Questions are transcribed from NTA's official question paper. Diagrams are described in words; open the official PDF to see them. Every answer was solved twice independently and is shown only when both agree. Where they did not, we say so.
The figure given below shows an $LCR$ series circuit with two switches $S_1$ and $S_2$. When switch $S_1$ is closed keeping $S_2$ open, the phase difference ($\phi$) between the current and source voltage is $30°$ and phase difference is $60°$ when $S_2$ is closed keeping $S_1$ open. The value of $(3L_1 - L_2)$ is ______ H. [Figure: An LCR circuit with $C = 100\mu F$, $R$, two inductors $L_1$ and $L_2$ with switches $S_1$ and $S_2$, and voltage source $v = V_0 \sin(300t)$]
(1) $\frac{9}{2}$
(2) $\frac{2}{9}$
(3) $\frac{1}{3}$
(4)
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Answer: Option (2)
Key idea: Phase difference in LCR circuits and impedance analysis
An inductor of 10 mH, capacitor of 0.1 $\mu$F and a resistor of 100 $\Omega$ are connected in series across an $a.c$ power supply 220 V, 70 Hz. The power factor of the given circuit is 0.5. The difference in the inductive reactance and capacitance reactance is $\sqrt{3}$ u $\Omega$. The value of u is ______.
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Answer: 100
Key idea: Power factor in RLC series circuit and impedance calculation
An a.c. source of angular frequency $\omega$ is connected across a resistor $R$ and a capacitor $C$ in series. The current is observed as $I$. Now the frequency of the source is changed to $\omega/4$, (keeping the voltage unchanged) the current is found to be $I/3$. The ratio of resistance to reactance at frequency $\omega$ is
(1) $\sqrt{\frac{6}{7}}$
(2) $\sqrt{\frac{3}{5}}$
(3) $\sqrt{\frac{7}{8}}$
(4)
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Answer: Option (3)
Key idea: RC Circuit Impedance and Frequency Dependence
For RC series circuit: $Z = \sqrt{R^2 + X_C^2}$ where $X_C = \frac{1}{\omega C}$
At frequency $\omega$: $V = I\sqrt{R^2 + \frac{1}{\omega^2 C^2}}$
At frequency $\omega/4$: Reactance becomes $X_C' = 4X_C$, so $V = \frac{I}{3}\sqrt{R^2 + \frac{16}{\omega^2 C^2}}$
Since voltage is constant: $\sqrt{R^2 + \frac{1}{\omega^2 C^2}} = \frac{1}{3}\sqrt{R^2 + \frac{16}{\omega^2 C^2}}$
A series LCR circuit with $R = 20$ $\Omega$, $L = 1.6$ H and $C = 40$ $\mu$F is connected to a variable frequency a.c. source. The inductive reactance at resonant frequency is __________ $\Omega$.
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Answer: 200
Key idea: Resonance in LCR circuits - At resonance, inductive reactance equals capacitive reactance
A LCR series circuit driven with $E_{rms} = 90$ V at frequency $f_d = 30$ Hz has resistance $R = 80$ $\Omega$, an inductance with inductive reactance $X_L = 20.0$ $\Omega$ and capacitance with capacitive reactance $X_C = 80.0$ $\Omega$. The power factor of the circuit is_____.
(1) 0.8
(2) 0.54
(3) 0.9
(4) 0.5
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Answer: Option (1)
Key idea: Power factor in LCR series circuit
Identify the given values: $R = 80\,\Omega$, $X_L = 20\,\Omega$, $X_C = 80\,\Omega$
Calculate the net reactance: $X_L - X_C = 20 - 80 = -60\,\Omega$
