Archaea that live in hot springs belong to which group?

If $y = \cos^{-1}\left(\frac{a^2-x^2}{a^2+x^2}\right) + \sin^{-1}\left(\frac{2ax}{a^2+x^2}\right)$, then $\frac{dy}{dx}$ is equal to

What are $X$ and $Y$ respectively in the following reaction?
$Z$-product $\stackrel{Y}{\longleftarrow} 2$-butyne $\stackrel{X}{\longrightarrow} E$-product

One energy difference between the states $n = 2$ and $n = 3$ is $E \, eV$ in a hydrogen atom. The ionization potential of the $H$ atom is ........... $E$.

The voltage of an $ac$ source varies with time according to the equation $V = 100 \sin(100 \pi t) \cos(100 \pi t)$,where $t$ is in $seconds$ and $V$ is in $volts$. Then:

Which one of the following orders presents the correct sequence of the increasing basic nature of the given oxides?