GATE 2026 EE Q.42: solution and answer

GATE 2026 · EEQ.42 · 2 marks · MCQ

An electrical component has voltage drop $v = V_{m} sin (ω t)$ , when the current through it is $i = I_{m} sin (ω t - θ)$ .

What is the average power dissipated over a half cycle corresponding to $ω$ ?

(A)
$0$
(B)
$V_{m} I_{m} cos θ$
(C)
$\frac{V _{m} I _{m}}{2} cos θ$
(D)
$\frac{V _{m} I _{m}}{4} cos θ$

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Answer and solution

ANSWER

The average power over a half cycle is $\frac{V _{m} I _{m}}{2} cos θ$ , which is option (C).

Official answer: CSource: Official answer key, GATE 2026 EE (IIT Guwahati)

Instantaneous power: $p (t) = v i = V_{m} I_{m} sin (ω t) sin (ω t - θ)$ .
Apply the product-to-sum identity $sin A sin B = \frac{1}{2} [cos (A - B) - cos (A + B)]$ with $A = ω t$ and $B = ω t - θ$ , so $A - B = θ$ and $A + B = 2 ω t - θ$ : $p (t) = \frac{V _{m} I _{m}}{2} [cos θ - cos (2 ω t - θ)]$
So $p (t)$ is a constant $\frac{V _{m} I _{m}}{2} cos θ$ plus a term that oscillates at $2 ω$ . The period of that term is $\frac{2 π}{2 ω} = \frac{π}{ω}$ .
A half cycle of $ω$ lasts $\frac{T}{2} = \frac{π}{ω}$ , which is exactly one full period of the $2 ω$ term. That term therefore averages to zero over the half cycle, wherever the half cycle starts.
Average power: $P_{avg} = \frac{ω}{π} \int_{0}^{π / ω} p (t) d t = \frac{V _{m} I _{m}}{2} cos θ + 0 = \frac{V _{m} I _{m}}{2} cos θ$
Check: this is the familiar $V_{rms} I_{rms} cos θ$ with $V_{rms} = \frac{V _{m}}{2}$ and $I_{rms} = \frac{I _{m}}{2}$ . The answer is option (C).

KEY IDEA

For sinusoidal $v$ and $i$ at the same frequency, $p (t)$ is a constant $\frac{V _{m} I _{m}}{2} cos θ$ plus a ripple at $2 ω$ . The ripple's period is $\frac{π}{ω}$ , exactly one half cycle of $ω$ . So averaging over a half cycle gives the same result as averaging over a full cycle: $P = V_{rms} I_{rms} cos θ$ .

WATCH OUT FOR

Three traps. Picking (B) means dropping the $\frac{1}{2}$ from the product-to-sum identity, which is the same as treating peak values as rms. Picking (A) means assuming anything sinusoidal averages to zero over a half cycle; $v$ and $i$ on their own do, but their product carries a constant $cos θ$ term. Picking (D) means thinking a half cycle halves the average. It does not, because $p (t)$ repeats every half cycle.

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GATE 2026 EE Q.42: solution and answer

Answer and solution

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