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:

The inductance $L$,capacitance $C$,and resistance $R$ are the values of the components connected in series to an $AC$ source of angular frequency $\omega$. The inductive and capacitive reactances are $X_L$ and $X_C$ respectively. If the circuit is purely resistive,then

$A$ series $LCR$ circuit with $L = \frac{100}{\pi} \text{ mH}$,$C = \frac{10^{-3}}{\pi} \text{ F}$,and $R = 10 \ \Omega$ is connected across an $AC$ source of $220 \text{ V}, 50 \text{ Hz}$ supply. The power factor of the circuit would be . . . . . . .

$A$ 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 $LCR$ circuit contains inductance $5 \ mH$,capacitance $2 \ \mu F$,and resistance $10 \ \Omega$. If the frequency of the $A.C.$ source is varied,what is the frequency at which maximum power is dissipated?

$A$ series $LCR$ circuit is connected across a source of alternating emf of changing frequency and resonates at frequency $f_0$. Keeping capacitance constant,if the inductance $(L)$ is increased by $\sqrt{3}$ times and resistance is increased $(R)$ by $1.4$ times,the resonant frequency now is