The equation of state of some gases can be expressed as $(P + \frac{a}{V^2}) = \frac{b\theta}{l}$,where $P$ is the pressure,$V$ is the volume,$\theta$ is the absolute temperature,and $a$ and $b$ are constants. The dimensional formula of $a$ is

If energy is given by $U = \frac{A\sqrt{x}}{x^2 + B}$,then the dimensions of $AB$ are:

If velocity $v$,acceleration $A$,and force $F$ are chosen as fundamental quantities,then the dimensional formula of angular momentum in terms of $v, A$,and $F$ would be

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?
$(A) E = \frac{2 E_0 L}{L_0}$
$(B) E = E_0 e^{-\frac{2 L}{L_0}}$
$(C) E = 2 L e^{-\frac{L}{E_0}}$
$(D) E = 2 \left( \frac{E_0}{L_0} \right) e^{-\frac{L}{L_0}}$

If the unit of force is $100\,N$,unit of length is $10\,m$ and unit of time is $100\,s$,what is the unit of mass in this system of units?

$A$ famous relation in physics relates 'moving mass' $m$ to the 'rest mass' $m_{0}$ of a particle in terms of its speed $v$ and the speed of light $c$. (This relation first arose as a consequence of special relativity due to Albert Einstein). $A$ boy recalls the relation almost correctly but forgets where to put the constant $c$. He writes:
$m = \frac{m_{0}}{(1 - v^{2})^{1/2}}$
Guess where to put the missing $c$.