For an $L-C-R$ $AC$ circuit,the resonance frequency is $5000 \ Hz$ and the frequencies at half-power points are $4950 \ Hz$ and $5050 \ Hz$. What will be the $Q$-factor?

$A$ resistor of $100 \Omega$,an inductor of self-inductance $(\frac{4}{\pi^2}) \text{ H}$,and a capacitor of unknown capacitance are connected in series to an $A$.$C$. source of $200 \text{ V}, 50 \text{ Hz}$. When the current and voltage are in phase,the capacitance and the power dissipated are respectively:

$A$ series $L-C-R$ circuit containing a resistance of $120 \Omega$ has an angular frequency of $4 \times 10^5 \ rad \ s^{-1}$. At resonance,the voltage across the resistance and the inductor are $60 \ V$ and $40 \ V$ respectively. The value of the inductance is: (in $mH$)

$A$ series $LCR$ circuit with $L=0.5 \text{ H}$ and $R=10 \Omega$ is connected to an $AC$ supply with $rms$ voltage and frequency equal to $200 \text{ V}$ and $\frac{150}{\pi} \text{ Hz}$, respectively. The magnitude of the capacitance is varied so that the current amplitude in the circuit becomes maximum. The $rms$ voltage difference across the inductor is (in $\text{ V}$)

Given below are two statements:
Statement $I$: Maximum power is dissipated in a circuit containing an inductor, a capacitor, and a resistor connected in series with an $AC$ source, when resonance occurs.
Statement $II$: Maximum power is dissipated in a circuit containing a pure resistor due to zero phase difference between current and voltage.
In the light of the above statements, choose the correct answer from the options given below:

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: