Question

Let P, Q, R, S and T are five sets about the quadratic equation

(a – 5)$x^2$ – 2ax + (a – 4) = 0, a $\ne$ 5 such that

P : All values of ‘a’ for which the product of roots of given quadratic equation is positive.

Q : All values of ‘a’ for which the product of roots of given quadratic equation is negative.

R : All values of ‘a’ for which the product of real roots of given quadratic equation is positive.

S : All values of ‘a’ for which the roots of given quadratic are real.

T : All values of ‘a’ for which the given quadratic equation has complex roots.

Which statement is correct regarding sets, P, Q and R ?

• A. $P\cap Q=\psi$
• B. $R\subseteq P$
• C. $P\cup Q=R^{\ast }-$ {4,5} (where' R*' is the set of real numbers)
• D. All of the above

Answer 1

The correct option is D

All of the above

(a) $P\cap Q=\varphi$
(b) $R\subseteq P$
(c) P $\cup Q=aϵ\left\{\right(-\infty ,4)\cup (5,\infty \left)\right\}\cup (4,5)$
= $R^{\ast }-\{4,5\}$