In a series $LR$ circuit with $X_L = R$, the power factor is $P_1$. If a capacitor of capacitance $C$ with $X_C = X_L$ is added to the circuit, the power factor becomes $P_2$. The ratio of $P_1$ to $P_2$ will be:

In a series $L-C-R$ circuit,$C = 2 \mu F$,$L = 1 \ mH$,and $R = 10 \ \Omega$. What is the ratio of energies stored in the inductor and the capacitor when the maximum current flows in the circuit?

An $AC$ circuit has $R = 100 \, \Omega$,$C = 2 \, \mu F$,and $L = 80 \, mH$ connected in series. The quality factor of the circuit is $.......$

$A$ telegraph line of length $100 \, km$ has a capacity of $0.01 \, \mu F/km$ and it carries an alternating current at $0.5 \, kHz$. If minimum impedance is required, then the value of the inductance that needs to be introduced in series is . . . . . . $mH$. (Take $\pi = \sqrt{10}$)

In an $LCR$ circuit,the inductance is changed from $L$ to $9 L$. For the same resonant frequency,the capacitance should be changed from $C$ to:

$A$ series $LCR$ circuit with $R = 20 \ \Omega$,$L = 1.6 \ \text{H}$ and $C = 40 \ \mu\text{F}$ is connected to a variable frequency a.c. source. The inductive reactance at resonant frequency is . . . . . . $\Omega$.