The resonant frequency of a circuit is $f$. If the capacitance is made $4$ times the initial value,then the resonant frequency will become

Suppose the frequency of the source in the previous example can be varied.
$(a)$ What is the frequency of the source at which resonance occurs?
$(b)$ Calculate the impedance,the current,and the power dissipated at the resonant condition.

For a given series $LCR$ circuit,it is found that maximum current is drawn when the value of the variable capacitance is $2.5 \ nF$. If a resistance of $200 \ \Omega$ and a $100 \ mH$ inductor are being used in the given circuit,the frequency of the $AC$ source is $... \times 10^3 \ Hz$. (Given $\pi^2 = 10$)

In a series $LR$ circuit with $X_L = R$, the 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:

$A$ series combination of a resistor of resistance $100\,\Omega$,an inductor of inductance $1\,H$,and a capacitor of capacitance $6.25\,\mu F$ is connected to an $AC$ source. The quality factor of the circuit will be $.............$.

The output voltage (taken across the resistance) of an $L-C-R$ series resonant circuit falls to half its peak value at a frequency of $200 \,Hz$ and again reaches the same value at $800 \,Hz$. The bandwidth of this circuit is ............. $\,Hz$.