Units & Dimensions question from JEE Main 2021 (25 Feb, Shift 2)

Answer: (c)

1. $\frac{e^2}{4\pi\varepsilon_0 r}$ is an energy, so $\frac{e^2}{4\pi\varepsilon_0}$ has dimension energy $\times$ length.
2. $hc = (\text{energy}\times\text{time})\times(\text{length/time}) =$ energy $\times$ length.
3. The ratio is dimensionless (it is the fine-structure constant $\approx 1/137$).
4. So the answer is (c) $[M^0L^0T^0]$.