If dimensions of critical velocity ${v_c}$ of a liquid flowing through a tube are expressed as $\eta^{x}\rho^yr^{z}$, where $\eta, \rho~\text{and}~r$ are the coefficient of viscosity of the liquid, the density of the liquid, and the radius of the tube respectively, then the values of ${x},$ ${y},$ and ${z},$ respectively, will be:

If force ($F$), velocity ($\mathrm{v}$), and time ($T$) are taken as fundamental units, the dimensions of mass will be:

The dimensions of ${\left({\mu }_0{\epsilon }_0\right)}^{-1/2}$ are:

1.  $[$ $L^{-1}T$ $]$

2.  $[$ $L{T}^{-1}$ $]$

3.  $[$ $L^{1/2}{T}^{1/2}$ $]$

4. $[$ $L^{1/2}{T}^{-1/2}$ $]$

The dimensions of ${\left({\mathrm{\mu }}_0{\mathrm{\epsilon }}_0\right)}^{-1/2}$ are:

1.  $\left[{\mathrm{L}}^{-1}\right]$ $\mathrm{T}$

2.  $\left[{\mathrm{LT}}^{-1}\right]$

3.  $\left[{\mathrm{L}}^{-1/2}\right]$ ${\mathrm{T}}^{1/2}$

4.  $\left[{\mathrm{L}}^{-1/2}\right]$ ${\mathrm{T}}^{-1/2}$

The density of a material in a CGS system of units is $4~\text{grams/cm}^3$. In a system of units in which the unit of length is $10~\text{cm}$ and the unit of mass is $100~\text{grams}$, the value of the density of the material will be:
1. $0.04$
2. $0.4$
3. $40$
4. $400$

The dimensions of $\frac{1}{2}{\mathrm{\epsilon }}_0{\mathrm{E}}^2$ where ${\epsilon }_0$ is the permittivity of free space and E is the electric field, are:

1. [ML²T^-2]

2. [ML^-1T^-2]

3.  [ML²T^-1]

4.  [MLT^-1]

A student measures the distance traversed in free fall of a body, initially at rest in a given time. He uses this data to estimate $g,$ the acceleration due to gravity. If the maximum percentage errors in the measurement of the distance and the time are $e_1$ and $e_2$ respectively, the percentage error in the estimation of $g$ is:
1. $e_1+2e_2$
2. $e_1+e_2$
3. $e_1-2e_2$
4. $e_2-e_1$