Let \(a, b, c, p, q\) be real numbers. Suppose \(\alpha, \beta\) are the roots of the equation \(x^{2}+2 p x+q=0\) and \(\alpha, \frac{1}{\beta}\) are the roots of the equation \(a x^{2}+2 b x+c=0\), where \(\beta^{2} \notin\{-1,0,1\}\). STATEMENT-1 : \(\left(p^{2}-q\right)\left(b^{2}-a c\right) \geq 0\) and STATEMENT-2 : \(b \neq p a\) or \(c \neq q a\)

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

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

(C) STATEMENT-1 is True, STATEMENT-2 is False

(D) STATEMENT-1 is False, STATEMENT-2 is True

JEE Advanced · 2008 · 13 Apr 2008 · Paper 2 · Quadratic Equations and Inequalities

Let

a, b, c, p, q

be real numbers. Suppose

α, β

are the roots of the equation

x^{2} + 2 p x + q = 0

and

α, \frac{1}{β}

are the roots of the equation

a x^{2} + 2 b x + c = 0

, where

β^{2} \in / {- 1, 0, 1}

.

STATEMENT-1 :

(p^{2} - q) (b^{2} - a c) \geq 0

and
STATEMENT-2 :

b \neq = p a

c \neq = q a

(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✓ Correct
(C)STATEMENT-1 is True, STATEMENT-2 is False
(D)STATEMENT-1 is False, STATEMENT-2 is True

Answer: (B)

α, β

are roots of

x^{2} + 2 p x + q = 0

, then

α + β = - 2 p

and

α β = q

. If

α, \frac{1}{β}

are roots of

a x^{2} + 2 b x + c = 0

, then

α + \frac{1}{β} = - \frac{2 b}{a}

and

\frac{α}{β} = \frac{c}{a}

. If the roots were imaginary,

β = \overset{α}{ˉ}

and

\frac{1}{β} = \overset{α}{ˉ}

would imply

β = \frac{1}{β}

, contradicting

β^{2} \neq = 1

. So roots are real, meaning both discriminants are non-negative:

p^{2} - q \geq 0

and

b^{2} - a c \geq 0

, so STATEMENT-1 is true. For STATEMENT-2, if

β = 1

, then

α = q

and

c = q a

, contradicting the given. Similarly,

b = p a

leads to contradiction. So STATEMENT-2 is true but not a correct explanation.

JEE Advanced · 2008 · 13 Apr 2008 · Paper 2 · Single correct option