The velocity of a freely falling body changes according to $g^p{h}^q$ where g is acceleration due to gravity and h is the height. The values of p and q are:

1. $1, \frac{1}{2}$

2. $\frac{1}{2}, \frac{{\displaystyle 1}}{2}$

3. $\frac{1}{2}, 1$

4. 1, 1

Which one of the following does not have the same dimensions

1. Work and energy

2. Angle and strain

3. Relative density and refractive index

4. Planck constant and energy

Which one has the dimensions different from the remaining three

1. Power

2. Work

3. Torque

4. Energy

A small steel ball of radius r is allowed to fall under gravity through a column of a viscous liquid of coefficient of viscosity $\eta$. After some time the velocity of the ball attains a constant value known as terminal velocity $v_T$. The terminal velocity depends on (i) the mass of the ball m, (ii) $\eta$, (iii) r and (iv) acceleration due to gravity g. Which of the following relations is dimensionally correct?

1. $v_T\propto \frac{mg}{\eta r}$

2. $v_T\propto \frac{\eta r}{mg}$

3. $v_T\propto \eta rmg$

4. $v_T\propto \frac{mgr}{\eta }$ 

The quantity $X=\frac{{\epsilon }_{0}LV}{t}; {\epsilon }_0$ is the permittivity of free space, L is length, V is the potential difference and t is time. The dimensions of X are the same as that of:

1. Resistance

2. Charge

3. Voltage

4. Current

${\mu }_0$ and ${\epsilon }_0$ denote the permeability and permittivity of free space, the dimensions of ${\mu }_0{\epsilon }_0$ are:

1. $L{T}^{-1}$

2. $L^{-2}{T}^2$

3. $M^{-1}{L}^{-3}{Q}^2{T}^2$

4. $M^{-1}{L}^{-3}{I}^2{T}^2$

The expression $\left[M{L}^2{T}^{-2}\right]$ represents

1. Pressure

2. Kinetic energy

3. Momentum

4. Power

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

1. $M^1{L}^4{T}^{-2}$

2. $M^2{L}^{-2}{T}^{-1}$

3. $M^2{L}^{-2}{T}^{-2}$

4. $M^1{L}^{-2}{T}^{-1}$

The dimensions of CV² matches with the dimensions of 

1. L²I

2. L²I ²

3. LI²

4. $\frac{1}{LI}$

The Martians use force (F), acceleration (A) and time (T) as their fundamental physical quantities. The dimensions of length on Martians system are 

1. [FT²]

2. [${\mathrm{F}}^{-1}{\mathrm{T}}^2$]

3. [${\mathrm{F}}^{-1}{\mathrm{A}}^2{\mathrm{T}}^{-1}$]

4. [AT²]