The GATE Grind

GATE 2026 IN – Question 19

Signals and Systems · Signals and linear time-invariant systems · 1 mark · Multiple choice

Given $x(t)=4\sin(15\pi t)+7\cos(4\pi t)$, where $t$ is in seconds, the fundamental period of $x(t)$ is ________ seconds.

  1. 15
  2. 60
  3. 30
  4. 2

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Show answer and explanation

Correct answer: (D) 2

Explanation

The component frequencies are $f_1=15\pi/(2\pi)=7.5\ \mathrm{Hz}$ and $f_2=4\pi/(2\pi)=2\ \mathrm{Hz}$. A common period must satisfy $7.5T=m$ and $2T=n$ for integers $m,n$.

The smallest positive solution has $m=15$, $n=4$, giving $T=2\ \mathrm s$. Thus the fundamental period is **2 seconds (D)**.