The motion of a particle varies with time according to the relation $y = a(\sin \omega \,t + \cos \omega \,t)$,then

How many amplitudes does a Simple Harmonic Oscillator $(SHO)$ cover in half of its time period?

The function of time representing a simple harmonic motion with a period of $\frac{\pi}{\omega}$ is :

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.

$A$ particle is executing simple harmonic motion with an amplitude $A$ and time period $T$. The displacement of the particle after $2T$ time from its initial position 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):
$(a)$ $\sin \omega t - \cos \omega t$
$(b)$ $\sin^3 \omega t$
$(c)$ $3 \cos (\pi/4 - 2 \omega t)$
$(d)$ $\cos \omega t + \cos 3 \omega t + \cos 5 \omega t$
$(e)$ $\exp(-\omega^2 t^2)$
$(f)$ $1 + \omega t + \omega^2 t^2$