Which of the following statements is true?

When a positively charged particle enters a uniform magnetic field with uniform velocity, its trajectory can be:
$(1)$ a straight line
$(2)$ a circle
$(3)$ a helix

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.

The acceleration of an electron at a moment in a magnetic field $\vec{B} = 2\hat{i} + 3\hat{j} + 4\hat{k}$ is $\vec{a} = x\hat{i} - 2\hat{j} + \hat{k}$. The value of $x$ is

At $t = 0$,a positively charged particle of mass $m$ is projected from the origin with velocity $u_0$ at an angle $37^o$ from the $x$-axis as shown in the figure. $A$ constant magnetic field $\vec{B_0} = B_0 \hat{j}$ is present in space. After a time interval $t_0$,the velocity of the particle may be:

The radius of curvature of the path of a charged particle moving in a static uniform magnetic field is