Question

The quadratic equation (2x − a) (2x − c) + λ(x − 2b) (x − 2d) = 0, (where 0 < 4a < 4b < c < 4d) has 

(A) A root between b and d for all λ

(B) A root between b and d for all −ve λ

(C) A root between b and d for all +ve λ

(D) None of these

Answer 1

Correct option  (A) A root between b and d for all λ

Let f(x) = (2x − a) (2x − c) + l(x − 2b)(x − 2d).

Clearly f(b)⋅f(d) < 0 ∀ λ.

Hence, a root of given equation will lie between b and d.