A particle of mass m having negative charge $-q$ is projected at an angle $\theta$ with x-axis. There exists a uniform electric field E and a uniform magnetic field B along x-axis. The particle will return to its initial point after n complete  revolutions in time t. What is n here?

Let particle completes n revolutions in time t, then t = nT

$\Rightarrow \mathrm{t}=\frac{\mathrm{n}2\mathrm{\pi m}}{\mathrm{qB}}$

If particle has to return to its initial point, then displacement along x-axis during this time should be zero.

$\begin{array}{l}{\mathrm{S}}_{\mathrm{x}}={\mathrm{u}}_{\mathrm{x}}\mathrm{t}+\frac{1}{2}{\mathrm{a}}_{\mathrm{x}}{\mathrm{t}}^2\\ \Rightarrow 0=\mathrm{v}\cos \mathrm{\theta t}-\frac{1}{2}\frac{\mathrm{qE}}{\mathrm{m}}{\mathrm{t}}^2\\ \Rightarrow \mathrm{v}\cos \mathrm{\theta }=\frac{\mathrm{qE}}{2\mathrm{m}}\mathrm{t}\\ \Rightarrow \mathrm{v}\cos \mathrm{\theta }=\frac{\mathrm{qE}}{2\mathrm{m}}\times \frac{\mathrm{n}2\mathrm{\pi m}}{\mathrm{qB}}\Rightarrow \mathrm{n}=\frac{\mathrm{Bvcos\theta }}{\mathrm{\pi E}}\end{array}$