When $100\, V$ $DC$ is applied across a solenoid,a current of $1\, A$ flows in it. When $100\, V$ $AC$ is applied across the same coil,the current drops to $0.5\, A$. If the frequency of the $AC$ source is $50\, Hz$,the impedance and inductance of the solenoid are:

$A$ circuit containing an $80 \, mH$ inductor and a $60 \, \mu F$ capacitor in series is connected to a $230 \, V, 50 \, Hz$ supply. The resistance of the circuit is negligible.
$(a)$ Obtain the current amplitude and $rms$ values.
$(b)$ Obtain the $rms$ values of potential drops across each element.
$(c)$ What is the average power transferred to the inductor?
$(d)$ What is the average power transferred to the capacitor?
$(e)$ What is the total average power absorbed by the circuit? ('Average' implies 'averaged over one cycle'.)

An inductance coil has a resistance of $100 \Omega$. When an $A.C.$ signal of frequency $100 \ Hz$ is applied to the coil,the voltage leads the current by $45^{\circ}$. The inductance of the coil in henry is $\left[\sin 45^{\circ}=\cos 45^{\circ}=\frac{1}{\sqrt{2}}\right]$

$A$ lamp is connected in series with a capacitor and an $AC$ source. What happens if the capacitance of the capacitor is reduced?

$A$ $10\, V, 60\, W$ bulb is to be connected to a $100\, V, 50\, Hz$ $AC$ source. To operate the bulb at its rated power,an induction coil is connected in series with it. Calculate the required self-inductance of the coil.

$A$ resistor of $450 \Omega$ and an inductor are connected in series to an ac source of frequency $\frac{75}{\pi} \text{ Hz}$. If the power factor of the circuit is $0.6$,then the inductance connected in the circuit is: