In the circuit shown, the ratio of the quality factor $(Q)$ and the band width $(BW)$ is (Given: band width = $R/L$)

$A$ resistor of $100 \Omega$, an inductor of $\frac{25}{\pi^2} \text{ mH}$ and a capacitor of $0.1 \mu\text{F}$ are connected in series to an $AC$ source. The impedance of the circuit is minimum for a frequency of (in $\text{ kHz}$)

Match List-$I$ with List-$II$ :

List-$I$	List-$II$
$(a)$ $\omega L > \frac{1}{\omega C}$	$(i)$ Current is in phase with $emf$
$(b)$ $\omega L = \frac{1}{\omega C}$	$(ii)$ Current lags behind the applied $emf$
$(c)$ $\omega L < \frac{1}{\omega C}$	$(iii)$ Maximum current occurs
$(d)$ Resonant frequency	$(iv)$ Current leads the $emf$

Choose the correct answer from the options given below :

$A$ $10 \, \Omega$ resistor, a $5 \, \text{mH}$ inductor, and a $10 \, \mu\text{F}$ capacitor are connected in series. When a suitable frequency alternating current source is connected to this combination, the circuit resonates. If the resistance is halved, the resonance frequency:

An inductance $L$ having a resistance $R$ is connected to an alternating source of angular frequency $\omega$. The Quality factor $Q$ of the inductance is

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