For a domestic AC supply of $220 \,V$ at $50 \,cps$, the potential difference between the terminals of a two-pin electric outlet in a room is (in volt) given by

The frequency of $ac$ mains in India is.......$Hz$

An alternating voltage is given by : $e = e_1\, \sin \omega t + e_2\, \cos \omega t$. Then  the root mean square value of voltage is given by :-

A resistance of $40 \,\Omega$ is connected to a source of alternating current rated $220\, V , 50 Hz$. Find the time taken by the current to change from its maximum value to $ms$ value

The instantaneous voltages at three terminals marked $\mathrm{X}, \mathrm{Y}$ and $\mathrm{Z}$ are given by

$V_x=V_0 \sin \omega t$ $V_y=V_0 \sin \left(\omega t+\frac{2 \pi}{3}\right) \text { and }$ $V_z=V_0 \sin \left(\omega t+\frac{4 \pi}{3}\right)$

An ideal voltmeter is configured to read rms value of the potential difference between its terminals. It is connected between points $\mathrm{X}$ and $\mathrm{Y}$ and then between $\mathrm{Y}$ and $\mathrm{Z}$. The reading(s) of the voltmeter will be

$[A]$ $\mathrm{V}_{\mathrm{XY}}^{\mathrm{mms}}=\mathrm{V}_0 \sqrt{\frac{3}{2}}$

$[B]$ $\mathrm{V}_{\mathrm{YZ}}^{\mathrm{mms}}=\mathrm{V}_0 \sqrt{\frac{1}{2}}$

$[C]$ $\mathrm{V}_{\mathrm{XY}}^{\mathrm{mms}}=\mathrm{V}_0$

$[D]$ independent of the choice of the two terminals

Find the time required for $50\,Hz$ alternating current to change its value from zero to maximum value.