$A$ series $LCR$ circuit with $L = \frac{100}{\pi} \text{ mH}$,$C = \frac{10^{-3}}{\pi} \text{ F}$,and $R = 10 \ \Omega$ is connected across an $AC$ source of $220 \text{ V}, 50 \text{ Hz}$ supply. The power factor of the circuit would be . . . . . . .

The frequency of $AC$ at which a $16 \mu F$ capacitor and a $\frac{10}{\pi^2} \ mH$ inductor will have the same reactance is: (in $kHz$)

In a series $LR$ circuit,$X_{L} = R$ and the power factor of the circuit is $P_{1}$. When a capacitor with capacitance $C$ such that $X_{L} = X_{C}$ is put in series,the power factor becomes $P_{2}$. The ratio $\frac{P_{1}}{P_{2}}$ is

In an $LCR$ circuit,the resonating frequency is $500 \,kHz$. If the value of $L$ is doubled and the value of $C$ is decreased to $\frac{1}{8}$ times its initial value,then the new resonating frequency in $kHz$ will be .......

In the adjoining $AC$ circuit,the voltmeter whose reading will be zero at resonance is

$A$ resistor of $100 \text{ } \Omega$, an inductor of self-inductance $(4/\pi^2) \text{ H}$, and a capacitor of unknown capacity are connected in series to an a.c. source of $200 \text{ V}$ and $50 \text{ Hz}$. When the current and voltage are in phase, the value of the capacity is: (in $\text{ } \mu\text{F}$)