If two bodies of masses 2 kg and 3 kg are moving at right angles with velocities \(20 \mathrm{~m} \mathrm{~s}^{-1}\) and \(10 \mathrm{~m} \mathrm{~s}^{-1}\) respectively, then the velocity of the centre of mass of the system of the two bodies is

Two identical charged spheres separated by a distance repel each other with a force $F$. If $10 \%$ of electrons are transferred from one sphere to the other than the force between them becomes

If the magnetic field inside a solenoid is B, then the magnetic energy stored in it per unit volume is
(c - speed of light in vacuum and \(\varepsilon_0\) is permittivity of free space)

A magnetic field intensity at the centre of a circular wire of radius \(0.1 \mathrm{~m}\) carrying a current \(0.2 \mathrm{~A}\) is

A \(50 \mathrm{~g}\) ice cube at \(-10^{\circ} \mathrm{C}\) is added to \(200 \mathrm{~g}\) of water at \(30^{\circ} \mathrm{C}\). The final temperature of the mixture is (specific heat of water \(=1 \mathrm{cal} \mathrm{g}^{-1}{ }^{\circ} \mathrm{C}^{-1}\), latent heat of fusion of ice \(=80 \mathrm{cal} \mathrm{g}^{-1}\) specific heat of ice \(=0.5 \mathrm{cal} \mathrm{g}^{-1}\) \({ }^{\circ} \mathrm{C}^{-1}\) )

In S.I. system, the total energy of the free surface of a liquid drop is \(2 \pi\) times the surface tension of the liquid. The diameter of the drop is __________ .

The rms speed of oxygen molecule at a certain absolute temperature is ' \(v\) '. If the absolute temperature is doubled and the oxygen molecules dissociate into atomic oxygen, then the rms speed would be

\(\cos \frac{\pi}{2^2} \cdot \cos \frac{\pi}{2^3} \cdot \cos \frac{\pi}{2^4} \cdots \cos \frac{\pi}{2^{10}}=\)

The line \(x+y+2=0\) intersect the circle \(x^2+y^2+4 x-\) \(4 y-4=0\) in two points \(A\) and \(B\). Let \(S \equiv x^2+y^2+2 g x+\) \(2 \mathrm{fy}+\mathrm{c}=0\) be a different circle passing through the points \(A\) and \(B\). If the distance of the centre of \(S=0\) from \(A B\) is \(\sqrt{2}\), then \(\mathrm{g}+\mathrm{f}+\mathrm{c}=\)

If \(\alpha, \beta, \gamma\) are the roots of \(x^3+a x^2+b x+c=0\) then \(\sum \frac{1}{\alpha}=\ldots\), if \(\alpha, \beta, \gamma\) are non-zero

Below shown molecules are

An observer and a source emitting sound of frequency \(120 \mathrm{~Hz}\) are on the \(X\)-axis. The observer is stationary while the source of sound is in motion given by the equation \(x=3 \sin \omega t\) ( \(x\) is in metres and \(t\) is in seconds). If the difference between the maximum and minimum frequencies of the sound observed by the observers is \(22 \mathrm{~Hz}\), then the value of \(\omega\) is ( ppeed of sound in air \(=330 \mathrm{~ms}^{-1}\) )

When ' \(\mathrm{X}\) ' \(\mathrm{g}\) of graphite is completely burnt in a bomb calorimeter in excess of \(\mathrm{O}_2\) at \(298 \mathrm{~K}\) and \(1 \mathrm{~atm}\) pressure as given in the equation
\(\mathrm{C}\) (graphite) \(+\mathrm{O}_2(\mathrm{~g}) \rightarrow \mathrm{CO}_2(\mathrm{~g})\)
The temperature of calorimeter raised from \(298 \mathrm{~K}\) to \(302 \mathrm{~K}\). If the heat capacity of the calorimeter and molar enthalpy change for the reaction at \(1 \mathrm{~atm}\) and \(298 \mathrm{~K}\) are \(20.7 \mathrm{~kJ} \mathrm{~K}^{-1}\) and \(-248.4 \mathrm{~kJ} \mathrm{~mol}^{-1}\), ' \(\mathrm{X}^{\prime}\) in \(\mathrm{g}\) is