A particle of mass m and charge q , moving with velocity v enters...

$\overrightarrow{\mathbf{V}} \perp \overrightarrow{\mathbf{B}}$ in region II. Therefore, path of particle is circle in region II. Particle enters in region III if, radius of circular path, $r>1$ or $\frac{m v}{B q}>1$ or $v>\frac{B q l}{m}$ If $v=\frac{B q l}{m}, r=\frac{m v}{B q}=l$, particle will turn back and path length will be maximum as shown in figure in region II. If particle returns to region I, time spent in region II will be $t=\frac{T}{2}=\frac{\pi m}{B q}$, which is independent of $v$. $\therefore$ correct options are (a), (c) and (d).