$A$ resistor of $50 \Omega$,an inductor of self-inductance $(\frac{2}{\pi^2}) \text{ H}$,and a capacitor of unknown capacity are connected in series to an $A$.$C$. source of $100 \text{ V}, 50 \text{ Hz}$. When the voltage and current are in phase,the value of the capacitance is: (in $\mu \text{F}$)

$A$ series $LCR$ circuit is connected to a $45 \sin (\omega t) \text{ V}$ source. The resonant angular frequency of the circuit is $10^5 \text{ rad s}^{-1}$ and current amplitude at resonance is $I_0$. When the angular frequency of the source is $\omega = 8 \times 10^4 \text{ rad s}^{-1}$, the current amplitude in the circuit is $0.05 I_0$. If $L = 50 \text{ mH}$, match each entry in List-$I$ with an appropriate value from List-$II$ and choose the correct option.

List-$I$	List-$II$
$(P)$ $I_0$ in $\text{mA}$	$(1)$ $44.4$
$(Q)$ The quality factor of the circuit	$(2)$ $18$
$(R)$ The bandwidth of the circuit in $\text{rad s}^{-1}$	$(3)$ $400$
$(S)$ The peak power dissipated at resonance in $\text{Watt}$	$(4)$ $2250$
	$(5)$ $500$

The frequencies at which the current amplitude in an $LCR$ series circuit becomes $\frac{1}{\sqrt{2}}$ times its maximum value are $212\,rad\,s^{-1}$ and $232\,rad\,s^{-1}$. The value of resistance in the circuit is $R = 5\,\Omega$. The self-inductance in the circuit is $.........\,mH$.

In an $L-C-R$ circuit,the capacitance is changed from $C$ to $2C$. For the resonant frequency to remain unchanged,the inductance should be changed from $L$ to:

$A$ $110 \; V, 50 \; Hz, AC$ source is connected in the circuit (as shown in figure). The current through the resistance $55 \; \Omega$,at resonance in the circuit,will be $\dots \; A$.

An $AC$ circuit has $R = 100 \, \Omega$,$C = 2 \, \mu F$,and $L = 80 \, mH$ connected in series. The quality factor of the circuit is $.......$