Which one of the following is a simple harmonic motion?

The displacement of a particle in $S.H.M.$ varies according to the relation $x=4(\cos \pi t+\sin \pi t)$. The amplitude of the particle is

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):
$(a)$ $\sin \omega t - \cos \omega t$
$(b)$ $\sin^3 \omega t$
$(c)$ $3 \cos (\pi/4 - 2 \omega t)$
$(d)$ $\cos \omega t + \cos 3 \omega t + \cos 5 \omega t$
$(e)$ $\exp(-\omega^2 t^2)$
$(f)$ $1 + \omega t + \omega^2 t^2$

Two particles executing $SHM$ along a straight line have the same amplitude $A$ and time period $T$. At $t=0$,one particle is at a displacement $+A$ and another is at a displacement $-\frac{A}{2}$ and they are approaching towards each other. They cross each other after a time.

If a $SHM$ is represented by $\frac{d^2x}{dt^2} + ax = 0$,its time period is:

Which of the following expressions corresponds to simple harmonic motion along a straight line,where $x$ is the displacement and $a, b, c$ are positive constants?