2 s3 a1 a2 Value 30 8 15 P Only use the tableau below if you need to correct the above tableau. b.v. x y z s1 s2 s3 a1 a2 Value 30 8 15 P 18 DO NOT WRITE IN THIS AREA DO NOT WRITE IN THIS AREA DO NOT WRITE IN THIS AREA Question 6 continued _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ 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_____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ (Total for Question 6 is 12 marks) 20 DO NOT WRITE IN THIS AREA DO NOT WRITE IN THIS AREA DO NOT WRITE IN THIS AREA 7. _____________________________________________________________________________________ _____________________________________________________________________________________ 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THIS AREA DO NOT WRITE IN THIS AREA Question 7 continued Turn to page 24 for a spare copy of these axes if you need to correct your work. 0 10 20 30 40 50 60 70 80 90 100 Diagram 1 0 20 40 60 80 100 120 140 160 y x 22 DO NOT WRITE IN THIS AREA DO NOT WRITE IN THIS AREA DO NOT WRITE IN THIS AREA Question 7 continued _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ 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160 y x (Total for Question 7 is 12 marks) TOTAL FOR PAPER IS 75 MARKS

13 Turn over Question 6 continued _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________

14 Question 6 continued _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________

15 Turn over Question 6 continued _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ (Total for Question 6 is 10 marks)

15 Turn over Question 7 continued _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ (Total for Question 7 is 8 marks)

2 P61186A Answer ALL questions. Write your answers in the answer book provided. 1. Table 1 shows the cost, in pounds, of transporting one unit of stock from each of four supply points, A, B, C and D, to each of four demand points, P, Q, R and S. It also shows the stock held at each supply point and the stock required at each demand point. A minimum cost solution is required. P Q R S Supply A 15 14 17 11 23 B 10 9 16 12 42 C 11 13 8 10 18 D 15 13 16 17 19 Demand 25 45 12 20 Table 1 Table 2 shows an initial solution given by the north-west corner method. P Q R S A 23 B 2 40 C 5 12 1 D 19 Table 2 (a) Taking DQ as the entering cell, use the stepping-stone method to find an improved solution. Make your method clear. (2) (b) Perform one further iteration of the stepping-stone method to obtain an improved solution. You must make your method clear by stating the Ɣ shadow costs Ɣ improvement indices Ɣ route Ɣ entering cell and exiting cell. (4) (c) Determine whether the solution obtained from this second iteration is optimal, giving a reason for your answer. (3) (d) State the cost of the solution found in (b). (1) (Total for Question 1 is 10 marks)

3 P61186A Turn over 2. Four workers, Ted (T), Harold (H), James (J) and Margaret (M), are to be assigned to four tasks, 1, 2, 3 and 4. Each worker must be assigned to just one task and each task must be done by just one worker. The profit, in pounds, resulting from allocating each worker to each task, is shown in the table below. The profit is to be maximised. 1 2 3 4 T 103 97 74 80 H 201 155 145 155 J 111 80 77 92 M 203 188 137 184 (a) Reducing rows first, use the Hungarian algorithm to obtain an allocation that maximises the total profit. You must make your method clear and show the table after each stage. (6) (b) Determine the resulting total profit. (1) (Total for Question 2 is 7 marks) 4 P61186A 3. B 24 24 14 –24 –24 –30 –17 –25 29 20 19 27 32 17 D C E F 19 G A H S I T J 13 28 13 x 18 Figure 1 In Figure 1 the weight of arc SB is denoted by x where x . 0 (a) Explain why Dijkstra’s algorithm cannot be used on the directed network in Figure 1. (1) It is given that the minimum weight route from S to T passes through B. (b) Use dynamic programming to find (i) the range of possible values of x (ii) the minimum weight route from S to T. (12) (Total for Question 3 is 13 marks) 5 P61186A Turn over 4. Player B Option X Option Y Option Z Option P 3 –2 0 Player A Option Q –4 4 –2 Option R 1 2 –1 A two person zero-sum game is represented by the pay-off matrix for player A shown above. (a) Verify that there is no stable solution to this game. (2) Player A intends to make a random choice between options P, Q and R, choosing option P with probability p1, option Q with probability p2 and option R with probability p3 Player A wants to find the optimal values of p1 , p2 and p3 using the Simplex algorithm. Player A formulates the following linear programming problem for the game, writing the constraints as inequalities. Maximise P = V subject to V . 3p1 – 4p2 + p3 V . –2p1 + 4p2 + 2p3 V . –2p2 – p3 p1 + p2 + p3 - 1 p1 . 0, p2 . 0, p3 . 0, V . 0 (b) Correct the errors made by player A in the linear programming formulation of the game, giving reasons for your answer. (3) (c) Write down an initial Simplex tableau for the corrected linear programming problem. (3) The Simplex algorithm is used to solve the corrected linear programming problem. The optimal values are p1 = 0.6, p2 = 0 and p3= 0.4 (d) Calculate the value of the game to player A. (2) (e) Determine the optimal strategy for player B, making your reasoning clear. (4) (Total for Question 4 is 14 marks)

6 P61186A 5. An increasing sequence {un} for n  ` is such that the difference between the nth term of {un} and the mean of the previous two terms of {un} is always 6 (a) Show that, for n . 3 2un – un – 1 – un – 2 = 12 (2) Given that u1 = 2 and u2 = 8 (b) find the solution of this second order recurrence relation to obtain an expression for un in terms of n. (7) (c) Show that as n o ’, un o kn where k is a constant to be determined. You must give reasons for your answer. (2) (Total for Question 5 is 11 marks) 7 P61186A Turn over 6. B H C2 C2 C1 C1 D E F C G A S T (3, 6) (5, 7) (4, 7) (5, 8) (6, 8) (5, 6) (4, 7) (17, 20) (10, 12) (10, 14) (2, 4) (13, 17) (8, 11) (7, 9) (0, 4) (3, 5) (3, 5) (1, 2) (4, 4) (3, 5) Figure 2 Figure 2 shows a capacitated, directed network. The network represents a system of pipes through which fluid flows from a source, S, to a sink, T. The numbers (l, u) on each arc represent, in litres per second, the lower capacity, l, and the upper capacity, u, of the corresponding pipe. Two cuts C1 and C2 are shown. (a) Find the capacity of (i) cut C1 (ii) cut C2 (2) (b) Explain why the arcs AE and CE cannot be at their upper capacities simultaneously. (1) (c) Explain why a flow of 31 litres per second through the system is not possible. (1) (d) Hence determine a minimum feasible flow and a maximum feasible flow through the system. You must draw these feasible flows on the diagrams in the answer book and give reasons to justify your answer. You should not apply the labelling procedure to find these flows. (4) (Total for Question 6 is 8 marks)

8 P61186A 7. Aisha is deciding whether or not to play a game. The game involves rolling three fair six-sided dice, which have faces numbered from 1 to 6 If the total score on the three dice is 16 or more then she wins a prize. If the total score is 15 or less then she loses and will have to pay the person running the game £3 (a) Given that the prize is £15 (i) draw a decision tree to model Aisha’s possible decisions and the possible outcomes (ii) determine Aisha’s optimal EMV and state the optimal strategy indicated by the decision tree. (6) The utility function of the game to Aisha is u(m) = 1 – e −m 500 where m is the amount of money that Aisha has available. Given that Aisha has exactly £3 and that the prize is now £x (b) find the expected utility to Aisha of playing the game in the form a b x c 1 500 − ⎛ ⎝ ⎜ ⎜ ⎞ ⎠ ⎟ ⎟ − + e ( ) where a, b and c are integers to be found. (2) Aisha decides to use the expected utilities to determine whether she should play the game or not. (c) Find the minimum prize for which Aisha would consider playing the game. (4) (Total for Question 7 is 12 marks) TOTAL FOR PAPER IS 75 MARKS Please check the examination details below before entering your candidate information Candidate surname Other names Total Marks Centre Number Candidate Number Pearson Edexcel Level 3 GCE Answer Book Do not return the question paper with the answer book Further Mathematics Advanced Paper 4D: Decision Mathematics 2 Paper Reference 9FM0/4D Tuesday 25 June 2019 Morning (Time: 1 hour 30 minutes) P61186A ©2019 Pearson Education Ltd. 1/1/1/1/1/ Turn over 2 DO NOT WRITE IN THIS AREA DO NOT WRITE IN THIS AREA DO NOT WRITE IN THIS AREA 1. P Q R S Supply A 15 14 17 11 23 B 10 9 16 12 42 C 11 13 8 10 18 D 15 13 16 17 19 Demand 25 45 12 20 Table 1 P Q R S A B C D P Q R S A B C D P Q R S A B C D P Q R S A B C D ____________________________________________________________________________ ____________________________________________________________________________ ____________________________________________________________________________ ____________________________________________________________________________ ____________________________________________________________________________ ____________________________________________________________________________ ____________________________________________________________________________ ____________________________________________________________________________ ____________________________________________________________________________ ____________________________________________________________________________ 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____________________________________________________________________________ ____________________________________________________________________________ ____________________________________________________________________________ ____________________________________________________________________________ 3 Turn over DO NOT WRITE IN THIS AREA DO NOT WRITE IN THIS AREA DO NOT WRITE IN THIS AREA Question 1 continued P Q R S A B C D P Q R S A B C D P Q R S A B C D P Q R S A B C D P Q R S A B C D (Total for Question 1 is 10 marks) ____________________________________________________________________________ ____________________________________________________________________________ ____________________________________________________________________________ ____________________________________________________________________________ ____________________________________________________________________________ ____________________________________________________________________________ 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____________________________________________________________________________ ____________________________________________________________________________ 4 DO NOT WRITE IN THIS AREA DO NOT WRITE IN THIS AREA DO NOT WRITE IN THIS AREA 2. 1 2 3 4 T 103 97 74 80 H 201 155 145 155 J 111 80 77 92 M 203 188 137 184 1 2 3 4 T H J M 1 2 3 4 T H J M 1 2 3 4 T H J M 1 2 3 4 T H J M ____________________________________________________________________________ ____________________________________________________________________________ ____________________________________________________________________________ ____________________________________________________________________________ ____________________________________________________________________________ ____________________________________________________________________________ ____________________________________________________________________________ ____________________________________________________________________________ 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(a) __________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ Stage State Action Destination Value 7 Turn over DO NOT WRITE IN THIS AREA DO NOT WRITE IN THIS AREA DO NOT WRITE IN THIS AREA Question 3 continued Stage State Action Destination Value _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ (Total for Question 3 is 13 marks) 8 DO NOT WRITE IN THIS AREA DO NOT WRITE IN THIS AREA DO NOT WRITE IN THIS AREA 4. Player B Option X Option Y Option Z Option P 3 –2 0 Player A Option Q –4 4 –2 Option R 1 2 –1 _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ 9 Turn over DO NOT WRITE IN THIS AREA DO NOT WRITE IN THIS AREA DO NOT WRITE IN THIS AREA Question 4 continued b.v. Value P Only use the tableau below if you need to correct your initial tableau. b.v. Value P _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ 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_____________________________________________________________________________________ 11 Turn over DO NOT WRITE IN THIS AREA DO NOT WRITE IN THIS AREA DO NOT WRITE IN THIS AREA Question 4 continued _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ 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_____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ (Total for Question 4 is 14 marks) 12 DO NOT WRITE IN THIS AREA DO NOT WRITE IN THIS AREA DO NOT WRITE IN THIS AREA 5. _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ 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_____________________________________________________________________________________ 15 Turn over DO NOT WRITE IN THIS AREA DO NOT WRITE IN THIS AREA DO NOT WRITE IN THIS AREA Question 5 continued _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ 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_____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ (Total for Question 5 is 11 marks) 16 DO NOT WRITE IN THIS AREA DO NOT WRITE IN THIS AREA DO NOT WRITE IN THIS AREA 6. (a) (i) Capacity of cut C1 = ____________________________________________________________ (ii) Capacity of cut C2 = ____________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ 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_____________________________________________________________________________________ _____________________________________________________________________________________ 17 Turn over DO NOT WRITE IN THIS AREA DO NOT WRITE IN THIS AREA DO NOT WRITE IN THIS AREA Question 6 continued _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ Minimum feasible flow B H D E F C G A S T Maximum feasible flow B H D E F C G A S T _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ (Total for Question 6 is 8 marks) 18 DO NOT WRITE IN THIS AREA DO NOT WRITE IN THIS AREA DO NOT WRITE IN THIS AREA 7. _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ 19 Turn over DO NOT WRITE IN THIS AREA DO NOT WRITE IN THIS AREA DO NOT WRITE IN THIS AREA Question 7 continued _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ 20 DO NOT WRITE IN THIS AREA DO NOT WRITE IN THIS AREA DO NOT WRITE IN THIS AREA Question 7 continued _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ (Total for Question 7 is 12 marks) TOTAL FOR PAPER IS 75 MARKS

2 Answer ALL questions. Write your answers in the spaces provided. 1. Given 2 4 1 2 2 x y × = express y as a function of x. (3) _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________

3 Turn over Question 1 continued _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ (Total for Question 1 is 3 marks)

4 2. The speed of a small jet aircraft was measured every 5 seconds, starting from the time it turned onto a runway, until the time when it left the ground. The results are given in the table below with the time in seconds and the speed in m sí . Time (s) 0 5 10 15 20 25 Speed (m sí ) 2 5 10 18 28 42 Using all of this information, (a) estimate the length of runway used by the jet to take off. (3) Given that the jet accelerated smoothly in these 25 seconds, (b) explain whether your answer to part (a) is an underestimate or an overestimate of the length of runway used by the jet to take off. (1) _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________

5 Turn over Question 2 continued _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ (Total for Question 2 is 4 marks) 6 3. O 5 cm B A 40° Figure 1 Figure 1 shows a sector AOB of a circle with centre O, radius 5 cm and angle AOB = 40° The attempt of a student to find the area of the sector is shown below. Area of sector = 1 2 U࣠2ș = 1 2 u 5 2 u 40 = 500 cm2 (a) Explain the error made by this student. (1) (b) Write out a correct solution. (2) _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________

7 Turn over Question 3 continued _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ (Total for Question 3 is 3 marks) 8 4. y x O S C2 C1 Figure 2 The curve C1 with parametric equations x ௘FRV t , y = 4 2sin t , 0 - t 2ʌ meets the circle C2 with equation x2 + y2 = 66 at four distinct points as shown in Figure 2. Given that one of these points, S, lies in the 4th quadrant, find the Cartesian coordinates of S. (6) _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________

9 Turn over Question 4 continued _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ (Total for Question 4 is 6 marks) 10 5. y O x į[ y P 4 9 y x = Figure 3 Figure 3 shows a sketch of the curve with equation y x = The point P(x, y) lies on the curve. The rectangle, shown shaded on Figure 3, has height y and width į[ Calculate 9 0 4 lim δx x x δx → =∑ (3) _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________

16 7. A small factory makes bars of soap. On any day, the total cost to the factory, £y, of making x bars of soap is modelled to be the sum of two separate elements: Ɣ a fixed cost Ɣ a cost that is proportional to the number of bars of soap that are made that day (a) Write down a general equation linking y with x, for this model. (1) The bars of soap are sold for £2 each. On a day when 800 bars of soap are made and sold, the factory makes a profit of £500 On a day when 300 bars of soap are made and sold, the factory makes a loss of £80 Using the above information, (b) show that y = 0.84x + 428 (3) (c) With reference to the model, interpret the significance of the value 0.84 in the equation. (1) Assuming that each bar of soap is sold on the day it is made, (d) find the least number of bars of soap that must be made on any given day for the factory to make a profit that day. (2) _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________

17 Turn over Question 7 continued _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________

18 Question 7 continued _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________

19 Turn over Question 7 continued _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ (Total for Question 7 is 7 marks)

20 8. (i) Find the value of 20 1 2 4 × ⎛ ⎝⎜ ⎞ ⎠⎟ = ∞ ∑ r r (3) (ii) Show that log5 1 48 2 1 2 n n n + + ⎛ ⎝⎜ ⎞ ⎠⎟= =∑ (3) _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________

21 Turn over Question 8 continued _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________

22 Question 8 continued _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________

23 Turn over Question 8 continued _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ (Total for Question 8 is 6 marks)

24 9. A research engineer is testing the effectiveness of the braking system of a car when it is driven in wet conditions. The engineer measures and records the braking distance, d metres, when the brakes are applied from a speed of V ௘NP Kí . Graphs of d against V and log10 d against log10 V were plotted. The results are shown below together with a data point from each graph. d (30, 20) O V O log10 d í log10 V Figure 5 Figure 6 (a) Explain how Figure 6 would lead the engineer to believe that the braking distance should be modelled by the formula d = kV n where k and n are constants with k | 0.017 (3) Using the information given in Figure 5, with k = 0.017 (b) find a complete equation for the model giving the value of n to 3 significant figures. (3) Sean is driving this car at 60 km hí in wet conditions when he notices a large puddle in the road 100 m ahead. It takes him 0.8 seconds to react before applying the brakes. (c) Use your formula to find out if Sean will be able to stop before reaching the puddle. (3) _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________

25 Turn over Question 9 continued _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________

26 Question 9 continued _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________

27 Turn over Question 9 continued _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ (Total for Question 9 is 9 marks) 28 10. A B O Figure 7 Figure 7 shows a sketch of triangle OAB. The point C is such that OC | = 2OA |. The point M is the midpoint of AB. The straight line through C and M cuts OB at the point N. Given OA | = a and OB | = b (a) Find CM | in terms of a and b (2) (b) Show that ON | = 3 2 2 λ ⎛ ⎞ − ⎜ ⎟ ⎝ ⎠ a + 1 2 Ȝb , where Ȝ is a scalar constant. (2) (c) Hence prove that ON : NB = 2 : 1 (2) _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________

29 Turn over Question 10 continued _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________

30 Question 10 continued _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________

31 Turn over Question 10 continued _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ (Total for Question 10 is 6 marks) 32 11. y x C O Figure 8 Figure 8 shows a sketch of the curve C with equation y = x x , x 0 (a) Find, by firstly taking logarithms, the x coordinate of the turning point of C. (6ROXWLRQV EDVHG HQWLUHO\ RQ JUDSKLFDO RU QXPHULFDO PHWKRGV DUH QRW DFFHSWDEOH ) (5) The point P(Į, 2) lies on C. (b) Show that 1.5 < Į < 1.6 (2) A possible iteration formula that could be used in an attempt to find Į is x x n n xn + − = 1 1 2 Using this formula with x1 = 1.5 (c) find x4 to 3 decimal places, (2) (d) describe the long-term behaviour of xn (2) _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________

33 Turn over Question 11 continued _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________

34 Question 11 continued _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________

35 Turn over Question 11 continued _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ (Total for Question 11 is 11 marks)

36 12. (a) Prove cos3 sin θ θ + sin3 cos θ θ { ௘FRW ș ș z (90n)°, n  ] (4) (b) Hence solve, for 90° ș 180°, the equation cos3 sin θ θ + sin3 cos θ θ = 4 giving any solutions to one decimal place. (3) _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________

37 Turn over Question 12 continued _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________

39 Turn over Question 12 continued _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ (Total for Question 12 is 7 marks) 40 13. r m r m h m Figure 9 [A sphere of radius r has volume 4 3 ʌU࣠3 and surface area 4ʌU࣠2] A manufacturer produces a storage tank. The tank is modelled in the shape of a hollow circular cylinder closed at one end with a hemispherical shell at the other end as shown in Figure 9. The walls of the tank are assumed to have negligible thickness. The cylinder has radius r metres and height h metres and the hemisphere has radius r metres. The volume of the tank is 6 m3. (a) Show that, according to the model, the surface area of the tank, in m2, is given by 12 r + 5 3 ʌU࣠2 (4) The manufacturer needs to minimise the surface area of the tank. (b) Use calculus to find the radius of the tank for which the surface area is a minimum. (4) (c) Calculate the minimum surface area of the tank, giving your answer to the nearest integer. (2) _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________

41 Turn over Question 13 continued _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________

42 Question 13 continued _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________

43 Turn over Question 13 continued _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ (Total for Question 13 is 10 marks)

44 14. (a) Use the substitution u í h to show that œ dh h 4 − í OQ௘|௘ í h |௘í h + k where k is a constant (6) A team of scientists is studying a species of slow growing tree. The rate of change in height of a tree in this species is modelled by the differential equation d d h t t h = − ( ) 0 25 4 20 . where h is the height in metres and t is the time, measured in years, after the tree is planted. (b) Find, according to the model, the range in heights of trees in this species. (2) One of these trees is one metre high when it is first planted. According to the model, (c) calculate the time this tree would take to reach a height of 12 metres, giving your answer to 3 significant figures. (7) _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________

45 Turn over Question 14 continued _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________

46 Question 14 continued _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________

47 Turn over Question 14 continued _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________

48 Question 14 continued _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ (Total for Question 14 is 15 marks) TOTAL FOR PAPER IS 100 MARKS

11864 10 (a) Find ∫ x– 1 2ln x dx [7] [Turn over 11864 11864 (b) Using the substitution u2 = x2 + 4 , or otherwise, find the exact value of 0 √x2 + 4 √5 ∫ dx x3 [8] [Turn over 11864

11864 11 The graphs of the curves y = sin 2x and y = cos 2x are shown in Fig. 3 below. The curves intersect at the points A and B. 0 −1 B x y 1 R 3π 4 π 2 π 4 A Fig. 3 (i) Show that the x-coordinates of A and B are π 8 and 5π 8 [4] [Turn over 11864 The top section of a trophy is a flat metal sheet modelled in the shape of the shaded region R. (ii) Calculate the area of this region. [6] 11864 A circle has the equation x2 + y2 = 4 The base of the trophy can be modelled as the solid formed when the area bounded by this circle, the y-axis and the line x = 1 is rotated through 2π radians about the x-axis. (iii) Find the exact volume of the trophy base. [6] [Turn over 11864

11864 12 (a) (i) Prove that the sum of n terms of an arithmetic progression with first term a and last term l is Sn = 1 2 n(a + l) [4] [Turn over 11864 The first term of an arithmetic progression is 7 and the last term is 79 The sum of the progression is 1075 (ii) Find the number of terms. [3] (iii) Find the common difference. [3] 11864 (b) A salesman receives a bonus at the end of each year and decides to invest this money in a savings account. At the end of Year 1 he invests £400 At the end of Year 2 he invests a further £400 and receives 2% interest on the first year’s £400 At the end of Year 3 he invests a further £400 and 2% interest is added to the total sum of money which he has accumulated during the first two years. (i) Show that he has £1,224.16 in his account at the end of Year 3 [4] [Turn over 11864 (ii) Assuming that the man continues to invest in this way, form and sum a series to prove that he will have £20 000(1.02n − 1) in his account at the end of n years. [6] 11864 (iii) Hence find the least number of years until his investment exceeds £7,000 [4] THIS IS THE END OF THE QUESTION PAPER 11864 BLANK PAGE DO NOT WRITE ON THIS PAGE Permission to reproduce all copyright material has been applied for. In some cases, efforts to contact copyright holders may have been unsuccessful and CCEA will be happy to rectify any omissions of acknowledgement in future if notified. Examiner Number DO NOT WRITE ON THIS PAGE 11864/4 For Examiner’s use only Question Number Marks 1 2 3 4 5 6 7 8 9 10 11 12 Total Marks

page 04 MARKS Total marks — 100 Attempt ALL questions 1. (a) Differentiate . (b) Given , find . Simplify your answer. (c) For evaluate . 2. Matrix A is defined by where . (a) Given that the determinant of A is 3, find the value of p. Matrix B is defined by where . (b) Find AB. (c) Explain why AB does not have an inverse. 2 3 3 3 2 1

page 05 MARKS 3. The function is defined by . The graph of is shown in the diagram. y −a O a x (a) State whether is odd, even or neither. Give a reason for your answer. (b) Sketch the graph of . 4. (a) Express in the form , where p, q and r are integers. (b) Hence express with partial fractions. [Turn over 1 1 1 3

38 14. A curve has equation y = g(x). Given that ● g(x) is a cubic expression in which the coefficient of x3 is equal to the coefficient of x ● the curve with equation y = g(x) passes through the origin ● the curve with equation y = g(x) has a stationary point at (2, 9) (a) find g(x), (7) (b) prove that the stationary point at (2, 9) is a maximum. (2) _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________

page 06 MARKS 5. For and , find (a) (b) . 6. A spherical balloon of radius r cm, r > 0, deflates at a constant rate of 60 cm3 s−1. Calculate the rate of change of the radius with respect to time when r = 3. The volume of a sphere is given by . 7. (a) Find an expression for ( ) 1 6 13 n r r = + ∑ in terms of n. (b) Hence, or otherwise, find ( ) 20 1 6 13 r p r = + + ∑ . 8. Find the particular solution of the differential equation given that y = 0 and , when x = 0. 2 2 3 1 2 5

page 07 MARKS 9. (a) Write down and simplify the general term in the binomial expansion of , where d is a constant. (b) Given that the coefficient of is −70 000, find the value of d. 10. A curve is defined implicitly by the equation . (a) Find an expression for in terms of x and y. (b) There are two points where the tangent to the curve has equation x = k, . Find the values of k. 11. Let n be a positive integer. (a) Find a counterexample to show that the following statement is false. is always a prime number. (b) (i) Write down the contrapositive of: If is even then n is odd. (ii) Use the contrapositive to prove that if is even then n is odd. 12. Express 23111 in base 7. [Turn over 3 2 3 2 1 1 3 3

page 08 MARKS 13. An electronic device contains a timer circuit that switches off when the voltage, V, reaches a set value. The rate of change of the voltage is given by , where k is a constant, t is the time in seconds, and 0 ≤ V < 12. Given that V = 2 when t = 0, express V in terms of k and t. 14. Prove by induction that ( ) ! ! 1 1 n r r r n = = + − ∑ 1 for all positive integers n. 15. The equations of two planes are given below. (a) Verify that the line of intersection, L1, of these two planes has parametric equations (b) Let π3 be the plane with equation . Calculate the acute angle between the line L1 and the plane π3. (c) L2 is the line perpendicular to π3 passing through P (1, 3, −2). Determine whether or not L1 and L2 intersect. 5 5 2 3 4

page 09 MARKS 16. (a) Use integration by parts to find the exact value of ( ) 1 2 4 0 2 1 x x x e dx − + ∫ . (b) A solid is formed by rotating the curve with equation ( ) 2 4 1 x y x e = − between x = 0 and x = 1 through 2π radians about the x-axis. Find the exact value of the volume of this solid. 17. The first three terms of a sequence are given by (a) When x = 11, show that the first three terms form the start of a geometric sequence, and state the value of the common ratio. (b) Given that the entire sequence is geometric for x = 11 (i) state why the associated series has a sum to infinity (ii) calculate this sum to infinity. (c) There is a second value for x that also gives a geometric sequence. For this second sequence (i) show that (ii) find the first three terms (iii) state the value of S2n and justify your answer. [Turn over for next question 5 3 2 1 2 2 2 1

page 10 MARKS 18. The complex number w has been plotted on an Argand diagram, as shown below. O w a Re Im (a) Express w in (i) Cartesian form (ii) polar form. (b) The complex number z1 is a root of z3 = w, where for integers k and m. Given that a = 4, (i) use de Moivre’s theorem to obtain the values of k and m, and (ii) find the remaining roots. [END OF QUESTION PAPER] 1 3 4 2 page 11 [BLANK PAGE] DO NOT WRITE ON THIS PAGE page 12 [BLANK PAGE] DO NOT WRITE ON THIS PAGE

page 03 MARKS Attempt ALL questions Total marks — 70 1. Find the x-coordinates of the stationary points on the curve with equation = − + y x x 4 3 1 2 6 2 . 2. The equation ( ) + − + = x k x 2 5 1 0 has equal roots. Determine the possible values of k. 3. Circle C1 has equation + − − − = 2 2 6 2 26 0 x y x y . Circle C2 has centre (4,-2). The radius of C2 is equal to the radius of C1. Find the equation of circle C2. 4. A sequence is generated by the recurrence relation + = + n n c u mu 1 , where the first three terms of the sequence are 6, 9 and 11. (a) Find the values of m and c. (b) Hence, calculate the fourth term of the sequence. 5. (a) Show that the points A(1,5,-3), B(4,-1,0) and C(8,-9,4) are collinear. (b) State the ratio in which B divides AC. 4 3 2 3 1 3 1 [Turn over

page 04 MARKS 6. Given that ( ) , , find = ≠ − dy y x dx x 5 1 1 3 1 3 . 7. The line, L, makes an angle of 30° with the positive direction of the x-axis. Find the equation of the line perpendicular to L, passing through (0,-4). 8. The graphs of = + + 2 2 3 y x x and = + + 2 2 1 y x x are shown below. O y x = + + 2 2 3 y x x = + + 2 2 1 y x x The graphs intersect at the points where and = − = 1 2 x x . (a) Express the shaded area, enclosed between the curves, as an integral. (b) Evaluate the shaded area. 3 4 1 3

page 05 MARKS 9. Vectors u and v have components and + ⎛ ⎞ ⎛ ⎞ ⎜ ⎟ ⎜ ⎟ − − ∈ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎝ ⎠ ⎝ ⎠ p p p R 2 16 2 3 , 4 6 . (a) (i) Find an expression for u.v. (ii) Determine the values of p for which u and v are perpendicular. (b) Determine the value of p for which u and v are parallel. 10. The diagram shows the graphs with equations ( ) ( ) and = = + y f x y kf x a. y x ( ) = + y kf x a ( ) = y f x O (2,5) (0,3) (2,-1) (a) State the value of a. (b) Find the value of k. 11. Evaluate cos π π ⎛ ⎞ − ⎜ ⎟ ⎝ ⎠ ⌠⎮⌡ x dx 9 0 3 6 . 1 3 2 1 1 4 [Turn over

page 06 MARKS 12. Functions f and g are defined by • ( ) , = > f x x x 1 where 0 • ( ) , = − ∈\ g x x x 5 where . (a) Determine an expression for ( ) ( ) f g x . (b) State the range of values of x for which ( ) ( ) f g x is undefined. 13. Triangles ABC and ADE are both right angled. Angles p and q are as shown in the diagram. B D A p q 1 1 1 E C 5 10 (a) Determine the value of (i) cos p (ii) cos q. (b) Hence determine the value of ( ) sin + p q . 14. (a) Evaluate log log + 10 10 4 2 5. (b) Solve ( ) , . log log − − = ≥ x x 2 2 7 2 3 5 1 2 1 1 1 3 3 3

page 04 MARKS 3. E,ABCD is a rectangular based pyramid. AB = p, AD = q and AE = r. F C B E D p r q A (a) Express BE in terms of p and r. Point F divides BC in the ratio 3:1. (b) Express vector EF in terms of p, q and r. 4. In a forest, the population of a species of mouse is falling by 2·7% each year. To increase the population scientists plan to release 30 mice into the forest at the end of March each year. (a) un is the estimated population of mice at the start of April, n years after the population was first estimated. It is known that un and un+1 satisfy the recurrence relation 1 n n u au b + = + . State the values of a and b. The scientists continue to release this species of mouse each year. (b) (i) Explain why the estimated population of mice will stabilise in the long term. (ii) Calculate the long term population to the nearest hundred. 1 2 1 1 2

page 05 MARKS 5. The diagram below shows the graph of a cubic function ( ) y g x = , with stationary points at and = − = 2 4 x x . O 4 −2 x y ( ) y g x = On the diagram in your answer booklet, sketch the graph of ( ) y g x ′ = . 6. (a) Express cos sin 2 3 ° ° x x − in the form ( ) cos ° k x a + where 0 0 360 k a > ≤ < and . (b) Hence solve cos sin ° ° 2 3 3 0 360 x x x − = ≤ < for . 7. (a) Express 2 6 24 25 x x − + − in the form ( ) . p x q r + + 2 (b) Given that ( ) 2 2 f x x x x = − + − + 3 2 12 5 9, show that ( ) f x is strictly decreasing for all . x∈\ [Turn over 2 4 3 3 3

page 06 MARKS 8. A function, f, is given by ( ) 3 8 f x x = + . The domain of f is , 1 1000 x x ≤ ≤ ∈\ . The inverse function, f -1, exists. (a) Find ( ) 1 f x − . (b) State the domain of f -1. 9. Electricity on a spacecraft can be produced by a type of nuclear generator. The electrical power produced by this generator can be modelled by · 0 0079 120 t tP e− = where Pt is the electrical power produced, in watts, after t years. (a) Determine the electrical power initially produced by the generator. (b) Calculate how long it takes for the electrical power produced by the generator to reduce by 15%. 10. (a) Show that ( ) 3 x + is a factor of 4 3 2 3 10 8 6 x x x x + + − −. (b) Hence, or otherwise, factorise 4 3 2 3 10 8 6 x x x x + + − − fully. 3 1 1 4 2 5

page 07 MARKS 11. A manufacturer of chocolates is launching a new product in novelty shaped cardboard boxes. CHOCOLATES The box is a cuboid with a cuboid shaped tunnel through it. • The height of the box is h centimetres • The top of the box is a square of side 3x centimetres • The end of the tunnel is a square of side x centimetres • The volume of the box is 2000 cm3 h x 3x (a) Show that the total surface area, A cm2, of the box is given by 2 4000 16x A x = + . (b) To minimise the cost of production, the surface area, A, of the box should be as small as possible. Find the minimum value of A. [Turn over 3 6

page 08 MARKS 12. Two variables, x and y, are connected by the equation x y ab = . The graph of log4 y against x is a straight line as shown. x (0,-1) (3,8) log4 y O Find the values of a and b. 13. For a function, f, defined on the set of real numbers, , it is known that • the rate of change of f with respect to x is given by 2 3 16 11 x x − + • the graph with equation ( ) y f x = crosses the x-axis at (7,0). Express ( ) f x in terms of x. 14. The vectors u and v are such that • 4 = u • 5 = v • ( ) . 21 + = u u v Determine the size of the angle between the vectors u and v. 5 5 4

page 09 MARKS 15. A circle has centre C(8,12). The point P(5,13) lies on the circle as shown. O T C(8,12) P(5,13) y x (a) Find the equation of the tangent at P. The tangent from P meets the y-axis at the point T. (b) (i) State the coordinates of T. (ii) Find the equation of the circle that passes through the points C, P and T. [END OF QUESTION PAPER] 3 1 3 page 10 [BLANK PAGE] DO NOT WRITE ON THIS PAGE page 11 [BLANK PAGE] DO NOT WRITE ON THIS PAGE page 12 [BLANK PAGE] DO NOT WRITE ON THIS PAGE Write your answers clearly in the spaces provided in the answer booklet. The size of the space provided for an answer is not an indication of how much to write. You do not need to use all the space. Additional space for answers is provided at the end of the answer booklet. If you use this space you must clearly identify the question number you are attempting. Use blue or black ink. Before leaving the examination room you must give your answer booklet to the Invigilator; if you do not, you may lose all the marks for this paper. FOR OFFICIAL USE X847/76/02 Mathematics Paper 2 Answer booklet Fill in these boxes and read what is printed below. Full name of centre Town Forename(s) Surname Number of seat Day Month Year Scottish candidate number © Mark Date of birth National Qualications 2019 THURSDAY, 2 MAY 11:00 AM – 12:45 PM A/PB H page 02 DO NOT WRITE IN THIS MARGIN QUESTION NUMBER 1.(a) 1.(b) 1.(c) page 03 2. DO NOT WRITE IN THIS MARGIN QUESTION NUMBER 3.(a) 3.(b) page 04 DO NOT WRITE IN THIS MARGIN QUESTION NUMBER 4.(a) 4.(b) (i) 4.(b) (ii) 5. An additional diagram, if required, can be found on page 16. y x –2 –3 –4 –5 –6 –7 –8 –1 4 5 6 7 8 3 2 1 O page 05 DO NOT WRITE IN THIS MARGIN QUESTION NUMBER 6.(a) 6.(b) page 06 DO NOT WRITE IN THIS MARGIN QUESTION NUMBER 7.(a) 7.(b) page 07 DO NOT WRITE IN THIS MARGIN QUESTION NUMBER 8.(a) 8.(b) page 08 DO NOT WRITE IN THIS MARGIN QUESTION NUMBER 9.(a) 9.(b) page 09 DO NOT WRITE IN THIS MARGIN QUESTION NUMBER 10.(a) 10.(b) page 10 DO NOT WRITE IN THIS MARGIN QUESTION NUMBER 11.(a) page 11 DO NOT WRITE IN THIS MARGIN QUESTION NUMBER 11.(b) page 12 DO NOT WRITE IN THIS MARGIN QUESTION NUMBER 12. page 13 DO NOT WRITE IN THIS MARGIN QUESTION NUMBER 13. 14. page 14 DO NOT WRITE IN THIS MARGIN QUESTION NUMBER 15.(a) 15.(b) (i) page 15 DO NOT WRITE IN THIS MARGIN QUESTION NUMBER 15.(b) (ii) page 16 DO NOT WRITE IN THIS MARGIN QUESTION NUMBER Additional diagram for question 5. y x –2 –3 –4 –5 –6 –7 –8 –1 4 5 6 7 8 3 2 1 O page 17 DO NOT WRITE IN THIS MARGIN ENTER NUMBER OF QUESTION ADDITIONAL SPACE FOR ANSWERS page 18 DO NOT WRITE IN THIS MARGIN ENTER NUMBER OF QUESTION ADDITIONAL SPACE FOR ANSWERS page 19 DO NOT WRITE IN THIS MARGIN ENTER NUMBER OF QUESTION ADDITIONAL SPACE FOR ANSWERS page 20 For marker’s use Question no Marks/Grades

page 22 Notes: 4. For a numerical approach award 0/6. 5. •6 can be awarded for 2 32 4000 x x − = . 6. For candidates who integrate any term at the •5 stage, only •6 is available on follow through for setting their ‘derivative’ to 0. 7. •7, •8 and •9 are only available for working with a derivative which contains an index 2 ≤−. 8. 3 4000 32 must be simplified at •7 or •8 for •7 to be awarded. 9. •8 is not available to candidates who consider a value of 0 x ≤ in the neighbourhood of 5. 10. •9 is still available in cases where a candidate’s table of signs does not lead legitimately to a minimum at •8. 11. •8 and •9 are not available to candidates who state that the minimum exists at a negative value of x . See Candidates C and D. For the table of signs for a derivative, accept: x 5 − 5 5 + ( ) A x ′ − 0 + Shape or slope x → 5 → ( ) A x ′ − 0 + Shape or slope Arrows are taken to mean ‘in the neighbourhood of’ x a 5 b ( ) A x ′ − 0 + Shape or slope Where 0 5 a < < and 5 b > • For this question do not penalise the omission of ‘x’ or the word ‘shape’/‘slope’. • Stating values of ( ) A x ′ in the table is an acceptable alternative to writing ‘+’ or ‘−’ signs. Values must be checked for accuracy. • The only acceptable variations of ( ) A x ′ are: A′, dA dx and 2 32 4000 x x − − . Commonly Observed Responses: Candidate A – differentiating over multiple lines •4 ^ ( ) 1 32 4000 A x x x− ′ = + ( ) 2 32 4000 A x x x− ′ = − •5 8 2 32 4000 0 x x − − = •6 9 1 Candidate B – differentiating over multiple lines 2 1 16 4000 A x x − = + •4 9 ( ) 1 32 4000 A x x x− ′ = + ( ) 2 32 4000 A x x x− ′ = − •5 8 2 32 4000 0 x x − − = •6 9 1 Candidate C – only considers 5 2 1 16 4000 A x x − = + •4 9 2 32 4000 0 A x x − ′ = − = •5 9 •6 9 5 x = ± •7 8 x → 5 → A′ − 0 + ∴ minimum •8 9 1 1200 or min value 1200 A = = •9 9 1 Candidate D – considers 5 and negative 5 in separate tables 2 1 16 4000 A x x − = + •4 9 2 32 4000 0 A x x − ′ = − = •5 9 •6 9 5 x = ± •7 8 x → 5 → x → −5 → A′ − 0 + A′ − 0 + ∴ minimum when 5 x = •8 9 1 1200 or min value 1200 A = = •9 9 1 Ignore incorrect working in second table page 23 Question Generic scheme Illustrative scheme Max mark 12. Method 1 •1 state linear equation •2 introduce logs •3 use laws of logs •4 use laws of logs •5 state a and b Method 1 •1 log4 3 1 y x = − •2 log log log 4 4 4 3 4 4 y x = − •3 log log log 3 4 4 4 4 4 x y = − •4 log log 3 4 4 4 4 x y ⎛ ⎞ = ⎜ ⎟ ⎝ ⎠ or log log 1 3 4 4 4 4 x y − = •5 , 1 64 4 a b = = 5 Method 2 •1 state linear equation •2 convert to exponential form •3 use laws of indices •4 state a •5 state b Method 2 •1 log4 3 1 y x = − •2 3 1 4 x y − = •3 1 3 4 4 x y − = •4 1 4 a = •5 64 b = 5 Method 3 •1 introduce logs to x y ab = •2 use laws of logs •3 interpret intercept •4 interpret gradient •5 state a and b Method 3 The equations at •1, •2, •3 and •4 must be stated explicitly. •1 log log 4 4 x y ab = •2 log log log 4 4 4 y a x b = + •3 log4 1 a −= •4 log4 3 b = •5 , 1 64 4 a b = = 5 page 24 Question Generic scheme Illustrative scheme Max mark Method 4 •1 interpret point on log graph •2 convert from log to exponential form •3 interpret point and convert •4 substitute into x y ab = and evaluate a •5 substitute other point into x y ab = and evaluate b Method 4 •1 log4 3 8 x y = = and •2 8 3 4 x y = = and •3 log4 0 1 x y = =− and 1 0 4 x y − = = and •4 1 0 1 4 4 ab a −= ⇒ = •5 8 3 1 4 64 4 b b = ⇒ = 5 Notes: 1. In any method, marks may only be awarded within a valid strategy using x y ab = . 2. Accept 1 64 4 x y ⋅ = for •5. 3. Markers must identify the method which best matches the candidates approach; they must not mix and match between methods. 4. Penalise the omission of base 4 at most once in any method. 5. Do not accept 1 4 a − = . Commonly Observed Responses: page 25 Question Generic scheme Illustrative scheme Max mark 13. •1 interpret information given •2 integrate any two terms •3 complete integration •4 interpret information given and substitute •5 process for c and state expression for ( ) f x •1 ( ) 2 3 16 11 f x x x ′ = − + or ( ) ( ) 2 3 16 11 f x x x dx = − + ∫ •2 eg ... 3 2 3 16 3 2 x x − •3 11x c …+ + •4 3 2 0 7 8 7 11 7 c = − × + × + •5 ( ) 3 2 8 11 28 f x x x x = − + − 5 Notes: 1. For candidates who make no attempt to integrate to find ( ) f x award 0/5. 2. Do not penalise the omission of ( ) f x or dx or the appearance of +c at •1. 3. If any two terms have been integrated correctly •1 may be implied by •2. 4. For candidates who omit c + , only •1 and •2 are available. 5. For candidates who differentiate any term, •3 •4 and •5 are not available. 6. Candidates must attempt to integrate both terms containing x for •4 and •5 to be available. See Candidate B. 7. Accept 3 2 8 11 28 y x x x = − + − at •5 since ( ) y f x = is defined in the question. 8. Candidates must simplify coefficients in their final line of working for the last mark available in that line of working to be awarded. Commonly Observed Responses: Candidate A – incomplete substitution ( ) 3 2 8 11 f x x x x c = − + + •1 9 •2 9 •3 9 ( ) 3 2 7 8 7 11 7 f x c = −× + × + •4 ^ 28 c = − ( ) 3 2 8 11 28 f x x x x = − + − •5 9 1 Candidate B - partial integration ( ) 3 2 8 11 f x x x c = − + + •1 9 •2 9 •3 8 3 2 0 7 8 7 11 c = − × + + •4 9 1 38 c = ( ) 3 2 8 49 f x x x = − + •5 9 1 page 26 Question Generic scheme Illustrative scheme Max mark 14. •1 expand •2 evaluate u.u •3 determine equation in cos θ •4 evaluate angle •1 . . + uu uv •2 16 •3 cos 20 5 θ = or cos 5 20 θ = •4 75 5 ⋅…° or 1 31 ⋅ … radians 4 Notes: 1. Do not accept 2 u for •1, however •2, •3 and •4 are still available. 2. Where there is no evidence for •1, then •2, •3 and •4 are not available, however see Candidates C and D. 3. Where candidates use 4 ≠ u , then •3 and •4 are not available. 4. Where there is no evidence of using 2 u , •3 is not available. See Candidate A. 5. Do not penalise omission of units in final answer. 6. Ignore the appearance of 284 5⋅°. 7. Accept answers which round to 76° or 1·3 radians. Commonly Observed Responses: Candidate A ( ) . . . + = + u u v uu u v •1 9 cos 4 20 21 θ + = •2 8 cos 17 20 θ = •3 9 2 31 7 θ = ⋅…° •4 9 1 Candidate B . 16 21 + = u v •1 9 •2 9 . 5 = uv cos 5 20 θ = •3 9 75 5 θ = ⋅° •4 9 Candidate C – missing working . 16 = u u •2 9 . 21 16 = − u v cos 5 20 θ = •1 9 •3 9 75 5 θ = ⋅° •4 9 Candidate D – missing working 21 16 5 − = •1 ^ cos 5 20 θ = •2 9 •3 9 75 5 θ = ⋅° •4 9 page 27 Question Generic scheme Illustrative scheme Max mark 15. (a) •1 find gradient of radius •2 state gradient of tangent •3 state equation of tangent •1 1 3 − •2 3 •3 3 2 y x = − 3 Notes: 1. Do not accept 3 2 1 y x = − for •3. 2. •3 is only available as a consequence of trying to find and use a perpendicular gradient. 3. At •3 accept, 3 2 0 y x − + = or any other rearrangement of the equation where the constant terms have been simplified. Commonly Observed Responses: (b) (i) •4 find coordinates of T •4 (0,−2) 1 (ii) •5 find midpoint CT •6 find radius of circle with diameter CT •7 state equation of circle •5 (4,5) •6 65 stated or implied by •7 •7 ( ) ( ) 2 2 4 5 65 x y − + − = 3 Notes: 4. Answers in part (b)(i) must be consistent with answers from part (a). 5. Accept 0, 2 x y = = − for •4. 6. ( ) ( ) ( ) 2 2 2 4 5 65 x y − + − = does not gain •7. 7. •7 is not available to candidates who use a line other than CT as the diameter of the circle. Commonly Observed Responses: [END OF MARKING INSTRUCTIONS]

24 (A420U30-1) © WJEC CBAC Ltd. 24 10. A rectangular coil rotates at a constant angular velocity within a uniform magnetic field of 0.114 T. The coil has 270 turns and cross-sectional area 420 cm2. The diagram below is a simplified 3D diagram of the coil. N S axis of rotation This second diagram is a 2D representation of the coil looking along the axis of rotation. normal coil uniform B-field of 0.114 T The flux linkage of the coil for the angles θ = –5° and θ = +5° is 1.29 Wb turn in each case. normal θ –5° +5° uniform B-field of 0.114 T (A420U30-1) Turn over. 25 © WJEC CBAC Ltd. 25 (a) Show clearly how this value for the flux linkage is obtained. [2] (b) Explain why the induced emf is zero when θ = 0. [1] (c) The flux linkage of the coil for the angles θ = 85° and θ = 95° are 0.11 Wb turn and –0.11 Wb turn respectively. The coil rotates 10° in a time of 5.8 ms. Calculate the mean induced emf when rotating between θ = 85° and θ = 95°. [2] Examiner only 5 26 (A420U30-1) © WJEC CBAC Ltd. 26 BLANK PAGE PLEASE DO NOT WRITE ON THIS PAGE (A420U30-1) Turn over. 27 © WJEC CBAC Ltd. 27 SECTION B: OPTIONAL TOPICS Option A – Alternating Currents Option B – Medical Physics Option C – The Physics of Sports Option D – Energy and the Environment Answer the question on one topic only. Place a tick (√) in one of the boxes above, to show which topic you are answering. You are advised to spend about 25 minutes on this section. 28 (A420U30-1) © WJEC CBAC Ltd. 28 Examiner only Option A – Alternating Currents 11. (a) Derive the expression for the resonance frequency, f0, of a series RCL circuit. [3] (b) Consider the following RCL circuit. Variable frequency a.c. supply, Vrms = 15 V 28 Ω 24 mH 8.2 nF (i) Calculate the rms current at the resonance frequency (f0). [1] f0 = 1 2π LC (A420U30-1) Turn over. 29 © WJEC CBAC Ltd. 29 Examiner only (ii) Calculate the rms current at twice the resonance frequency (2f0). [4] (iii) Calculate the Q factor of the RCL circuit. [2] (iv) Sketch a graph of the rms current in the RCL circuit versus applied frequency of the a.c. supply on the axes provided. Label this graph 28 Ω. [3] rms current / A Frequency f0 2f0 (v) The 28 Ω resistor is replaced by a 56 Ω resistor. On the same axes, sketch and label a second graph showing the rms current versus frequency for the new circuit. [2] 30 (A420U30-1) © WJEC CBAC Ltd. 30 Examiner only (c) Morgan claims that the rms output pd (Vout) in the following circuit is greater than 4.25 V when the frequency is greater than 82.5 Hz but less than 4.25 V below 82.5 Hz. Investigate whether or not Morgan is correct. [5] Vout 28 Ω 54 mH Vin = 6.0 V (rms) (Variable frequency) 20 (A420U30-1) Turn over. 31 © WJEC CBAC Ltd. 31 BLANK PAGE PLEASE DO NOT WRITE ON THIS PAGE 32 (A420U30-1) © WJEC CBAC Ltd. 32 Examiner only Option B – Medical Physics 12. (a) (i) Sketch graphs to show how the intensity of X-rays from an X-ray tube varies with wavelength for a tube operating at two different voltages. Label the main features of the graphs and indicate which curve represents the higher voltage. [4] Wavelength Intensity (ii) The lower voltage tube operates at 20 kV. Determine the velocity with which the electrons strike the target. [2] (iii) Calculate the minimum wavelength of the X-ray photons produced by these electrons. [2] (A420U30-1) Turn over. 33 © WJEC CBAC Ltd. 33 (b) (i) Ultrasound can be used to carry out either an amplitude scan (A-scan) or a brightness scan (B-scan). Explain which of these two methods you would use to determine the depth of the tumour. Justify your answer. [2] (ii) An ultrasound scan can be used to indicate the thickness of fat on a person’s body. Typically fat has a density of 930 kg m–3 and an acoustic impedance of 1.35 × 106 kg m–2 s–1. If the time delay for the ultrasound pulse is 0.040 ms. Determine the thickness of fat. [3] Examiner only 34 (A420U30-1) © WJEC CBAC Ltd. 34 Examiner only (c) (i) Explain two properties of a radioactive isotope used as a tracer in medicine. [2] (ii) A small volume of Human Serum Albumin (HSA) labelled with iodine-125 of activity 160 Bq is injected into the bloodstream of a patient. A sample of 0.8 cm3 of blood was taken a few hours later and was found to have an activity of 0.025 Bq. If the half-life of iodine-125 is 60 days, calculate the volume of blood in the patient. State any assumptions you make. [3] (d) In a magnetic resonance imaging (MRI) scanner a large magnetic field of 1.5 T is used along with short pulses of radio waves. Dr Francis suggests that radio waves of wavelength approximately 5 m would be suitable for this MRI scanner. Determine whether or not Dr Francis is correct. [2] 20 (A420U30-1) Turn over. 35 © WJEC CBAC Ltd. 35 BLANK PAGE PLEASE DO NOT WRITE ON THIS PAGE 36 (A420U30-1) © WJEC CBAC Ltd. 36 Examiner only Option C – The Physics of Sports 13. (a) The diagram shows the forces acting on an athlete’s arm as she holds a discus. Calculate the magnitude of the tension, T, provided by the deltoid muscle. [3] 18° 0.14 m 0.35 m 0.80 m Tension, T Wdiscus = 19.6 N Warm = 39.0 N Deltoid (A420U30-1) Turn over. 37 © WJEC CBAC Ltd. 37 (b) (i) Define angular acceleration. [1] (ii) When thrown, the discus experiences an angular acceleration. It accelerates from rest to 2.3 revolutions per second in a time of 0.27 s. Calculate the angular acceleration of the discus. [2] (iii) Calculate the mean torque applied to the discus of mass 2.0 kg and radius 11 cm. The moment of inertia is given by the equation I = . [3] mr2 2 Examiner only 38 (A420U30-1) Examiner only © WJEC CBAC Ltd. (c) (i) Calculate the maximum height attained by the discus if it is thrown with a velocity of 24 m s–1 and an angle of 38° from a height of 1.2 m. Ignore the effects of air on the motion of the discus. [4] 38 38° 1.2 m 24 m s–1 (A420U30-1) Turn over. 39 Examiner only © WJEC CBAC Ltd. 39 (ii) Taking the effects of air into account, evaluate whether the horizontal distance travelled by the discus will increase, decrease or remain approximately the same. The diagram shows the speed of air relative to the upper and lower surfaces of the discus. The density of air = 1.3 kg m–3. [5] Speed of air = 21 m s–1 Speed of air = 22 m s–1 (iii) A wind tunnel is used to examine the motion of the discus in a wind speed of 20 m s–1. Calculate the factor by which the drag force increases if the speed of the wind is increased to 30 m s–1 and all other factors are kept constant. [2] 20 40 (A420U30-1) Examiner only © WJEC CBAC Ltd. 40 Option D – Energy and the Environment 14. (a) (i) Solar energy resources are considered to be renewable resources. State what is meant by a renewable energy resource. [1] (ii) The proton-proton chain is a set of reactions that take place in our Sun and can be summarised in the following equation. Use data to show that the percentage mass loss is approximately 0.7%. [2] Mass of H = 1.007 28 u Mass of He = 4.001 51 u Mass of e = 0.000 55 u Mass of νe = 0.000 00 u (iii) The Sun can be assumed to have come to the end of its life when it has lost 0.7 % of its mass to radiated energy. Estimate the lifetime of the Sun in years. Take the mass of the Sun to be 2.0 × 1030 kg and assume it to have a constant power output of 3.8 × 1026 W. [2] H + + He 41 1 2 0 –1e 4 2 2νe 1 1 4 2 0 –1 (A420U30-1) Turn over. 41 © WJEC CBAC Ltd. 41 Examiner only (b) The power output, P, from a photovoltaic (PV) cell of surface area, A, can be calculated using the equation: P = μAIcosθ where μ is the conversion efficiency of the cell, I is the intensity of solar radiation and θ is the angle between the normal and the incident sunlight. 20° 60° PV cell θ Diagram not to scale A factory decides to install rooftop PV cells at an angle of 20° to the horizontal. At midday when the Sun’s elevation is 60° the solar radiation incident upon the surface of the Earth has an intensity of 600 W m–2. An individual PV cell has a conversion efficiency of 20 % and is found to produce a power output of 150 W. (i) Show that the area of the PV cell is approximately 1.3 m2. [2] (ii) The factory roof covers an area of 3.6 × 104 m2 and the factory owner plans for the installation to produce a mean power output of 4.0 MW. It is suggested that the company should install 27 500 of these PV cells. Discuss whether or not you believe this to be suitable. [3] (A420U30-1) 42 © WJEC CBAC Ltd. 42 (c) (i) Describe what is meant by the enrichment of uranium and explain why it is necessary. [2] (ii) Gaseous diffusion was one of the original methods used to enrich uranium fuel. Uranium hexafluoride gas composing of 235UF6 and 238UF6 moves from a region of high pressure to a region of low pressure through a porous membrane. The lighter and faster 235UF6 molecules diffuse through the membrane at a greater rate. The first stage of the enrichment process is shown below. Porous membrane During this stage, the mixture is enriched by a factor that can be calculated using: Use the equation and data below to show that the uranium hexafluoride gas would need to complete more than 450 stages if it is to increase the concentration of uranium-235 from 0.7% to 5%. [3] Molar mass of 235UF6 = 349 g mol–1 Molar mass of 238UF6 = 352 g mol–1 (iii) State an alternative method for the enrichment of uranium. [1] Further enrichment Gas enriched in 235UF6 Gas depleted of 235UF6 235UF6 238UF6 Gas mixture of 235UF6 and 238UF6 Examiner only enrichment factor = molar mass of 238UF6 molar mass of 235UF6 43 (A420U30-1) © WJEC CBAC Ltd. 43 (d) A method of reducing CO2 emissions involves the use of fuel cells to power cars. The basic layout for a proton exchange membrane fuel cell is shown in the diagram below. Electric current Unused gases out Electrolyte Anode Cathode Air in Fuel in Excess fuel out H2O H2 O2 (i) Describe the purpose of the electrolyte and state the waste product from the reaction. [2] (ii) In order for the fuel cell to be considered to have no CO2 emissions, the sourcing of the hydrogen fuel must be considered. Two possible sources are: • Electrolysis of water – where electrical energy is used to split water molecules to create hydrogen and oxygen; • Reforming fossil fuels – where steam at high temperature is reacted with a fossil fuel to separate the hydrogen from the carbon in a hydrocarbon. Discuss their likely impact on CO2 emissions. [2] END OF PAPER Examiner only 20 Turn over. e– e– e– e– H+ H+ (A420U30-1) 44 © WJEC CBAC Ltd. 44 Question number Additional page, if required. Write the question number(s) in the left-hand margin. Examiner only

2 1. A system of three equations is defined by kx + 3y –   z = 3 3x –   y +   z = –k –16x – ky – kz = k where k is a positive constant. Given that there is no unique solution to all three equations, (a) show that k = 2 (2) Using k = 2 (b) determine whether the three equations are consistent, justifying your answer. (3) (c) Interpret the answer to part (b) geometrically. (1) _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________

3 Turn over Question 1 continued _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ (Total for Question 1 is 6 marks)

4 2. Given that 1 1 2 1 2 2 3i 39 2 arg( ) 4 = + = = z z z π z z where z1 and z2 are complex numbers, (a) write z1 in the form r(cos θ + i sin θ) Give the exact value of r and give the value of θ in radians to 4 significant figures. (2) (b) Find z2 giving your answer in the form a + ib where a and b are integers. (6) _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________

7 Turn over Question 3 continued _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ (Total for Question 3 is 5 marks) 8 4. All units in this question are in metres. A lawn is modelled as a plane that contains the points L (–2, –3, –1), M (6, –2, 0) and N (2, 0, 0), relative to a fixed origin O. (a) Determine a vector equation of the plane that models the lawn, giving your answer in the form r = a + λb + μc (3) (b) (i) Show that, according to the model, the lawn is perpendicular to the vector 1 2 10 −         (ii) Hence determine a Cartesian equation of the plane that models the lawn. (4) There are two posts set in the lawn. There is a washing line between the two posts. The washing line is modelled as a straight line through points at the top of each post with coordinates P (–10, 8, 2) and Q (6, 4, 3). (c) Determine a vector equation of the line that models the washing line. (2) (d) State a limitation of one of the models. (1) The point R (2, 5, 2.75) lies on the washing line. (e) Determine, according to the model, the shortest distance from the point R to the lawn, giving your answer to the nearest cm. (2) Given that the shortest distance from the point R to the lawn is actually 1.5 m, (f) use your answer to part (e) to evaluate the model, explaining your reasoning. (1) _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________

9 Turn over Question 4 continued _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________

10 Question 4 continued _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________

11 Turn over Question 4 continued _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ (Total for Question 4 is 13 marks) 12 5. r (r + 2) (r + 1) Figure 2 A block has length (r + 2) cm, width (r + 1) cm and height r cm, as shown in Figure 2. In a set of n such blocks, the first block has a height of 1 cm, the second block has a height of 2 cm, the third block has a height of 3 cm and so on. (a) Use the standard results for r r r r n r n r n 3 1 2 1 1 = = = ∑ ∑ ∑ , and to show that the total volume, V, of all n blocks in the set is given by V n n n n = + + + 4 1 2 3 ( )( )( ) n  1 (5) Given that the total volume of all n blocks is (n4 + 6n3 – 11 710) cm3 (b) determine how many blocks make up the set. (2) _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________

13 Turn over Question 5 continued _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________

14 Question 5 continued _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________

15 Turn over Question 5 continued _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ (Total for Question 5 is 7 marks)

16 6. (i) A = −     2 4 a a b where a and b are non-zero constants. Given that the matrix A is self-inverse, (a) determine the value of b and the possible values for a. (5) The matrix A represents a linear transformation M. Using the smaller value of a from part (a), (b) show that the invariant points of the linear transformation M form a line, stating the equation of this line. (3) (ii) P = −     p p p 2 1 3 where p is a positive constant. The matrix P represents a linear transformation U. The triangle T has vertices at the points with coordinates (1, 2), (3, 2) and (2, 5). The area of the image of T under the linear transformation U is 15 (a) Determine the value of p. (4) The transformation V consists of a stretch scale factor 3 parallel to the x‑axis with the y-axis invariant followed by a stretch scale factor –2 parallel to the y‑axis with the x-axis invariant. The transformation V is represented by the matrix Q. (b) Write down the matrix Q. (2) Given that U followed by V is the transformation W, which is represented by the matrix R, (c) find the matrix R. (2) _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________

17 Turn over Question 6 continued _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________

18 Question 6 continued _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________

19 Turn over Question 6 continued _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ (Total for Question 6 is 16 marks) 20 7. f(z) = z4 + az3 + bz2 + cz + d where a, b, c and d are real constants. The equation f(z) = 0 has complex roots z1, z2, z3 and z4 When plotted on an Argand diagram, the points representing z1, z2, z3 and z4 form the vertices of a square, with one vertex in each quadrant. Given that z1 = 2 + 3i, determine the values of a, b, c and d. (6) _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________

21 Turn over Question 7 continued _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ (Total for Question 7 is 6 marks)

22 8. Prove by induction that, for n ∈ + f(n) = 2n + 2 + 32n + 1 is divisible by 7 (6) _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________

23 Turn over Question 8 continued _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ (Total for Question 8 is 6 marks)

24 9. The cubic equation 3x3 + x2 – 4x + 1 = 0 has roots α, β, and γ. Without solving the cubic equation, (a) determine the value of 1 1 1 α β γ + + (3) (b) find a cubic equation that has roots 1 1 1 , and α β γ , giving your answer in the form x3 + ax2 + bx + c = 0, where a, b and c are integers to be determined. (3) _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________

25 Turn over Question 9 continued _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ (Total for Question 9 is 6 marks)

26 10. Given that there are two distinct complex numbers z that satisfy { } 3 : 3 5i 2 :arg( 2) 4 π z z r z z   − − = ∩ − =     determine the exact range of values for the real constant r. (7) _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________

4 2. Mary, Jahil and Dawn are judging the cakes in a village show. They have 5 features to consider and each feature is awarded up to 5 points. The total score the judges gave each cake are given in the table below. Cake A B C D E F G H I Mary 19 17 23 10 21 15 12 8 14 Jahil 22 18 21 10 24 20 16 12 15 Dawn 9 11 6 18 9 15 13 20 13 (a) Calculate Spearman’s rank correlation coefficient between Mary’s scores and Jahil’s scores. (4) (b) Calculate Spearman’s rank correlation coefficient between Jahil’s scores and Dawn’s scores. (3) The judges discussed their interpretation of the points system and agreed that the first prize should go to cake C. (c) Explain how different interpretations of the points system could give rise to the results in part (a) and part (b). (2) _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________

5 Turn over Question 2 continued _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________

6 Question 2 continued _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________

7 Turn over Question 2 continued _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ (Total for Question 2 is 9 marks)

8 3. The continuous random variable X has cumulative distribution function F( ) x px k x x x x = −      0 1 4 4 9 9 < >   where p and k are constants. (a) Find the value of p and the value of k. (4) Given that E(X) = 119 18 (b) show that Var(X) = 2.05 to 3 significant figures. (6) (c) Write down the mode of X. (1) (d) Find the exact value of the constant a such that P(X  a) =  7 27 (3) _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________

9 Turn over Question 3 continued _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________

10 Question 3 continued _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________

11 Turn over Question 3 continued _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ (Total for Question 3 is 14 marks)

12 4. Some students are investigating the strength of wire by suspending a weight at the end of the wire. They measure the diameter of the wire, d mm, and the weight, w  grams, when the wire fails. Their results are given in the following table. These 14 points are plotted on page 13 Not yet plotted d 0.5 0.6 0.7 0.8 0.9 1.1 1.3 1.6 2 2.4 2.8 3.3 3.5 3.9 4.5 4.6 4.8 5.4 w 1.2 1.7 2.3 3.0 3.8 5.6 7.7 11.6 18 25.9 34.9 47.4 52.7 63.9 81 83.6 89.9 109.4 The first 14 points are plotted on the axes on page 13. (a) On the axes on page 13, complete the scatter diagram for these data. (1) (b) Use your calculator to write down the equation of the regression line of w on d. (2) (c) With reference to the scatter diagram, comment on the appropriateness of using this linear regression model to make predictions for w for different values of d between 0.5 and 5.4 (1) The product moment correlation coefficient for these data is r = 0.987 (to 3 significant figures). (d) Calculate the residual sum of squares (RSS) for this model. (2) Robert, one of the students, suggests that the model could be improved and intends to find the equation of the line of regression of w on u, where u = d 2 He finds the following statistics Swu = 5721.625 Suu = 1482.619 ∑u = 157.57 (e) By considering the physical nature of the problem, give a reason to support Robert’s suggestion. (1) (f) Find the equation of the regression line of w on u. (3) (g) Find the residual sum of squares (RSS) for Robert’s model. (2) (h) State, giving a reason based on these calculations, which of these models better describes these data. (1) (i) Hence estimate the weight at which a piece of wire with diameter 3 mm will fail. (1)

13 Turn over Question 4 continued w d 0 100 90 80 70 60 50 40 30 20 10 0 1 2 3 4 5

14 Question 4 continued _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________

15 Turn over Question 4 continued _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________

16 Question 4 continued _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ (Total for Question 4 is 14 marks) TOTAL FOR FURTHER STATISTICS 2 IS 40 MARKS END

3 P62675A Turn over 2. Four workers, A, B, C and D, are each to be assigned to one of four tasks, P, Q, R and S. Each worker must be assigned to one task, and each task must be done by exactly one worker. Worker C cannot be assigned to task Q. The amount, in pounds, that each worker would earn when assigned to each task is shown in the table below. P Q R S A 72 98 59 84 B 67 87 68 86 C 70 – 62 79 D 78 93 64 81 The Hungarian algorithm is to be used to find the maximum total amount that can be earned by the four workers. (a) Explain how the table should be modified so that the Hungarian algorithm may be applied. (2) (b) Modify the table so that the Hungarian algorithm may be applied. (1) (c) Reducing rows first, use the Hungarian algorithm to obtain an allocation that maximises the total earnings. You should explain how any initial row and column reductions were made and also how you determined if the table was optimal at each stage. (6) (Total for Question 2 is 9 marks)

4 P62675A 3. Two teams, A and B, each have three team members. One member of Team A will compete against one member of Team B for 10 rounds of a competition. None of the rounds can end in a draw. Table 1 shows, for each pairing, the expected number of rounds that the member of Team A will win minus the expected number of rounds that the member of Team B will win. These numbers are the scores awarded to Team A. This competition between Teams A and B is a zero‑sum game. Each team must choose one member to play. Each team wants to choose the member who will maximise its score. Team B Paul Qaasim Rashid Team A Mischa 4 –6 2 Noel 0 –2 6 Olive –6 2 0 Table 1 (a) (i) Find the number of rounds that Team A expects to win if Team A chooses Mischa and Team B chooses Paul. (ii) Find the number of rounds that Team B expects to win if Team A chooses Noel and Team B chooses Qaasim. (2) Table 1 models this zero-sum game. (b) (i) Find the play‑safe strategies for the game. (ii) Explain how you know that the game is not stable. (4) (c) Determine which team member Team B should choose if Team B thinks that Team A will play safe. Give a reason for your answer. (1) At the last minute, Rashid is ill and is therefore unavailable for selection by Team B. (d) Find the best strategy for Team B, defining any variables you use. (7) (Total for Question 3 is 14 marks) 4. A sequence {un}, where n 1, satisfies the recurrence relation 2un = un – 1 – kn2 where 4u2 – u0 = 27k2 and k is a non‑zero constant. Show that, as n becomes large, un can be approximated by a quadratic function of the form an2 + bn + c where a, b and c are constants to be determined. (8) (Total for Question 4 is 8 marks) TOTAL FOR DECISION MATHEMATICS 2 IS 40 MARKS END Please check the examination details below before entering your candidate information Candidate surname Other names Total Marks Centre Number Candidate Number Pearson Edexcel Level 3 GCE D2 Answer Book Do not return the question paper with the answer book. Further Mathematics Advanced Subsidiary Further Mathematics options 28: Decision Mathematics 2 (Part of option K only) Afternoon Paper Reference 8FM0/28 Thursday 14 May 2020 P62675A ©2020 Pearson Education Ltd. 1/1/1/1/1/1/1/ Turn over 2 DO NOT WRITE IN THIS AREA DO NOT WRITE IN THIS AREA DO NOT WRITE IN THIS AREA 1. S A D B C C1 C2 F E T 15 15 29 26 23 y 26 23 17 7 18 15 13 10 8 5 37 32 18 18 12 12 7 0 14 x 12 10 C1 C2 Figure 1 _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ 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_____________________________________________________________________________________ _____________________________________________________________________________________ 3 Turn over DO NOT WRITE IN THIS AREA DO NOT WRITE IN THIS AREA DO NOT WRITE IN THIS AREA Question 1 continued _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ 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_____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ (Total for Question 1 is 9 marks) 4 DO NOT WRITE IN THIS AREA DO NOT WRITE IN THIS AREA DO NOT WRITE IN THIS AREA 2. P Q R S A 72 98 59 84 B 67 87 68 86 C 70 – 62 79 D 78 93 64 81 _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ You may not need to use all of these tables. _ ________________________________________________ _ ________________________________________________ _ ________________________________________________ _ ________________________________________________ _ ________________________________________________ _ ________________________________________________ _ ________________________________________________ _ ________________________________________________ _ ________________________________________________ _ ________________________________________________ _ ________________________________________________ _ ________________________________________________ _ ________________________________________________ _ ________________________________________________ _ ________________________________________________ _ ________________________________________________ _ ________________________________________________ _ ________________________________________________ _ ________________________________________________ _ ________________________________________________ _ ________________________________________________ _ ________________________________________________ _ ________________________________________________ _ ________________________________________________ P Q R S A B C D P Q R S A B C D P Q R S A B C D 5 Turn over DO NOT WRITE IN THIS AREA DO NOT WRITE IN THIS AREA DO NOT WRITE IN THIS AREA Question 2 continued _ ________________________________________________ _ ________________________________________________ _ ________________________________________________ _ ________________________________________________ _ ________________________________________________ _ ________________________________________________ _ ________________________________________________ _ ________________________________________________ _ ________________________________________________ _ ________________________________________________ _ ________________________________________________ _ ________________________________________________ _ ________________________________________________ _ ________________________________________________ _ ________________________________________________ _ ________________________________________________ _ ________________________________________________ _ ________________________________________________ _ ________________________________________________ _ ________________________________________________ _ ________________________________________________ _ ________________________________________________ _ ________________________________________________ _ ________________________________________________ _ ________________________________________________ _ ________________________________________________ _ ________________________________________________ _ ________________________________________________ _ ________________________________________________ _ ________________________________________________ _ ________________________________________________ _ ________________________________________________ _ ________________________________________________ _ ________________________________________________ (Total for Question 2 is 9 marks) P Q R S A B C D P Q R S A B C D P Q R S A B C D P Q R S A B C D P Q R S A B C D 6 DO NOT WRITE IN THIS AREA DO NOT WRITE IN THIS AREA DO NOT WRITE IN THIS AREA 3. Team B Paul Qaasim Rashid Team A Mischa 4 –6 2 Noel 0 –2 6 Olive –6 2 0 Table 1 _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ 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_____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ 7 Turn over DO NOT WRITE IN THIS AREA DO NOT WRITE IN THIS AREA DO NOT WRITE IN THIS AREA Question 3 continued _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ 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DO NOT WRITE IN THIS AREA Question 3 continued _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ 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_____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ 9 Turn over DO NOT WRITE IN THIS AREA DO NOT WRITE IN THIS AREA DO NOT WRITE IN THIS AREA Question 3 continued _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ 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_____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ (Total for Question 3 is 14 marks) 10 DO NOT WRITE IN THIS AREA DO NOT WRITE IN THIS AREA DO NOT WRITE IN THIS AREA 4. _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ 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_____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ 11 Turn over DO NOT WRITE IN THIS AREA DO NOT WRITE IN THIS AREA DO NOT WRITE IN THIS AREA Question 4 continued 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_____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ 12 DO NOT WRITE IN THIS AREA DO NOT WRITE IN THIS AREA DO NOT WRITE IN THIS AREA Question 4 continued _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ 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_____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ (Total for Question 4 is 8 marks) TOTAL FOR DECISION MATHEMATICS 2 IS 40 MARKS

2 1. A curve has equation y = 2x3 – 4x + 5 Find the equation of the tangent to the curve at the point P(2, 13). Write your answer in the form y = mx + c, where m and c are integers to be found. Solutions relying on calculator technology are not acceptable. (5) _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________

3 Turn over Question 1 continued _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ (Total for Question 1 is 5 marks)

16 8. The temperature, θ °C, of a cup of tea t minutes after it was placed on a table in a room, is modelled by the equation θ = 18 + 65e −t 8 t  0 Find, according to the model, (a) the temperature of the cup of tea when it was placed on the table, (1) (b) the value of t, to one decimal place, when the temperature of the cup of tea was 35 °C. (3) (c) Explain why, according to this model, the temperature of the cup of tea could not fall to 15 °C. (1) (8, 50) (0, 94) μ O l t Figure 2 The temperature, μ °C, of a second cup of tea t minutes after it was placed on a table in a different room, is modelled by the equation μ = A + Be −t 8 t  0 where A and B are constants. Figure 2 shows a sketch of μ against t with two data points that lie on the curve. The line l, also shown on Figure 2, is the asymptote to the curve. Using the equation of this model and the information given in Figure 2 (d) find an equation for the asymptote l. (4) _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________

17 Turn over Question 8 continued _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________

18 Question 8 continued _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________

19 Turn over Question 8 continued _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ (Total for Question 8 is 9 marks) 20 9. P(c, d) y O x Figure 3 Figure 3 shows part of the curve with equation y = 3 cos x°. The point P(c, d) is a minimum point on the curve with c being the smallest negative value of x at which a minimum occurs. (a) State the value of c and the value of d. (1) (b) State the coordinates of the point to which P is mapped by the transformation which transforms the curve with equation y = 3 cos x° to the curve with equation (i) y = 3 cos  x°     4 (ii) y = 3 cos (x – 36)° (2) (c) Solve, for 450°  θ < 720°, 3 cos θ = 8 tan θ giving your solution to one decimal place. In part (c) you must show all stages of your working. Solutions relying entirely on calculator technology are not acceptable. (5)

21 Turn over Question 9 continued _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________

22 Question 9 continued _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________

23 Turn over Question 9 continued _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ (Total for Question 9 is 8 marks) 24 10. g(x) = 2x3 + x2 – 41x – 70 (a) Use the factor theorem to show that g(x) is divisible by (x – 5). (2) (b) Hence, showing all your working, write g(x) as a product of three linear factors. (4) The finite region R is bounded by the curve with equation y = g(x) and the x‑axis, and lies below the x‑axis. (c) Find, using algebraic integration, the exact value of the area of R. (4) _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________

25 Turn over Question 10 continued _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________

26 Question 10 continued _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________

27 Turn over Question 10 continued _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ (Total for Question 10 is 10 marks)

28 11. (i) A circle C1 has equation x2 + y2 + 18x – 2y + 30 = 0 The line l is the tangent to C1 at the point P(–5, 7). Find an equation of l in the form ax + by + c = 0, where a, b and c are integers to be found. (5) (ii) A different circle C2 has equation x2 + y2 – 8x + 12y + k = 0 where k is a constant. Given that C2 lies entirely in the 4th quadrant, find the range of possible values for k. (4) _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________

29 Turn over Question 11 continued _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________

30 Question 11 continued _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________

31 Turn over Question 11 continued _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ (Total for Question 11 is 9 marks)

32 12. An advertising agency is monitoring the number of views of an online advert. The equation log10 V = 0.072t + 2.379 1  t  30, t ∈  is used to model the total number of views of the advert, V, in the first t days after the advert went live. (a) Show that V = abt where a and b are constants to be found. Give the value of a to the nearest whole number and give the value of b to 3 significant figures. (4) (b) Interpret, with reference to the model, the value of ab. (1) Using this model, calculate (c) the total number of views of the advert in the first 20 days after the advert went live. Give your answer to 2 significant figures. (2) _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________

33 Turn over Question 12 continued _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________

34 Question 12 continued _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________

35 Turn over Question 12 continued _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ (Total for Question 12 is 7 marks)

36 13. (a) Prove that for all positive values of a and b 4a b b a +  4 (4) (b) Prove, by counter example, that this is not true for all values of a and b. (1) _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________

37 Turn over Question 13 continued _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ (Total for Question 13 is 5 marks)

2 1. At time t = 0, a small ball is projected vertically upwards with speed U m s–1 from a point A that is 16.8 m above horizontal ground. The speed of the ball at the instant immediately before it hits the ground for the first time is 19 m s–1 The ball hits the ground for the first time at time t = T seconds. The motion of the ball, from the instant it is projected until the instant just before it hits the ground for the first time, is modelled as that of a particle moving freely under gravity. The acceleration due to gravity is modelled as having magnitude 10 m s–2 Using the model, (a) show that U = 5 (2) (b) find the value of T, (2) (c) find the time from the instant the ball is projected until the instant when the ball is 1.2 m below A. (4) (d) Sketch a velocity‑time graph for the motion of the ball for 0  t  T, stating the coordinates of the start point and the end point of your graph. (2) In a refinement of the model of the motion of the ball, the effect of air resistance on the ball is included and this refined model is now used to find the value of U. (e) State, with a reason, how this new value of U would compare with the value found in part (a), using the initial unrefined model. (1) (f) Suggest one further refinement that could be made to the model, apart from including air resistance, that would make the model more realistic. (1) _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________

3 Turn over Question 1 continued _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________

4 Question 1 continued _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________

5 Turn over Question 1 continued _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ (Total for Question 1 is 12 marks) 6 2. P (4m) Q (3m) Figure 1 One end of a string is attached to a small ball P of mass 4m. The other end of the string is attached to another small ball Q of mass 3m. The string passes over a fixed pulley. Ball P is held at rest with the string taut and the hanging parts of the string vertical, as shown in Figure 1. Ball P is released. The string is modelled as being light and inextensible, the balls are modelled as particles, the pulley is modelled as being smooth and air resistance is ignored. (a) Using the model, find, in terms of m and g, the magnitude of the force exerted on the pulley by the string while P is falling and before Q hits the pulley. (8) (b) State one limitation of the model, apart from ignoring air resistance, that will affect the accuracy of your answer to part (a). (1) _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________

7 Turn over Question 2 continued _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________

8 Question 2 continued _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________

9 Turn over Question 2 continued _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ (Total for Question 2 is 9 marks)

10 3. A particle P moves along a straight line such that at time t seconds, t  0, after leaving the point O on the line, the velocity, v m s–1, of P is modelled as v = (7 – 2t)(t + 2) (a) Find the value of t at the instant when P stops accelerating. (4) (b) Find the distance of P from O at the instant when P changes its direction of motion. (5) In this question, solutions relying on calculator technology are not acceptable. _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________

11 Turn over Question 3 continued _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________

12 Question 3 continued _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ (Total for Question 3 is 9 marks) TOTAL FOR MECHANICS IS 30 MARKS

2 1. The curve C has equation y = 31 sinh x − 2 sinh 2x        x ∈  Determine, in terms of natural logarithms, the exact x coordinates of the stationary points of C. (7) _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________

3 Turn over Question 1 continued _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ (Total for Question 1 is 7 marks)

4 2. In an Argand diagram, the points A and B are represented by the complex numbers −3 + 2i and 5 − 4i respectively. The points A and B are the end points of a diameter of a circle C. (a) Find the equation of C, giving your answer in the form |z − a| = b        a ∈ ,  b ∈  (3) The circle D, with equation |z − 2 − 3i| = 2 , intersects C at the points representing the complex numbers z1 and z2 (b) Find the complex numbers z1 and z2 (6) _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________

5 Turn over Question 2 continued _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________

6 Question 2 continued _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________

7 Turn over Question 2 continued _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ (Total for Question 2 is 9 marks)

8 3. A scientist is investigating the concentration of antibodies in the bloodstream of a patient following a vaccination. The concentration of antibodies, x, measured in micrograms (μg) per millilitre (ml) of blood, is modelled by the differential equation 100 d d 2 2 x t + 60 d d x t + 13x = 26 where t is the number of weeks since the vaccination was given. (a) Find a general solution of the differential equation. (4) Initially, •  there are no antibodies in the bloodstream of the patient •  the concentration of antibodies is estimated to be increasing at 10 μg/ml per week (b) Find, according to the model, the maximum concentration of antibodies in the bloodstream of the patient after the vaccination. (8) A second dose of the vaccine has to be given to try to ensure that it is fully effective. It is only safe to give the second dose if the concentration of antibodies in the bloodstream of the patient is less than 5 μg/ml. (c) Determine whether, according to the model, it is safe to give the second dose of the vaccine to the patient exactly 10 weeks after the first dose. (2) _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________

9 Turn over Question 3 continued _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________

10 Question 3 continued _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________

11 Turn over Question 3 continued _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ (Total for Question 3 is 14 marks)

Turn over 13 Question 4 continued _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________

16 5. (a) y = tan−1 x Assuming the derivative of tan x , prove that d d y x x = + 1 1 2 (3) f ( x ) = x tan−1 4x (b) Show that f d ( ) tan tan x x Ax x Bx C x k ∫ = + + + − − 2 1 1 4 4 where k is an arbitrary constant and A, B and C are constants to be determined. (5) (c) Hence find, in exact form, the mean value of f( x ) over the interval 0 3 4 ,       (2) _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________

17 Turn over Question 5 continued _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________

18 Question 5 continued _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________

19 Turn over Question 5 continued _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ (Total for Question 5 is 10 marks) 20 6. M = −         k 5 7 1 1 1 2 1 1 where k is a constant (a) Given that k ≠ 4, find, in terms of k, the inverse of the matrix M. (4) (b) Find, in terms of p, the coordinates of the point where the following planes intersect. 2x + 5y + 7z = 1 x +   y +   z = p 2x +   y −   z = 2 (3) (c) (i) Find the value of q for which the following planes intersect in a straight line. 4x + 5y + 7z = 1 x +   y +   z = q 2x +   y −   z = 2 (ii) For this value of q, determine a vector equation for the line of intersection. (7) _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________

21 Turn over Question 6 continued _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________

22 Question 6 continued _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________

23 Turn over Question 6 continued _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ (Total for Question 6 is 14 marks) 24 7. y x O A C1 B C2 Figure 1 A student wants to make plastic chess pieces using a 3D printer. Figure 1 shows the central vertical cross-section of the student’s design for one chess piece. The plastic chess piece is formed by rotating the region bounded by the y-axis, the x-axis, the line with equation x = 1, the curve C1 and the curve C2 through 360° about the y-axis. The point A has coordinates (1, 0.5) and the point B has coordinates (0.5, 2.5) where the units are centimetres. The curve C1 is modelled by the equation x = a y b + 0.5  y  2.5 (a) Determine the value of a and the value of b according to the model. (2) The curve C2 is modelled to be an arc of the circle with centre (0, 3). (b) Use calculus to determine the volume of plastic required to make the chess piece according to the model. (9) _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________

25 Turn over Question 7 continued _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________

26 Question 7 continued _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________

27 Turn over Question 7 continued _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________

28 Question 7 continued _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ (Total for Question 7 is 11 marks) TOTAL FOR PAPER IS 75 MARKS

2 1. Use l’Hospital’s Rule to show that 2 sin (e cos(3 ) e) 3 lim tan(2 ) 2 π x x x x → − − = − (5) _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________

3 Turn over Question 1 continued _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ (Total for Question 1 is 5 marks) 4 2. y 3 1 −1 O x Figure 1 Figure 1 shows a sketch of the vertical cross-section of the entrance to a tunnel. The width at the base of the tunnel entrance is 2 metres and its maximum height is 3 metres. The shape of the cross-section can be modelled by the curve with equation y = f ( x ) where f ( x ) = 3 cos 2 2 π x              x ∈ [−1, 1] A wooden door of uniform thickness 85 mm is to be made to seal the tunnel entrance. Use Simpson’s rule with 6 intervals to estimate the volume of wood required for this door, giving your answer in m3 to 4 significant figures. (6) _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________

5 Turn over Question 2 continued _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ (Total for Question 2 is 6 marks)

6 3. The points A, B and C, with position vectors a = 3i − 2j + k , b = i + 4j + 5k and c = −2i + 3j + 3k respectively, lie on the plane Π (a) Find AB → × AC → (3) (b) Find an equation for Π in the form r.n = p (2) The point D has position vector 8i + 7j + 5k (c) Determine the volume of the tetrahedron ABCD (4) _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________

7 Turn over Question 3 continued _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________

8 Question 3 continued _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________

9 Turn over Question 3 continued _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ (Total for Question 3 is 9 marks) 10 4. f (x) = x4 sin(2 x) Use Leibnitz’s theorem to show that the coefficient of (x − π)8 in the Taylor series expansion of f (x) about π is 3 315 aπ bπ + where a and b are integers to be determined. (8) The Taylor series expansion of about x = k is given by f ( ) x f ( ) f ( ) ( )f ( ) ( ) f ( ) ( ) f ( ) r x k x k k x k k x k r k r = + − + − + + − +   ' '' 2 2! ! ( ) … …         _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________

11 Turn over Question 4 continued _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________

12 Question 4 continued _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________

13 Turn over Question 4 continued _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ (Total for Question 4 is 8 marks)

16 6. A physics student is studying the movement of particles in an electric field. In one experiment, the distances in micrometres of two moving particles, A and B, from a fixed point O are modelled by d t d t t A B = − = − + 5 31 3 25 8 2 respectively, where t is the time in seconds after motion begins. (a) Use algebra to find the range of time for which particle A is further away from O than particle B is from O. (8) It was recorded that the distance of particle B from O was less than the distance of particle A from O for approximately 4 seconds. (b) Use this information to assess the validity of the model. (2) _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________

17 Turn over Question 6 continued _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________

18 Question 6 continued _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________

19 Turn over Question 6 continued _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ (Total for Question 6 is 10 marks)

20 7. The points P(9p2, 18p) and Q(9q2, 18q) , p ≠ q , lie on the parabola C with equation y2 = 36x The line l passes through the points P and Q (a) Show that an equation for the line l is ( p + q) y = 2(x + 9pq) (3) The normal to C at P and the normal to C at Q meet at the point A. (b) Show that the coordinates of A are (9( p2 + q2 + pq + 2) , −9pq( p + q)) (7) Given that the points P and Q vary such that l always passes through the point (12, 0) (c) find, in the form y2 = f ( x ) , an equation for the locus of A, giving f ( x ) in simplest form. (4) _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________

21 Turn over Question 7 continued _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________

22 Question 7 continued _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________

23 Turn over Question 7 continued _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ (Total for Question 7 is 14 marks) 24 8. f ( ) sin cos x x x = + − 3 13 6 5 Using the substitution t = tan x 2     (a) show that f ( x ) can be written in the form 3 1 2 3 1 6 2 2 ( ) ( ) + + + t t (3) (b) Hence solve, for 0 < x < 2π , the equation f ( ) x = 3 7 giving your answers to 2 decimal places where appropriate. (5) (c) Use the result of part (a) to show that 4 3 3 3 9 3 3 f ( )d arctan arctan 3 3 π π x x K π       − + = − +             ∫ where K is a constant to be determined. (8) _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________

25 Turn over Question 8 continued _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________

26 Question 8 continued _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________

27 Turn over Question 8 continued _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________

28 Question 8 continued _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ (Total for Question 8 is 16 marks) TOTAL FOR PAPER IS 75 MARKS

2 1. A particle P of mass 0.5 kg is moving with velocity (4i + 3j) m s–1 when it receives an impulse J N s. Immediately after receiving the impulse, P is moving with velocity (–i + 6j) m s–1. (a) Find the magnitude of J. (4) The angle between the direction of the impulse and the direction of motion of P immediately before receiving the impulse is α° (b) Find the value of α (3) _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________

3 Turn over Question 1 continued _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ (Total for Question 1 is 7 marks)

4 2. A truck of mass 1200 kg is moving along a straight horizontal road. At the instant when the speed of the truck is v m s–1, the resistance to the motion of the truck is modelled as a force of magnitude (900 + 9v) N. The engine of the truck is working at a constant rate of 25 kW. (a) Find the deceleration of the truck at the instant when v = 25 (4) Later on, the truck is moving up a straight road that is inclined at an angle θ to the horizontal, where sin θ = 1 20 At the instant when the speed of the truck is v m s–1, the resistance to the motion of the truck from non‑gravitational forces is modelled as a force of magnitude (900 + 9v) N. When the engine of the truck is working at a constant rate of 25 kW the truck is moving up the road at a constant speed of V m s–1. (b) Find the value of V. (5) _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________

5 Turn over Question 2 continued _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________

6 Question 2 continued _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________

7 Turn over Question 2 continued _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ (Total for Question 2 is 9 marks)

8 3. Two particles, A and B, have masses 3m and 4m respectively. The particles are moving in the same direction along the same straight line on a smooth horizontal surface when they collide directly. Immediately before the collision the speed of A is 2u and the speed of B is u. The coefficient of restitution between A and B is e. (a) Show that the direction of motion of each of the particles is unchanged by the collision. (8) After the collision with A, particle B collides directly with a third particle, C, of mass 2m, which is at rest on the surface. The coefficient of restitution between B and C is also e. (b) Show that there will be a second collision between A and B. (6) _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________

9 Turn over Question 3 continued _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________

10 Question 3 continued _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________

10 5. The curve with equation y = 3 × 2 x meets the curve with equation y = 15 − 2 x+1 at the point P. Find, using algebra, the exact x coordinate of P. (4) _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________

11 Turn over Question 3 continued _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ (Total for Question 3 is 14 marks) 12 4. [In this question, i and j are perpendicular unit vectors in a horizontal plane.] (i + 6j) m s–1 (7i + 2j) m s–1 A B Figure 1 Figure 1 represents the plan view of part of a smooth horizontal floor, where AB represents a fixed smooth vertical wall. A small ball of mass 0.5 kg is moving on the floor when it strikes the wall. Immediately before the impact the velocity of the ball is (7i + 2j) m s–1. Immediately after the impact the velocity of the ball is (i + 6j) m s–1. The coefficient of restitution between the ball and the wall is e. (a) Show that AB is parallel to (2i + 3j). (4) (b) Find the value of e. (5) _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________

13 Turn over Question 4 continued _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________

14 Question 4 continued _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________

15 Turn over Question 4 continued _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ (Total for Question 4 is 9 marks)

16 5. A smooth uniform sphere P has mass 0.3 kg. Another smooth uniform sphere Q, with the same radius as P, has mass 0.2 kg. The spheres are moving on a smooth horizontal surface when they collide obliquely. Immediately before the collision the velocity of P is (4i + 2j) m s–1 and the velocity of Q is (–3i + j) m s–1. At the instant of collision, the line joining the centres of the spheres is parallel to i. The kinetic energy of Q immediately after the collision is half the kinetic energy of Q immediately before the collision. (a) Find (i) the velocity of P immediately after the collision, (ii) the velocity of Q immediately after the collision, (iii) the coefficient of restitution between P and Q, carefully justifying your answers. (11) (b) Find the size of the angle through which the direction of motion of P is deflected by the collision. (3) _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________

17 Turn over Question 5 continued _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________

18 Question 5 continued _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________

19 Turn over Question 5 continued _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ (Total for Question 5 is 14 marks)

20 6. A light elastic string with natural length l and modulus of elasticity kmg has one end attached to a fixed point A on a rough inclined plane. The other end of the string is attached to a package of mass m. The plane is inclined at an angle θ to the horizontal, where tan θ =  5 12 The package is initially held at A. The package is then projected with speed 6gl up a line of greatest slope of the plane and first comes to rest at the point B, where AB = 3l. The coefficient of friction between the package and the plane is 1 4 By modelling the package as a particle, (a) show that k = 15 26 (6) (b) find the acceleration of the package at the instant it starts to move back down the plane from the point B. (5) _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________

21 Turn over Question 6 continued _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________

22 Question 6 continued _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________

23 Turn over Question 6 continued _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ (Total for Question 6 is 11 marks) 24 7. v m s–1 1 2 α α A C D B Figure 2 Figure 2 represents the plan view of part of a horizontal floor, where AB and CD represent fixed vertical walls, with AB parallel to CD. A small ball is projected along the floor towards wall AB. Immediately before hitting wall AB, the ball is moving with speed v m s–1 at an angle α to AB, where 0 < α < 2 π The ball hits wall AB and then hits wall CD. After the impact with wall CD, the ball is moving at angle  1 2 α to CD. The coefficient of restitution between the ball and wall AB is  2 3 The coefficient of restitution between the ball and wall CD is also 2 3 The floor and the walls are modelled as being smooth. The ball is modelled as a particle. (a) Show that tan  1 2 α       = 1 3 (7) (b) Find the percentage of the initial kinetic energy of the ball that is lost as a result of the two impacts. (4) _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________

25 Turn over Question 7 continued _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________

26 Question 7 continued _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________

27 Turn over Question 7 continued _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________

28 Question 7 continued _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ (Total for Question 7 is 11 marks) TOTAL FOR PAPER IS 75 MARKS

2 P62678A 1. The table below shows the lengths, in km, of the roads in a network connecting seven towns, A, B, C, D, E, F and G. A B C D E F G A – 24 – 22 35 – – B 24 – 25 27 – – – C – 25 – 33 31 36 26 D 22 27 33 – – 42 – E 35 – 31 – – 37 29 F – – 36 42 37 – 40 G – – 26 – 29 40 – (a) By adding the arcs from vertex D along with their weights, complete the drawing of the network on Diagram 1 in the answer book. (2) (b) Use Kruskal’s algorithm to find a minimum spanning tree for the network. You should list the arcs in the order that you consider them. In each case, state whether you are adding the arc to your minimum spanning tree. (3) (c) State the weight of the minimum spanning tree. (1) (Total for Question 1 is 6 marks) 3 P62678A Turn over 2. A(6) B(7) C(5) G(4) H(3) D(4) E(5) I(4) F(3) K(5) L(6) J(6) Figure 1 The network in Figure 1 shows the activities that need to be undertaken to complete a project. Each activity is represented by an arc and the duration, in hours, of the corresponding activity is shown in brackets. (a) Explain why each of the dummy activities is required. (2) (b) Complete the table in the answer book to show the immediately preceding activities for each activity. (2) (c) (i) Complete Diagram 1 in the answer book to show the early event times and the late event times. (ii) State the minimum completion time for the project. (iii) State the critical activities. (6) Each activity requires one worker. Each worker is able to do any of the activities. Once an activity is started it must be completed without interruption. (d) On Grid 1 in the answer book, draw a resource histogram to show the number of workers required at each time when each activity begins at its earliest possible start time. (3) (e) Determine whether or not the project can be completed in the minimum possible time using fewer workers than the number indicated by the resource histogram in (d). You must justify your answer with reference to the resource histogram and the completed Diagram 1. (2) (Total for Question 2 is 15 marks) 4 P62678A 3. A B C D E 1 7 8 3 6 4 10 1 Figure 2 Direct roads between five villages, A, B, C, D and E, are shown in Figure 2. The weight on each arc is the time, in minutes, it takes to travel along the corresponding road. The road from D to C is one-way as indicated by the arrow on the corresponding arc. Floyd’s algorithm is to be used to find the complete network of shortest times between the five villages. (a) Set up initial time and route matrices. (2) The matrices after two iterations of Floyd’s algorithm are shown below. Time matrix Route matrix A B C D E A – 8 4 7 18 B 8 – 3 15 10 C 4 3 – 11 6 D 7 15 1 – 1 E 18 10 6 1 – A B C D E A A B C D B B A B C A E C A B C A E D A A C D E E B B C D E (b) Perform the next two iterations of Floyd’s algorithm that follow from the tables above. You should show the time and route matrices after each iteration. (4) The final time matrix after completion of Floyd’s algorithm is shown below. Final time matrix A B C D E A – 7 4 7 8 B 7 – 3 10 9 C 4 3 – 7 6 D 5 4 1 – 1 E 6 5 2 1 – 5 P62678A Turn over (c) (i) Use the nearest neighbour algorithm, starting at A, to find a Hamiltonian cycle in the complete network of shortest times. (ii) Find the time taken for this cycle. (iii) Interpret the cycle in terms of the actual villages visited. (3) (Total for Question 3 is 9 marks) 6 P62678A 4. y O 7 – 6 – 5 – 4 – 3 – 2 – 1 – 1 2 3 4 5 x 2y = 5x 6x + 5y = 30 R y = x + 1 – – – – – – – Figure 3 Figure 3 shows the constraints of a linear programming problem in x and y, where R is the feasible region. (a) Write down the inequalities that define R. (2) The objective is to maximise P, where P = 3x + y (b) Obtain the exact value of P at each of the three vertices of R and hence find the optimal vertex, V. (4) The objective is changed to maximise Q, where Q = 3x + ay. Given that a is a constant and the optimal vertex is still V, (c) find the range of possible values of a. (4) (Total for Question 4 is 10 marks)

7 P62678A Turn over 5. The nine distinct numbers in the following list are to be packed into bins of size 50 23        17        19        x        24        8        18        10        21 When the first-fit bin packing algorithm is applied to the numbers in the list it results in the following allocation. Bin 1:    23        17          8 Bin 2:    19          x        10 Bin 3:    24        18 Bin 4:    21 (a) Explain why 13 < x < 21 (3) The same list of numbers is to be sorted into descending order. A bubble sort, starting at the left-hand end of the list, is to be used to obtain the sorted list. After the first complete pass the list is 23        19        17        24        x        18        10        21        8 (b) Using this information, write down the smallest interval that must contain x, giving your answer as an inequality. (2) When the first-fit decreasing bin packing algorithm is applied to the nine distinct numbers it results in the following allocation. Bin 1:    24        23 Bin 2:    21        19        10 Bin 3:    18        17          x Bin 4:      8 Given that only one of the bins is full and that x is an integer, (c) calculate the value of x. You must give reasons for your answer. (2) (Total for Question 5 is 7 marks) 8 P62678A 6. C G E D B A F 19 24 65 42 21 y 15 24 34 45 x 31 Figure 4 [The total weight of the network is 320 + x + y] (a) State, with justification, whether the graph in Figure 4 is Eulerian, semi-Eulerian or neither. (2) The weights on the arcs in Figure 4 represent distances. The weight on arc EF is x where 12 < x < 26 and the weight on arc DG is y where 0 < y < 10 An inspection route of minimum length that traverses each arc at least once is found. The inspection route starts and finishes at A and has a length of 409 It is also given that the length of the shortest route from F to G via A is 140 (b) Using appropriate algorithms, find the value of x and the value of y. (9) (Total for Question 6 is 11 marks)

7 Turn over Question 2 continued _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ (Total for Question 2 is 9 marks) 8 3. M = 1 2 2 4 1 1 2 3 k − −         where k is a constant. (a) Show that, in terms of k, a characteristic equation for M is given by λ3 – (2k + 13)λ + 5(k + 6) = 0 (3) Given that det M = 5 (b) (i) find the value of k (ii) use the Cayley‑Hamilton theorem to find the inverse of M. (7) _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________

9 Turn over Question 3 continued _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________

10 Question 3 continued _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________

11 Turn over Question 3 continued _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ (Total for Question 3 is 10 marks) 12 4. 2.35 m O C 30 cm Initial line Figure 1 Figure 2 Figure 1 shows a sketch of a design for a road speed bump of width 2.35 metres. The speed bump has a uniform cross‑section with vertical ends and its length is 30 cm. A side profile of the speed bump is shown in Figure 2. The curve C shown in Figure 2 is modelled by the polar equation r = 30(1 – θ2) 0  θ  1 The units for r are centimetres and the initial line lies along the road surface, which is assumed to be horizontal. Once the speed bump has been fixed to the road, the visible surfaces of the speed bump are to be painted. Determine, in cm2, the area that is to be painted, according to the model. (10) _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________

13 Turn over Question 4 continued _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________

14 Question 4 continued _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________

15 Turn over Question 4 continued _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ (Total for Question 4 is 10 marks)

16 5. A transformation T from the z‑plane to the w‑plane is given by w = 1 3 2 − + z z i z ≠ –2i The circle with equation ½ z + i ½ = 3 is mapped by T onto the circle C. (a) Show that the equation for C can be written as 3½ w + 3 ½ = ½ 1 + (3 – w)i ½ (4) (b) Hence find (i) a Cartesian equation for C, (ii) the centre and radius of C. (6) _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________

17 Turn over Question 5 continued _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________

18 Question 5 continued _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________

19 Turn over Question 5 continued _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ (Total for Question 5 is 10 marks) 20 6. 1 2 3 4 5 6 Figure 3 Figure 3 shows a plane shape made up of a regular hexagon with an equilateral triangle joined to each edge and with alternate equilateral triangles shaded. The symmetries of this shape are the rotations and reflections of the plane that preserve the shape and its shading. The symmetries of the shape can be represented by permutations of the six vertices labelled 1 to 6 in Figure 3. The set of these permutations with the operation of composition form a group, G. (a) Describe geometrically the symmetry of the shape represented by the permutation 1 2 3 4 3 4 5 6 5 6 1 2     (2) (b) Write down, in similar two‑line notation, the remaining elements of the group G. (4) (c) Explain why each of the following statements is false, making your reasoning clear. (i) G has a subgroup of order 4 (ii) G is cyclic. (2) Diagram 1, on page 23, shows an unshaded shape with the same outline as the shape in Figure 3. (d) Shade the shape in Diagram 1 in such a way that the group of symmetries of the resulting shaded shape is isomorphic to the cyclic group of order 6 (2) _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________

21 Turn over Question 6 continued _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________

22 Question 6 continued _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________

23 Turn over Question 6 continued _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ Diagram 1 Spare copy of Diagram 1 Only use this diagram if you need to redraw your answer to part (d). (Total for Question 6 is 10 marks) 24 7. In = ∫(4 – x2) –n dx n > 0 (a) Show that, for n > 0 I x n x n n I n n n + = − + − 1 2 8 4 2 1 8 ( ) (5) (b) Find I2 (3) _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________

25 Turn over Question 7 continued _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________

26 Question 7 continued _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________

27 Turn over Question 7 continued _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ (Total for Question 7 is 8 marks)

28 8. The four digit number n = abcd satisfies the following properties: (1) n ≡ 3 (mod 7) (2) n is divisible by 9 (3) the first two digits have the same sum as the last two digits (4) the digit b is smaller than any other digit (5) the digit c is even (a) Use property (1) to explain why 6a + 2b + 3c + d ≡ 3 (mod 7) (2) (b) Use properties (2), (3) and (4) to show that a + b = 9 (4) (c) Deduce that c ≡ 5(a – 1) (mod 7) (2) (d) Hence determine the number n, verifying that it is unique. You must make your reasoning clear. (4) _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________

29 Turn over Question 8 continued _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________

30 Question 8 continued _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________

31 Turn over Question 8 continued _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________

2 1 Gina receives a large number of packages from two companies, A and B. She believes that the variance of the weights of packages from company A is greater than the variance of the weights of packages from company B. Gina takes a random sample of 7 packages from company A and an independent random sample of 10 packages from company B. Her results are summarised below a = 300 Saa = 145 496 b = 233.4 Sbb = 56 364.4 [You may assume that the weights of packages from the two companies are normally distributed.] Test Gina’s belief. Use a 5% level of significance and state your hypotheses clearly. (6) _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________

3 Turn over Question 1 continued _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ (Total for Question 1 is 6 marks)

4 2 Jemima makes jam to sell in a local shop. The jam is sold in jars and the weight of jam in a jar is normally distributed. Jemima takes a random sample of 8 of her jars of jam and weighs the contents of each jar, x grams. Her results are summarised as follows ∑ x = 3552 ∑ x2 = 1 577 314 (a) Calculate a 95% confidence interval for the mean weight of jam in a jar. (5) The labels on the jars state that the average contents weigh 440 grams. (b) State, giving a reason, whether or not Jemima should be concerned about the labels on her jars of jam. (1) _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________

5 Turn over Question 2 continued _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ (Total for Question 2 is 6 marks)

6 3 Below are 3 sketches from some students of the residuals from their linear regressions of y on x. Residuals 0 x I Residuals 0 x III Residuals 0 x II For each sketch you should state, giving your reason, (i) whether or not the sketch is feasible and if it is feasible (ii) whether or not the sketch suggests a linear or a non-linear relationship between y and x. (6) _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________

7 Turn over Question 3 continued _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ (Total for Question 3 is 6 marks)

8 4 A biased coin has a probability p of landing on heads, where 0 < p < 1 Simon spins the coin n times and the random variable X represents the number of heads. Taruni spins the coin m times, m ≠ n, and the random variable Y represents the number of heads. Simon and Taruni want to combine their results to find unbiased estimators of p. Simon proposes the estimator S = X Y m n + + and Taruni proposes T = 1 2 X n Y m +     (a) Show that both S and T are unbiased estimators of p. (3) (b) Prove that, for all values of m and n, S is the better estimator. (4) _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________

9 Turn over Question 4 continued _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________

10 Question 4 continued _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________

11 Turn over Question 4 continued _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ (Total for Question 4 is 7 marks) 12 5 y x O 2π Figure 1 The random variable X has probability density function f (x) and Figure 1 shows a sketch of f (x) where f (x) = (1 cos ) 0 2 0 otherwise k x x π −    (a) Show that k = 1 2π (3) The random variable Y ~ N ( μ, σ 2) and E(Y ) = E(X ) The probability density function of Y is g ( y), where g ( y) = 2 1 2 1 e 2 y μ σ σ π −   −               − ∞ < y < ∞ Given that g ( μ) = f ( μ) (b) find the exact value of σ (3) (c) Calculate the error in using P 3 2 2 π π Y       < < as an approximation to P 3 2 2 π π X       < < (4) _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________

13 Turn over Question 5 continued _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________

14 Question 5 continued _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________

15 Turn over Question 5 continued _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ (Total for Question 5 is 10 marks)

16 6 A new employee, Kim, joins an existing employee, Jiang, to work in the quality control department of a company producing steel rods. Each day a random sample of rods is taken, their lengths measured and a 95% confidence interval for the mean length of the rods, in metres, is calculated. It is assumed that the lengths of the rods produced are normally distributed. Kim took a random sample of 25 rods and used the t distribution to obtain a 95% confidence interval of (1.193, 1.367) for the mean length of the rods. Jiang commented that this interval was a little wider than usual and explained that they usually assume that the standard deviation does not change and can be taken as 0.175 metres. (a) Test, at the 10% level of significance, whether or not Kim’s sample suggests that the standard deviation is different from 0.175 metres. State your hypotheses clearly. (9) Using Kim’s sample and the normal distribution with a standard deviation of 0.175 metres, (b) find a 95% confidence interval for the mean length of the rods. (3) _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________

17 Turn over Question 6 continued _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ (Total for Question 6 is 12 marks)

14 Question 4 continued _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________

18 7 Fence panels come in two sizes, large and small. The lengths of the large panels are normally distributed with mean 198 cm and standard deviation 5 cm. The lengths of the small panels are normally distributed with mean 74 cm and standard deviation 3 cm. (a) Find the probability that the total length of a random sample of 3 large panels is greater than the total length of a random sample of 8 small panels. (6) One large panel and one small panel are selected at random. (b) Find the probability that the length of the large panel is more than 8 3 times the length of the small panel. (5) Rosa needs 1000 cm of fencing. The large panels cost £80 each and the small panels cost £30 each. Rosa’s plan is to buy 5 large panels and measure the total length. If the total length is less than 1000 cm she will then buy one small panel as well. (c) Calculate whether or not the expected cost of Rosa’s plan is cheaper than simply buying 14 small panels. (6) _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________

19 Turn over Question 7 continued _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________

20 Question 7 continued _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________

21 Turn over Question 7 continued _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ (Total for Question 7 is 17 marks)

22 8 A circle, centre O, has radius x cm, where x is an observation from the random variable X which has a rectangular distribution on [0, π] (a) Find the probability that the area of the circle is greater than 10 cm2 (3) (b) State, giving a reason, whether the median area of the circle is greater or less than 10 cm2 (1) The triangle OAB is drawn inside the circle with OA and OB as radii of length x cm and angle AOB x radians. (c) Use algebraic integration to find the expected value of the area of triangle OAB. Give your answer as an exact value. (7) _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________

23 Turn over Question 8 continued _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________

24 Question 8 continued _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ (Total for Question 8 is 11 marks) TOTAL FOR PAPER IS 75 MARKS

2 1. Three particles of masses 3m, 4m and 2m are placed at the points (−2, 2) , (3, 1) and ( p, p) respectively. The value of p is such that the distance of the centre of mass of the three particles from the point (0, 0) is as small as possible. Find the value of p. (7) _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________

3 Turn over Question 1 continued _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ (Total for Question 1 is 7 marks) 4 2. y x y = 8e−x R ln 2 O ln 5 A C D B Figure 1 Figure 2 A uniform plane figure R, shown shaded in Figure 1, is bounded by the x-axis, the line with equation x = ln 5 , the curve with equation y = 8e−x and the line with equation x = ln 2 . The unit of length on each axis is one metre. The area of R is 2.4 m2 The centre of mass of R is at the point with coordinates (x, y) . (a) Use algebraic integration to show that y = 1.4 (4) Figure 2 shows a uniform lamina ABCD, which is the same size and shape as R. The lamina is freely suspended from C and hangs in equilibrium with CB at an angle θ° to the downward vertical. (b) Find the value of θ (6) _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________

5 Turn over Question 2 continued _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________

6 Question 2 continued _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________

7 Turn over Question 2 continued _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ (Total for Question 2 is 10 marks)

8 3. A particle P of mass 0.5 kg is moving along the positive x-axis in the direction of x increasing. At time t seconds (t  0) , P is x metres from the origin O and the speed of P is v m s−1. The resultant force acting on P is directed towards O and has magnitude kv2 N, where k is a positive constant. When x = 1 , v = 4 and when x = 2 , v = 2 (a) Show that v = ab x , where a and b are constants to be found. (6) The time taken for the speed of P to decrease from 4 m s−1 to 2 m s−1 is T seconds. (b) Show that T = 1 4 2 ln (4) _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________

9 Turn over Question 3 continued _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________

10 Question 3 continued _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________

11 Turn over Question 3 continued _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ (Total for Question 3 is 10 marks) 12 4. O r 4 3 r Figure 3 A uniform solid cylinder of base radius r and height 4 3 r has the same density as a uniform solid hemisphere of radius r. The plane face of the hemisphere is joined to a plane face of the cylinder to form the composite solid S shown in Figure 3. The point O is the centre of the plane face of S. (a) Show that the distance from O to the centre of mass of S is 73 72 r (4) The solid S is placed with its plane face on a rough horizontal plane. The coefficient of friction between S and the plane is μ. A horizontal force P is applied to the highest point of S. The magnitude of P is gradually increased. (b) Find the range of values of μ for which S will slide before it starts to tilt. (5) _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________

13 Turn over Question 4 continued _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________

14 Question 4 continued _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________

2 1. (a) Find the first four terms, in ascending powers of x, of the binomial expansion of 1 8 1 2 + ( ) x giving each term in simplest form. (3) (b) Explain how you could use x = 1 32 in the expansion to find an approximation for 5 There is no need to carry out the calculation. (2) _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________

Turn over 3 Question 1 continued _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ (Total for Question 1 is 5 marks)

15 Turn over Question 4 continued _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ (Total for Question 4 is 9 marks) 16 5. 40 cm P O A 60 cm Figure 4 A particle P of mass 0.75 kg is attached to one end of a light inextensible string of length 60 cm. The other end of the string is attached to a fixed point A that is vertically above the point O on a smooth horizontal table, such that OA = 40 cm. The particle remains in contact with the table, with the string taut, and moves in a horizontal circle with centre O, as shown in Figure 4. The particle is moving with a constant angular speed of 3 radians per second. (a) Find (i) the tension in the string, (ii) the normal reaction between P and the table. (7) The angular speed of P is now gradually increased. (b) Find the angular speed of P at the instant P loses contact with the table. (4) _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________

17 Turn over Question 5 continued _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________

18 Question 5 continued _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________

19 Turn over Question 5 continued _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ (Total for Question 5 is 11 marks) 20 6. O v P α l Figure 5 A particle P of mass m is attached to one end of a light inextensible string of length l. The other end of the string is attached to a fixed point O. The particle is held with the string taut and OP horizontal. The particle is then projected vertically downwards with speed u, where u2 = 9 5 gl . When OP has turned through an angle α and the string is still taut, the speed of P is v, as shown in Figure 5. At this instant the tension in the string is T. (a) Show that T = 3mg sin α + 9 5 mg (6) (b) Find, in terms of g and l, the speed of P at the instant when the string goes slack. (3) (c) Find, in terms of l, the greatest vertical height reached by P above the level of O. (4) _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________

21 Turn over Question 6 continued _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________

22 Question 6 continued _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________

23 Turn over Question 6 continued _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ (Total for Question 6 is 13 marks)

24 7. A light elastic spring has natural length l and modulus of elasticity 4mg . A particle P of mass m is attached to one end of the spring. The other end of the spring is attached to a fixed point A. The point B is vertically below A with AB = 7 4 l. The particle P is released from rest at B. (a) Show that P moves with simple harmonic motion with period π l g (7) (b) Find, in terms of m, l and g, the maximum kinetic energy of P during the motion. (3) (c) Find the time within each complete oscillation for which the length of the spring is less than l. (5) _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________

25 Turn over Question 7 continued _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________

26 Question 7 continued _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________

27 Turn over Question 7 continued _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________

28 Question 7 continued _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ (Total for Question 7 is 15 marks) TOTAL FOR PAPER IS 75 MARKS

2 P62682RA 1. Four workers, A, B, C and D, are to be assigned to four tasks, 1, 2, 3 and 4. Each worker must be assigned to exactly one task and each task must be done by exactly one worker. Worker A cannot do task 3 and worker B cannot do task 4 The table below shows the profit, in pounds, that each worker would earn if assigned to each of the tasks. 1 2 3 4 A 29 20 – 23 B 32 30 28 – C 35 32 34 25 D 29 31 27 30 (a) Reducing rows first, use the Hungarian algorithm to obtain an allocation that maximises the total profit. You must make your method clear and show the table after each stage. (7) (b) Determine the resulting total profit. (1) (Total for Question 1 is 8 marks)

3 P62682RA Turn over 2. Jenny can choose one of three options, A, B or C, when playing a game. The profit, in pounds, associated with each outcome and their corresponding probabilities are shown on the decision tree in Figure 1. A C B 0.6 0.4 0.75 0.25 0.8 0.2 350 −140 260 −190 220 −230 Figure 1 (a) Calculate the optimal EMV to determine Jenny’s best course of action. You must make your working clear. (3) For a profit of £ x, Jenny’s utility is given by 1 400 − −e x (b) Using expected utility as the criterion for the best course of action, determine what Jenny should do now to maximise her profit. You must make your working clear. (4) (Total for Question 2 is 7 marks)

4 P62682RA 3. Table 1 shows the cost, in pounds, of transporting one unit of stock from each of four supply points, A, B, C and D, to three sales points, P, Q and R. It also shows the number of units held at each supply point and the number of units required at each sales point. A minimum cost solution is required. P Q R Supply A 25 24 17 42 B 7 12 14 68 C 13 11 20 25 D 16 15 13 40 Demand 59 72 44 Table 1 Table 2 shows an initial solution given by the north-west corner method. P Q R A 42 B 17 51 C 21 4 D 40 Table 2 (a) Taking AR as the entering cell, use the stepping-stone method to find an improved solution. Make your method clear. (2) (b) Perform one further iteration of the stepping-stone method to obtain an improved solution. You must make your method clear by stating ●  shadow costs ●  improvement indices ●  route ●  entering cell and exiting cell. (4) (c) Determine whether the solution obtained from this second iteration is optimal, giving the reason for your answer. (3) (d) Formulate this situation as a linear programming problem. You must define your decision variables and make the objective function and constraints clear. (6) (e) Explain why the Simplex algorithm cannot be used to solve transportation linear programming problems such as that formulated in (d). (1) (Total for Question 3 is 16 marks)

12 6. (a) Express sin x + 2 cos x in the form R sin (x + α) where R and α are constants, R > 0 and 0 < α < 2 π Give the exact value of R and give the value of α in radians to 3 decimal places. (3) The temperature, θ °C , inside a room on a given day is modelled by the equation θ = 5 + sin 3 12   −     πt + 2 cos 3 12   −     πt 0  t < 24 where t is the number of hours after midnight. Using the equation of the model and your answer to part (a), (b) deduce the maximum temperature of the room during this day, (1) (c) find the time of day when the maximum temperature occurs, giving your answer to the nearest minute. (3) _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________

Turn over 13 Question 6 continued _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________

14 Question 6 continued _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________

Turn over 15 Question 6 continued _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ (Total for Question 6 is 7 marks) 16 7. y x O l C R 25 (−2, 13) Figure 1 Figure 1 shows a sketch of a curve C with equation y = f (x) and a straight line l. The curve C meets l at the points (−2, 13) and (0, 25) as shown. The shaded region R is bounded by C and l as shown in Figure 1. Given that ● f (x) is a quadratic function in x ● (−2, 13) is the minimum turning point of y = f (x) use inequalities to define R. (5) _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________

Turn over 17 Question 7 continued _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________

18 Question 7 continued _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________

Turn over 19 Question 7 continued _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ (Total for Question 7 is 5 marks)

20 8. A new smartphone was released by a company. The company monitored the total number of phones sold, n, at time t days after the phone was released. The company observed that, during this time, the rate of increase of n was proportional to n Use this information to write down a suitable equation for n in terms of t. (You do not need to evaluate any unknown constants in your equation.) (2) _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________

Turn over 21 Question 8 continued _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ (Total for Question 8 is 2 marks) 22 9. y x O C Figure 2 Figure 2 shows a sketch of the curve C with equation y = f (x) where f (x) = 4(x2 − 2)e−2x        x   (a) Show that f ʹ(x) = 8(2 + x − x2)e−2x (3) (b) Hence find, in simplest form, the exact coordinates of the stationary points of C. (3) The function g and the function h are defined by g (x) = 2f (x)            x   h (x) = 2f (x) − 3      x  0 (c) Find (i) the range of g (ii) the range of h (3) _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________

Turn over 23 Question 9 continued _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________

24 Question 9 continued _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________

Turn over 25 Question 9 continued _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ (Total for Question 9 is 9 marks)

26 10. (a) Use the substitution x = u2 + 1 to show that 3 1 3 2 1 6 3 2 5 10 d d x x x u u u p q − ( ) + − ( ) = + ( ) ∫ ∫ where p and q are positive constants to be found. (4) (b) Hence, using algebraic integration, show that 3 1 3 2 1 5 10 dx x x a − ( ) + − ( ) = ∫ ln where a is a rational constant to be found. (6) _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________

Turn over 27 Question 10 continued _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________

28 Question 10 continued _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________

Turn over 29 Question 10 continued _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ (Total for Question 10 is 10 marks) 30 11. y C2 B A O C1 x Figure 3 Circle C1 has equation x2 + y2 = 100 Circle C2 has equation (x − 15)2 + y2 = 40 The circles meet at points A and B as shown in Figure 3. (a) Show that angle AOB = 0.635 radians to 3 significant figures, where O is the origin. (4) The region shown shaded in Figure 3 is bounded by C1 and C2 (b) Find the perimeter of the shaded region, giving your answer to one decimal place. (4) _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________

Turn over 31 Question 11 continued _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________

32 Question 11 continued _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________

Turn over 33 Question 11 continued _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ (Total for Question 11 is 8 marks) 34 12. In this question you must show all stages of your working. Solutions relying entirely on calculator technology are not acceptable. (a) Show that cosec θ − sin θ ≡ cos θ cot θ            θ ≠ (180n)°      n   (3) (b) Hence, or otherwise, solve for 0 < x < 180° cosec x − sin x = cos x cot (3x − 50°) (5) _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________

Turn over 35 Question 12 continued _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________

36 Question 12 continued _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________

Turn over 37 Question 12 continued _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ (Total for Question 12 is 8 marks)

38 13. A sequence of numbers a1, a2, a3, … is defined by an+1 = k a a n n + ( ) 2 n   where k is a constant. Given that ● the sequence is a periodic sequence of order 3 ● a1 = 2 (a) show that k2 + k − 2 = 0 (3) (b) For this sequence explain why k ≠ 1 (1) (c) Find the value of ar r=∑ 1 80 (3) _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________

Turn over 39 Question 13 continued _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________

40 Question 13 continued _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________

Turn over 41 Question 13 continued _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ (Total for Question 13 is 7 marks)

42 14. A large spherical balloon is deflating. At time t seconds the balloon has radius r cm and volume V cm3 The volume of the balloon is modelled as decreasing at a constant rate. (a) Using this model, show that d d r t = −k r 2 where k is a positive constant. (3) Given that ● the initial radius of the balloon is 40 cm ● after 5 seconds the radius of the balloon is 20 cm ● the volume of the balloon continues to decrease at a constant rate until the balloon is empty (b) solve the differential equation to find a complete equation linking r and t. (5) (c) Find the limitation on the values of t for which the equation in part (b) is valid. (2) _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________

Turn over 43 Question 14 continued _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________

44 Question 14 continued _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________

Turn over 45 Question 14 continued _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ (Total for Question 14 is 10 marks)

46 15. The curve C has equation x2 tan y = 9            0 < y < 2 π (a) Show that d d y x = − + 18 81 4 x x (4) (b) Prove that C has a point of inflection at x = 27 4 (3) _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________

Turn over 47 Question 15 continued _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________

48 Question 15 continued _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________

Turn over 49 Question 15 continued _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ (Total for Question 15 is 7 marks)

50 16. Prove by contradiction that there are no positive integers p and q such that 4 p2 − q2 = 25 (4) _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________

Turn over 51 Question 16 continued _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________

52 Question 16 continued _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ (Total for Question 16 is 4 marks) TOTAL FOR PAPER IS 100 MARKS

2 1 The table below shows corresponding values of x and y for y = x x 1 + The values of y are given to 4 significant figures. x 0.5 1 1.5 2 2.5 y 0.5774 0.7071 0.7746 0.8165 0.8452 (a) Use the trapezium rule, with all the values of y in the table, to find an estimate for x x x 1 0 5 2 5 + ∫ d . . giving your answer to 3 significant figures. (3) (b) Using your answer to part (a), deduce an estimate for 9 1 0 5 2 5 x x x + ∫ d . . (1) Given that 9 1 0 5 2 5 x x x + ∫ d . . = 4.535 to 4 significant figures (c) comment on the accuracy of your answer to part (b). (1) _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________

Turn over 3 Question 1 continued _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ (Total for Question 1 is 5 marks)

4 2. Relative to a fixed origin, points P, Q and R have position vectors p, q and r respectively. Given that ● P, Q and R lie on a straight line ● Q lies one third of the way from P to R show that q = 1 3 2 ( ) r + p (3) _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________

Turn over 5 Question 2 continued _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ (Total for Question 2 is 3 marks)

6 3. (a) Given that 2 log (4 − x) = log (x + 8) show that x2 − 9x + 8 = 0 (3) (b) (i) Write down the roots of the equation x2 − 9x + 8 = 0 (ii) State which of the roots in (b)(i) is not a solution of 2 log (4 − x) = log (x + 8) giving a reason for your answer. (2) _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________

12 6. (a) Given that x x x Ax B C x x x 2 8 3 2 2 2 + − + ≡ + + + ∈ ≠−  find the values of the constants A, B and C (3) (b) Hence, using algebraic integration, find the exact value of x x x x 2 0 6 8 3 2 + − + ∫ d giving your answer in the form a + b ln 2 where a and b are integers to be found. (4) _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________

Turn over 13 Question 6 continued _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________

14 Question 6 continued _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________

Turn over 15 Question 6 continued _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ (Total for Question 6 is 7 marks) 16 7. y P O x Figure 1 Figure 1 shows a sketch of the curve C with equation y = 4 2 2x x x + − 4 ln x        x > 0 (a) Show that d d y x = 12 16 4 2x x x x x + − (4) The point P, shown in Figure 1, is the minimum turning point on C. (b) Show that the x coordinate of P is a solution of x = 4 3 12 2 3 −     x (3) (c) Use the iteration formula xn+1 = 4 3 12 2 3 −       xn with x1 = 2 to find (i) the value of x2 to 5 decimal places, (ii) the x coordinate of P to 5 decimal places. (3) _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________

Turn over 17 Question 7 continued _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________

18 Question 7 continued _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________

Turn over 19 Question 7 continued _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ (Total for Question 7 is 10 marks)

20 8. A curve C has equation y = f (x) Given that ● f ʹ (x) = 6x2 + ax − 23 where a is a constant ● the y intercept of C is −12 ● (x + 4) is a factor of f (x) find, in simplest form, f (x) (6) _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________

Turn over 21 Question 8 continued _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________

22 Question 8 continued _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________

Turn over 23 Question 8 continued _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ (Total for Question 8 is 6 marks)

24 9. A quantity of ethanol was heated until it reached boiling point. The temperature of the ethanol, θ °C, at time t seconds after heating began, is modelled by the equation θ = A − Be− 0.07t where A and B are positive constants. Given that ● the initial temperature of the ethanol was 18 °C ● after 10 seconds the temperature of the ethanol was 44 °C (a) find a complete equation for the model, giving the values of A and B to 3 significant figures. (4) Ethanol has a boiling point of approximately 78 °C (b) Use this information to evaluate the model. (2) _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________

Turn over 25 Question 9 continued _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ (Total for Question 9 is 6 marks) 26 10. In this question you must show all stages of your working. Solutions relying entirely on calculator technology are not acceptable. (a) Show that cos 3 A ≡ 4 cos3 A − 3 cos A (4) (b) Hence solve, for −90°  x  180°, the equation 1 − cos 3x = sin2 x (4) _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________

Turn over 27 Question 10 continued _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________

28 Question 10 continued _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________

Turn over 29 Question 10 continued _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ (Total for Question 10 is 8 marks) 30 11. y O P x Figure 2 Figure 2 shows a sketch of the graph with equation y = 2 | x + 4 | − 5 The vertex of the graph is at the point P, shown in Figure 2. (a) Find the coordinates of P. (2) (b) Solve the equation 3x + 40 = 2 | x + 4 | − 5 (2) A line l has equation y = ax, where a is a constant. Given that l intersects y = 2 | x + 4 | − 5 at least once, (c) find the range of possible values of a, writing your answer in set notation. (3) _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________

Turn over 31 Question 11 continued _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________

32 Question 11 continued _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________

Turn over 33 Question 11 continued _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ (Total for Question 11 is 7 marks) 34 12. y R O x Figure 3 The curve shown in Figure 3 has parametric equations x = 6 sin t        y = 5 sin 2t        0  t  2 π The region R, shown shaded in Figure 3, is bounded by the curve and the x-axis. (a) (i) Show that the area of R is given by 2 2 0 60sin cos d π t t t ∫ (3) (ii) Hence show, by algebraic integration, that the area of R is exactly 20 (3) y M O x N 4.2 Figure 4 Part of the curve is used to model the profile of a small dam, shown shaded in Figure 4. Using the model and given that ● x and y are in metres ● the vertical wall of the dam is 4.2 metres high ● there is a horizontal walkway of width MN along the top of the dam (b) calculate the width of the walkway. (5) _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________

Turn over 35 Question 12 continued _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________

36 Question 12 continued _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________

Turn over 37 Question 12 continued _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ (Total for Question 12 is 11 marks)

38 13. The function g is defined by g (x) = 3 7 2 ln( ) ln( ) x x − − x > 0      x ≠ k where k is a constant. (a) Deduce the value of k. (1) (b) Prove that gʹ (x) > 0 for all values of x in the domain of g. (3) (c) Find the range of values of a for which g (a) > 0 (2) _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________

Turn over 39 Question 13 continued _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________

40 Question 13 continued _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________

Turn over 41 Question 13 continued _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ (Total for Question 13 is 6 marks)

42 14. A circle C with radius r ● lies only in the 1st quadrant ● touches the x-axis and touches the y-axis The line l has equation 2x + y = 12 (a) Show that the x coordinates of the points of intersection of l with C satisfy 5x 2 + (2r − 48) x + (r 2 − 24r + 144) = 0 (3) Given also that l is a tangent to C, (b) find the two possible values of r, giving your answers as fully simplified surds. (4) _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________

Turn over 43 Question 14 continued _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________

44 Question 14 continued _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________

Turn over 45 Question 14 continued _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ (Total for Question 14 is 7 marks) 46 15. In this question you must show all stages of your working. Solutions relying entirely on calculator technology are not acceptable. A geometric series has common ratio r and first term a. Given r ≠ 1 and a ≠ 0 (a) prove that Sn = a r r n 1 1 − ( ) − (4) Given also that S10 is four times S5 (b) find the exact value of r. (4) _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________

Turn over 47 Question 15 continued _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________

48 Question 15 continued _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________

Turn over 49 Question 15 continued _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ (Total for Question 15 is 8 marks)

50 16. Use algebra to prove that the square of any natural number is either a multiple of 3 or one more than a multiple of 3 (4) _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________

Turn over 51 Question 16 continued _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________

52 Question 16 continued _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ (Total for Question 16 is 4 marks) TOTAL FOR PAPER IS 100 MARKS

2 1. The Venn diagram shows the probabilities associated with four events, A, B, C and D A 0.07 0.16 0.24 C B s r q p D (a) Write down any pair of mutually exclusive events from A, B, C and D (1) Given that P(B) = 0.4 (b) find the value of p (1) Given also that A and B are independent (c) find the value of q (2) Given further that P ( Bʹ | C ) = 0.64 (d) find (i) the value of r (ii) the value of s (4) _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________

Turn over 3 Question 1 continued _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ (Total for Question 1 is 8 marks)

4 2. A random sample of 15 days is taken from the large data set for Perth in June and July 1987. The scatter diagram in Figure 1 displays the values of two of the variables for these 15 days. 0 10 15 20 x y 0 Figure 1 (a) Describe the correlation. (1) The variable on the x-axis is Daily Mean Temperature measured in °C. (b) Using your knowledge of the large data set, (i) suggest which variable is on the y-axis, (ii) state the units that are used in the large data set for this variable. (2) Stav believes that there is a correlation between Daily Total Sunshine and Daily Maximum Relative Humidity at Heathrow. He calculates the product moment correlation coefficient between these two variables for a random sample of 30 days and obtains r = −0.377 (c) Carry out a suitable test to investigate Stav’s belief at a 5% level of significance. State clearly ● your hypotheses ● your critical value (3) On a random day at Heathrow the Daily Maximum Relative Humidity was 97% (d) Comment on the number of hours of sunshine you would expect on that day, giving a reason for your answer. (1)

Turn over 5 Question 2 continued _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________

8 3. Each member of a group of 27 people was timed when completing a puzzle. The time taken, x minutes, for each member of the group was recorded. These times are summarised in the following box and whisker plot. 0 10 20 30 40 50 60 70 Time (x minutes) (a) Find the range of the times. (1) (b) Find the interquartile range of the times. (1) For these 27 people Σ x = 607.5 and Σ x 2 = 17 623.25 (c) calculate the mean time taken to complete the puzzle, (1) (d) calculate the standard deviation of the times taken to complete the puzzle. (2) Taruni defines an outlier as a value more than 3 standard deviations above the mean. (e) State how many outliers Taruni would say there are in these data, giving a reason for your answer. (1) Adam and Beth also completed the puzzle in a minutes and b minutes respectively, where a > b. When their times are included with the data of the other 27 people ● the median time increases ● the mean time does not change (f) Suggest a possible value for a and a possible value for b, explaining how your values satisfy the above conditions. (3) (g) Without carrying out any further calculations, explain why the standard deviation of all 29 times will be lower than your answer to part (d). (1) _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________

Turn over 9 Question 3 continued _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________

10 Question 3 continued _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________

Turn over 11 Question 3 continued _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ (Total for Question 3 is 10 marks)

12 4. The discrete random variable D has the following probability distribution d 10 20 30 40 50 P (D = d ) k 10 k 20 k 30 k 40 k 50 where k is a constant. (a) Show that the value of k is 600 137 (2) The random variables D1 and D2 are independent and each have the same distribution as D. (b) Find P (D1 + D2 = 80) Give your answer to 3 significant figures. (3) A single observation of D is made. The value obtained, d, is the common difference of an arithmetic sequence. The first 4 terms of this arithmetic sequence are the angles, measured in degrees, of quadrilateral Q (c) Find the exact probability that the smallest angle of Q is more than 50° (5) _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________

Turn over 13 Question 4 continued _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________

14 Question 4 continued _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________

Turn over 15 Question 4 continued _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ (Total for Question 4 is 10 marks)

16 5. A health centre claims that the time a doctor spends with a patient can be modelled by a normal distribution with a mean of 10 minutes and a standard deviation of 4 minutes. (a) Using this model, find the probability that the time spent with a randomly selected patient is more than 15 minutes. (1) Some patients complain that the mean time the doctor spends with a patient is more than 10 minutes. The receptionist takes a random sample of 20 patients and finds that the mean time the doctor spends with a patient is 11.5 minutes. (b) Stating your hypotheses clearly and using a 5% significance level, test whether or not there is evidence to support the patients’ complaint. (4) The health centre also claims that the time a dentist spends with a patient during a routine appointment, T minutes, can be modelled by the normal distribution where T ~ N(5, 3.52) (c) Using this model, (i) find the probability that a routine appointment with the dentist takes less than 2 minutes (1) (ii) find P (T < 2 | T > 0) (3) (iii) hence explain why this normal distribution may not be a good model for T. (1) The dentist believes that she cannot complete a routine appointment in less than 2 minutes. She suggests that the health centre should use a refined model only including values of  T > 2 (d) Find the median time for a routine appointment using this new model, giving your answer correct to one decimal place. (5) _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________

Turn over 17 Question 5 continued _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________

18 Question 5 continued _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________

Turn over 19 Question 5 continued _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________

20 Question 5 continued _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ (Total for Question 5 is 15 marks) TOTAL FOR STATISTICS IS 50 MARKS

2 1. A rough plane is inclined to the horizontal at an angle α , where tan α = 3 4 A brick P of mass m is placed on the plane. The coefficient of friction between P and the plane is μ Brick P is in equilibrium and on the point of sliding down the plane. Brick P is modelled as a particle. Using the model, (a) find, in terms of m and g, the magnitude of the normal reaction of the plane on brick P (2) (b) show that μ = 3 4 (4) For parts (c) and (d), you are not required to do any further calculations. Brick P is now removed from the plane and a much heavier brick Q is placed on the plane. The coefficient of friction between Q and the plane is also 3 4 (c) Explain briefly why brick Q will remain at rest on the plane. (1) Brick Q is now projected with speed 0.5 m s−1 down a line of greatest slope of the plane. Brick Q is modelled as a particle. Using the model, (d) describe the motion of brick Q, giving a reason for your answer. (2) _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________

Turn over 3 Question 1 continued _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________

4 Question 1 continued _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________

Turn over 5 Question 1 continued _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ (Total for Question 1 is 9 marks)

Turn over 15 Question 4 continued _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________

6 2. A particle P moves with acceleration (4i − 5j) m s−2 At time t = 0, P is moving with velocity (−2i + 2j) m s−1 (a) Find the velocity of P at time t = 2 seconds. (2) At time t = 0, P passes through the origin O. At time t = T seconds, where T > 0, the particle P passes through the point A. The position vector of A is (λi − 4.5j)m relative to O, where λ is a constant. (b) Find the value of T. (4) (c) Hence find the value of λ (2) _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________

Turn over 7 Question 2 continued _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ (Total for Question 2 is 8 marks)

8 3. (i) At time t seconds, where t  0 , a particle P moves so that its acceleration a m s−2 is given by a = (1 − 4t) i + (3 − t2) j At the instant when t = 0, the velocity of P is 36i m s−1 (a) Find the velocity of P when t = 4 (3) (b) Find the value of t at the instant when P is moving in a direction perpendicular to i (3) (ii) At time t seconds, where t  0 , a particle Q moves so that its position vector r metres, relative to a fixed origin O, is given by r = (t2 − t) i + 3t j Find the value of t at the instant when the speed of Q is 5 m s−1 (6) _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________

Turn over 9 Question 3 continued _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________

10 Question 3 continued _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________

Turn over 11 Question 3 continued _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ (Total for Question 3 is 12 marks) 12 4. A C α 6a 4a B Figure 1 A ladder AB has mass M and length 6a. The end A of the ladder is on rough horizontal ground. The ladder rests against a fixed smooth horizontal rail at the point C. The point C is at a vertical height 4a above the ground. The vertical plane containing AB is perpendicular to the rail. The ladder is inclined to the horizontal at an angle α , where sin α = 4 5  , as shown in Figure 1. The coefficient of friction between the ladder and the ground is μ. The ladder rests in limiting equilibrium. The ladder is modelled as a uniform rod. Using the model, (a) show that the magnitude of the force exerted on the ladder by the rail at C is 9 25 Mg (3) (b) Hence, or otherwise, find the value of μ. (7) _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________

Turn over 13 Question 4 continued _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________

14 Question 4 continued _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________

Turn over 15 Question 4 continued _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ (Total for Question 4 is 10 marks) 16 5. N 45° O 25 m A U m s−1 100 m Figure 2 A small ball is projected with speed U m s−1 from a point O at the top of a vertical cliff. The point O is 25 m vertically above the point N which is on horizontal ground. The ball is projected at an angle of 45° above the horizontal. The ball hits the ground at a point A, where AN = 100 m, as shown in Figure 2. The motion of the ball is modelled as that of a particle moving freely under gravity. Using this initial model, (a) show that U = 28 (6) (b) find the greatest height of the ball above the horizontal ground NA. (3) In a refinement to the model of the motion of the ball from O to A, the effect of air resistance is included. This refined model is used to find a new value of U. (c) How would this new value of U compare with 28, the value given in part (a)? (1) (d) State one further refinement to the model that would make the model more realistic. (1) _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________

Turn over 17 Question 5 continued _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________

18 Question 5 continued _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________

Turn over 19 Question 5 continued _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________

20 Question 5 continued _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ (Total for Question 5 is 11 marks) TOTAL FOR MECHANICS IS 50 MARKS

2 Answer ALL questions. Write your answers in the spaces provided. 1 The weights at 3 months of age, in kg, of baby girls born in the UK are modelled by the normal distribution N(5.8, 0.742). Rose has recently looked up the weights of a sample of 14 baby girls weighed at 3 months of age in her clinic. The mean weight of the baby girls in her sample is 6.3 kg. (a) Calculate a 95% confidence interval for the mean weight at 3 months of age of baby girls in Rose’s clinic. (3) _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________

3 Turn over Question 1 continued (b) Explain whether your answer to part (a) provides significant evidence that the mean weight of baby girls at 3 months of age in Rose’s clinic is different from 5.8 kg. (2) _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ (c) State two assumptions needed for the confidence interval you calculated in part (a) to be valid. (2) _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ (Total for Question 1 is 7 marks)

4 2 A survey investigated whether shoppers prefer to buy cucumbers wrapped in plastic or unwrapped. Of these shoppers, 75% preferred to buy unwrapped cucumbers. Julia had expected the percentage preferring to buy unwrapped cucumbers to be higher and decided to do a survey of her gardening club. Julia took a random sample of 45 gardeners and found that 36 of them preferred to buy unwrapped cucumbers. Investigate whether the percentage of gardeners preferring to buy unwrapped cucumbers is higher than 75%. Make any necessary assumptions. (7) _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ (Total for Question 2 is 7 marks)

5 Turn over 3 Simon, an educational researcher, wanted to test the effectiveness of a computer program designed to improve the numerical skills of children in Key Stage 1 (KS1). Simon recorded the marks in two similar numerical tests taken by a random selection of KS1 children. One test was taken before and one was taken after the children used the computer program. For each child, he calculated the value of test mark after – test mark before Simon used his calculator to conduct a paired t-test. The calculator output is shown in Figure 1. 1 - Sample tTest µ > 0 t = 2.619047619 p = 0.007520838 x = 5.5 Sx = 10.5 n = 25 Figure 1 (a) Use the information given in the calculator output in Figure 1 to investigate whether there is evidence that children did better in the test of numerical skills, on average, after using the computer program. (4) _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________

16 Question 4 continued _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ (Total for Question 4 is 11 marks) TOTAL FOR FURTHER MECHANICS 2 IS 40 MARKS

6 Question 3 continued (b) Explain the meaning of a Type I error in a hypothesis test. Write down the probability of a Type I error in Simon’s test in part (a). (2) _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ (c) Make one comment on the reliability of your conclusion to Simon’s test in part (a). (1) _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ (Total for Question 3 is 7 marks)

7 Turn over 4 Professional tennis players’ prize money is published online. Martina wanted to do an analysis by nationality of top players (those who earned at least $1 million). She randomly selected a year since 2000 and obtained the data for that year to use in her investigation. She then aggregated the results for men and women by region of the world, as shown in Figure 2. Men Women Region of the world Australia and South Africa 4 1 Eastern Europe 11 19 Far East 2 3 South America 3 0 USA and Canada 8 5 Western Europe 29 10 Total 57 38 (Data source: Association of Tennis Professionals and the Women’s Tennis Association) Figure 2 Martina decides to test the hypothesis that, among top players, there is no association between region of the world and sex of player. Martina combines Australia and South Africa, Far East and South America into a single class called Other Regions of the World. (a) Explain why Martina needs to combine regions to carry out her test. You must give numeric justifications. (3) _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________

8 Question 4 continued (b) Complete Martina’s hypothesis test. (9) _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________

9 Turn over Question 4 continued _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ (c) Describe, with numerical justification, the nature of any association found between region of the world and sex of player in your conclusion to the test in part (b). (2) _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ (Total for Question 4 is 14 marks)

10 5 GCSE performance at schools in England can be compared using Progress 8. A higher score in Progress 8 means a better performance. South West London schools are believed to have ‘well above average’ scores in Progress 8. Provisional scores for summer 2018 for a random sample of 10 state schools in South West London are given in Figure 3, together with the average for all English state schools. State Schools in South West London Progress 8 scores Summer 2018 A 1.19 B 1.09 C 0.98 D 0.73 E 0.63 F 0.28 G 0.24 H 0.23 I −0.07 J −0.09 Average for all English state schools −0.02 (Source: © Crown Copyright) Figure 3 Davinder has tested a hypothesis. The first two lines of his working are given in Figure 4. X ~ B (10, 0.5) P (X ≤ 2) = 0.0547 Figure 4

11 Turn over Question 5 continued (a) Name Davinder’s hypothesis test. State his hypotheses. Justify why his value for P (X  2) does not provide sufficient evidence that the average performance of schools in South West London is better than the average for all English state schools. (3) _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ Davinder wants to use a Wilcoxon signed-rank test to investigate whether the average performance of schools in South West London is better than the average for all English state schools. (b) Using the data given in Figure 3 and making any necessary assumptions, conduct the Wilcoxon signed-rank test for Davinder. What can he conclude? (6) _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________

12 Question 5 continued _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ Davinder’s test in part (a) could lead to a different conclusion from the Wilcoxon signed‑rank test in part (b). (c) If Davinder obtains different conclusions, how should Davinder choose between the two tests? (2) _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ (Total for Question 5 is 11 marks)

Turn over 9 Question 3 continued _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________

13 Turn over 6 A formula meal is a calorie controlled drink. The results of a clinical trial were published in September 2018. The trial compared weight loss of adult patients assigned randomly either to Diet A, based only on formula meals, or to Diet B, based only on conventional meals. (a) Define a double blind trial. Explain why it was not possible for this clinical trial to be double blind. (2) _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ The mean weight loss after one year for each group is shown in Figure 5, together with the sample standard deviation of weight loss for each group. Sample sizes for each diet are also given. Diet A Diet B Mean weight loss, x, kg 10.7 3.1 Sample standard deviation of weight loss, s, kg 9.6 7.0 Sample size, n 104 95 (Source: British Medical Journal) Figure 5 (b) Conduct a hypothesis test to determine whether the mean weight loss of patients assigned to Diet A is different from that of patients assigned to Diet B. (7) _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________

14 Question 6 continued _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ (c) Explain why it is not necessary to make an assumption about the distributions of the mean weight losses to ensure the validity of your test in part (b). (1) _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ (d) State one necessary assumption you made to ensure the validity of your test in part (b). (1) _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________

15 Turn over Question 6 continued To make Diet A medically worthwhile, patients assigned to Diet A would need to lose at least 4 kg more than those assigned to Diet B. (e) Adapt your hypothesis test to determine whether the mean weight loss of patients assigned to Diet A is at least 4 kg more than that of patients assigned to Diet B. (6) _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ (Total for Question 6 is 17 marks)

16 7 Daniel, a car fleet manager, has collected data about the durability of four brands of car tyre. He recorded the distance, X thousand miles, travelled by each worn out tyre replaced during April 2019. He calls this distance ‘tyre life’. Daniel has calculated summary statistics for tyre life, as given in Figure 6. Brand A Brand B Brand C Brand D ∑ x 190.078 184.401 133.191 146.858 ∑ x2 5236.989 4414.446 3068.984 2787.855 n 7 8 6 8 Figure 6 (a) Use ANOVA to investigate whether there is a difference between the mean tyre lives of the four brands, A, B, C and D. You should make any necessary assumptions. (11) _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________

17 Turn over Question 7 continued _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________

18 Question 7 continued Daniel has also compared tyre lives for brands A, B, C and D by plotting a diagram, as shown in Figure 7. Tyre life (thousand miles) 40 35 30 25 20 15 10 5 0 Brand A Brand C Brand B Brand D Figure 7

19 Turn over Question 7 continued (b) Discuss briefly the validity of your test in part (a). You should consider both the information in Figure 7 and the method used by Daniel to collect the data. (3) _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ Question 7 continues on the next page

20 Question 7 continued Daniel’s summary statistics and diagram, Figure 7, are based on the tyre lives for front wheel tyres. He has additional data relating to the tyre lives for rear wheel tyres. Daniel knows that rear wheel tyres can be expected to have longer lives than front wheel tyres. (c) (i) Explain how Daniel could use a Randomised Block Design to include samples of the four tyre brands and front and rear wheel tyres. (2) _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ (ii) Name a hypothesis test that Daniel can use with a Randomised Block Design to investigate whether there is a difference between the mean tyre lives of the four brands, A, B, C and D. (1) _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ (Total for Question 7 is 17 marks) TOTAL FOR PAPER IS 80 MARKS

Turn over 9 Question 4 continued _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ (Total for Question 4 is 5 marks) 10 5. O Q P(2, 10) y x Figure 1 Figure 1 shows part of the curve with equation y = 3x 2 – 2 The point P(2, 10) lies on the curve. (a) Find the gradient of the tangent to the curve at P. (2) The point Q with x coordinate 2 + h also lies on the curve. (b) Find the gradient of the line PQ, giving your answer in terms of h in simplest form. (3) (c) Explain briefly the relationship between part (b) and the answer to part (a). (1) _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________

2 Answer ALL questions. Write your answers in the spaces provided. 1 Teru is looking at some graphs from an official UK government publication that show patterns of employment for civil servants (central government employees) over time. Figure 1 shows the gender balance of each civil service grade between 2010–2017. Most junior Most senior 2010 2017 2010 2017 2010 2010 2017 2017 AO/AA EO SEO/HEO Grades 6 and 7 2010 2017 SCS 50% 57% 58% 56% 56% 46% 48% 40% 45% 34% 41% Male Female Male Female Male Female Male Female Male Female (Source: https://www.instituteforgovernment.org.uk/publications/whitehall-monitor-2018) Figure 1 Teru says that Figure 1 shows that the total number of female civil servants has definitely increased between 2010 and 2017. (a) Explain whether Teru is correct or not. (2) _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________

3 Turn over Question 1 continued (b) Describe two important trends shown in Figure 1. (2) _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________

4 Question 1 continued Teru has another graph, Figure 2, which is taken from the same official UK government publication. It shows the percentage of civil servants from an ethnic minority for 1988 to 2017. Figure 2 also shows the percentage of senior civil servants (SCS) from an ethnic minority between 2006 and 2017. These were the only years for which records were available for SCS. Ethnicity Senior civil servants All grades UK population (Census 2011) 20% 15% 10% 5% 0% 1988 1992 1996 2000 2004 2008 2012 2016 (Source: https://www.instituteforgovernment.org.uk/publications/whitehall-monitor-2018) Figure 2 One additional fact, A, is also shown on Figure 2 by a short dotted line. This is: A The percentage of the UK population from an ethnic minority in 2011 taken from the UK Census. (c) Give one reason why the additional fact A, included on Figure 2, provides useful information. (1) _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________

5 Turn over Question 1 continued (d) Write a short report for a news website on the pattern of employment of civil servants from an ethnic minority, based on the information in Figure 2. Make three distinct comments in an appropriate style. (4) _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ (Total for Question 1 is 9 marks)

6 2 Snowy tree crickets are insects that live in various states of the USA. Male snowy tree crickets make a noise called a chirp by rubbing their wings together. Thomas made a study of snowy tree crickets living in the wild in California. When he encountered a snowy tree cricket, he recorded the rate of the cricket’s chirp (X chirps per minute) and the air temperature (Y °C). His results are shown in Figure 3. Snowy tree cricket Rate of chirp x Air temperature y A 50 9.2 B 50 9.4 C 110 15.6 D 127 17.8 E 129 18.3 F 133 18.6 G 154 19.7 H 159 19.7 (Data source: Annals of the Entomological Society of America, 55(3), 303–322.) Figure 3 (a) Use the data in Figure 3 to calculate Pearson’s product moment correlation coefficient for rate of a snowy tree cricket’s chirp and air temperature. (1) _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ (b) Conduct an appropriate hypothesis test using the value you calculated in part (a). You should make any necessary assumptions. (4) _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________

7 Turn over Question 2 continued _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ It has been suggested that the rate of a snowy tree cricket’s chirp can be used as a reliable means of estimating air temperature. (c) Comment on this suggestion with reference to your answers in parts (a) and (b). (2) _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________

8 Question 2 continued Juliana entered the data in Figure 3 in a spreadsheet and calculated the following least squares regression line for air temperature (Y ° C) on rate of snowy tree cricket’s chirp (X chirps per minute) in the form y = a + bx. y = 4.44 + 0.102x (d) Interpret, in context, the values of a and b in Juliana’s least squares regression line. (3) _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________

9 Turn over Question 2 continued A graph of the data in Figure 3 with Juliana’s least squares regression line is shown in Figure 4. H Rate of chirp (x per minute) 40 60 80 100 120 140 160 180 Temperature (y ° C) 22 20 18 16 14 12 10 8 Figure 4 (e) Calculate the residual for cricket H. Give your answer to 2 significant figures. (2) _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________

10 Question 2 continued (f) Make a comment about the validity of using Juliana’s regression line to estimate air temperature from the rate of a snowy tree cricket’s chirp: (i) with regard to the residuals of the points plotted in Figure 4, (1) _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ (ii) with regard to any other piece of evidence. (1) _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________

11 Turn over Question 2 continued (g) Make one suggestion for the next stage of the research. Explain how your suggestion would improve the validity of the results. (2) _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ (Total for Question 2 is 16 marks)

12 3 People buy and sell shares in companies to try to make money. Share prices can change daily and the daily percentage change of a share price can be calculated. Chidi has bought some shares in the company BASF. He has a sample of the daily percentage change of the BASF share price, summarised below: Sample size Mean Standard deviation 251 –0.163% 1.37% Using his sample and the null hypothesis H0: μ = 0, he performs a parametric hypothesis test to find out whether there is evidence that the BASF share price has a negative daily percentage change, on average. (a) Complete Chidi’s hypothesis test. You do not need to conclude in context. (5) _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________

13 Turn over Question 3 continued Chidi’s friend, Jason, discovers that Chidi’s sample is, in fact, the daily percentage change of the BASF share price for all of the 251 days in 2018 on which shares were bought and sold. Jason makes two criticisms of Chidi’s decision to perform his hypothesis test on this data: A Chidi did not need to do a hypothesis test to find out whether the average daily percentage change of the BASF share price was negative in 2018. B Chidi cannot use the outcome of his hypothesis test to reliably make a judgement about the average daily percentage change of the BASF share price in the future. (b) Explain why each of Jason’s criticisms is correct. (2) _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________

24 6 Betsan surveyed a sample of 22 knitters. 12 of the knitters said they regularly attended knitting groups where they shared their hobby with others. 10 said they did not. The knitters were asked to rate how calming they found knitting on a scale from 0 ‘not at all’ to 100 ‘extremely’. Betsan decides to perform a Wilcoxon rank-sum test to find out whether knitters who attend knitting groups find knitting more calming, on average, than knitters who do not attend knitting groups. Betsan’s data are shown in Figure 8 along with a partial ranking of the ratings. Attends knitting groups Does not attend knitting groups Calmness rating Rank Calmness rating Rank 35 3 25 1 50 6 30 2 58 8 37 4 62 10 40 5 69 12 55 7 75 14 60 9 79 15 65 11 85 17 70 13 90 80 16 95 93 98 100 Figure 8 (Data source: adapted from DOI:10.4276/030802213X13603244419077) (a) Complete the ranking in Figure 8. (1) _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________

14 Question 3 continued Chidi looks at the daily price of BASF shares over several years in the past and decides that he will keep his shares. He wants to apply a statistical model to his complete 2018 dataset of the daily percentage change of the BASF share price so that he can make comparisons between 2018 and other years. He uses statistical software to plot a histogram of the daily percentage change of the BASF share price for 2018 and marks the mean value with a vertical line, Figure 5. 80 60 40 20 0 Daily percentage change of BASF share price in 2018 –7.5 –5.0 –2.5 0.0 2.5 5.0 Frequency (Data source: https://www.basf.com) Figure 5 As well as calculating the mean of –0.163% and the standard deviation of 1.37%, Chidi finds that: 183 of the 251 values in his dataset are within 1 standard deviation of the mean 241 of the 251 values in his dataset are within 2 standard deviations of the mean (c) Comment on the validity of using the model N(–0.163, 1.372) for the daily percentage change of the BASF share price in 2018. Use Chidi’s calculations above and Figure 5 to support your comments. (4) _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________

15 Turn over Question 3 continued _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ (Total for Question 3 is 11 marks)

16 4 Donal works in the head office of a large law firm. The law firm employs both qualified solicitors and ‘agents’ to resolve legal cases. Each legal case is assigned to an agent for one month to find out if it can be resolved easily. If not, the legal case has to be assigned to a solicitor. Donal is writing a report on the use of work targets for agents at the law firm. To collect data from his colleagues, he used the following sampling method. • Randomly select 3 of the law firm’s 10 offices. • Take a random sample of 20% of the agents from each of the 3 offices selected. Donal visited the 3 offices selected to collect detailed information from the agents in his sample. (a) Write down the name of Donal’s sampling method. (1) _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ (b) For Donal’s sampling method write down (i) two advantages, (2) _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________

17 Turn over Question 4 continued (ii) one disadvantage. (1) _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ At the start of the month each agent is assigned 40 legal cases. From past experience, Maxine, a manager, claims that each agent resolves 40% of their legal cases by the end of the month. (c) If Maxine’s claim is correct, explain why a binomial model might be appropriate for the number of legal cases resolved by an agent by the end of the month. (3) _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________

18 Question 4 continued Agents have the target of resolving at least 15 of their 40 cases by the end of the month. Assume that the model B(40, 0.4) is appropriate for the number of cases resolved by an agent by the end of the month. (d) Calculate the probability that an agent selected at random does not meet their target by the end of any given month. (1) _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ Agents are assessed quarterly (every three months). Those agents that do not meet the target for all three months in the quarter are given a formal warning. (e) Calculate the probability that an agent selected at random will be given a formal warning at the end of any given quarter. (2) _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________

19 Turn over Question 4 continued Donal obtained records for the number of legal cases resolved by the end of the month for each of the 90 agents in his sample. These are shown in Figure 6. Number of resolved cases Frequency Probability x  11 9 0.071 12  x  14 26 0.246 15  x  17 36 18  x  20 17 x  21 2 Total 90 Figure 6 (f) Use a χ2 goodness-of-fit test to investigate whether Donal’s records imply that the model B(40, 0.4) is indeed appropriate for the number of legal cases resolved by an agent by the end of the month. You may use the empty columns in Figure 6 for your working, if you require them. (10) _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________

20 Question 4 continued _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________

21 Turn over Question 4 continued (g) Explain the meaning of a Type II error in the context of this question. (2) _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ (Total for Question 4 is 22 marks)

22 5 A palaeontologist studies fossils. A “dig” is a site where palaeontologists try to find fossils. Sharon is a palaeontologist interested in studying an extinct rhinoceros called a teleoceras that lived in North America in the Miocene era (about 23 million to 5 million years ago). She has been offered two places on digs, one in California and the other in Montana. Both are new digs on sites dating from the Miocene era. Sharon has obtained some information about previous Miocene era fossil digs in California and Montana. This is shown in Figure 7. Number of digs Number of digs where teleoceras fossils were found Location California 947 12 Montana 75 4 (Data source: https://paleobiodb.org/) Figure 7 Using the information in Figure 7, carry out a hypothesis test of proportion. Use the results of your hypothesis test to advise Sharon whether she should accept the place on the dig in California or the place on the dig in Montana. (10) _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________

23 Turn over Question 5 continued _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ (Total for Question 5 is 10 marks)

25 Turn over Question 6 continued (b) Complete Betsan’s Wilcoxon rank-sum test. (7) _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ ______________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________

26 Question 6 continued _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ (c) State two reasons why Betsan’s choice to use a Wilcoxon rank-sum test rather than the equivalent parametric hypothesis test might have been appropriate. (2) _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________

27 Question 6 continued Betsan also asked the knitters how happy knitting makes them feel on a rating of 0 ‘not at all’ to 100 ‘extremely’. She performed another Wilcoxon rank-sum test on the results to find out whether knitters who attend knitting groups find knitting makes them happier, on average, than knitters who do not attend knitting groups. The result of her test was to reject H0 at the 5% significance level. Betsan claims that her test shows that knitting in a group makes people happier than knitting on your own. (d) Suggest two other possible explanations for Betsan’s results. Assume that none of the knitters lied to Betsan and that she did not make any mistakes in her calculations. (2) _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ (Total for Question 6 is 12 marks) TOTAL FOR PAPER IS 80 MARKS 28 BLANK PAGE Every effort has been made to contact copyright holders to obtain their permission for the use of copyright material. Pearson Education Ltd. will, if notified, be happy to rectify any errors or omissions and include any such rectifications in future editions.

Turn over 3 Question 1 continued _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ (Total for Question 1 is 7 marks)

4 2. The cubic equation 9x3 – 5x2 + 4x + 7 = 0 has roots α, β and γ. Without solving the equation, find the cubic equation whose roots are (3α – 2), (3β – 2) and (3γ – 2), giving your answer in the form aw3 + bw2 + cw + d = 0, where a, b, c and d are integers to be determined. (5) _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________

Turn over 5 Question 2 continued _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ (Total for Question 2 is 5 marks)

6 3. (a) Use the standard results for summations to show that for all positive integers n r n =∑ 1 (5r – 2)2 = 1 6 n(an2 + bn + c) where a, b and c are integers to be determined. (5) (b) Hence determine the value of k for which r k =∑ 1 (5r – 2)2 = 94k2 (4) _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________

Turn over 7 Question 3 continued _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ (Total for Question 3 is 9 marks) 8 4. M = 2 1 4 2 2 4 1 2 k − −         N = k − − − − −         7 6 10 2 20 24 3 2 1 where k is a constant. (a) Determine, in simplest form in terms of k, the matrix M N. (2) (b) Given that k = 5 (i) write down M N (ii) hence write down M–1 (2) (c) Solve the simultaneous equations 2x +   y + 4z = 2 5x + 2y – 2z = 3 4x +   y – 2z = –1 (2) (d) Interpret the answer to part (c) geometrically. (1) _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________

Turn over 9 Question 4 continued _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________

10 Question 4 continued _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________

Turn over 11 Question 4 continued _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ (Total for Question 4 is 7 marks) 12 5. –5 –4 –3 –2 –1 0 –1 1 Re Im 1 2 3 Figure 1 Figure 1 shows an Argand diagram. The set P, of points that lie within the shaded region including its boundaries, is defined by P = {z ∈  : a  ½ z + b + ci ½  d} where a, b, c and d are integers. (a) Write down the values of a, b, c and d. (3) The set Q is defined by Q = {z ∈  : a  ½ z + b + ci ½  d} ∩ {z ∈  : ½ z – i ½  ½ z – 3i ½} (b) Determine the exact area of the region defined by Q, giving your answer in simplest form. (7) _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________

Turn over 13 Question 5 continued _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________

14 Question 5 continued _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________

Turn over 15 Question 5 continued _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ (Total for Question 5 is 10 marks)

16 6. A mining company has identified a mineral layer below ground. The mining company wishes to drill down to reach the mineral layer and models the situation as follows. With respect to a fixed origin O,  the ground is modelled as a horizontal plane with equation z = 0  the mineral layer is modelled as part of the plane containing the points A(10, 5, –50), B(15, 30, –45) and C(–5, 20, –60), where the units are in metres (a) Determine an equation for the plane containing A, B and C, giving your answer in the form r . n = d (5) (b) Determine, according to the model, the acute angle between the ground and the plane containing the mineral layer. Give your answer to the nearest degree. (3) The mining company plans to drill vertically downwards from the point (5, 12, 0) on the ground to reach the mineral layer. (c) Using the model, determine, in metres to 1 decimal place, the distance the mining company will need to drill in order to reach the mineral layer. (2) (d) State a limitation of the assumption that the mineral layer can be modelled as a plane. (1) _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________

Turn over 17 Question 6 continued _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________

18 Question 6 continued _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________

Turn over 19 Question 6 continued _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ (Total for Question 6 is 11 marks) 20 7. f(z) = z4 – 6z3 + pz2 + qz + r where p, q and r are real constants. The roots of the equation f(z) = 0 are α, β, γ and δ where α = 3 and β = 2 + i Given that γ is a complex root of f(z) = 0 (a) (i) write down the root γ, (ii) explain why δ must be real. (2) (b) Determine the value of δ. (2) (c) Hence determine the values of p, q and r. (3) (d) Write down the roots of the equation f(–2z) = 0 (2) _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________

Turn over 21 Question 7 continued _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________

22 Question 7 continued _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________

Turn over 23 Question 7 continued _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ (Total for Question 7 is 9 marks)

Turn over 27 Question 8 continued _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ (Total for Question 8 is 9 marks) 28 9. y x A B C D E O Figure 2 Figure 2 shows the vertical cross‑section, AOBCDE, through the centre of a wax candle. In a model, the candle is formed by rotating the region bounded by the y‑axis, the line OB, the curve BC, and the curve CD through 360° about the y‑axis. The point B has coordinates (3, 0) and the point C has coordinates (5, 15). The units are in centimetres. The curve BC is represented by the equation y = 225 2025 2 x a − 3  x < 5 where a is a constant. (a) Determine the value of a according to this model. (2) The curve CD is represented by the equation y = 16 – 0.04x2 0  x < 5 (b) Using algebraic integration, determine, according to the model, the exact volume of wax that would be required to make the candle. (9) (c) State a limitation of the model. (1) When the candle was manufactured, 700 cm3 of wax were required. (d) Use this information and your answer to part (b) to evaluate the model, explaining your reasoning. (1)

Turn over 29 Question 9 continued _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________

30 Question 9 continued _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________

Turn over 31 Question 9 continued _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________

32 Question 9 continued _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ (Total for Question 9 is 13 marks) TOTAL FOR CORE PURE MATHEMATICS IS 80 MARKS

2 1. Use algebra to determine the values of x for which x x x x − ( ) > − 1 1 giving your answer in set notation. (6) _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________

Turn over 3 Question 1 continued _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ (Total for Question 1 is 6 marks)

4 2. The variables x and y satisfy the differential equation d d d d 2 2 2 15 3 2 y x y x y x + − = where y = 1 at x = 0 and where y = 2 at x = 0.1 Use the approximations d d and d d 2 2 1 1 2 2 y x y y y h y x n n n n n       − + ( )       + − ≈ ≈ y y h n n + − − ( ) 1 1 2 with h = 0.1 to find an estimate for the value of y when x = 0.3 (6) _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________

Turn over 5 Question 2 continued _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________

6 Question 2 continued _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________

Turn over 7 Question 2 continued _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ (Total for Question 2 is 6 marks)

8 3. On a particular day, the depth of water in a river estuary at a specific location is modelled by the equation sin cos 0 7 (I) x x D x π       = + +           2 3 6 3 3 where the depth of water is D metres at time x hours after midnight on that day. (a) Write down the depth of water at midnight, according to the model. (1) Using the substitution t x =       tan 6 (b) show that equation (I) can be re-written as D t t t = + + + 3 4 9 1 2 2 (3) (c) Hence determine, according to the model, the time after midnight when the depth of water is 5 metres for the first time. Give your answer to the nearest minute. (5) _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________

Turn over 9 Question 3 continued _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________

10 Question 3 continued _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________

Turn over 11 Question 3 continued _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ (Total for Question 3 is 9 marks)

12 4. With respect to a fixed origin O, the points A, B and C have position vectors given by OA → = 18i − 14j − 2k    OB → = −7i − 5j + 3k    OC → = −2i − 9j − 6k The points O, A, B and C form the vertices of a tetrahedron. (a) Show that the area of the triangular face ABC of the tetrahedron is 108 to 3 significant figures. (3) (b) Find the volume of the tetrahedron. (4) An oak wood block is made in the shape of the tetrahedron, with centimetres taken for the units. The density of oak is 0.85 g cm−3 (c) Determine the mass of the block, giving your answer in kg. (2) _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________

Turn over 13 Question 4 continued _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________

14 Question 4 continued _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________

Turn over 15 Question 4 continued _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ (Total for Question 4 is 9 marks)

16 5. The point P (ap2, 2ap), where a is a positive constant, lies on the parabola with equation y2 = 4ax The normal to the parabola at P meets the parabola again at the point Q (aq2, 2aq) (a) Show that q p p = − − 2 2 (5) (b) Hence show that PQ ka p p n 2 2 4 2 1 = + ( ) where k and n are integers to be determined. (5) _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________

Turn over 17 Question 5 continued _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________

18 Question 5 continued _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________

Turn over 19 Question 5 continued _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________

2 1. Flobee sells tomato seeds in packets, each containing 40 seeds. Flobee advertises that only 4% of its tomato seeds do not germinate. Amodita is investigating the germination of Flobee’s tomato seeds. She plants 125 packets of Flobee’s tomato seeds and records the number of seeds that do not germinate in each packet. Number of seeds that do not germinate 0 1 2 3 4 5 6 or more Frequency 15 35 38 22 10 5 0 Amodita wants to test whether the binomial distribution B(40, 0.04) is a suitable model for these data. The table below shows the expected frequencies, to 2 decimal places, using this model. Number of seeds that do not germinate 0 1 2 3 4 5 or more Expected Frequency 24.42 40.70 r 17.45 6.73 s (a) Calculate the value of r and the value of s (2) (b) Stating your hypotheses clearly, carry out the test at the 5% level of significance. You should state the number of degrees of freedom, critical value and conclusion clearly. (6) Amodita believes that Flobee should use a more realistic value for the percentage of their tomato seeds that do not germinate. She decides to test the data using a new model B(40, p) (c) Showing your working, suggest a more realistic value for p (2) _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________

Turn over 3 Question 1 continued _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________

4 Question 1 continued _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________

Turn over 5 Question 1 continued _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ (Total for Question 1 is 10 marks)

6 2. Rowan and Alex are both check-in assistants for the same airline. The number of passengers, R, checked in by Rowan during a 30-minute period can be modelled by a Poisson distribution with mean 28 (a) Calculate P(R  23) (1) The number of passengers, A, checked in by Alex during a 30-minute period can be modelled by a Poisson distribution with mean 16, where R and A are independent. A randomly selected 30-minute period is chosen. (b) Calculate the probability that exactly 42 passengers in total are checked in by Rowan and Alex. (2) The company manager is investigating the rate at which passengers are checked in. He randomly selects 150 non-overlapping 60-minute periods and records the total number of passengers checked in by Rowan and Alex, in each of these 60-minute periods. (c) Using a Poisson approximation, find the probability that for at least 25 of these 60-minute periods Rowan and Alex check in a total of fewer than 80 passengers. (4) On a particular day, Alex complains to the manager that the check-in system is working slower than normal. To see if the complaint is valid the manager takes a random 90-minute period and finds that the total number of people Rowan checks in is 67 (d) Test, at the 5% level of significance, whether or not there is evidence that the system is working slower than normal. You should state your hypotheses and conclusion clearly and show your working. (4) _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________

Turn over 7 Question 2 continued _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________

8 Question 2 continued _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________

Turn over 9 Question 2 continued _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ (Total for Question 2 is 11 marks)

10 3. The discrete random variable X has probability distribution x −3 −2 −1 0 2 5 P(X = x) 0.3 0.15 0.1 0.15 0.1 0.2 (a) Find E(X ) (1) Given that Var(X ) = 8.79 (b) find E(X 2) (2) The discrete random variable Y has probability distribution y −2 −1 0 1 2 P(Y = y) 3a a b a c where a, b and c are constants. For the random variable Y P(Y  0) = 0.75      and      E(Y 2 + 3) = 5 (c) Find the value of a, the value of b and the value of c (5) The random variable W = Y − X where Y and X are independent. The random variable T = 3W – 8 (d) Calculate P (W > T) (4) _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________

Turn over 11 Question 3 continued _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________

12 Question 3 continued _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________

Turn over 13 Question 3 continued _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ (Total for Question 3 is 12 marks)

14 4. Charlie carried out a survey on the main type of investment people have. The contingency table below shows the results of a survey of a random sample of people. Main type of investment Bonds Cash Stocks Age 25 – 44 a b − e e 45 – 75 c d − 59 59 (a) Find an expression, in terms of a, b, c and d, for the difference between the observed and the expected value (O − E ) for the group whose main type of investment is Bonds and are aged 45 – 75 Express your answer as a single fraction in its simplest form. (4) Given that O E E − ( ) ∑ 2 = 9.62 for this information, (b) test, at the 5% level of significance, whether or not there is evidence of an association between the age of a person and the main type of investment they have. You should state your hypotheses, critical value and conclusion clearly. You may assume that no cells need to be combined. (3) _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________

Turn over 15 Question 4 continued _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________

16 Question 4 continued _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ (Total for Question 4 is 7 marks) TOTAL FOR FURTHER STATISTICS 1 IS 40 MARKS

Turn over 3 Question 1 continued _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________

4 Question 1 continued _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________

Turn over 5 Question 1 continued _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ (Total for Question 1 is 7 marks) 6 2. u Figure 2 A particle of mass em is at rest on a smooth horizontal plane between two smooth fixed parallel vertical walls, as shown in the plan view in Figure 2. The particle is projected along the plane with speed u towards one of the walls and strikes the wall at right angles. The coefficient of restitution between the particle and each wall is e and air resistance is modelled as being negligible. Using the model, (a) find, in terms of m, u and e, an expression for the total loss in the kinetic energy of the particle as a result of the first two impacts. (3) Given that e can vary such that 0 < e < 1 and using the model, (b) find the value of e for which the total loss in the kinetic energy of the particle as a result of the first two impacts is a maximum, (4) (c) describe the subsequent motion of the particle. (2) _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________

Turn over 7 Question 2 continued _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________

8 Question 2 continued _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________

Turn over 9 Question 2 continued _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ (Total for Question 2 is 9 marks)

10 3. The total mass of a cyclist and his bicycle is 100 kg. In all circumstances, the magnitude of the resistance to the motion of the cyclist from non-gravitational forces is modelled as being kv2 N, where v m s−1 is the speed of the cyclist. The cyclist can freewheel, without pedalling, down a slope that is inclined to the horizontal at an angle α, where sin α = 1 35 , at a constant speed of V m s−1 When he is pedalling up a slope that is inclined to the horizontal at an angle β, where sin β = 1 70 , and he is moving at the same constant speed V m s−1 , he is working at a constant rate of P watts. (a) Find P in terms of V. (7) If he pedals and works at a rate of 35 V watts on a horizontal road, he moves at a constant speed of U m s−1 (b) Find U in terms of V. (4) _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________

Turn over 11 Question 3 continued _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________

12 Question 3 continued _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________

Turn over 13 Question 3 continued _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ (Total for Question 3 is 11 marks)

14 4. Two particles, P and Q, have masses m and em respectively. The particles are moving on a smooth horizontal plane in the same direction along the same straight line when they collide directly. The coefficient of restitution between P and Q is e, where 0 < e < 1 Immediately before the collision the speed of P is u and the speed of Q is eu. (a) Show that the speed of Q immediately after the collision is u. (6) (b) Show that the direction of motion of P is unchanged by the collision. (3) The magnitude of the impulse on Q in the collision is 2 9 mu (c) Find the possible values of e. (4) _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________

Turn over 15 Question 4 continued _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________

16 Question 4 continued _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ (Total for Question 4 is 13 marks) TOTAL FOR FURTHER MECHANICS 1 IS 40 MARKS

Turn over 3 Question 1 continued _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ (Total for Question 1 is 6 marks) 4 2. A B P 3a a Figure 2 A small smooth ring P, of mass m, is threaded onto a light inextensible string of length 4a. One end of the string is attached to a fixed point A on a smooth horizontal table. The other end of the string is attached to a fixed point B which is vertically above A. The ring moves in a horizontal circle with centre A and radius a, as shown in Figure 2. The ring moves with constant angular speed 2 3 g a about AB. The string remains taut throughout the motion. (a) Find, in terms of m and g, the magnitude of the normal reaction between P and the table. (6) The angular speed of P is now gradually increased. (b) Find, in terms of a and g, the angular speed of P at the instant when it loses contact with the table. (3) (c) Explain how you have used the fact that P is smooth. (1) _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________

Turn over 5 Question 2 continued _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________

6 Question 2 continued _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________

Turn over 7 Question 2 continued _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ (Total for Question 2 is 10 marks) 8 3. A B C D E F G H I J 8a 5a 2a 2a 2a 2a 2a 2a Figure 3 The uniform lamina ABCDEFGHIJ is shown in Figure 3. The lamina has AJ = 8a , AB = 5a and BC = DE = EF = FG = GH = HI = IJ = 2a. All the corners are right angles. (a) Show that the distance of the centre of mass of the lamina from AJ is 49 26 a (5) A light inextensible rope is attached to the lamina at A and another light inextensible rope is attached to the lamina at B. The lamina hangs in equilibrium with both ropes vertical and AB horizontal. The weight of the lamina is W. (b) Find, in terms of W, the tension in the rope attached to the lamina at B. (3) The rope attached to B breaks and subsequently the lamina hangs freely in equilibrium, suspended from A. (c) Find the size of the angle between AJ and the downward vertical. (5) _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________

Turn over 9 Question 3 continued _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________

10 Question 3 continued _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________

Turn over 11 Question 3 continued _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ (Total for Question 3 is 13 marks)

12 4. A particle P moves on the x-axis. At time t seconds, t  0 , P is x metres from the origin O and moving with velocity v m s−1 in the direction of x increasing, where v = 5 sin 2t When t = 0 , x = 1 and P is at rest. (a) Find the magnitude and direction of the acceleration of P at the instant when P is next at rest. (4) (b) Show that 1  x  6 (4) (c) Find the total time, in the first 4π seconds of the motion, for which P is more than 3 metres from O (3) _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________

P66794A 4 3. Donald plans to bake and sell cakes. The three types of cake that he can bake are brownies, flapjacks and muffins. Donald decides to bake 48 brownies and muffins in total. Donald decides to bake at least 5 brownies for every 3 flapjacks. At most 40% of the cakes will be muffins. Donald has enough ingredients to bake 60 brownies or 45 flapjacks or 35 muffins. Donald plans to sell each brownie for £1.50, each flapjack for £1 and each muffin for £1.25 He wants to maximise the total income from selling the cakes. Let x represent the number of brownies, let y represent the number of flapjacks and let z represent the number of muffins that Donald will bake. Formulate this as a linear programming problem in x and y only, stating the objective function and listing the constraints as simplified inequalities with integer coefficients. You should not attempt to solve the problem. (Total for Question 3 is 9 marks) P66794A 5 4. B 2 25 25 E 5 56 56 A 1 0 0 C 3 35 37 35 F 6 63 63 D 4 42 45 42 G 7 66 74 70 68 66 Figure 2 Dijkstra’s algorithm has been applied to the network in Figure 2. A working value has only been replaced at a node if the new working value is smaller. (a) State the length of the shortest path from A to G. (1) (b) Complete the table in the answer book giving the weight of each arc listed. (Note that arc CE and arc EF are not in the table.) (3) (c) State the shortest path from A to G. (1) It is now given that •  when Prim’s algorithm, starting from A, is applied to the network, the order in which the arcs are added to the tree is AB, BC, CD, CE, EF and FG •  the weight of the corresponding minimum spanning tree is 80 •  the shortest path from A to F via E has weight 67 (d) Determine the weight of arc CE and the weight of arc EF, making your reasoning clear. (3) (Total for Question 4 is 8 marks) TOTAL FOR DECISION MATHEMATICS 1 IS 40 MARKS END P66794A 6 BLANK PAGE P66794A 7 BLANK PAGE P66794A 8 BLANK PAGE Turn over Pearson Edexcel Level 3 GCE Candidate surname Other names Total Marks Centre Number Candidate Number Please check the examination details below before entering your candidate information Paper reference P66794A ©2021 Pearson Education Ltd. 1/1/1/1/1/ Answer Book Do not return the question paper with the answer book. Further Mathematics Advanced Subsidiary Further Mathematics options 27: Decision Mathematics 1 (Part of options D, F, H and K) 8FM0/27 2 1. _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________

Turn over 3 Question 1 continued _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________

4 Question 1 continued _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________

Turn over 5 Question 1 continued _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ (Total for Question 1 is 11 marks)

6 2. (a) Activity Immediately preceding activities Activity Immediately preceding activities Activity Immediately preceding activities A G M B H N C I P D J Q E K F L (b) A(7) D(5) J(11) E(4) G(6) K(16) N(x) B(8) P(5) H(4) C(6) F(7) I(8) L(10) M(6) Q(9) Key: Early event time Late event time Diagram 1 _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________

Turn over 7 Question 2 continued _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ 0 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30 32 34 36 38 40 42 Grid 1 (Total for Question 2 is 12 marks) 8 3. _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________

Turn over 9 Question 3 continued _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ (Total for Question 3 is 9 marks) 10 4. Vertex Order of labelling Final value Working values Key: B 2 25 25 E 5 56 56 A 1 0 0 C 3 35 37 35 F 6 63 63 D 4 42 45 42 G 7 66 74 70 68 66 (a) length of the shortest path from A to G:___________________________________________________ _____________________________________________________________________________________ (b) Arc Weight AB AC AD BC BE CD CF CG DG EG FG _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________

Turn over 11 Question 4 continued (c) shortest path from A to G:_ ____________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ (d)___________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________

12 Question 4 continued _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ (Total for Question 4 is 8 marks) TOTAL FOR DECISION MATHEMATICS 1 IS 40 MARKS

P66795A 2 Write your answers in the answer book provided. 1. Five workers, A, B, C, D and E, are available to complete four tasks, P, Q, R and S. Each task must be assigned to exactly one worker and each worker can do at most one task. Worker B cannot be assigned to task R. The amount, in pounds, that each worker will earn if they are assigned to each task is shown in the table below. P Q R S A 55 56 58 57 B 60 61 – 64 C 59 60 62 63 D 64 66 71 69 E 65 68 72 66 The Hungarian algorithm is to be used to find the maximum total amount that can be earned by the five workers. (a) Explain how the table should be modified to allow the Hungarian algorithm to be used, giving reasons for your answer. (2) (b) Reducing rows first, use the Hungarian algorithm to obtain the maximum possible total earnings. You should explain how any initial row and column reductions were made and how you determined if the table was optimal at each stage. (7) (Total for Question 1 is 9 marks) P66795A 3 Turn over 2. A G C S D T F B E H 30 25 60 13 17 41 24 2 0 32 13 19 1 5 36 64 28 22 49 15 66 24 47 21 31 14 19 8 25 17 35 74 15 28 20 11 56 32 19 37 Figure 1 Figure 1 shows a capacitated, directed network. The number on each arc represents the capacity of that arc. The numbers in circles represent an initial flow. (a) State the value of the initial flow. (1) (b) Obtain the capacity of the cut that passes through the arcs AG, CG, GF, FT, FH and EH. (1) (c) Complete the initialisation of the labelling procedure on Diagram 1 in the answer book by entering values along SD, BD, BE and GF. (2) (d) Use the labelling procedure to find a maximum flow through the network. You must list each flow-augmenting route you use, together with its flow. (3) (e) Use the answer to part (d) to add a maximum flow pattern to Diagram 2 in the answer book. (1) (f) Prove that your answer to part (e) is optimal. (3) (Total for Question 2 is 11 marks) P66795A 4 3. In your answer to this question you must show detailed reasoning. A two-person zero-sum game is represented by the following pay-off matrix for player A. B plays X B plays Y A plays Q 4 −3 A plays R 2 −1 A plays S −3 5 A plays T −1 3 (a) Verify that there is no stable solution to this game. (2) Player B plays their option X with probability p. (b) Use a graphical method to find the optimal value of p and hence find the best strategy for player B. (6) (c) Find the value of the game to player A. (1) (d) Hence find the best strategy for player A. (2) (Total for Question 3 is 11 marks)

Turn over 11 Question 5 continued _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ (Total for Question 5 is 6 marks) 12 6. In this question you should show all stages of your working. Solutions relying on calculator technology are not acceptable. (a) Using algebra, find all solutions of the equation 3x 3 – 17x 2 – 6x = 0 (3) (b) Hence find all real solutions of 3(y – 2) 6 – 17(y – 2) 4 – 6(y – 2) 2 = 0 (3) _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________

P66795A 5 4. Sarah takes out a mortgage of £155 000 to buy a house. Interest is added each month on the outstanding balance at a constant rate of r % each month. Sarah makes fixed monthly repayments to reduce the amount owed. Each month, interest is added, and then her monthly repayment is used to reduce the outstanding amount owed. The recurrence relationship for the amount of the mortgage outstanding after n + 1 months is modelled by un +1 = 1.0025un − x            n  0 where £un is the amount of the mortgage outstanding after n months and £x is the monthly repayment. (a) State the value of r. (1) (b) Solve the recurrence relation to find an expression for un in terms of x and n. (5) Given that the mortgage will be paid off in exactly 30 years, (c) determine, to 2 decimal places, the least possible value of x. (3) (Total for Question 4 is 9 marks) TOTAL FOR DECISION MATHEMATICS 2 IS 40 MARKS END P66795A 6 BLANK PAGE P66795A 7 BLANK PAGE P66795A 8 BLANK PAGE Turn over Pearson Edexcel Level 3 GCE Candidate surname Other names Total Marks Centre Number Candidate Number Please check the examination details below before entering your candidate information Paper reference P66795A ©2021 Pearson Education Ltd. 1/1/1/1/1/ 8FM0/28 Answer Book Do not return the question paper with the answer book. Further Mathematics Advanced Subsidiary Further Mathematics options 28: Decision Mathematics 2 (Part of option K only)

2 1. (a) _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ P Q R S A 55 56 58 57 B 60 61 – 64 C 59 60 62 63 D 64 66 71 69 E 65 68 72 66 You may not need to use all of these tables or all the rows / columns in each table. _ ____________________________________________ _ ____________________________________________ _ ____________________________________________ _ ____________________________________________ _ ____________________________________________ _ ____________________________________________ _ ____________________________________________ _ ____________________________________________ _ ____________________________________________ _ ____________________________________________ _ ____________________________________________ _ ____________________________________________ _ ____________________________________________ _ ____________________________________________ _ ____________________________________________ _ ____________________________________________ _ ____________________________________________ _ ____________________________________________ _ ____________________________________________ _ ____________________________________________ _ ____________________________________________ _ ____________________________________________ _ ____________________________________________ P Q R S A B C D E P Q R S A B C D E P Q R S A B C D E

Turn over 3 Question 1 continued _ ____________________________________________ _ ____________________________________________ _ ____________________________________________ _ ____________________________________________ _ ____________________________________________ _ ____________________________________________ _ ____________________________________________ _ ____________________________________________ _ ____________________________________________ _ ____________________________________________ _ ____________________________________________ _ ____________________________________________ _ ____________________________________________ _ ____________________________________________ _ ____________________________________________ _ ____________________________________________ _ ____________________________________________ _ ____________________________________________ _ ____________________________________________ _ ____________________________________________ _ ____________________________________________ _ ____________________________________________ _ ____________________________________________ _ ____________________________________________ _ ____________________________________________ _ ____________________________________________ _ ____________________________________________ _ ____________________________________________ _ ____________________________________________ _ ____________________________________________ _ ____________________________________________ _ ____________________________________________ _ ____________________________________________ _ ____________________________________________ _ ____________________________________________ (Total for Question 1 is 9 marks) P Q R S A B C D E P Q R S A B C D E P Q R S A B C D E P Q R S A B C D E P Q R S A B C D E 4 2. _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ (c) A G C S D T F B E H 6 24 4 1 1 0 8 0 3 10 28 13 10 49 15 1 60 30 13 17 0 1 7 64 0 32 19 7 25 36 4 0 Diagram 1

Turn over 5 Question 2 continued _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ (e) A G C S D T F B E H Diagram 2 _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ (Total for Question 2 is 11 marks) 6 3. B plays X B plays Y A plays Q 4 −3 A plays R 2 −1 A plays S −3 5 A plays T −1 3 _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________

Turn over 7 Question 3 continued _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________

8 Question 3 continued _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________

Turn over 9 Question 3 continued _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ 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_____________________________________________________________________________________ (Total for Question 3 is 11 marks) 10 4. _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ 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Turn over 11 Question 4 continued _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________

12 Question 4 continued _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ (Total for Question 4 is 9 marks) TOTAL FOR DECISION MATHEMATICS 2 IS 40 MARKS

Turn over 3 Question 1 continued _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ (Total for Question 1 is 3 marks) 4 2. In this question you should show all stages of your working. Solutions relying on calculator technology are not acceptable. Given 9 3 81 1 2 x y − + = express y in terms of x, writing your answer in simplest form. (3) _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________

Turn over 5 Question 2 continued _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ (Total for Question 2 is 3 marks)

6 3. Find 3 4 2 4 3 x x x − ∫ d writing your answer in simplest form. (4) _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________

Turn over 7 Question 3 continued _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ (Total for Question 3 is 4 marks)

8 4. [In this question the unit vectors i and j are due east and due north respectively.] A stone slides horizontally across ice. Initially the stone is at the point A(–24i – 10j) m relative to a fixed point O. After 4 seconds the stone is at the point B(12i + 5j) m relative to the fixed point O. The motion of the stone is modelled as that of a particle moving in a straight line at constant speed. Using the model, (a) prove that the stone passes through O, (2) (b) calculate the speed of the stone. (3) _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________

Turn over 13 Question 6 continued _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ (Total for Question 6 is 6 marks)

14 7. A parallelogram PQRS has area 50 cm 2 Given • PQ has length 14 cm • QR has length 7 cm • angle SPQ is obtuse find (a) the size of angle SPQ, in degrees, to 2 decimal places, (3) (b) the length of the diagonal SQ, in cm, to one decimal place. (2) _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________

Turn over 15 Question 7 continued _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ (Total for Question 7 is 5 marks) 16 8. g(x) = (2 + ax) 8 where a is a constant Given that one of the terms in the binomial expansion of g(x) is 3402x 5 (a) find the value of a. (4) Using this value of a, (b) find the constant term in the expansion of 1 1 2 4 8 +     + x ax ( ) (3) _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________

Turn over 17 Question 8 continued _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ (Total for Question 8 is 7 marks)

18 9. Find the value of the constant k, 0 < k < 9, such that 6 20 9 x x k d = ∫ (4) _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________

Turn over 19 Question 9 continued _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ (Total for Question 9 is 4 marks)

20 10. A student is investigating the following statement about natural numbers. “ n 3 – n is a multiple of 4 ” (a) Prove, using algebra, that the statement is true for all odd numbers. (4) (b) Use a counterexample to show that the statement is not always true. (1) _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________

Turn over 21 Question 10 continued _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________

22 Question 10 continued _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________

Turn over 23 Question 10 continued _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ (Total for Question 10 is 5 marks)

24 11. The owners of a nature reserve decided to increase the area of the reserve covered by trees. Tree planting started on 1st January 2005. The area of the nature reserve covered by trees, A km 2, is modelled by the equation A = 80 – 45e ct where c is a constant and t is the number of years after 1st January 2005. Using the model, (a) find the area of the nature reserve that was covered by trees just before tree planting started. (1) On 1st January 2019 an area of 60 km 2 of the nature reserve was covered by trees. (b) Use this information to find a complete equation for the model, giving your value of c to 3 significant figures. (4) On 1st January 2020, the owners of the nature reserve announced a long-term plan to have 100 km 2 of the nature reserve covered by trees. (c) State a reason why the model is not appropriate for this plan. (1) _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________

Turn over 25 Question 11 continued _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ (Total for Question 11 is 6 marks) 26 12. In this question you should show all stages of your working. Solutions relying entirely on calculator technology are not acceptable. (i) Solve, for 0 < θ  450°, the equation 5 cos 2 θ = 6 sin θ giving your answers to one decimal place. (5) (ii) (a) A student’s attempt to solve the question “ Solve, for –90° < x < 90°, the equation 3 tan x – 5 sin x = 0 ” is set out below. 3 tan x – 5 sin x = 0 3 sin cos x x – 5 sin x = 0 3 sin x – 5 sin x cos x = 0 3 – 5 cos x = 0 cos x = 3 5 x = 53.1° Identify two errors or omissions made by this student, giving a brief explanation of each. (2) The first four positive solutions, in order of size, of the equation cos (5α + 40°) = 3 5 are α1 , α2 , α3 and α4 (b) Find, to the nearest degree, the value of α4 (2) _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________

Turn over 27 Question 12 continued _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________

28 Question 12 continued _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________

Turn over 29 Question 12 continued _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ (Total for Question 12 is 9 marks) 30 13. 2.25 log10 h O log10 m Figure 2 The resting heart rate, h, of a mammal, measured in beats per minute, is modelled by the equation h = pm q where p and q are constants and m is the mass of the mammal measured in kg. Figure 2 illustrates the linear relationship between log10 h and log10 m The line meets the vertical log10 h axis at 2.25 and has a gradient of –0.235 (a) Find, to 3 significant figures, the value of p and the value of q. (3) A particular mammal has a mass of 5 kg and a resting heart rate of 119 beats per minute. (b) Comment on the suitability of the model for this mammal. (3) (c) With reference to the model, interpret the value of the constant p. (1) _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________

Turn over 31 Question 13 continued _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________

32 Question 13 continued _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________

Turn over 33 Question 13 continued _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ (Total for Question 13 is 7 marks)

34 14. A curve C has equation y = f(x) where f(x) = –3x 2 + 12x + 8 (a) Write f(x) in the form a(x + b) 2 + c where a, b and c are constants to be found. (3) The curve C has a maximum turning point at M. (b) Find the coordinates of M. (2) R l O x y M C Figure 3 Figure 3 shows a sketch of the curve C. The line l passes through M and is parallel to the x-axis. The region R, shown shaded in Figure 3, is bounded by C, l and the y-axis. (c) Using algebraic integration, find the area of R. (5) _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________

Turn over 37 Question 14 continued _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ (Total for Question 14 is 10 marks) 38 15. P l O x y N C Figure 4 Figure 4 shows a sketch of a circle C with centre N(7, 4) The line l with equation y = 1 3 x is a tangent to C at the point P. Find (a) the equation of line PN in the form y = mx + c, where m and c are constants, (2) (b) an equation for C. (4) The line with equation y = 1 3 x + k, where k is a non-zero constant, is also a tangent to C. (c) Find the value of k. (3) _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________

Turn over 39 Question 15 continued _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________

40 Question 15 continued _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________

Turn over 41 Question 15 continued _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ (Total for Question 15 is 9 marks)

42 16. The curve C has equation y = f(x) where f(x) = ax 3 + 15x 2 – 39x + b and a and b are constants. Given • the point (2, 10) lies on C • the gradient of the curve at (2, 10) is –3 (a) (i) show that the value of a is –2 (ii) find the value of b. (4) (b) Hence show that C has no stationary points. (3) (c) Write f(x) in the form (x – 4)Q(x) where Q(x) is a quadratic expression to be found. (2) (d) Hence deduce the coordinates of the points of intersection of the curve with equation y = f(0.2x) and the coordinate axes. (2) _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________

Turn over 43 Question 16 continued _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________

44 Question 16 continued _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ (Total for Question 16 is 11 marks) TOTAL FOR PAPER IS 100 MARKS

Turn over 3 Question 1 continued _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ (Total for Question 1 is 2 marks)

4 2. The partially completed table and partially completed histogram give information about the ages of passengers on an airline. There were no passengers aged 90 or over. Age (x years) 0  x < 5 5  x < 20 20  x < 40 40  x < 65 65  x < 80 80  x < 90 Frequency 5 45 90 1 10 20 30 40 50 60 70 80 90 x y 0 0 Frequency density Age (years) (a) Complete the histogram. (3) (b) Use linear interpolation to estimate the median age. (4) An outlier is defined as a value greater than Q3 + 1.5 × interquartile range. Given that Q1 = 27.3 and Q3 = 58.9 (c) determine, giving a reason, whether or not the oldest passenger could be considered as an outlier. (2)

Turn over 5 Question 2 continued _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________

6 Question 2 continued _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________

Turn over 7 Question 2 continued _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ (Total for Question 2 is 9 marks)

8 3. Helen is studying one of the qualitative variables from the large data set for Heathrow from 2015. She started with the data from 3rd May and then took every 10th reading. There were only 3 different outcomes with the following frequencies Outcome A B C Frequency 16 2 1 (a) State the sampling technique Helen used. (1) (b) From your knowledge of the large data set (i) suggest which variable was being studied, (ii) state the name of outcome A. (2) George is also studying the same variable from the large data set for Heathrow from 2015. He started with the data from 5th May and then took every 10th reading and obtained the following Outcome A B C Frequency 16 1 1 Helen and George decided they should examine all of the data for this variable for Heathrow from 2015 and obtained the following Outcome A B C Frequency 155 26 3 (c) State what inference Helen and George could reliably make from their original samples about the outcomes of this variable at Heathrow, for the period covered by the large data set in 2015. (1) _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________

Turn over 9 Question 3 continued _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ (Total for Question 3 is 4 marks)

10 4. A nursery has a sack containing a large number of coloured beads of which 14% are coloured red. Aliya takes a random sample of 18 beads from the sack to make a bracelet. (a) State a suitable binomial distribution to model the number of red beads in Aliya’s bracelet. (1) (b) Use this binomial distribution to find the probability that (i) Aliya has just 1 red bead in her bracelet, (ii) there are at least 4 red beads in Aliya’s bracelet. (3) (c) Comment on the suitability of a binomial distribution to model this situation. (1) After several children have used beads from the sack, the nursery teacher decides to test whether or not the proportion of red beads in the sack has changed. She takes a random sample of 75 beads and finds 4 red beads. (d) Stating your hypotheses clearly, use a 5% significance level to carry out a suitable test for the teacher. (4) (e) Find the p‑value in this case. (1) _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________

Turn over 11 Question 4 continued _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________

12 Question 4 continued _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________

Turn over 13 Question 4 continued _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ (Total for Question 4 is 10 marks)

10 Question 3 continued _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________

14 5. Two bags, A and B, each contain balls which are either red or yellow or green. Bag A contains 4 red, 3 yellow and n green balls. Bag B contains 5 red, 3 yellow and 1 green ball. A ball is selected at random from bag A and placed into bag B. A ball is then selected at random from bag B and placed into bag A. The probability that bag A now contains an equal number of red, yellow and green balls is p. Given that p > 0, find the possible values of n and p. (5) _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________

Turn over 15 Question 5 continued _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________

16 Question 5 continued _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ (Total for Question 5 is 5 marks) TOTAL FOR STATISTICS IS 30 MARKS

2 1. At time t = 0, a small stone is thrown vertically upwards with speed 14.7 m s−1 from a point A. At time t = T seconds, the stone passes through A, moving downwards. The stone is modelled as a particle moving freely under gravity throughout its motion. Using the model, (a) find the value of T, (2) (b) find the total distance travelled by the stone in the first 4 seconds of its motion. (4) (c) State one refinement that could be made to the model, apart from air resistance, that would make the model more realistic. (1) _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________

Turn over 3 Question 1 continued _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ (Total for Question 1 is 7 marks)

4 2. A particle P moves along a straight line. At time t seconds, the velocity v m s−1 of P is modelled as v = 10t − t 2 − k        t  0 where k is a constant. (a) Find the acceleration of P at time t seconds. (2) The particle P is instantaneously at rest when t = 6 (b) Find the other value of t when P is instantaneously at rest. (4) (c) Find the total distance travelled by P in the interval 0  t  6 (4) _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________

Turn over 5 Question 2 continued _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________

6 Question 2 continued _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________

Turn over 7 Question 2 continued _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ (Total for Question 2 is 10 marks) 8 3. P(2m) h 2h Q(5m) Figure 1 A ball P of mass 2m is attached to one end of a string. The other end of the string is attached to a ball Q of mass 5m. The string passes over a fixed pulley. The system is held at rest with the balls hanging freely and the string taut. The hanging parts of the string are vertical with P at a height 2h above horizontal ground and with Q at a height h above the ground, as shown in Figure 1. The system is released from rest. In the subsequent motion, Q does not rebound when it hits the ground and P does not hit the pulley. The balls are modelled as particles. The string is modelled as being light and inextensible. The pulley is modelled as being small and smooth. Air resistance is modelled as being negligible. Using this model, (a) (i) write down an equation of motion for P, (ii) write down an equation of motion for Q, (4) (b) find, in terms of h only, the height above the ground at which P first comes to instantaneous rest. (7) (c) State one limitation of modelling the balls as particles that could affect your answer to part (b). (1) In reality, the string will not be inextensible. (d) State how this would affect the accelerations of the particles. (1)

Turn over 9 Question 3 continued _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________

10 Question 3 continued _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________

Turn over 11 Question 3 continued _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________

12 Question 3 continued _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ (Total for Question 3 is 13 marks) TOTAL FOR MECHANICS IS 30 MARKS

2 1. The transformation P is an enlargement, centre the origin, with scale factor k, where k > 0 The transformation Q is a rotation through angle θ degrees anticlockwise about the origin. The transformation P followed by the transformation Q is represented by the matrix M = − − −     4 4 3 4 3 4 (a) Determine (i) the value of k, (ii) the smallest value of θ (4) A square S has vertices at the points with coordinates (0, 0), (a, –a), (2a, 0) and (a, a) where a is a constant. The square S is transformed to the square S′ by the transformation represented by M. (b) Determine, in terms of a, the area of S′ (2) _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________

Turn over 3 Question 1 continued _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________

4 Question 1 continued _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________

Turn over 5 Question 1 continued _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ (Total for Question 1 is 6 marks)

6 2. (a) Use the Maclaurin series expansion for cos x to determine the series expansion of cos2 3 x    in ascending powers of x, up to and including the term in x 4 Give each term in simplest form. (2) (b) Use the answer to part (a) and calculus to find an approximation, to 5 decimal places, for 2 2 6 1 cos d 3          ∫ π π x x x (3) (c) Use the integration function on your calculator to evaluate 2 2 6 1 cos d 3          ∫ π π x x x Give your answer to 5 decimal places. (1) (d) Assuming that the calculator answer in part (c) is accurate to 5 decimal places, comment on the accuracy of the approximation found in part (b). . (1) _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________

Turn over 7 Question 2 continued _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________

8 Question 2 continued _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________

Turn over 9 Question 2 continued _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ (Total for Question 2 is 7 marks)

Turn over 21 Question 6 continued _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ (Total for Question 6 is 14 marks)

10 3. The cubic equation ax 3 + bx 2 – 19x – b = 0 where a and b are constants, has roots α, β and γ The cubic equation w 3 – 9w 2 – 97w + c = 0 where c is a constant, has roots (4α – 1), (4β – 1) and (4γ – 1) Without solving either cubic equation, determine the value of a, the value of b and the value of c. (6) _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________

Turn over 11 Question 3 continued _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________

12 Question 3 continued _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________

Turn over 13 Question 3 continued _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ (Total for Question 3 is 6 marks)

14 4. (i) A is a 2 by 2 matrix and B is a 2 by 3 matrix. Giving a reason for your answer, explain whether it is possible to evaluate (a) AB (b) A + B (2) (ii) Given that −         − − −         5 3 1 0 0 0 5 0 2 12 1 1 11 3 a b a b = λI where a, b and λ are constants, (a) determine • the value of λ • the value of a • the value of b (b) Hence deduce the inverse of the matrix −         5 3 1 0 0 a b a b (3) (iii) Given that M = 1 1 1 0 sin cos 0 cos2 sin 2 θ θ θ θ         where 0  θ < π determine the values of θ for which the matrix M is singular. (4) _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________

Turn over 15 Question 4 continued _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________

16 Question 4 continued _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________

Turn over 17 Question 4 continued _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ (Total for Question 4 is 9 marks)

18 5. (i) Evaluate the improper integral 2 1 1 2 e d ∞ − ∫ x x (3) (ii) The air temperature, θ °C, on a particular day in London is modelled by the equation θ = 8 – 5 sin  12 π t       – cos  6 π t       0  t  24 where t is the number of hours after midnight. (a) Use calculus to show that the mean air temperature on this day is 8 °C, according to the model. (3) Given that the actual mean air temperature recorded on this day was higher than 8 °C, (b) explain how the model could be refined. (1) _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________

Turn over 19 Question 5 continued _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ (Total for Question 5 is 7 marks)

20 6. A tourist decides to do a bungee jump from a bridge over a river. One end of an elastic rope is attached to the bridge and the other end of the elastic rope is attached to the tourist. The tourist jumps off the bridge. At time t seconds after the tourist reaches their lowest point, their vertical displacement is x metres above a fixed point 30 metres vertically above the river. When t = 0 • x = –20 • the velocity of the tourist is 0 m s –1 • the acceleration of the tourist is 13.6 m s –2 In the subsequent motion, the elastic rope is assumed to remain taut so that the vertical displacement of the tourist can be modelled by the differential equation 5k d d 2 2 x t + 2k d d x t + 17x = 0 t  0 where k is a positive constant. (a) Determine the value of k (2) (b) Determine the particular solution to the differential equation. (7) (c) Hence find, according to the model, the vertical height of the tourist above the river 15 seconds after they have reached their lowest point. (2) (d) Give a limitation of the model. (1) _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________

Turn over 21 Question 6 continued _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________

22 Question 6 continued _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________

Turn over 23 Question 6 continued _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ (Total for Question 6 is 12 marks)

24 7. The plane Π has equation 3 1 2 3 2 0 2 1 1 λ μ −         = + +         r where λ and μ are scalar parameters. (a) Show that vector 2i + 3j – 4k is perpendicular to Π. (2) (b) Hence find a Cartesian equation of Π. (2) The line l has equation r = −        + −         4 5 2 1 6 3 t where t is a scalar parameter. The point A lies on l. Given that the shortest distance between A and Π is 2 29 (c) determine the possible coordinates of A. (4) _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________

Turn over 25 Question 7 continued _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________

26 Question 7 continued _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________

Turn over 27 Question 7 continued _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ (Total for Question 7 is 8 marks)

Turn over 9  Question 4 continued _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________

28 8. Two different colours of paint are being mixed together in a container. The paint is stirred continuously so that each colour is instantly dispersed evenly throughout the container. Initially the container holds a mixture of 10 litres of red paint and 20 litres of blue paint. The colour of the paint mixture is now altered by • adding red paint to the container at a rate of 2 litres per second • adding blue paint to the container at a rate of 1 litre per second • pumping fully mixed paint from the container at a rate of 3 litres per second. Let r litres be the amount of red paint in the container at time t seconds after the colour of the paint mixture starts to be altered. (a) Show that the amount of red paint in the container can be modelled by the differential equation d 2 d r r t α = − where α is a positive constant to be determined. (2) (b) By solving the differential equation, determine how long it will take for the mixture of paint in the container to consist of equal amounts of red paint and blue paint, according to the model. Give your answer to the nearest second. (6) It actually takes 9 seconds for the mixture of paint in the container to consist of equal amounts of red paint and blue paint. (c) Use this information to evaluate the model, giving a reason for your answer. (1) _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________

Turn over 29 Question 8 continued _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________

30 Question 8 continued _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________

Turn over 31 Question 8 continued _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ (Total for Question 8 is 9 marks)

32 9. (a) Use a hyperbolic substitution and calculus to show that x x x x x x k 2 2 2 1 1 2 1 − = − +    + ∫ d arcosh where k is an arbitrary constant. (6) y O x = 3 R C x Figure 1 Figure 1 shows a sketch of part of the curve C with equation y = 4 15 x arcosh x x  1 The finite region R, shown shaded in Figure 1, is bounded by the curve C, the x‑axis and the line with equation x = 3 (b) Using algebraic integration and the result from part (a), show that the area of R is given by 1 15 17 3 2 2 6 2 ln + ( ) −   (5) _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________

Turn over 33 Question 9 continued _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________

34 Question 9 continued _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________

Turn over 35 Question 9 continued _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________

36 Question 9 continued _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ (Total for Question 9 is 11 marks) TOTAL FOR PAPER IS 75 MARKS

2 1. Given that 1 2 3 cos isin 3 3 2 cos isin 12 12 π π z π π z       = +                   = −             (a) write down the exact value of (i) | z1z2 | (ii) arg (z1z2) (2) Given that w = z1z2 and that arg (wn) = 0 , where n  + (b) determine (i) the smallest positive value of n (ii) the corresponding value of | wn | (3) _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________

Turn over 3 Question 1 continued _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ (Total for Question 1 is 5 marks) 4 2. A = 4 2 5 3 −     The matrix A represents the linear transformation M. Prove that, for the linear transformation M, there are no invariant lines. (5) _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________

Turn over 5 Question 2 continued _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ (Total for Question 2 is 5 marks) 6 3. f (x) = arcsin x            −1  x  1 (a) Determine the first two non-zero terms, in ascending powers of x, of the Maclaurin series for f (x) , giving each coefficient in its simplest form. (4) (b) Substitute x = 1 2 into the answer to part (a) and hence find an approximate value for π Give your answer in the form p q where p and q are integers to be determined. (2) _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________

Turn over 7 Question 3 continued _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ (Total for Question 3 is 6 marks)

8 4. In this question you may assume the results for r r n 3 1 =∑ , r r n 2 1 =∑ and r r n =∑ 1 (a) Show that the sum of the cubes of the first n positive odd numbers is n2 (2n2 − 1) (5) The sum of the cubes of 10 consecutive positive odd numbers is 99 800 (b) Use the answer to part (a) to determine the smallest of these 10 consecutive positive odd numbers. (4) _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________

Turn over 9 Question 4 continued _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________

10 Question 4 continued _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________

Turn over 11 Question 4 continued _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ (Total for Question 4 is 9 marks)

12 5. The curve C has equation y = arccos  1 2 x                −2  x  2 (a) Show that C has no stationary points. (3) The normal to C, at the point where x = 1 , crosses the x-axis at the point A and crosses the y-axis at the point B. Given that O is the origin, (b) show that the area of the triangle OAB is 1 54 (p 3 + qπ + r 3 π2)  where p, q and r are integers to be determined. (5) _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________