GATE 2025 EE Q.30: solution and answer

Official answer: 7Source: Official answer key, GATE 2025 EE (IIT Roorkee)

1. Period. The cosines have frequencies $1,2,3\text{Hz}$ ($\omega =2\pi ,4\pi ,6\pi \text{rad/s}$). The fundamental frequency is their GCD, $f_0=1\text{Hz}$, so $T=1\text{s}$ and the three cosines are the 1st, 2nd and 3rd harmonics.

2. What is asked. $\frac{1}{T}{\int }_0^T|x(t){|}^{2}dt$ is the average power $P$ of the periodic signal. Since $x(t)$ is real, $|x(t){|}^2=x^2(t)$.

3. Parseval's theorem (trigonometric form). For $x(t)=a_0+{\sum }_k{A}_k\cos (k{\omega }_{0}t+{\varphi }_k)$, the DC term and the harmonics are mutually orthogonal over one period, so every cross term integrates to zero and $$P=a_0^2+\sum _k\frac{A_k^2}{2}.$$

4. DC term: $a_0=1$, contributing $1^2=1$.

5. Each harmonic: $A_k=2$, contributing $\frac{2^2}{2}=2$; the three harmonics give $3\times 2=6$.

6. Total: $$P=1+2+2+2=7.$$

7. Check with exponential coefficients: $2\cos k{\omega }_{0}t=e^{jk{\omega }_{0}t}+e^{-jk{\omega }_{0}t}$, so $c_0=1$ and $c_{\pm 1}=c_{\pm 2}=c_{\pm 3}=1$. Then $P={\sum }_k|c_k{|}^2=1+6\times 1=7$.