If an ac main supply is given to be $220 V$. What would be the average e.m.f. during a positive half cycle.....$V$

The power is transmitted from a power house on high voltage $ac$ because

The voltage of an $ac$ source varies with time according to the equation $V = 100\sin \,\left( {100\pi t} \right)\cos \,\left( {100\pi t} \right)$ where $t$ is in seconds and $V$ is in volts. Then

The direct current which would give the same heating effect in an equal constant resistance as the current shown in figure, i.e. the $r.m.s.$ current, is.....$A$

The average value of potential difference $V$ shown in figure is

Three alternating voltage sources $V_1$ = $3 sin \omega t $ volt , $V_2= 5  sin(\omega t + \phi _1)$ volt and $V_3 = 5  sin(\omega t -\phi_2 )$ volt connected across a resistance $R= \sqrt {\frac{7}{3}} \Omega $ as shown in the figure (where $ \phi_1$  and $ \phi_2$   corresponds to $30^o $ and $127^o $ respectively). Find the peak current (in Amp) through the resistor