The equation of motion of a particle is x =a cos(α t)2. The motion is:

1. periodic but not oscillatory

2. periodic and oscillatory

3. oscillatory but not periodic

4. neither periodic nor oscillatory

3. Hint: The displacement of the particle is the same after a time period in a periodic motion.
Step 1: Find the position of the particle after one time period.
As the given equation is,
                            x =a cos(α t)2 is a cosine function. Hence, it is an oscilatory motion.
Now, putting t+T in place of t,
                      x(t+T)=a cos[α(t+T)]2                         [x(t)=acos(at)2]
                               =a cos [α2t2+α2T2+2αtT ]x(t)
where, T is supposed as period of the function ω(t).
Hence, it is not periodic.