Two charges $q_1$ and $q_2$ are placed $30\,cm$ apart,as shown in the figure. $A$ third charge $q_3$ is moved along the arc of a circle of radius $40\,cm$ from $C$ to $D$. The change in the potential energy of the system is $\frac{q_3}{4\pi \epsilon_0}k$,where $k$ is:

The height of water in a tank is $H$. The range of the liquid emerging out from a hole in the wall of the tank at a depth $\frac{3H}{4}$ from the upper surface of water will be

$\int_0^\pi \frac{x \tan x}{\sec x+\tan x} d x=$

Let $z = a - \frac{i}{2}$,where $a \in R$. Then $|i + z|^2 - |i - z|^2$ is equal to

Photoelectric emission is observed from a metallic surface for frequencies $v_1$ and $v_2$ of the incident light rays $(v_1 > v_2)$. If the maximum values of kinetic energy of the photoelectrons emitted in the two cases are in the ratio of $1 : k$,then the threshold frequency of the metallic surface is

The angle between the lines $x \cos \alpha_1 + y \sin \alpha_1 = p_1$ and $x \cos \alpha_2 + y \sin \alpha_2 = p_2$ is: