The electric field at the centre $O$ of a semicircle of radius $a$ having a linear charge density $\lambda$ is given by:

Six charges are placed around a regular hexagon of side length $a$ as shown in the figure. Five of them have charge $q$,and the remaining one has charge $x$. The perpendicular from each charge to the nearest hexagon side passes through the center $O$ of the hexagon and is bisected by the side.
Which of the following statement$(s)$ is(are) correct in $SI$ units?
$(A)$ When $x=q$,the magnitude of the electric field at $O$ is zero.
$(B)$ When $x=-q$,the magnitude of the electric field at $O$ is $\frac{q}{6 \pi \epsilon_0 a^2}$.
$(C)$ When $x=2q$,the potential at $O$ is $\frac{7q}{4 \sqrt{3} \pi \epsilon_0 a}$.
$(D)$ When $x=-3q$,the potential at $O$ is $\frac{3q}{4 \sqrt{3} \pi \epsilon_0 a}$.

Two point charges $q_1$ and $q_2 (=q_1/2)$ are placed at points $A(0, 1)$ and $B(1, 0)$ as shown in the figure. The electric field vector at point $P(1, 1)$ makes an angle $\theta$ with the $x$-axis,then the angle $\theta$ is

In the given figure, find the distance from point $A$ where the electric field is zero (in $cm$).

Electric field in a certain region is given by $\overrightarrow{E} = (\frac{A}{x^2} \hat{i} + \frac{B}{y^3} \hat{j})$. The $SI$ units of $A$ and $B$ are:

Two point charges $q_1 = \sqrt{10} \, \mu C$ and $q_2 = -25 \, \mu C$ are placed on the $x$-axis at $x = 1 \, m$ and $x = 4 \, m$ respectively. The electric field (in $V/m$) at a point $y = 3 \, m$ on the $y$-axis is,[ take $\frac{1}{4\pi\varepsilon_0} = 9 \times 10^9 \, Nm^2C^{-2}$ ]