The dimensions of physical quantity $X$ in the equation $\text{Force} = \frac{X}{\text{Density}}$ are given by:

In the equation $x = a + bt + ct^2$,where $x$ is in meters and $t$ is in seconds,find the units of $a, b,$ and $c$.

Planck's constant $h$,speed of light $c$,and gravitational constant $G$ are used to form a unit of length $L$ and a unit of mass $M$. Then the correct option$(s)$ is(are):
$(A)$ $M \propto \sqrt{c}$
$(B)$ $M \propto \sqrt{G}$
$(C)$ $L \propto \sqrt{h}$
$(D)$ $L \propto \sqrt{G}$

If $L$,$C$,and $R$ denote the inductance,capacitance,and resistance respectively,the dimensional formula for $C^{2}LR$ is

Due to an explosion underneath water, a bubble started oscillating. If this oscillation has a time period $T$, which is proportional to $p^\alpha S^\beta E^\gamma$, where $p$ is static pressure, $S$ is the density of water, and $E$ is the total energy of the explosion, determine $\alpha, \beta$, and $\gamma$.

$A$ modified gravitational potential is given by $V = -\frac{GM}{r} + \frac{A}{r^2}$. If the constant $A$ is expressed in terms of gravitational constant $G$, mass $M$, and velocity of light $c$, then from dimensional analysis, $A$ is: