In domestic electric mains supply,the voltage and the current are

In an $LCR$ circuit $R = 100 \ \Omega$. When capacitance $C$ is removed,the current lags behind the voltage by $\pi /3$. When inductance $L$ is removed,the current leads the voltage by $\pi /3$. The impedance of the circuit is......$\Omega$.

In the circuit shown,$L = 1 \mu H$,$C = 1 \mu F$,and $R = 1 k\Omega$. They are connected in series with an $a.c.$ source $V = V_0 \sin \omega t$ as shown. Which of the following options is/are correct?
[$A$] The frequency at which the current will be in phase with the voltage is independent of $R$.
[$B$] At $\omega \sim 0$,the current flowing through the circuit becomes nearly zero.
[$C$] At $\omega \gg 10^6 \text{ rad } s^{-1}$,the circuit behaves like a capacitor.
[$D$] The current will be in phase with the voltage if $\omega = 10^6 \text{ rad } s^{-1}$.

The figure shows a system of an inductor and a parallel plate capacitor made of $2$ parallel circular plates of area $A$,filled with a dielectric liquid of dielectric constant $K$. $A$ small leak develops in the capacitor,and the liquid starts to fill the inductor of the same dimensions having $n$ turns per unit length. Find the ratio of the magnitude of the initial reactance to the final reactance of the circuit after the liquid fills the inductor completely.
Given: $\omega^2 A^2 n^2 = c^2$
$\omega \rightarrow$ angular frequency of $AC$
$c \rightarrow$ speed of light
$\mu_r \rightarrow$ relative permeability of the liquid

Find the peak current and resonant frequency of the following circuit (as shown in the figure).

In the circuit shown in the figure, the source frequency is $\omega = 2000 \, rad/s$. The current in the circuit will be: