$A$ bulb and a capacitor are in series with an $ac$ source. On increasing the frequency,how will the glow of the bulb change?

In a circuit,a metal filament lamp is connected in series with a capacitor of capacitance $C \mu F$ across a $200 V, 50 Hz$ supply. The power consumed by the lamp is $500 W$ while the voltage drop across it is $100 V$. Assume that there is no inductive load in the circuit. Take $rms$ values of the voltages. The magnitude of the phase-angle (in degrees) between the current and the supply voltage is $\varphi$.
Assume,$\pi \sqrt{3} \approx 5$.
$(1)$ The value of $C$ is . . . . . .
$(2)$ The value of $\varphi$ is
Give the answers of the questions $(1)$ and $(2)$:

$A$ circuit when connected to an $A.C.$ source of $12 \; V$ gives a current of $0.2 \; A$. The same circuit when connected to a $D.C.$ source of $12 \; V$,gives a current of $0.4 \; A$. The circuit is

In the figure,if $A$ and $B$ are identical bulbs,which bulb glows brighter?

In an $LCR$ series circuit,the source voltage is $120 \ V$,the voltage across the inductor is $50 \ V$,and the voltage across the resistor is $40 \ V$. Determine the voltage across the capacitor.

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 $\left[\sin 45^{\circ}=\cos 45^{\circ}=\frac{1}{\sqrt{2}}\right]$