If the charge on the capacitor is increased by $2 \ C$,the energy stored in it increases by $21 \%$. The total original charge on the capacitor is: (in $C$)

$A$ light wave has a frequency of $4 \times 10^{14} ~Hz$ and a wavelength of $5 \times 10^{-7} ~m$ in a medium. The refractive index of the medium is:

$A$ bob of mass $m$ attached to an inextensible string of length $l$ is suspended from a vertical support. The bob rotates in a horizontal circle with an angular speed $\omega \ rad/s$ about the vertical axis. About the point of suspension:

If a circle $C$ passing through the point $(4, 0)$ touches the circle $x^2 + y^2 + 4x - 6y = 12$ externally at the point $(1, -1)$,then the radius of $C$ is

The value of $(0.05)^{\log_{\sqrt{20}}(0.1 + 0.01 + 0.001 + \dots)}$ is

Two particles $P$ and $Q$ start from the origin and execute simple harmonic motion along the $X$-axis with the same amplitude but with periods $3 \ s$ and $6 \ s$,respectively. The ratio of the velocities of $P$ and $Q$ when they meet is