In a series $LCR$ circuit,$R = 200 \, \Omega$ and the voltage and frequency of the main supply are $220 \, V$ and $50 \, Hz$ respectively. On taking out the capacitance from the circuit,the current lags behind the voltage by $30^\circ$. On taking out the inductor from the circuit,the current leads the voltage by $30^\circ$. The power dissipated in the $LCR$ circuit is......$W$.

Given below are two statements:
Statement-$I$: The reactance of an $ac$ circuit is zero. It is possible that the circuit contains a capacitor and an inductor.
Statement-$II$: In an $ac$ circuit, the average power delivered by the source never becomes zero.
In the light of the above statements, choose the correct answer from the options given below:

If $C, R, L$ and $I$ denote capacity, resistance, inductance and electric current respectively, the quantities having the same dimensions of time are :
$(1)$ $C R$
$(2)$ $\frac{L}{R}$
$(3)$ $\sqrt{L C}$
$(4)$ $L I^2$

In a series $LR$ circuit with $X_L = R$. Power factor is $P_1$. If a capacitor of capacitance $C$ with $X_c = X_L$ is added to the circuit the power factor becomes $P_2$. The ratio of $P_1$ to $P_2$ will be :

$A$ circuit with an electrical load having impedance $Z$ is connected with an $AC$ source as shown in the diagram. The source voltage varies in time as $V(t) = 300 \sin (400 t) \text{ V}$,where $t$ is time in seconds. List-$I$ shows various options for the load. The possible currents $i(t)$ in the circuit as a function of time are given in List-$II$. Choose the option that describes the correct match between the entries in List-$I$ to those in List-$II$.

List-$I$	List-$II$
$(P)$ Resistor $R = 30 \ \Omega$	$(1)$ $i(t) = 5 \sin(400t)$
$(Q)$ Resistor $R = 30 \ \Omega$ and Inductor $L = 100 \text{ mH}$	$(2)$ $i(t) = 6 \sin(400t + 53^{\circ})$
$(R)$ Capacitor $C = 50 \ \mu\text{F}$,Resistor $R = 30 \ \Omega$,and Inductor $L = 25 \text{ mH}$	$(3)$ $i(t) = 10 \sin(400t)$
$(S)$ Capacitor $C = 50 \ \mu\text{F}$,Resistor $R = 60 \ \Omega$,and Inductor $L = 125 \text{ mH}$	$(4)$ $i(t) = 20 \sin(400t - 90^{\circ})$
	$(5)$ $i(t) = 6 \sin(400t - 53^{\circ})$

$A$ $ 100 \,W $ bulb is connected to an $ AC $ source of $ 220 \,V, 50 \,Hz $. Then the current flowing through the bulb is