Let a, b, c, p, q be real numbers. Suppose α, β are the roots of the equation x^2 + 2px + q = 0 and , 1 are the roots of the equation ax^2 + 2bx + c = 0, where ^2 -1, 0, 1 . STATEMENT-1 : (p^2 - q)(b^2 - ac) 0 and STATEMENT-2 : b ≠ pa or c ≠ qa

JEE Advanced 2008 Mathematics question

Let a, b, c, p, q be real numbers. Suppose α, β are the roots of the equation and are the roots of the equation , where . STATEMENT-1 : and STATEMENT-2 : b ≠ 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-1Correct
(C)STATEMENT-1 is True, STATEMENT-2 is False
(D)STATEMENT-1 is False, STATEMENT-2 is True

Answer: (B)

Solution

If the roots α, β were non-real, then β = ᾱ; the second equation also has real coefficients, so its non-real roots α and are conjugates: , giving β² = 1, which is excluded. So α, β are real, and then α and are real too: both discriminants are non-negative,

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Question ID VK-037615