If $f(x) = \frac{x}{1+|x|}$, for $x \in \mathbb{R}$, then $f^{\prime}(0)$ is equal to :

$A$ rotating body has angular momentum $L$. If its frequency of rotation is halved and its rotational kinetic energy is doubled,what will be its new angular momentum?

The path difference between the two waves $y_1 = a_1 \sin \left( \omega t - \frac{2\pi x}{\lambda} \right)$ and $y_2 = a_2 \cos \left( \omega t - \frac{2\pi x}{\lambda} + \phi \right)$ is

The velocities of three particles of masses $20 \ g$,$30 \ g$ and $50 \ g$ are $10 \hat{i}$,$10 \hat{j}$ and $10 \hat{k}$,respectively. The velocity of the centre of mass of the three particles is

If $m$ and $M$ respectively denote the minimum and maximum of $f(x)=(x-1)^2+3$ for $x \in [-3, 1]$,then the ordered pair $(m, M)$ is equal to

The variance of $20$ observations is $5$. If each observation is multiplied by $2$,then the new variance of the resulting observations is: