$A$ system has basic dimensions as density $[D]$,velocity $[V]$,and area $[A]$. The dimensional representation of force in this system is:

Young's modulus of elasticity $Y$ is expressed in terms of three derived quantities,namely,the gravitational constant $G$,Planck's constant $h$,and the speed of light $c$,as $Y = c^\alpha h^\beta G^\gamma$. Which of the following is the correct option?

The dimensions of $\frac{\alpha}{\beta}$ in the equation $F = \frac{\alpha - t^2}{\beta v^2}$,where $F$ is force,$v$ is velocity,and $t$ is time,are ..........

$A$ gas bubble formed from an explosion under water oscillates with a period $T$,proportional to $P^{a} d^{b} E^{c}$,where $P$ is the static pressure,$d$ is the density of water,and $E$ is the energy of explosion. Then $a, b, c$ are,respectively:

$A$ charged particle of charge $e$ and mass $m$ is moving in an electric field $\vec{E}$ and magnetic field $\vec{B}$. Construct dimensionless quantities and quantities of dimension $T^{-1}$.

Which of the following relations can be derived using dimensional analysis?