$A$ circuit has a resistance of $12 \; \Omega$ and an impedance of $15 \; \Omega$. The power factor of the circuit will be

$A$ series $LCR$ circuit is connected to an a.c. source of $230 \ V, 50 \ Hz$. The circuit contains a resistance of $80 \ \Omega$,an inductor having inductive reactance $70 \ \Omega$,and a capacitor of capacitive reactance $130 \ \Omega$. The power factor of the circuit is $x$. The value of $x$ is

An inductor and a resistor of $25 \Omega$ are connected in series to an ac source of voltage $100 \sin (100 \pi t) \ V$. If the impedance of the circuit is $50 \Omega$,the average power dissipated per cycle in the circuit is: (in $W$)

In an $LCR$ series circuit,an alternating e.m.f. '$e$' and current '$i$' are given by the equations $e = 160 \sin(100t) \text{ V}$ and $i = 250 \sin(100t + \frac{\pi}{3}) \text{ mA}$. The average power dissipated in the circuit is: (in $\text{ W}$)

$(a)$ 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.

Current in the circuit is wattless, if