In a series $LCR$ circuit,$R = 18 \ \Omega$ and the impedance $Z = 33 \ \Omega$. An $RMS$ voltage of $220 \ V$ is applied across the circuit. The true power consumed in the $AC$ circuit is: (in $W$)

For an $AC$ circuit $V = 15 \sin \omega t$ and $I = 20 \cos \omega t$,the average power consumed in this circuit is ..... $W$.

When a voltage $V = V_0 \cos \omega t$ is applied across a resistor of resistance $R$,the average power dissipated per cycle in the resistor is given by

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

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

$(a)$ For circuits used for transporting electric power, a low power factor implies large power loss in transmission. Explain.
$(b)$ Power factor can often be improved by the use of a capacitor of appropriate capacitance in the circuit. Explain.