Which of the following is a synthetic auxin?

If $y=\tan ^{-1}\left(\frac{\sqrt{1+a^2 x^2}-1}{a x}\right)$,then $\left(1+a^2 x^2\right) y^{\prime \prime}+2 a^2 x y^{\prime}$ is equal to

Four massless springs whose force constants are $2k, 2k, k$ and $2k$ respectively are attached to a mass $M$ kept on a frictionless plane (as shown in the figure). If the mass $M$ is displaced in the horizontal direction,then the frequency of oscillation of the system is:

Acidity of diprotic acids in aqueous solutions increases in the order

Which of the following taxonomical aids is being described by the given statements $(i-iv)$?
$(i)$. They are generally established in educational institutes.
$(ii)$. They have collections of preserved plant and animal specimens for study and reference.
$(iii)$. Insects are preserved in boxes after collecting,killing and pinning.
$(iv)$. They often have collections of skeletons of animals too.

$\sim (p \vee q) \vee (\sim p \wedge q)$ is logically equivalent to