A new unit of length is chosen such that the speed of light in vacuum is unity. What is the distance between the Sun and the Earth in terms of the new unit if light takes $8\; min$ and $20\; s$ to cover this distance?

A function $f(\theta )$ is defined as $f(\theta )\, = \,1\, - \theta  + \frac{{{\theta ^2}}}{{2!}} - \frac{{{\theta ^3}}}{{3!}} + \frac{{{\theta ^4}}}{{4!}} + ...$ Why is it necessary for  $f(\theta )$  to be a dimensionless quantity ?

Force $(F)$ and density $(d)$ are related as $F\, = \,\frac{\alpha }{{\beta \, + \,\sqrt d }}$ then dimension of $\alpha $ are

The formula $X = 5YZ^2$,  $X$ and $Z$ have dimensions of capacitance and magnetic field respectively. What are the dimensions of $Y$ in $SI$ units?

The dimensions of Stefan-Boltzmann's constant $\sigma$ can be written in terms of Planck's constant $h$, Boltzmann's constant $k_B$ and the speed of light $c$ as $\sigma=h^\alpha k_B^\beta c^\gamma$. Here,

In the relation $P = \frac{\alpha }{\beta }{e^{ - \frac{{\alpha Z}}{{k\theta }}}}$ $P$ is pressure, $Z$ is the distance, $k$ is Boltzmann constant and $\theta$ is the temperature. The dimensional formula of $\beta$ will be