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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(c)$y = a(\cos \omega \,t + \sin \omega \,t)$$ = a\sqrt 2 \left[ {\frac{1}{{\sqrt 2 }}\cos \omega \,t + \frac{1}{{\sqrt 2 }}\sin \omega \,t} \right]$
$ = a\sqrt 2 [\sin 45^\circ \cos \omega \,t + \cos 45^\circ \sin \omega \,t]$
$ = a\sqrt 2 \sin (\omega \,t + 45^\circ )$
$ = a\sqrt 2 [\sin 45^\circ \cos \omega \,t + \cos 45^\circ \sin \omega \,t]$
$ = a\sqrt 2 \sin (\omega \,t + 45^\circ )$
==> Amplitude $ = a\sqrt 2 $
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