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).

$A$ group of electric lamps having a total power rating of $1000 \text{ W}$ is supplied by an $AC$ voltage $E = 200 \sin(310t + 60^\circ)$. Then the $rms$ value of the circuit current is

Power dissipated in an $LCR$ series circuit connected to an $A.C.$ source of $e.m.f.$ $\varepsilon$ is

$A$ $LCR$ series circuit driven with $V_{rms} = 90 \text{ V}$ at frequency $f_d = 30 \text{ Hz}$ has resistance $R = 80 \text{ }\Omega$, an inductance with inductive reactance $X_L = 20.0 \text{ }\Omega$ and capacitance with capacitive reactance $X_C = 80.0 \text{ }\Omega$. The power factor of the circuit is . . . . . . .

The potential differences across the resistance,capacitance,and inductance are $80 \, V$,$40 \, V$,and $100 \, V$ respectively in an $L-C-R$ circuit. The power factor of this circuit is:

Power factor of an ideal choke coil (i.e.,$R=0$) is