If the power factor changes from $0.5$ to $0.25$ because impedance changes from $Z_1$ to $Z_2$, then $Z_1 = x Z_2$. The value of $x$ is (Resistance remains constant).

An e.m.f. $e=E_0 \cos \omega t$ is applied to a circuit containing $L, C$ and $R$ in series where $X_L=3 R$ and $X_C=R$. The average power dissipated in the circuit is

An alternating voltage of $V = V_{0} \sin \omega t$ is applied across a circuit. As a result, a current $I = I_{0} \sin (\omega t - \frac{\pi}{2})$ flows in it. The power consumed per cycle is . . . . . . .

The power factor of an $AC$ circuit containing peak current $2 \ A$ and peak voltage $1 \ V$ is $1/2$. The angle between voltage and current is: (in $^{\circ}$)

The instantaneous values of current and potential difference in an alternating circuit are $i = \sin \omega t$ and $E = 100 \cos \omega t$ respectively. The $r.m.s.$ value of the wattless current (in $A$) in the circuit is:

An $AC$ voltage $V=20 \sin 200 \pi t$ is applied to a series $LCR$ circuit which drives a current $I=10 \sin (200 \pi t + \frac{\pi}{3})$. The average power dissipated is: (in $W$)