page 05 MARKS 7. The complex number 3 z i   is a root of 2 6 0 z z a   , where a is a real number. (a) State the second root of 2 6 0 z z a   . (b) Hence, or otherwise, find the value of a. The expression 2 6 z z a   is a factor of 3 2 20 z z z b   , where b is a real number. (c) Find the value of b. 8. (a) Differentiate ln x x x  with respect to x. (b) Hence find the general solution of the differential equation ln x dy y x x dx    . 9. The matrix A is given by 3 2 = 0 1 A        . Prove by induction that 3 1 3 0 1 n n n A        , n N  . 10. Solve the differential equation sin cos 2 2 4 4 9 13 d y dy y x x dx dx     given that 5 y  and 0 dy dx  when 0 x . 11. A curve defined parametrically has the following properties: • 1 tan 2 x t   •   2 6 1 4 dy t t dx   • 5 y  when 1 t . Find y in terms of t. 1 2 1 2 4 5 9 4 [Turn over