(a)
The given function is:

$\begin{array}{l}y=\sin \omega t-\cos \omega t\\ \mathrm{y}=\sqrt{2}[\frac{1}{\sqrt{2}}\sin \omega t-\frac{1}{\sqrt{2}}\cos \omega t]\\ \mathrm{y}=\sqrt{2}[\sin \omega t\times \cos \frac{\pi }{4}-\cos \omega t\times \sin \frac{\pi }{4}]\\ \mathrm{y}=\sqrt{2}\sin (\omega t-\frac{\pi }{4})\end{array}$
$\mathrm{This} \mathrm{function} \mathrm{represents} \mathrm{SHM} \mathrm{as} \mathrm{it} \mathrm{can} \mathrm{be} \mathrm{written} \mathrm{in} \mathrm{the} \mathrm{form}:$
$\mathrm{y}=\mathrm{Asin}\left(\mathrm{\omega t}+\mathrm{\varphi }\right)$
$\mathrm{Its} \mathrm{period} \mathrm{is}: \frac{2\mathrm{\pi }}{\mathrm{\omega }}$

$\left(\mathrm{b}\right)$
$\mathrm{The} \mathrm{given} \mathrm{function} \mathrm{is}:$
$y={\sin }^3\mathrm{\omega t}$
$y=\frac{1}{2}\left[3 \mathrm{sin\omega t}-\sin 3\mathrm{\omega t}\right]$

The terms sinωt and sin3ωt individually represent simple harmonic motion (SHM).
However, the superposition of two SHM is periodic and but not simple harmonic.

(c)

The given function is:

$\mathrm{y}=3\cos \left[\frac{\mathrm{\pi }}{4}-2\mathrm{\omega t}\right]$
$\mathrm{y}=3\cos \left[2\mathrm{\omega t}-\frac{\mathrm{\pi }}{4}\right]$
$\mathrm{This} \mathrm{function} \mathrm{represents} \mathrm{simple} \mathrm{harmonic} \mathrm{motion} \mathrm{because} \mathrm{it} \mathrm{can}$
$\mathrm{be} \mathrm{written} \mathrm{in} \mathrm{the} \mathrm{form}:$
$\mathrm{y}=\mathrm{Acos}\left(\mathrm{\omega t}-\mathrm{\varphi }\right)$
$\mathrm{Its} \mathrm{period} \mathrm{is}: \mathrm{T}= \frac{2\mathrm{\pi }}{2\mathrm{\omega }}=\frac{\mathrm{\pi }}{\mathrm{\omega }}$

(d)
The given function is:

$\mathrm{y}=\mathrm{cos\omega t}+\cos 3\mathrm{\omega t}+\cos 5\mathrm{\omega t}$.

Each individual cosine function represents SHM. However, the superposition of three simple harmonic motions is periodic, but not simple harmonic.

(e)

The given function is:

$\mathrm{y}={\mathrm{e}}^{\left(-{\mathrm{\omega }}^2{\mathrm{t}}^2\right)}$

It is an exponential function. Exponential functions do not repeat themselves. Therefore, it is a non-periodic motion.

The given function is:

$\mathrm{y}=1+\mathrm{\omega t}+{\mathrm{\omega }}^2{\mathrm{t}}^2$ 

It is a non-periodic function as it does not repeat itself.