The GATE Grind

GATE 2020 EE – Question 12

Signals and Systems · Linear time invariant and causal systems · 1 mark · Multiple choice

Which of the following is true for all possible non-zero choices of integers $m,n;\ m\ne n$, or all possible non-zero choices of real numbers $p,q;\ p\ne q$, as applicable?

  1. $\dfrac1\pi\displaystyle\int_0^\pi\sin m\theta\sin n\theta\,d\theta=0$
  2. $\dfrac1{2\pi}\displaystyle\int_{-\pi/2}^{\pi/2}\sin p\theta\sin q\theta\,d\theta=0$
  3. $\dfrac1{2\pi}\displaystyle\int_{-\pi}^{\pi}\sin p\theta\cos q\theta\,d\theta=0$
  4. $\displaystyle\lim_{\alpha\to\infty}\dfrac1{2\alpha}\int_{-\alpha}^{\alpha}\sin p\theta\sin q\theta\,d\theta=0$

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Correct answer: (C) $\dfrac1{2\pi}\displaystyle\int_{-\pi}^{\pi}\sin p\theta\cos q\theta\,d\theta=0$

Explanation

In C, $\sin p\theta\cos q\theta$ is an odd function of $\theta$, so its integral over the symmetric interval $[-\pi,\pi]$ is zero for all $p,q$. In the other options the integrals are not zero for all choices of $p,q$ (for example B over a half-period, and D for $p=q$ type limits or non-orthogonal choices).