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