In an $A.C.$ circuit containing $L, C, R$ in series,the ratio of true power to apparent power is ($Z=$ impedance of the circuit and $R$ is the resistance).

$A$ series $LCR$ circuit is connected to an $AC$ source of $220\,V, 50\,Hz$. The circuit contains a resistance $R = 80\,\Omega$,an inductor of inductive reactance $X_L = 70\,\Omega$,and a capacitor of capacitive reactance $X_C = 130\,\Omega$. The power factor of the circuit is $\frac{x}{10}$. The value of $x$ is:

The potential difference $V$ and the current $I$ flowing through an instrument in an $AC$ circuit of frequency $f$ are given by $V = 5 \cos(\omega t)$ volts and $I = 2 \sin(\omega t)$ amperes (where $\omega = 2 \pi f$). The power dissipated in the instrument is......$W$

The coefficient of induction of a choke coil is $0.1\,H$ and resistance is $12\,\Omega$. If it is connected to an alternating current source of frequency $60\,Hz$,then the power factor will be:

Can the instantaneous power output of an $AC$ source ever be negative? Can the average power output be negative?

$(a)$ 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.