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

$A$ force is represented by $F = ax^2 + bt^{1/2}$,where $x$ is distance and $t$ is time. The dimensions of $b^2/a$ are:

The velocity $v$ of a particle at time $t$ is given by $v = at + \frac{b}{t + c}$,where $a, b,$ and $c$ are constants. What are the dimensions of $a, b,$ and $c$ respectively?

$A$ calorie is a unit of heat or energy and it equals about $4.2 \; J$ where $1 \; J = 1 \; kg \; m^2 \; s^{-2}$. Suppose we employ a system of units in which the unit of mass equals $\alpha \; kg$, the unit of length equals $\beta \; m$, and the unit of time equals $\gamma \; s$. Show that a calorie has a magnitude $4.2 \; \alpha^{-1} \beta^{-2} \gamma^2$ in terms of the new units.

Frequency $(n)$ is a function of density $(\rho)$,length $(a)$,and surface tension $(T)$. Then its value is:

If $A, B$ and $C$ are three different physical quantities with different dimensional formulae, then the combination which can never give a proper physical quantity is