A particle moves such that its position from origin is given by (...

1. Analyzing the Statements: - (A): Particle performs S.H.M. of amplitude $3, m$ and frequency ($\frac{\omega}{\pi}$). - False: The amplitude is actually ($\frac{3}{2},$m) (not $3, m$), and frequency is derived from (\cos (2 $\omega$t)) which would be ($\frac{2 \omega}{2 \pi}=\frac{\omega}{\pi}$), but the amplitude is incorrect. (B): Particle performs oscillations but no S.H.M. - False: The particle does perform oscillatory motion about a mean position with a periodic nature (but not simple harmonic due to the (\sin ^2) term). (C): Particle performs S.H.M. about mean position at $x=7, m$. - False: The mean position is at (x=$\frac{7}{2},$m(\operatorname{not} 7, m)). (D): Particle performs oscillations of amplitude 3, $m$. - False: The amplitude is ($\frac{3}{2},$m) (not $3, m$).