GATE 2025 IN GA Q.3: solution and answer

Official answer: ASource: Official answer key, GATE 2025 IN (IIT Roorkee)

1. Multiply both sides by $e^x$, which is never zero: $$P{e}^{2x}=Q\text{for every real }x.$$ The right side is a constant, so the left side must also be constant for every $x$.

2. Put $x=0$. Since $e^0=1$, this gives $P=Q$.

3. Substitute $Q=P$ back: $P{e}^{2x}-P=0$, that is, $$P\left(e^{2x}-1\right)=0\text{for all }x.$$

4. Take any $x\ne 0$, for example $x=1$. Then $e^2-1\ne 0$, so $P=0$, and hence $Q=P=0$. Option (A) holds.

5. Check the other options. (B): $P=Q=1$ would need $e^x=e^{-x}$ for all $x$, which fails at $x=1$. (C): $P=1,Q=-1$ would need $e^x=-e^{-x}$, which is impossible because the left side is positive and the right side is negative. (D): $\frac{P}{Q}=0$ needs $P=0$ with $Q\ne 0$, but $P=0$ forces $Q=0$, so $\frac{P}{Q}$ is not even defined.

6. So the only true statement is (A): $P=Q=0$.