Two particles $A$ and $B$ initially at rest move towards each other under a mutual force of attraction. At the instant when the velocity of $A$ is $v$ and that of $B$ is $2v$,the velocity of the centre of mass of the system is

If $f(x) = \cot^{-1} \left( \frac{x^x - x^{-x}}{2} \right)$,then $f'(1)$ is equal to

If $4$ moles of an ideal monoatomic gas at temperature $400 ~K$ is mixed with $2$ moles of another ideal monoatomic gas at temperature $700 ~K$,the temperature of the mixture is

For a ray of light,the critical angle is minimum when it travels from

$A$ uniform metal rod is used as a bar pendulum. If the room temperature rises by $10^{\circ} C$ and the coefficient of linear expansion of the metal of the rod is $2 \times 10^{-6} /^{\circ} C$,the period of the pendulum will have a percentage increase of

If the angle between the circles $x^2+y^2-2x-4y+c=0$ and $x^2+y^2-4x-2y+4=0$ is $60^{\circ}$,then $c$ is equal to