Consider the motion of a positive point charge in a region where there are simultaneous uniform electric and magnetic fields $E=E_0\hat{j}$ and $B=B_0\hat{j}$. At time $t=0$, this charge has velocity $v$ in the $xy-$plane making an angle $\theta$ with the $x$-axis. Which of the following option(s) is (are) correct for time $t>0$?

(a) If $\theta =0°$, the charge moves in a circular path in the $xz-$plane.

(b) If $\theta =0°$, the charge undergoes helical motion with constant pitch along the $y-$axis

(c) If $\theta =10°$, the charge undergoes helical motion with its pitch increasing with time along the $y-$axis.

(d) If $\theta =90°$, the charge undergoes linear but accelerated motion along the $y-$axis.

Magnetic field will rotate the particle in a circular path (in $xz-$plane which is perpendicular to $B$). Electric field will exert a constant force on the particle in positive $y-$direction. Therefore, resultant path is neither purely circular nor helical or the option (a) and (b) both are wrong.

(c) $v_{\perp }$ and $B$ will rotate the particle in a circular path in $xz-$plane (or perpendicular to $B$). Further, $v_{\parallel }$ and $E$ will move the particle (with increasing speed) along positive  $y-$axis (or along the axis above circular path). Therefore, the resultant path is helical with increasing pitch along the $y-$axis (or along $B$ and $E$). Therefore, option (c) is correct.

(d) 

Magnetic force is zero, as $\theta$ between $B$ and $v$ is zero. But, electric force will act in $y-$direction. Therefore, motion is 1-D uniformly accelerated (towards positive $y-$direction). Therefore, option (d) is also correct.