The distance between two plates of a capacitor is \(d\) and its capacitance is \(C _1\), when air is the medium between the plates. If a metal sheet of thickness \(\frac{2 d }{3}\) and of same area as plate is introduced between the plates, the capacitance of the capacitor becomes \(C _2\). The ratio \(\frac{ C _2}{ C _1}\) is:

A ball of mass \(10\, kg\) moving with a velocity \(10 \sqrt{3} m / s\) along the \(x-\)axis, hits another ball of mass \(20\, kg\) which is at rest. After the collision, first ball comes to rest while the second ball disintegrates into two equal pieces. One piece starts moving along \(y-\)axis with a speed of \(10\) \(m / s\). The second piece starts moving at an angle of \(30^{\circ}\) with respect to the \(x-\)axis. The velocity of the ball moving at \(30^{\circ}\) with \(x-\) axis is \(x\, m / s\). The configuration of pieces after collision is shown in the figure below. The value of \(x\) to the nearest integer is

\(\vec{A}\) is a vector quantity such that \(|\vec{A}|=\) nonzero constant. Which of the following expressions is true for \(\vec{A}\) \(?\)

A thin prism \(P_1\) with an angle \(6^{\circ}\) and made of glass of refractive index \(1.54\) is combined with another prism \(P _2\) made from glass of refractive index \(1.72\) to produce dispersion without average deviation. The angle of prism \(P_2\) is \(........{}^{\circ}\)

A ring is hung on a nail. It can oscillate, without slipping or sliding \((i)\) in its plane with a time period \(T_{1}\) and, \((ii)\) back and forth in a direction perpendicular to its plane, with a period \(T _{2}\). the ratio \(\frac{ T _{1}}{ T _{2}}\) will be 

If three moles of monoatomic gas \(\left(\gamma=\frac{5}{3}\right)\) is mixed with two moles of a diatomic gas \(\left(\gamma=\frac{7}{5}\right)\), the value of adiabatic exponent \(\gamma\) for the mixture is _______.

A positive charge \('q'\) of mass \('m'\) is moving along the \(+ x\) axis. We wish to apply a uniform magnetic field \(B\) for time \(\Delta t\) so that the charge reverses its direction crossing the \(y\) axis at a distance \(d.\) Then

A particle of charge \(-q\) and mass \(m\) moves in a circle of radius \(r\) around an infinitely long line charge of linear density \(+\lambda\). Then time period will be given as _______. (Consider \(k\) as Coulomb's constant)

The energy levels of an atom is shown is figure. Which one of these transitions will result in the emission of a photon of wavelength \(124.1\,nm\) ?Given \(\left( h =6.62 \times 10^{-34}\,Js \right)\)

A large number of water drops, each of radius \(r\), combine to have a drop of radius \(R\). If the surface tension is \(T\) and mechanical equivalent of heat is \(J ,\) the rise in heat energy per unit volume will be

The kinetic energy of emitted electron is \(E\) when the light incident on the metal has wavelength \(\lambda\). To double the kinetic energy, the incident light must have wavelength.

A thin tube sealed at both ends is \(100\, cm\) long. It lies horizontally, the middle \(20\, cm\) containing mercury and two equal ends containing air at standard atmospheric pressure . If the tube is now turned to a vertical position, by what amount will the mercury be displaced ? (Given : cross-section of the tube can be assumed to be uniform) ........ \(cm\)