An AC current is given by \( i = i_1\cos\omega t + i_2\sin\omega t \). The rms current is given by:
Answer & explanation
Correct answer: option 3
The correct answer is Option (3) → \( \displaystyle\frac{1}{\sqrt{2}}\left(i_1^{2} + i_2^{2}\right)^{\frac{1}{2}} \)
The rms value is the square root of the mean of \( i^{2} \) over one cycle. Squaring the given current:
\( i^{2} = i_1^{2}\cos^{2}\omega t + i_2^{2}\sin^{2}\omega t + 2i_1i_2\sin\omega t\cos\omega t \)
Over a full cycle \( \langle\cos^{2}\omega t\rangle = \langle\sin^{2}\omega t\rangle = \frac{1}{2} \), and \( \langle\sin\omega t\cos\omega t\rangle = \frac{1}{2}\langle\sin2\omega t\rangle = 0 \).
\( \displaystyle \langle i^{2}\rangle = \frac{i_1^{2}}{2} + \frac{i_2^{2}}{2} = \frac{i_1^{2} + i_2^{2}}{2} \)
\( \displaystyle i_{\mathrm{rms}} = \sqrt{\langle i^{2}\rangle} = \frac{1}{\sqrt{2}}\left(i_1^{2} + i_2^{2}\right)^{\frac{1}{2}} \)
Note: A similar question was also asked in CUET (UG) 2026 on 20-05-2026 (Slot 1).