If $x = \frac{1}{2} \left( \sqrt{3} + \frac{1}{\sqrt{3}} \right)$,then the value of $\frac{\sqrt{x^2 - 1}}{x - \sqrt{x^2 - 1}}$ is equal to

For a gas $\frac{R}{C_v} = 0.4$,where $R$ is the universal gas constant and $C_v$ is the molar specific heat at constant volume. The gas is made up of molecules which are

The internal energy of a monoatomic gas is $3 \ nRT$. $1$ mole of helium is kept in a cylinder having internal cross section area of $17 \ cm^2$ and fitted with a light movable frictionless piston. The gas is heated slowly by suppling $126 \ J$ heat. If the temperature rises by $4 \ ^\circ C$, then the piston will move $. . . . . . \ cm$.
(atmospheric pressure $= 10^5 \ Pa$)

The ratio of the magnetic field at the centre of a current-carrying circular coil to its magnetic moment is $x$. If the current and the radius are both doubled,the new ratio will become:

Four thin rods of same mass $M$ and same length $l$ form a square as shown in the figure. The moment of inertia of this system about an axis passing through the centre $O$ and perpendicular to its plane is

$A$ wheel is rotating at $900 \ r.p.m.$ about its axis. When the power is cut-off,it comes to rest in $1 \ minute$. The angular retardation in $rad/s^2$ is