Equations ${y_1} = A\sin \omega t$ and ${y_2} = \frac{A}{2}\sin \omega t + \frac{A}{2}\cos \omega t$ represent $S.H.M.$ The ratio of the amplitudes of the two motions is
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(d) ${y_2} = \frac{A}{2}\sin \omega \,t + \frac{A}{2}\cos \omega \,t$
${y_2} = \frac{A}{2}(\sin \omega \,t + \cos \omega \,t) = \frac{A}{2} \times \sqrt 2 \;[\sin (\omega \,t + {45^o})]$
${y_2} = \frac{A}{{\sqrt 2 }}\sin (\omega \,t + {45^o})$$ \Rightarrow \frac{{{A_1}}}{{{A_2}}} = \frac{A}{{A/\sqrt 2 }} = \sqrt 2 $
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