b
When a particle moves in a magnetic field of intensity B pointing downwards into the page and particle is moving with a speed of \(v\) on the plane of paper:

\(F = qvB\) (Force of charged particle in a magnetic field)

And we know that

\(F =\frac{ mv ^2}{ r } \quad\) (r is the radius of motion and \(m\) is mass of particle)

\(\Rightarrow qvB =\frac{ mv ^2}{ r }\)

\(\Rightarrow r =\frac{ mv }{ Bq }\)

Now as we know that

\(\omega=\frac{ V }{ r }\)

\(\Rightarrow \omega=\frac{ Bq }{ m }\)

Time period, \(T =\frac{2 \pi}{\omega}\)

\(\Rightarrow T =\frac{2 \pi m }{ Bq }\)

And this shows that it is independent of both radius and velocity.