d
(d) Let simple harmonic motions be represented by

\({y_1} = a\sin \,\left( {\omega \,t - \frac{\pi }{4}} \right)\); \({y_2} = a\sin \omega \,t\) and
\({y_3} = a\sin \,\left( {\omega \,t + \frac{\pi }{4}} \right)\).

On superimposing, resultant SHM will be \(y = a\;\left[ {\sin \,\left( {\omega \,t - \frac{\pi }{4}} \right) + \sin \omega \,t + \sin \,\left( {\omega \,t + \frac{\pi }{4}} \right)} \right]\)

\( = a\;\left[ {2\sin \omega \,t\cos \frac{\pi }{4} + \sin \omega \,t} \right]\)

\( = a\;[\sqrt 2 \sin \omega t + \sin \omega t] = a\;(1 + \sqrt 2 )\sin \omega \,t\)

Resultant amplitude =\((1 + \sqrt 2 )a\)

Energy is \(S.H.M.\) \(\propto\) (Amplitude)\(^2\)

 \(\frac{{{E_{{\rm{Resultant}}}}}}{{{E_{{\rm{Single}}}}}} = {\left( {\frac{A}{a}} \right)^2} = {(\sqrt 2 + 1)^2} = (3 + 2\sqrt 2 )\)

==> \({E_{{\rm{Resultant}}}} = (3 + 2\sqrt 2 ){E_{{\rm{Single}}}}\)