The work done by a gas molecule in an isolated system is given by $W = \alpha \beta^{2} e^{-\frac{x^{2}}{\alpha kT}}$,where $x$ is the displacement,$k$ is the Boltzmann constant,$T$ is the temperature,and $\alpha$ and $\beta$ are constants. Then the dimension of $\beta$ will be:

The time period of revolution of a satellite $(T)$ around the earth depends on the radius of the circular orbit $(R)$,mass of the earth $(M)$,and universal gravitational constant $(G)$. The expression for $T$,using dimensional analysis is ($K$ is a constant of proportionality):

$A$ 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$]$.

If momentum $(P)$,area $(A)$,and time $(T)$ are taken to be fundamental quantities,then the dimensional formula for energy is:

If $E$ and $E_0$ denote energies at time $t$ and $t_0$ respectively, and $L$ and $L_0$ denote distances from some point at $t$ and $t_0$ respectively, then which of the following equations can be declared to be incorrect on dimensional grounds?
$(A) E = \frac{2 E_0 L}{L_0}$
$(B) E = E_0 e^{-\frac{2 L}{L_0}}$
$(C) E = 2 L e^{-\frac{L}{E_0}}$
$(D) E = 2 \left( \frac{E_0}{L_0} \right) e^{-\frac{L}{L_0}}$

The velocity $v$ (in $cm/sec$) of a particle is given in terms of time $t$ (in $sec$) by the relation $v = at + \frac{b}{t + c}$. The dimensions of $a, b,$ and $c$ are: