$A$ proton (mass $m$ and charge $+e$) and an $\alpha$-particle (mass $4m$ and charge $+2e$) are projected with the same kinetic energy at right angles to a uniform magnetic field. Which one of the following statements will be true?

An electron is travelling along the $x$-direction. It encounters a magnetic field in the $y$-direction. Its subsequent motion will be

An $\alpha$-particle and a proton moving with the same kinetic energy enter a region of uniform magnetic field at right angles to the field. The ratio of the radii of the paths of $\alpha$-particle to that of the proton is

The figure shows a region of length $l$ with a uniform magnetic field of $0.3 \, T$ in it. $A$ proton enters the region with a velocity of $4 \times 10^{5} \, m/s$,making an angle of $60^{\circ}$ with the field. If the proton completes $10$ revolutions by the time it crosses the region shown,$l$ is close to....... $m$ (mass of proton $= 1.67 \times 10^{-27} \, kg$,charge of the proton $= 1.6 \times 10^{-19} \, C$)

Given below are two statements:
Statement $I$: The electric force changes the speed of the charged particle and hence changes its kinetic energy; whereas the magnetic force does not change the kinetic energy of the charged particle.
Statement $II$: The electric force accelerates the positively charged particle perpendicular to the direction of the electric field. The magnetic force accelerates the moving charged particle along the direction of the magnetic field.
In the light of the above statements,choose the most appropriate answer from the options given below:

Consider the motion of a positive point charge in a region where there are simultaneous uniform electric and magnetic fields $\vec{E}=E_0 \hat{j}$ and $\vec{B}=B_0 \hat{j}$. At time $t=0$,this charge has velocity $\vec{v}$ in the $x-y$ plane,making an angle $\theta$ with the $x$-axis. Which of the following option$(s)$ is(are) correct for time $t>0$?
$(A)$ If $\theta=0^{\circ}$,the charge moves in a circular path in the $x-z$ plane.
$(B)$ If $\theta=0^{\circ}$,the charge undergoes helical motion with constant pitch along the $y$-axis.
$(C)$ If $\theta=10^{\circ}$,the charge undergoes helical motion with its pitch increasing with time,along the $y$-axis.
$(D)$ If $\theta=90^{\circ}$,the charge undergoes linear but accelerated motion along the $y$-axis.