GATE CS 2025 Set 2 Q.3: Quantitative Aptitude

If $P e^x = Q e^{-x}$ for all real values of $x$ , which one of the following statements is true?

$P = Q = 0$

$P = Q = 1$

$P = 1; \; Q = -1$

$\frac{P}{Q} = 0$

Official answer and solution

Solution

Substitute two distinct real values of $x$ (such as $x = 0$ and $x = 1$ ) to solve for $P$ and $Q$ .

Since $P e^x = Q e^{-x}$ holds for all $x \in \mathbb{R}$ :

Substituting $x = 0$ gives $P(1) = Q(1) \implies P = Q$ .
Substituting $x = 1$ gives $P e = Q e^{-1} \implies P(e - e^{-1}) = 0$ .
Since $e - e^{-1} \neq 0$ , we must have $P = 0$ , and hence $P = Q = 0$ . Note that Option (D) $\frac{P}{Q} = 0$ is undefined when $Q = 0$ .

Final answer

Option (A)

Common trap

Choosing (D) $P/Q = 0$ without noticing that $Q = 0$ makes division by $Q$ undefined.

The answer is the one in the official GATE answer key. The solution is this site's own explanation.

Paper: GATE 2025 Computer Science and Information Technology - Session 2 (CS2)
Question: 3 of 65
Section: General Aptitude

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