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(C) Given the equation of motion: $y = a(\sin \omega t + \cos \omega t)$.Multiply and divide by $\sqrt{2}$:$y = a\sqrt{2} \left( \frac{1}{\sqrt{2}} \s

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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: $y = a(\sin \omega t + \cos \omega t)$.Multiply and divide 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 $\sin(A + B) = \sin A \cos B + \cos A \sin B$,we can write:$y = a\sqrt{2} (\sin \omega t \cos \frac{\pi}{4} + \cos \omega t \sin \frac{\pi}{4})$.$y = a\sqrt{2} \sin(\omega t + \frac{\pi}{4})$.This equation represents a Simple Harmonic Motion $(S.H.M.)$ of the form $y = A \sin(\omega t + \phi)$,where the amplitude $A = a\sqrt{2}$.

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

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