A solid sphere of mass 5 kg and radius 10 cm is kept in contact with another solid sphere of mass 10 kg and radius 20 cm. The moment of inertia of this pair of spheres about the tangent passing through the point of contact is __________ \( kg.m^{2} \).

The width of fringe is \(2\,mm\) on the screen in a double slits experiment for the light of wavelength of \(400\,nm\). The width of the fringe for the light of wavelength \(600\,nm\) will be \(..............\,mm\)

Match List \(I\) with List \(II\) :

List \(I\)	List \(II\)
\(A\) Isothermal Process	\(I\) Work done by the gas decreases internal energy
\(B\) Adiabatic Process	\(II\) No change in internal energy
\(C\) Isochoric Process	\(III\) The heat absorbed goes partly to increase internal energy and partly to do work
\(D\) Isobaric Process	\(IV\) No work is done on or by the gas

Choose the correct answer from the options given below :

A thermodynamic system is taken from an original state \(A\) to an intermediate state \(B\) by a linear process as shown in the figure. It's volume is then reduced to the original value from \(B\) to \(C\) by an isobaric process. The total work done by the gas from \(A\) to \(B\) and \(B\) to \(C\) would be :

A piece of bone of an animal from a ruin is found to have \(^{14}C\) activity of \(12\) disintegrations per minute per gm of its carbon content. The \(^{14}C\) activity of a living animal is \(16\) disintegrations per minute per gm. How long ago nearly did the animal die? ............\(years\) (Given halflife of \(^{14}C\) is \(t_{1/2} = 5760\,years\) )

In the first configuration (1) as shown in the figure, four identical charges \(\left(q_0\right)\) are kept at the corners \(A, B, C\) and \(D\) of square of side length 'a'. In the second configuration (2), the same charges are shifted to mid points \(\mathrm{G}, \mathrm{E}, \mathrm{H}\) and F, of the square, If \(\mathrm{K}=\frac{1}{4 \pi \varepsilon_0}\), the difference between the potential energies of configuration (2) and (1) is given by :

A block of metal weighing \(2 \,kg\) is resting on a frictionless plane (as shown in figure). It is struck by a jet releasing water at a rate of \(1 \,kgs ^{-1}\) and at a speed of \(10 \,ms ^{-1}\). Then, the initial acceleration of the block, in \(ms ^{-2}\), will be .................

Wires \({W}_{1}\) and \({W}_{2}\) are made of same material having the breaking stress of \(1.25 \times 10^{9} \,{N} / {m}^{2}\) \({W}_{1}\) and \({W}_{2}\) have cross-sectional area of \(8 \times 10^{-7}\, {m}^{2}\) and \(4 \times 10^{-7}\, {m}^{2}\), respectively. Masses of \(20 \,{kg}\) and \(10\, {kg}\) hang from them as shown in the figure. The maximum mass that can be placed in the pan without breaking the wires is \(.....{kg}\) (Use \(\left.{g}=10\, {m} / {s}^{2}\right)\)

A transverse wave is represented by  \(y = \frac{{10}}{\pi }\,\sin \,\left( {\frac{{2\pi }}{T}t - \frac{{2\pi }}{\lambda }x} \right)\) For what value of the wavelength the wave velocity is twice the maximum particle velocity ..... \(cm\) ?

A quantity \(f\) is given by \(f=\sqrt{\frac{{hc}^{5}}{{G}}}\) where \(c\) is speed of light, \(G\) universal gravitational constant and \(h\) is the Planck's constant. Dimension of \(f\) is that of