If force $=\frac{\mathrm{\alpha }}{{\mathrm{\beta }}^3+\mathrm{density}},$ then the dimensions of $\mathrm{\alpha }$ and $\mathrm{\beta }$ will be:

 

The notations $\mathrm{E}, \mathrm{m}, \mathrm{J} \mathrm{and} \mathrm{G}$ repersent energy, mass, angular momentum and gravitational constant respectively. $\mathrm{E} {\mathrm{J}}^2/ {\mathrm{m}}^5 {\mathrm{G}}^2$ has the dimensions of :

If force $\left(\mathrm{F}\right),$ velocity $\left(\mathrm{V}\right)$ and time $\left(\mathrm{T}\right)$ are taken as fundamental units, the dimensions of mass are :

The variation of pressure $\mathrm{P}$ with distance $\mathrm{z}$ is given by

                                                                   $\mathrm{P}=\frac{\mathrm{\alpha }}{\mathrm{\beta }}\mathrm{e}-\mathrm{\alpha z}/\mathrm{k\theta },$

where $\mathrm{\alpha },\mathrm{\beta }$ are constants, $\mathrm{k}$ is Boltzmann constant and $\mathrm{\theta }$ is temperature. The dimensional formula of $\mathrm{\beta }$ will be:

Choose the correct statement (s) :

The time dependence of a physical quantity $\mathrm{p}$ is given by $\mathrm{P}={\mathrm{P}}_0{\mathrm{e}}^{-\mathrm{\alpha }}{\mathrm{t}}^2,$ where $\mathrm{\alpha }$ is constant and $\mathrm{t}$ is time. The constant $\mathrm{\alpha } :$

The number of particles crossing per unit area perpendicular to $\mathrm{X}-\mathrm{axis}$per unit time is given by $\mathrm{n}=-\mathrm{D}\frac{{\mathrm{n}}_2-{\mathrm{n}}_1}{{\mathrm{x}}_2-{\mathrm{x}}_1}, \mathrm{where} {\mathrm{n}}_2\mathrm{and} {\mathrm{n}}_1$ are number of particle per unit volume at ${\mathrm{x}}_2 \mathrm{and} {\mathrm{x}}_1$ respectively. The dimension of $\mathrm{D}.$ called diffusion constant, are :