ISRO EC 2023 Q.4: solution and answer

Official answer: CSource: ISRO's revised and final answer key of 16 February 2024

1. The impulse $\delta (t)$ has the sifting property: ${\int }_{-\infty }^{\infty }\delta (t)f(t)dt=f(0)$ for any $f$ continuous at $t=0$.

2. Here $f(t)=\sin \left(\frac{t}{\sqrt{2}}\right)$, which is continuous everywhere, so the property applies.

3. Evaluate at the impulse location: $f(0)=\sin (0)=0$.

4. So the integral equals $0$, which is option (c).

5. Check on the other options: $\sin (t/\sqrt{2})$ at $t=0$ is $0$ whatever the factor $1/\sqrt{2}$ is, so the values $0.707$ and $1.414$ (which involve $\sqrt{2}$) cannot arise. The value $1$ would arise only if the integrand were $\cos \left(t/\sqrt{2}\right)$, since $\cos 0=1$.