Which of the following $statement(s)$ is/are correct?

The area (in square units) of the triangle formed by the points with polar coordinates $(1, 0)$,$(2, \frac{\pi}{3})$,and $(3, \frac{2\pi}{3})$ is

For $0 < x \leq \pi$,$\sinh ^{-1}(\cot x)$ is equal to

Factors affecting the Hardy-Weinberg principle are:
$(I)$ Gene flow
$(II)$ Genetic drift
$(III)$ Mutation
$(IV)$ Genetic recombination
$(V)$ Natural selection
The correct combination is:

If $b+c=3a$,then $\cot \frac{B}{2} \cot \frac{C}{2}$ is equal to :

Carbon,Silicon,and Germanium atoms have four valence electrons each. Their valence and conduction bands are separated by energy band gaps represented by $(E_g)_C$,$(E_g)_{Si}$,and $(E_g)_{Ge}$ respectively. Which one of the following relationships is true in their case?