d
(d) The instantaneous values of emf and current in inductive circuit are given by \(E = {E_0}\sin \omega t\) and \(i = {i_0}\sin \left( {\omega t - \frac{\pi }{2}} \right)\)respectively. So, \({P_{inst}} = Ei = {E_0}\sin \omega t \times {i_0}\sin \left( {\omega t - \frac{\pi }{2}} \right)\)

\( = {E_0}{i_0}\sin \omega t\left( {\sin \omega t\cos \frac{\pi }{2} - \cos \omega t\sin \frac{\pi }{2}} \right)\)

\( = {E_0}{i_0}\sin \omega t\;\cos \omega t\)

\( = \frac{1}{2}{E_0}{i_0}\sin 2\omega t\)             \((\sin 2\omega t = 2\sin \omega t\;\cos \omega t)\)Hence, angular frequency of instantaneous power is \(2\omega \).