Identify the halogen exchange reaction from the following.

The minimum value of $27 \tan ^2 \theta + 3 \cot ^2 \theta$ is

Let $f$ be defined on $D = \mathbb{R} - \{-1, 1\}$ by $f(x) = \frac{|x|}{1-|x|}$,then

If $\sin ^{-1}\left(\frac{x}{13}\right)+\operatorname{cosec}^{-1}\left(\frac{13}{12}\right)=\frac{\pi}{2}$,then the value of $x$ is

In a playground,there is a merry-go-round of mass $120 \ kg$ and radius $4 \ m$. The radius of gyration is $3 \ m$. $A$ child of mass $30 \ kg$ runs at a speed of $5 \ m/s$ tangent to the rim of the merry-go-round when it is at rest and then jumps on it. Neglecting friction,find the angular velocity of the merry-go-round and child in $rad/s$.

$\lim _{x \rightarrow 8} \frac{\sqrt{1+\sqrt{1+x}}-2}{x-8}$ is equal to