87. The polynomials ax³ + 3x² - 3 and 2x³ - 5x + a when divided by x - 4 leaves the remainders p and q respectively. Find the value of a, if 2p = q. 

Let f(x) = ax³ + 3x² - 3
and    g(x) = 2x³ - 5x + a
By remainder theorem,
f(4) = p ...(1) | x - 4 = 0 ⇒ x = 4
and    g (4) = q ...(2) | x - 4 = 0 ⇒ x = 4
(1) gives
a(4)³ + 3(4)² - 3 = p

⇒    64a + 48 - 3 = p
⇒    64a + 45 = p    ...(3)
(2) gives
2(4)³ - 5(4) + a = q
⇒    128 - 20 + a = q
⇒    108 + a = q    ...(4)
According to the question,

                 2p = q
           2(64a + 45) = 108 + a
           128a + 90 = 108 + a
           128a - a = 108 - 90
           127a  = 18