$\mathrm{M}$ grams of steam at  ${100}^{\circ }\mathrm{C}$ is mixed with $200\mathrm{g}$ of ice at its melting point in a thermally insulated container. If it produces liquid water at ${40}^{?}\mathrm{C}$ [heat of vaporization of water is $540\mathrm{c}\mathrm{a}\mathrm{l}/\mathrm{g}$ and heat of fusion of ice is $80\mathrm{c}\mathrm{a}\mathrm{l}/\mathrm{g}$ ], the value of $M$  is     

Two SHMs in same direction are represented by y1 = asinωt and y2 = asin(ωt + δ). When they superimpose, the initial phase difference of the resultant SHM is

The potential energy of a particle of mass 2 kg in motion along the x-axis is given by U = 2 (1 – cos 2x) J, where x is in meters. The period of small oscillations (in s) is

A child is standing with folded hands at the centre of a platform rotating about its central axis. The kinetic energy of the system is K. The child now stretches his arms so that the moment of inertia of the system becomes 3/2 of the initial moment of inertia. The kinetic energy of the system is now

If the absolute temperature of hot black body is raised by 2%, by how much percentage rate of heat energy radiated would be increased nearly?

Two moles of helium gas are taken over the cycle ABCDA as shown below in P-T diagram. Assuming the gas to be ideal, the work done on the gas in taking from A to B is