III 15 ( 15 marks) Use the Question 15 Writing Booklet. (a) Let P(x) = a,x" +a,_,x"-+...+ a, be a polynomial of degree n 21 where each a, E Z. (i) P(x) has the property that P(0), P(1), P(2),... are all prime numbers. If P(0) = p, where p is prime, show that wittily Listrins no P(kp) = P for all integers k 21. (ii) Deduce that no such polynomial P(x) exists with the given property. 3 (b) (i) Using de Moivre's theorem, show that 3 tan 50 = 5 tan 0 -10 tan 0 + tan 0 1-10 tan' 0 + 5 tan* 0 (ii) Hence show that tan and tan 27 5 are the roots of the 2 equation x -10x +5=0. (c) A particle is initially at rest on the number line at a position of x =1. The particle moves continuously along the number line according to the acceleration equation 4 8 * = (x - 2)2