Let us consider a system of units in which mass and angular momentum are dimensionless. If length has dimension of $L$, which of the following statement ($s$) is/are correct ?

$(1)$ The dimension of force is $L ^{-3}$

$(2)$ The dimension of energy is $L ^{-2}$

$(3)$ The dimension of power is $L ^{-5}$

$(4)$ The dimension of linear momentum is $L ^{-1}$

Write the dimensions of $a/b$ in the relation $P = \frac{{a - {t^2}}}{{bx}}$ , where $P$ is pressure, $x$ is the distance and $t$ is the time 

The equation $\frac{{dV}}{{dt}} = At - BV$ is describing the rate of change of velocity of a body falling from rest in a resisting medium. The dimensions of $A$ and $B$ are

To find the distance $d$ over which a signal can be seen clearly in foggy conditions, a railways engineer uses dimensional analysis and assumes that the distance depends on the mass density $\rho$ of the fog, intensity (power/area) $S$ of the light from the signal and its frequency $f$. The engineer find that $d$ is proportional to $S ^{1 / n}$. The value of $n$ is:

A massive black hole of mass $m$ and radius $R$ is spinning with angular velocity $\omega$. The power $P$ radiated by it as gravitational waves is given by $P=G c^{-5} m^{x} R^{y} \omega^{z}$, where $c$ and $G$ are speed of light in free space and the universal gravitational constant, respectively. Then,

The quantities $A$ and $B$ are related by the relation, $m = A/B$, where $m$ is the linear density and $A$ is the force. The dimensions of $B$ are of