The shortest distance of the point $(a, b, c)$ from the $x$-axis is

The sum of the infinite series ${\tan ^{ - 1}}\left( {\frac{2}{{1 - {1^2} + {1^4}}}} \right) + {\tan ^{ - 1}}\left( {\frac{4}{{1 - {2^2} + {2^4}}}} \right) + {\tan ^{ - 1}}\left( {\frac{6}{{1 - {3^2} + {3^4}}}} \right) + \dots$ is:

$A$ car starts from rest and moves with uniform acceleration $a$ on a straight road from time $t = 0$ to $t = T$. After that,a constant deceleration brings it to rest. In this process,the average speed of the car is

The potential energy of a particle of mass $1\, kg$ in motion along the $x-$axis is given by $U = 4(1 - \cos 2x)$,where $x$ is in $meters$. The period of small oscillation (in $seconds$) is

$A$ large tank filled with water to a height $h$ is to be emptied through a small hole at the bottom. The ratio of the time taken for the level of water to fall from $h$ to $\frac{h}{2}$ and from $\frac{h}{2}$ to zero is:

$A$ bar magnet of length $10 \, cm$ and having a pole strength equal to $10^{-3} \, Wb$ is kept in a magnetic field having magnetic induction $B$ equal to $4\pi \times 10^{-3} \, T$. It makes an angle of $30^{\circ}$ with the direction of the magnetic induction. The value of the torque acting on the magnet is: