In an $LCR$ circuit,the resonating frequency is $500 \,kHz$. If the value of $L$ is doubled and the value of $C$ is decreased to $\frac{1}{8}$ times its initial value,then the new resonating frequency in $kHz$ will be .......

$A$ 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 $mH$)

In a series $LCR$ circuit,$C = 2 \mu F$,$L = 1 \text{ mH}$,and $R = 10 \Omega$. The ratio of the energies stored in the inductor and the capacitor,when the maximum current flows in the circuit,is:

$A$ resistor of $50 \Omega$,an inductor of self-inductance $\left(\frac{3}{\pi^2}\right) H$,and a capacitor of unknown capacity are connected in series to an a.c. source of $100 \ V$ and $50 \ Hz$. When the voltage and current are in phase,the value of capacitance is (nearly):

In an $L-C-R$ series circuit,the value of only capacitance $C$ is varied. The resulting variation of resonance frequency $f_0$ as a function of $C$ can be represented as

An inductor of inductance $2\,\mu\text{H}$ is connected in series with a resistance,a variable capacitor,and an $AC$ source of frequency $7\,\text{kHz}$. The value of capacitance for which maximum current is drawn into the circuit is $\frac{1}{x}\text{ F}$,where the value of $x$ is $.........$. (Take $\pi = \frac{22}{7}$)