Three concentric spherical metallic shells \(X , Y\) and \(Z\) of radius \(a , b\) and c respectively \([ a < b < c ]\) have surface charge densities \(\sigma,-\sigma\) and \(\sigma\), respectively. The shells \(X\) and \(Z\) are at same potential. If the radii of \(X\) and \(Y\) are \(2\,cm\) and \(3\,cm\), respectively.The radius of shell \(Z\) is \(......cm\).

A monoatomic gas at pressure \(P\) and volume \(V\) is suddenly compressed to one eighth of its original volume. The final pressure at constant entropy will be \(.....P\)

A spherical surface of radius of curvature \(R\), separates air from glass (refractive index \(=1.5\) ). The centre of curvature is in the glass medium. A point object ' \(O\) ' placed in air on the optic axis of the surface, so that its real image is formed at ' \(I\) ' inside glass. The line OI intersects the spherical surface at P and \(\mathrm{PO}=\mathrm{PI}\). The distance PO equals to ________.

The fractional compression \(\left(\frac{\Delta \mathrm{V}}{\mathrm{V}}\right)\) of water at the depth of 2.5 km below the sea level is ______ \(\%\). Given, the Bulk modulus of water \(=2 \times 10^9 \mathrm{~N} \mathrm{~m}^{-2}\), density of water \(=10^3\) \(\mathrm{kg} \mathrm{m}^{-3}\), acceleration due to gravity \(=\mathrm{g}=10 \mathrm{~m} \mathrm{~s}^{-2}\).

Two identical bodies, projected with the same speed at two different angles cover the same horizontal range \(R\). If the time of flight of these bodies are \(5\) s and \(10\) s, respectively, then the value of \(R\) is ________ m. (Take \(g=10\) m/s\(^2\))

A small ball of mass \(M\) and density \(\rho\) is dropped in a viscous liquid of density \(\rho_0\). After some time, the ball falls with a constant velocity. What is the viscous force on the ball ?

The de-Broglie wavelength of a particle having kinetic energy \(E\) is \(\lambda\). How much extra energy must be given to this particle so that the de-Broglie wavelength reduces to \(75 \,\%\) of the initial value ?

A metal surface is illuminated by a radiation of wavelength \(4500 \; \mathring A\). The ejected photo-electron enters a constant magnetic field of \(2 \; mT\) making an angle of \(90^{\circ}\) with the magnetic field. If it starts revolving in a circular path of radius \(2 \; mm\), the work function of the metal is approximately ............. \(eV\)

If one wants to remove all the mass of the earth to infinity in order to break it up completely. The amount of energy that needs to be supplied will be \(\frac{x}{5}\, \frac{ GM ^{2}}{ R }\) where \(x\) is \(\quad\,..........\) (Round off to the Nearest Integer) (\(M\) is the mass of earth, \(R\) is the radius of earth, \(G\) is the gravitational constant)

A mixture of hydrogen and oxygen has volume \(2000 \; cm ^{3}\), temperature \(300 \; K\), pressure \(100 \; kPa\) and mass \(0.76 \; g\) The ratio of number of moles of hydrogen to number of moles of oxygen in the mixture will be

A cube having a side of 10 cm with unknown mass and 200 gm mass were hung at two ends of an uniform rigid rod of 27 cm long. The rod along with masses was placed on a wedge keeping the distance between wedge point and 200 gm weight as 25 cm. Initially the masses were not at balance. A beaker is placed beneath the unknown mass and water is added slowly to it. At given point the masses were in balance and half volume of the unknown mass was inside the water.
(Take the density of unknown mass is more than that of the water, the mass did not absorb water and water density is \(1 \mathrm{gm} / \mathrm{cm}^3\).) The unknown mass is ________ kg.

An electron, a proton and an alpha particle having the same kinetic energy are moving in circular orbits of radii \(r_e,r_p\) and \({r_\alpha }\) respectively in a uniform magnetic field \(B\). The relation between \(r_e,r_p\) and \(\;{r_\alpha }\) is

An ice cube of dimensions \(60\,cm \times 50\,cm \times 20\,cm\) is placed in an insulation box of wall thickness \(1\,cm\). The box keeping the ice cube at \(0^{\circ}\,C\) of temperature is brought to a room of temperature \(40^{\circ}\,C\). The rate of melting of ice is approximately. (Latent heat of fusion of ice is \(3.4 \times 10^{5}\,J\,kg ^{-1}\) and thermal conducting of insulation wall is \(0.05\,Wm ^{-10} C ^{-1}\) )