An expression for a dimensionless quantity $P$ is given by $P = \frac{\alpha}{\beta} \log_{e} \left( \frac{kt}{\beta x} \right)$,where $\alpha$ and $\beta$ are constants,$x$ is distance,$k$ is the Boltzmann constant,and $t$ is the temperature. Then the dimensions of $\alpha$ will be:

What is Dimensional Analysis? State the uses of Dimensional Analysis.

If $x$ and $a$ represent distance,then for what value of $n$ is the given equation dimensionally correct? The equation is $\int \frac{dx}{\sqrt{a^2 - x^n}} = \sin^{-1} \frac{x}{a}$.

Given below are two statements: One is labelled as Assertion $(A)$ and other is labelled as Reason $(R)$.
Assertion $(A)$: Time period of oscillation of a liquid drop depends on surface tension $(S)$,if density of the liquid is $\rho$ and radius of the drop is $r$,then $T = k \sqrt{\rho r^{3} / S}$ is dimensionally correct,where $k$ is dimensionless.
Reason $(R)$: Using dimensional analysis,we find that the $R.H.S.$ has different dimensions than that of the time period.

If the present units of length,time,and mass $(m, s, kg)$ are changed to $100 \; m, 100 \; s, 0.1 \; kg$,then:

In the equation,pressure $P = \frac{c - t^{2}}{DS}$,where $S$ and $t$ represent the distance and time respectively. The dimensions of $\left(\frac{D}{c}\right)$ are