$A$ physical quantity of the dimensions of length that can be formed out of $c, G$ and $\frac{e^2}{4\pi \varepsilon_0}$ is $[c$ is velocity of light,$G$ is the universal gravitational constant and $e$ is charge$]$.

Match List-$I$ with List-$II$.

List-$I$	List-$II$
$A$. Meter $(L)$	$I$. $\sqrt{\frac{hc}{G}}$
$B$. Second $(S)$	$II$. $\sqrt{\frac{Gh}{c^5}}$
$C$. Kilogram $(M)$	$III$. $\sqrt{\frac{L^2c^3}{Gh}}$
$D$. Kelvin $(K)$	$IV$. $\sqrt{\frac{Gh}{c^3}}$

where $h$ (Planck's constant), $G$ (gravitational constant) and $c$ (speed of light in vacuum) are fundamental units. Choose the correct answer from the options given below:

The equation of a stationary wave is given by $y = 2a \sin \left( \frac{2 \pi nt}{\lambda} \right) \cos \left( \frac{2 \pi x}{\lambda} \right)$. Which of the following statements is $NOT$ correct?

An artificial satellite is revolving around a planet of mass $M$ and radius $R$ in a circular orbit of radius $r$. From Kepler's third law about the period of a satellite around a common central body,the square of the period of revolution $T$ is proportional to the cube of the radius of the orbit $r$. Show using dimensional analysis that $T = \frac{k}{R}\sqrt{\frac{r^3}{g}}$,where $k$ is a dimensionless constant and $g$ is the acceleration due to gravity.

Due to an explosion underneath water, a bubble started oscillating. If this oscillation has a time period $T$, which is proportional to $p^\alpha S^\beta E^\gamma$, where $p$ is static pressure, $S$ is the density of water, and $E$ is the total energy of the explosion, determine $\alpha, \beta$, and $\gamma$.

Which of the following relations can be derived by the method of dimensional analysis?