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(C) Given the equation of motion: $y = a(\sin \omega \,t + \cos \omega \,t)$.We can rewrite this by multiplying and dividing by $\sqrt{2}$:$y = a\sqrt

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The motion is $S.H.M.$ with amplitude $a\sqrt{2}$

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(C) Given the equation of motion: $y = a(\sin \omega \,t + \cos \omega \,t)$.We can rewrite this by multiplying and dividing by $\sqrt{2}$:$y = a\sqrt{2} \left( \frac{1}{\sqrt{2}} \sin \omega \,t + \frac{1}{\sqrt{2}} \cos \omega \,t \right)$.Using the trigonometric identity $\sin(A+B) = \sin A \cos B + \cos A \sin B$,where $\sin 45^\circ = \frac{1}{\sqrt{2}}$ and $\cos 45^\circ = \frac{1}{\sqrt{2}}$:$y = a\sqrt{2} (\sin \omega \,t \cos 45^\circ + \cos \omega \,t \sin 45^\circ)$.$y = a\sqrt{2} \sin(\omega \,t + 45^\circ)$.This is the standard equation of Simple Harmonic Motion $(S.H.M.)$,which is $y = A \sin(\omega \,t + \phi)$,where the amplitude $A = a\sqrt{2}$.

The motion of a particle varies with time according to the relation $y = a(\sin \omega \,t + \cos \omega \,t)$,then

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