Which of the following functions represent `SHM` :- (i) `sin 2omegat` , (ii) `sin omegat + 2cos omegat` , (iii) `sinomegat + cos 2omegat`

A motion will be `S.H.M.` if acceleration `prop - y`
(i) As `y = sin 2omegat rArr v = (dy)/(dt) = 2omegacos 2 omegat`
`rArr` Acceleration `= (d^(2)y)/(dt^(2)) = -4omega^(2) sin 2omegat = - 4omega^(2)y`
so `y = sin 2omegat` represents `S.H.M.`
(ii) `y = sin omegat + 2 cos omegat rArr v = (dy)/(dt) = omegacost - 2omegasint`,
Accleration `= (dv)/(dt) = - omega^(2)sinomegat - 2omega^(2)cosomegat = - omega^(2)(sinomegat + 2cosomegat) = -omega^(2)y`
`:.` The given function represents `SHM`
(iii) `y = sin omegat + cos 2omegat rArr (dy)/(dt) = omegacosomegat - 2omegasin2omegat, (d^(2)y)/(dt^(2)) = - omega^(2)sint -4omega^(2)cos^(2)omegat = - omega^(2)(sinomegat + 4cos^(2)omegat)`
`(d^(2)y)/(dt^(2)) cancelprop (-y)` (Oscillatory nut `S.H.M.` not possible)