Two moles a monoatomic gas is mixed with six moles of a diatomic gas. The molar specific heat of the mixture at constant volume is _______.

When a rubber-band is stretched by a distance \(x\), it exerts restoring force of magnitude  \(F=ax+bx^2\) where \(a\) and \(b\) areconstants. The work done in stretching the unstretched rubber-band by \(L\) is

Identify the characteristics of an adiabatic process in a monoatomic gas.
(A) Internal energy is constant.
(B) Work done in the process is equal to the charge in internal energy.
(C) The product of temperature and volume is a constant.
(D) The product of pressure and volume is a constant.
(E) The work done to change the temperature from \(\mathrm{T}_1\) to \(\mathrm{T}_2\) is proportional to \(\left(\mathrm{T}_2-\mathrm{T}_1\right)\)
Choose the correct answer from the options given below :

A Carnot engine operates between two reservoirs of temperatures \(900\; \mathrm{K}\) and \(300 \;\mathrm{K}\) The engine performs \(1200\; \mathrm{J}\) of work per cycle. The heat energy (in \(\mathrm{J}\) ) delivered by the engine to the low temperature reservoir, in a cycle. is

Given below are two statements : Statements \(I\) : Astronomical unit \((Au)\). Parsec \((Pc)\) and Light year \((ly)\) are units for measuring astronomical distances. Statements \(II:\) \(Au < Parsec ( Pc ) < ly\) In the light of the above statements. choose the most appropriate answer from the options given below:

Consider a planet in some solar system which has a mass double the mass of earth and density equal to the average density of earth. If the weight of an object on earth is \({W}\), then weight of the same object on that planet will be

A ball projected from ground at an angle of \(45^o\) just clears a wall in front. If point of projection is \(4\,m\) from the foot of wall and ball strikes the ground at a distance of \(6\,m\) on the other side of the wall, the height of the walI is  ........ \( m\)

A free electron of \(2.6\, {eV}\) energy collides with a \({H}^{+}\) ion. This results in the formation of a hydrogen atom in the first excited state and a photon is released. Find the frequency of the emitted photon. \(\left({h}=6.6 \times 10^{-34}\, {J} {s}\right)\)

A proton and an electron are associated with same de-Broglie wavelength. The ratio of their kinetic energies is _______. (Assume \(\mathrm{h}=6.63 \times 10^{-34} \mathrm{~J} \mathrm{~s}, \mathrm{~m}_{\mathrm{e}}=9.0 \times 10^{-31} \mathrm{~kg}\) and \(\mathrm{m}_{\mathrm{p}}=1836\) times \(\mathrm{m}_{\mathrm{e}}\) )

Given below are two statements: Statement \(I:\) For a planet, if the ratio of mass of the planet to its radius increases, the escape velocity from the planet also increases. Statement \(II:\) Escape velocity is independent of the radius of the planet. In the light of above statements, choose the most appropriate answer from the options given below on :

A light rope is wound around a hollow cylinder of mass \(5\,kg\) and radius \(70\,cm\). The rope is pulled with a force of \(52.5\,N\). The angular acceleration of the cylinder will be.....\( {rad \,s}{ }^{-2}\).

An engine operates by taking \(n\,moles\) of an ideal gas through the cycle \(ABCDA\) shown in figure. The thermal efficiency of the engine is : (Take \(C_v =1 .5\, R\), where \(R\) is gas constant)