The time dependence of a physical quantity $P$ is given by $ P = P_0 exp^{(-\alpha t^{2})} $ where $\alpha$ is a constant and $t$ is time. The constant $\alpha$ 

If orbital velocity of planet is given by $v = {G^a}{M^b}{R^c}$, then

The quantity $X = \frac{{{\varepsilon _0}LV}}{t}$: ${\varepsilon _0}$ is the permittivity of free space, $L$ is length, $V$ is potential difference and $t$ is time. The dimensions of $X$ are same as that of

If dimensions of critical velocity $v_c$ of a liquid flowing through a tube are expressed as$ [\eta ^x \rho ^yr^z]$ where  $\eta ,\rho $ and $r $ are the coefficient of viscosity of liquid, density of liquid and radius of the tube respectively, then the values of $x, y$ and $z$ are given by

if Energy is given by $U = \frac{{A\sqrt x }}{{{x^2} + B}},\,$, then dimensions of $AB$ is

$A$ and $B$ possess unequal dimensional formula then following operation is not possible in any case:-