GATE 2026 IN, question 33: AC, power & resonance — solution and answer

Official key: 25 to 25 · Official answer key, GATE 2026 IN (IIT Guwahati)

1. Series circuit, one current. R and L carry the same current, so the current through the resistor is the circuit current. The impedance is $Z=R+j\omega L$.

2. Phase angle of a series RL circuit. With the source voltage as reference, $I=\frac{V}{Z}$, so the current lags the source voltage by the angle of $Z$: $$\varphi ={\tan }^{-1}\left(\frac{\omega L}{R}\right)$$

3. Use the 45° condition. $\varphi =45^{\circ }$ gives $\tan 45^{\circ }=1$, so $X_L=R$: $$\omega L=R\Rightarrow \omega =\frac{R}{L}=\frac{100\pi }{2}=50\pi \text{rad/s}$$

4. Convert angular frequency to hertz. $$f=\frac{\omega }{2\pi }=\frac{50\pi }{2\pi }=25\text{Hz}$$

5. Check. At $f=25\text{Hz}$, $X_L=2\pi (25)(2)=100\pi \Omega =R$. So $Z=100\pi \sqrt{2}\angle 45^{\circ }\Omega$ and the current lags the voltage by $45^{\circ }$, as the question says. The answer is exactly 25, so no rounding is needed.