The position vector of a particle moving in space is given by $\overrightarrow{\mathrm{r}}=(1+2\cos 2\mathrm{\omega t})\widehat{\mathrm{i}}+\left(3{\sin }^2\mathrm{\omega t}\right)\widehat{\mathrm{j}}+\left(3\mathrm{t}\right)\widehat{\mathrm{k}}$ in the ground frame. All the units are in SI. 
Choose the correct statement(s):

The amplitude of the SHM of the particle in frame S is $\frac{5}{2}\mathrm{m}$

The particle executes SHM in a frame S moving along the positive z–axis with a velocity of 3m/s.

The particle executes SHM in the ground frame about the mean position $\left(1,\frac{3}{2},3\mathrm{t}\right)$

The line of the SHM of the particle is parallel to the vector $\left(\frac{4}{5}\widehat{\mathrm{i}}-\frac{3}{5}\widehat{\mathrm{j}}\right)$