The number of quadratic polynomials a x 2 + 2 b x + c which satisfy the following conditions is k (i) a , b , c are distinct (ii) a , b , c ∈ 1,2 , 3,4 , … 2001,2002 (iii) x + 1 divides a x 2 + 2 b x + c . Then k 10 5 is equal to

Step-by-step solution

x = - 1 is the root of the quadratic equation a x 2 + 2 b x + c = 0 ⇒ a - 2 b + c = 0 ⇒ 2 b = a + c ⇒ a and c both must be even or odd. So, the total number of quadratic polynomials, k = C 2 1001 × 2 + C 2 1001 × 2 = C 1001 2 × 2 × 2 = 1001 × 1000 2 × 2 × 2 ⇒ k 10 5 = 20.02