Question

Let a, b, c be non-zero real numbers such that 

∫ (1 + cos⁸ x)(ax² + bx + c) dx , x∈[0,1]= ∫ (1 + cos⁸ x)(ax² + bx + c) dx ,x∈[0,2]

Then, the quadratic equation ax² + bx + c = 0 has

(a) no root in (0, 2)

(b) at least one root in (1, 2)

(c) a double root in (0, 2)

(d) two imaginary roots

Answer 1

Correct option (b) at least one root in (1, 2)

Explanation :

Consider,

Obviously, /(r) is continuous and differentiable in the interval [1, 2].

Also, f(1) = f(2) [given]

By Rolle's theorem there exist at least one point k ∈ (1, 2) such that f'(k) = 0.