GATE 2026 IN Q.19: solution and answer

Official answer: DSource: Official answer key, GATE 2026 IN (IIT Guwahati)

1. Find the period of each sinusoid with $T=\frac{2\pi }{\omega }$. For $4\sin (15\pi t)$, ${\omega }_1=15\pi$, so $T_1=\frac{2\pi }{15\pi }=\frac{2}{15}\text{s}$. For $7\cos (4\pi t)$, ${\omega }_2=4\pi$, so $T_2=\frac{2\pi }{4\pi }=\frac{1}{2}\text{s}$.

2. Check that the sum is periodic: $\frac{T_1}{T_2}=\frac{2/15}{1/2}=\frac{4}{15}$. This is rational, so $x(t)$ is periodic.

3. The fundamental period is the smallest $T$ that is a whole-number multiple of both, $T=m{T}_1=n{T}_2$. This needs $\frac{T_1}{T_2}=\frac{n}{m}=\frac{4}{15}$. The fraction $\frac{4}{15}$ is already in lowest terms (4 and 15 share no common factor), so the smallest choice is $m=15$, $n=4$.

4. $$T=15\times \frac{2}{15}=4\times \frac{1}{2}=2\text{s}$$

5. Check using frequencies: $f_1=7.5\text{Hz}$ and $f_2=2\text{Hz}$. Both are whole-number multiples of $0.5\text{Hz}$ ($7.5=15\times 0.5$, $2=4\times 0.5$), and $15$ and $4$ share no common factor, so the fundamental frequency is their GCD, $f_0=0.5\text{Hz}$. Then $T=1/f_0=2\text{s}$. This is option (D).