c
(c) \({y_1} = 10\sin \,\left( {3\pi t + \frac{\pi }{3}} \right)\) ...\((i)\)

and \({y_2} = 5[\sin 3\pi t + \sqrt 3 \cos \,3\pi t]\)

\( = 5 \times 2\,\left[ {\frac{1}{2} \times \sin 3\pi t + \frac{{\sqrt 3 }}{2} \times \cos 3\pi t} \right]\)

\( = 10\,\left[ {\cos \frac{\pi }{3}\sin 3\pi t + \sin \frac{\pi }{3}\cos \pi t} \right]\)

\( = 10\,\left[ {\sin \,\left( {3\pi t + \frac{\pi }{t}} \right)} \right]\) ...\( (ii)\)    ( \(\because\)  \(sin(A + B) = sinA \,cosB + cosA sinB)\) 

Comparing equation \( (i)\) and \((ii)\) we get ratio of amplitude \(1 : 1.\)