Study of which motion is required to understand many physical phenomena?

The displacement of a particle along the $x$-axis is given by $x = a \sin^2 \omega t$. The motion of the particle corresponds to:

What is constant in $S.H.M.$?

Which of the following functions of time represent $(a)$ simple harmonic,$(b)$ periodic but not simple harmonic,and $(c)$ non-periodic motion? Give period for each case of periodic motion ($\omega$ is any positive constant):
$(a)$ $\sin \omega t - \cos \omega t$
$(b)$ $\sin^3 \omega t$
$(c)$ $3 \cos (\pi/4 - 2 \omega t)$
$(d)$ $\cos \omega t + \cos 3 \omega t + \cos 5 \omega t$
$(e)$ $\exp(-\omega^2 t^2)$
$(f)$ $1 + \omega t + \omega^2 t^2$

Which of the following equations represents a simple harmonic motion? $(\omega$ is angular frequency, $A$ is amplitude of oscillation and $i = \sqrt{-1})$

$A$ particle is executing $SHM$ about $y=0$ along the $y$-axis. Its position at an instant is given by $y = (7 \, m) \sin(\pi t)$. Its average velocity for the time interval $0$ to $0.5 \, s$ is ........... $m/s$.