If the polynomial

Let, p (x) = ax³+3x²-3 and q (x) = 2x³-5x+a be the given polynomials.

Now,

R₁ = Remainder when p (x) is divided by (x – 4)

= p (4)

= a (4)³ + 3 (4)² – 3 [Therefore, p (x) = ax³+3x²-3]

= 64a + 48 – 3

R₁ = 64a + 45

And,

R₂ = Remainder when q (x) is divided by (x – 4)

= q (4)

= 2 (4)³ – 5 (4) + a [Therefore, q (x) = 2x³-5x+a]

= 128 – 20 + a

R₂ = 108 + a

(i) Given condition is,

R₁ = R₂

64a + 45 = 108 + a

63a – 63 = 0

63a = 63

a = 1

(ii) Given condition is R₁ + R₂ = 0

64a + 45 + 108 + a = 0

65a + 153 = 0

65a = -153

a =

(iii) Given condition is 2R₁ – R₂ = 0

2 (64a + 45) – (108 + a) = 0

128a + 90 – 108 – a

127a – 18 = 0

127a = 18

a =