A sample contains \(10^{-2}\, kg\) each of two substances A and \(B\) with half lives \(4 \,s\) and \(8 \,s\) respectively. The ratio of then atomic weights is \(1: 2\) The ratio of the amounts of \(A\) and \(B\) after \(16 \,s\) is \(\frac{x}{100}\). the value of \(x\) is........

In a medium the speed of light wave decreases to \(0.2\) times to its speed in free space The ratio of relative permittivity to the refractive index of the medium is \(x: 1\). The value of \(x\) is \(...........\) (Given speed of light in free space \(=3 \times 10^8 m s ^{-1}\) and for the given medium \(\mu_{ r }=1\) )

Steel ruptures when a shear of \(3 .5 \times 10^8\,\,N\,m^{-2}\) is applied. The force needed to punch a \(1\,cm\) diameter hole in a steel sheet \(0.3\,cm\) thick is nearly

An electric field, \(\overrightarrow{\mathrm{E}}=\frac{2 \hat{\mathrm{i}}+6 \hat{\mathrm{j}}+8 \hat{\mathrm{k}}}{\sqrt{6}}\) passes through the surface of \(4 \mathrm{~m}^2\) area having unit vector \(\hat{\mathrm{n}}=\left(\frac{2 \hat{\mathrm{i}}+\hat{\mathrm{j}}+\hat{\mathrm{k}}}{\sqrt{6}}\right)\). The electric flux for that surface is _______ \(\mathrm{Vm}\)

As shown in figure, when a spherical cavity (centred at \(\mathrm{O})\) of radius \(1\) is cut out of a uniform sphere of radius \(\mathrm{R} \text { (centred at } \mathrm{C}),\) the centre of mass of remaining (shaded) part of sphere is at \(G\), i.e, on the surface of the cavity. \(\mathrm{R}\) can be detemined by the equation

12 wires each having resistance \(2 \Omega\) are joined to form a cube. A battery of \(6 \mathrm{~V}\) emf is joined across point \(\mathrm{a}\) and \(\mathrm{c}\). The voltage difference between \(e\) and \(f\) is ______V.

A total of \(48 \mathrm{~J}\) heat is given to one mole of helium kept in a cylinder. The temperature of helium increases by \(2^{\circ} \mathrm{C}\). The work done by the gas is _______. (Given, \(\mathrm{R}=8.3 \mathrm{~J} \mathrm{~K}^{-1} \mathrm{~mol}^{-1}\).)

The torque of a force \(5 \hat{i}+3 \hat{j}-7 \hat{k}\) about the origin is \(\tau\). If the force acts on a particle whose position vector is \(2 \hat{i}+2 \hat{j}+\hat{k}\), then the value of \(\tau\) will be.

A thin prism with angle \( 5^{\circ} \) of refractive index 1.72 is combined with another prism of refractive index 1.9 to produce dispersion without deviation. The angle of second prism is ___________.

A \(12.5\, eV\) electron beam is used to bombard gaseous hydrogen at room temperature. It will emit

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

Given below are two statement  Statement \(-I\) : What \(\mu\) amount of an ideal gas undergoes adiabatic change from state \(\left( P _{1}, V _{1}, T _{1}\right)\) to state \(\left( P _{2}, V _{2}, T _{2}\right)\), the work done is \(W =\frac{1 R \left( T _{2}- T _{1}\right)}{1-\gamma}\), where \(\gamma=\frac{ C _{ P }}{ C _{ V }}\) and \(R =\) universal gas constant, Statement \(-II\) : In the above case. when work is done on the gas. the temperature of the gas would rise. Choose the correct answer from the options given below

Consider a finite insulated, uncharged conductor placed near a finite positively charged conductor. The uncharged body must have a potential