A$3.$ Referring to the network shown in Fig. A$3$a$,$ the voltage $v$ and current i are recorded, and their phasor diagrams are shown in Fig. A$3$b$.$ Given that the frequency of the network is $50$ Hz $.$

Fig. A$3$a

Fig. A$3$b

$($a$)$ Express the voltage $v$ in polar form.

$(1$ mark$)$

$($b$)$ Express the current $i$ in time domain form.

$(4$ marks$)$

$($c$)$ Determine the element of impedance Z involved is a capacitor, an inductor or a resistor. Support with a reason.

$(3$ marks$)$