A particle of mass m and charge q moves with speed v0 and enters, perpendicular to its boundary, a region of width d=mv0/2qB containing a uniform…

The orbit radius is $r=mv_0/(qB)$, so $d=r/2$. (A) The particle enters along the normal; geometry gives $\sin\theta=d/r=\tfrac12$, so $\theta=30^\circ$: true. (B) The time equals the arc angle over $\omega=qB/m$: $t=\dfrac{\pi/6}{qB/m}=\dfrac{\pi m}{6qB}$, not $\pi m/3qB$: false. (C) The magnetic force is always perpendicular to the velocity and does no work, so the speed and kinetic energy remain unchanged: false. (D) The particle turns back only if $d>r$. For $d=0.8r$ we get $\sin\theta=0.8$, so it still crosses the region and emerges from the far face deflected by about $53^\circ$: false.