As shown in the figure. a configuration of two equal point charges \(\left( q _0=+2 \mu C \right)\) is placed on an inclined plane. Mass of each point charge is \(20\,g\). Assume that there is no friction between charge and plane. For the system of two point charges to be in equilibrium (at rest) the height \(h = x \times 10^{-3}\,m\) The value of \(x\) is \(..........\).(Take \(\left.\frac{1}{4 \pi \varepsilon_0}=9 \times 10^9\,Nm ^2 C ^{-2}, g=10\,ms ^{-1}\right)\)

A poly-atomic molecule ( \(C_V=3 R, C_P=4 R\), where \(R\) is gas constant) goes from phase space point \(\mathrm{A}\left(\mathrm{P}_{\mathrm{A}}=10^5 \mathrm{~Pa}, \mathrm{~V}_{\mathrm{A}}=4 \times 10^{-6} \mathrm{~m}^3\right)\) to point \(\mathrm{B}\left(\mathrm{P}_{\mathrm{B}}=5 \times 10^4 \mathrm{~Pa}, \mathrm{~V}_{\mathrm{B}}=6 \times 10^{-6} \mathrm{~m}^3\right)\) to point \(\mathrm{C}\left(\mathrm{P}_{\mathrm{C}}=10^4\right.\) \(\left.\mathrm{Pa}, \mathrm{V}_{\mathrm{C}}=8 \times 10^{-6} \mathrm{~m}^3\right)\). A to B is an adiabatic path and \(B\) to \(C\) is an isothermal path. The net heat absorbed per unit mole by the system is :

The light waves from two coherent sources have same intensity \({I}_{1}={I}_{2}={I}_{0} .\) In interference pattern the intensity of light at minima is zero.What will be the intensity of light at maxima?

As shown in the diagram, when the incident ray is parallel to base of the prism, the emergent ray grazes along the second surface.

If refractive index of the material of prism is \(\sqrt{2}\), the angle \(\theta\) of prism is.

Two coils have mutual inductance \(0.002 \ \mathrm{H}\). The current changes in the first coil according to the relation \(\mathrm{i}=\mathrm{i}_0 \sin \omega \mathrm{t}\), where \(\mathrm{i}_0=5 \mathrm{~A}\) and \(\omega=50 \pi\) \(\mathrm{rad} / \mathrm{s}\). The maximum value of \(\mathrm{emf}\) in the second coil is \(\frac{\pi}{\alpha} \mathrm{V}\). The value of \(\alpha\) is_______.

The depth below the surface of sea to which a rubber ball be taken so as to decrease its volume by \(0.02 \%\) is ______ \(\mathrm{m}\). (Take density of sea water \(=10^3 \mathrm{kgm}^{-3}\), Bulk modulus of rubber \(=9 \times 10^8 \mathrm{Nm}^{-2}\), and \(\mathrm{g}=10 \mathrm{~ms}^{-2}\) )

A source of potential difference \(V\) is connected to the combination of two identical capacitors as shown in the figure. When key ' \(K\) ' is closed, the total energy stored across the combination is \(E _{1}\). Now key ' \(K\) ' is opened and dielectric of dielectric constant 5 is introduced between the plates of the capacitors. The total energy stored across the combination is now \(E _{2}\). The ratio \(E _{1} / E _{2}\) will be :

Let \(f(x)=4 \cos ^3 x+3 \sqrt{3} \cos ^2 x-10\). The number of points of local maxima of \(f\) in interval \((0,2 \pi)\) is:

In a Young's double slit experiment, the intensity at a point is \(\left(\frac{1}{4}\right)^{\text {th }}\) of the maximum intensity, the minimum distance of the point from the central maximum is _______ \(\mu \mathrm{m}\). (Given : \(\lambda=600 \mathrm{~nm}, d=1.0 \mathrm{~mm}, \mathrm{D}=1.0 \mathrm{~m}\) )

A body of mass \(8\,kg\) and another of mass \(2\, kg\) are moving with equal kinetic energy. The ratio of their respective momenta will be.

For a transparent prism, if the angle of minimum deviation is equal to its refracting angle, the refractive index n of the prism satisfies.

Given : \(f(x)=\left\{\begin{array}{ccc}{x} & {,} & {0 \leq x < \frac{1}{2}} \\ {\frac{1}{2}} & {,} & {x=\frac{1}{2}} \\ {1-x} & {,} & {\frac{1}{2} < x \leq 1}\end{array}\right.\) and \(g(x)=\left(x-\frac{1}{2}\right)^{2}, x \in R .\) Then the area (in sq. units) of the region bounded by the curves, \(y=f(x)\) and \(y=g(x)\) between the lines, \(2 \mathrm{x}=1\) and \(2 \mathrm{x}=\sqrt{3},\) is

A certain pressure \(P\) is applied to \(1\) litre of water and \(2\) litre of a liquid separately. Water gets compressed to \(0.01 \%\) whereas the liquid gets compressed to \(0.03 \%\). The ratio of Bulk modulus of water to that of the liquid is \(\frac{3}{x}\). The value of \(x\) is \(...........\)