(a) A ball is projected from a fixed point, O, on a horizontal plane, with velocity Y = 12 ms"', and passes through the point P at time t seconds after projection. The point O is on the horizontal plane vertically below P . The only force acting on the ball is gravity. " = ( 12 ) ms / O(0,0) (i) Show, by integration, that the position vector for the ball is given by 2 61 r = 121 - 2 (ii) It is given that OO =2 PQ Find the speed of the ball as it passes through point P. 3(b) By considering sin X , or otherwise, determine xcos x - sin x dx . x + sin x (c) Let f. (x) = x" + x ", where n E Z and x # 0. (i) Show that fill (x) = (x+x') f. (x)-f. (x). (ii) Using the recurrence from part 0, prove by mathematical induction that for all positive integers n, if x + x is rational, then x" + x"" is rational. iii) Deduce that, for all integers n, #V3+ V2 +