Answer
(a) stay the same
(b) III
Work Step by Step
(a) We know that $t=\frac{2\pi r}{v}$
But $v=\frac{Bqr}{m}$
$\implies t=\frac{2\pi r}{\frac{Bqr}{m}}=\frac{2\pi m}{Bq}$
This equation shows that the time is independent of the velocity of radius. Thus, the time spent remains the same.
(b) We know that the best explanation is option (III) -- that is, the time for an orbit in a magnetic field is independent of speed. Thus, the time the proton spends in each of magnetic regions is the same no matter what its speed.
