$A$ 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?

$A$ dense collection of equal number of electrons and positive ions is called neutral plasma. Certain solids containing fixed positive ions surrounded by free electrons can be treated as neutral plasma. Let $N$ be the number density of free electrons, each of mass $m$. When the electrons are subjected to an electric field, they are displaced relatively away from the heavy positive ions. If the electric field becomes zero, the electrons begin to oscillate about the positive ions with a natural angular frequency $\omega_p$, which is called the plasma frequency. To sustain the oscillations, a time-varying electric field needs to be applied that has an angular frequency $\omega$, where a part of the energy is absorbed and a part of it is reflected. As $\omega$ approaches $\omega_p$, all the free electrons are set to resonance together and all the energy is reflected. This is the explanation of high reflectivity of metals.
$1.$ Taking the electronic charge as $e$ and the permittivity as $\varepsilon_0$, use dimensional analysis to determine the correct expression for $\omega_p$.
$(A) \sqrt{\frac{N e}{m \varepsilon_0}}$ $(B) \sqrt{\frac{m \varepsilon_0}{N e}}$ $(C) \sqrt{\frac{N e^2}{m \varepsilon_0}}$ $(D) \sqrt{\frac{m \varepsilon_0}{N e^2}}$
$2.$ Estimate the wavelength at which plasma reflection will occur for a metal having the density of electrons $N \approx 4 \times 10^{27} \ m^{-3}$. Take $\varepsilon_0 \approx 10^{-11}$ and $m \approx 10^{-30}$, where these quantities are in proper $SI$ units.
$(A) 800 \ nm$ $(B) 600 \ nm$ $(C) 300 \ nm$ $(D) 200 \ nm$
Give the answer for question $1$ and $2$.

If the unit of force is $100\,N$,unit of length is $10\,m$ and unit of time is $100\,s$,what is the unit of mass in this system of units?

The density of a material in $SI$ units is $128 \ kg \ m^{-3}$. In a system of units where the unit of length is $25 \ cm$ and the unit of mass is $50 \ g$,the numerical value of the density of the material is:

In the equation $S = a + bt + ct^2$,where $S$ is measured in metres and $t$ is measured in seconds,the unit of $c$ is:

The entropy of any system is given by
$S = \alpha^{2} \beta \ln \left[\frac{\mu k R}{J \beta^{2}} + 3\right]$
Where $\alpha$ and $\beta$ are constants. $\mu, J, k$ and $R$ are the number of moles,mechanical equivalent of heat,Boltzmann constant,and gas constant respectively. [Take $S = \frac{dQ}{T}$].
Choose the incorrect option from the following: