Condition for real and distinct roots

CAT 2019 Slot 2 · Quantitative Ability · Medium · Linear & Quadratic equations

This is a medium Quantitative Ability question from the CAT 2019 Slot 2 paper. It tests Linear & Quadratic equations. The full answer key and a step-by-step explanation are below — try it yourself first, then reveal the solution.

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Let A be a real number. Then the roots of the equation $x^2 - 4x - \log_2 A = 0$ are real and distinct if and only if

A.
$A < 1/16$
B.
$A > 1/8$
C.
$A > 1/16$
D.
$A < 1/8$

Answer

Explanation

For the quadratic equation to have real and distinct roots, its discriminant must be strictly positive: $D = (-4)^2 - 4(1)(-\log_2 A) > 0$ $16 + 4\log_2 A > 0$ $\log_2 A > -4 \implies A > 2^{-4} = \frac{1}{16}$

Also $A > 0$ for $\log_2 A$ to be defined, which is inherently satisfied when $A > 1/16$ .

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