Question

Let a, b, c, p and q be real numbers. Suppose α,β are the roots of the equation x² + 2px + q = 0 and α,1/β are roots of the equation ax² + 2bx + c = 0, where β² ∉ , {−1 0,1}.

Statement-1 : (p² - ac) ≥ 0

Statement-2 : p ≠ pa or c ≠ qa

(A) Statement-1 is True, Statement-2 is True; Statement-2 is a correct explanation for Statement-1.

(B) Statement-1 is True, Statement-2 is True; Statement-2 is NOT a correct explanation for Statement-1. 

(C) Statement-1 is True, Statement-2 is False.

(D) Statement-1 is False, Statement-2 is True. 

Answer 1

Correct option (B) Statement-1 is True, Statement-2 is True; Statement-2 is NOT a correct explanation for Statement-1. 

Given that, α,β are the roots of equation

x² + 2px + q = 0

Therefore

α + β = -2p,αβ = q ....(1)

Since α, 1/β are the roots of equation.

ax² + 2bx + c = 0

We have

α + 1/β = -2/a, α/β = c/a

Using Eq. (1), if β = 1, then α = q.

Using Eq. (2), if b = 1, then α = c/a

So  α = q = c/α ⇒ c = qa (not possible)

Also

α + 1 = -2p= -2b/a ⇒ b = pa (not possible)

Therefore, Statement-2 is correct.

Now, if the roots are imaginary, we have

Therefore, roots are real in both equations. So

(4p² - 4q)( 4b² - 4ac) ≥ 0

⇒ (p² - q)(b² -ac) ≥ 0