Two charged conducting spheres of radii \(a\) and \(b\) are connected to each other by a conducting wire. The ratio of charges of the two spheres respectively is _______.

Number of molecules in a volume of \(4\, cm ^{3}\) of a perfect monoatomic gas at some temperature \(T\) and at a pressure of \(2\, cm\) of mercury is close to \(?\) (Given, mean kinetic energy of a molecule (at \(T\) ) is \(4 \times 10^{-14}\) erg, \(g=980\, cm / s ^{2}\), density of mercury \(=13.6\, g / cm ^{3}\))

Given below are two statement : Statement \(I :\) If the number of turns in the coil of a moving coil galvanometer is doubled then the current sensitivity becomes double. Statement \(II :\) Increasing current sensitivity of a moving coil galvanometer by only increasing the number of turns in the coil will also increase its voltage sensitivity in the same ratio: In the light of the above statement, choose the correct answer from the options given below :

Particle \(A\) of mass \(m _{1}\) moving with velocity \((\sqrt{3} \hat{i}+\hat{j})\, ms ^{-1}\) collides with another particle \(B\) of mass \(m _{2}\) which is at rest initially. Let \(\overrightarrow{ V }_{1}\) and \(\overrightarrow{ V }_{2}\) be the velocities of particles \(A\) and \(B\) after collision respectively. If \(m _{1}=2\, m _{2}\) and after collision \(\overrightarrow{ V }_{1}=(\hat{ i }+\sqrt{3} \hat{ j })\, ms ^{-1},\) the angle between \(\overrightarrow{ V }_{1}\) and \(\overrightarrow{ V }_{2}\) is\(......^o\)

When \(M_1\) gram of ice at \(-10\,^oC\) (specific heat \(= 0.5\, cal\, g^{-1}\,^oC^{-1}\)) is added to \(M_2\) gram of water at \(50\,^oC\), finally no ice is left and the water is at \(0\,^oC\). The value of latent heat of ice, in \(cal\, g^{-1}\) is

A string of length \(1\, m\) and mass \(5\, g\) is fixed at both ends. The tension in the string is \(8.0\, N\). The string is set into vibration using an external vibrator of frequency \(100\, Hz\). The separation between successive nodes on the string is close to .... \(cm\)

A wire of resistance R is bent into an equilateral triangle and an identical wire is bent into \(a\) square. The ratio of resistance between the two end points of an edge of the triangle to that of the square is:

A body of mass \(1 kg\) begins to move under the action of a time dependent force \(\vec{F}=\left(t \hat{i}+3 t^2 \hat{ j }\right) N\). where \(\hat{i}\) and \(\hat{j}\) are the unit vectors along \(x\) and \(y\) axis. The 10.Power developed by above force, at the time \(t=2 s\). will be \(..............W\).

If the wavelength of the first member of Lyman series of hydrogen is \(\lambda\). The wavelength of the second member will be _______.

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

List \(I\)	List \(II\)
\(A\) Spring constant	\(I\) \((T ^{-1})\)
\(B\) Angular speed	\(II\) \((MT ^{-2})\)
\(C\) Angular momentum	\(III\) \((ML ^2)\)
\(D\) Moment of Inertia	\(IV\) \((ML ^2 T ^{-1})\)

Choose the correct answer from the options given below

A body of mass \(m\) is moving in a circular orbit of radius \(R\) about a planet of mass \(M\). At some instant, it splits into two equal masses. The first mass moves in a circular orbit of radius \(\frac{R}{2}\) , and the other mass, in a circular orbit of radius \(\frac{3R}{2}\). The difference between the final and initial total energies is

The frequency of revolution of the electron in Bohr's orbit varies with \(n\), the principal quantum number as _______.