$A$ resistance $R$ draws power $P$ when connected to an $AC$ source. If an inductance is now placed in series with the resistance,such that the impedance of the circuit becomes $Z$,the power drawn will be

Alternating current is flowing in an $LR$ circuit with inductance $L$ and resistance $R$. The frequency of the source is $f = \omega / 2\pi$. Which of the following statements is correct?

In an $RL$ circuit,the resistance is $\pi \sqrt{3} \,\Omega$. The phase difference between the current and the voltage is $30^\circ$. The $ac$ frequency is $50 \,Hz$. The inductance is . . . . . . $Henry$.

In a circuit,a metal filament lamp is connected in series with a capacitor of capacitance $C \mu F$ across a $200 V, 50 Hz$ supply. The power consumed by the lamp is $500 W$ while the voltage drop across it is $100 V$. Assume that there is no inductive load in the circuit. Take $rms$ values of the voltages. The magnitude of the phase-angle (in degrees) between the current and the supply voltage is $\varphi$.
Assume,$\pi \sqrt{3} \approx 5$.
$(1)$ The value of $C$ is . . . . . .
$(2)$ The value of $\varphi$ is
Give the answers of the questions $(1)$ and $(2)$:

The power factor of an $R-L$ circuit is $\frac{1}{\sqrt{2}}$. If the frequency of $AC$ is doubled,the power factor will now be

For the series $LCR$ circuit,$R = \frac{X_L}{2} = 2 X_C$. The impedance of the circuit and the phase difference between $V$ and $I$ will be