A proton and an electron are associated with same de-Broglie wavelength. The ratio of their kinetic energies is _______. (Assume \(\mathrm{h}=6.63 \times 10^{-34} \mathrm{~J} \mathrm{~s}, \mathrm{~m}_{\mathrm{e}}=9.0 \times 10^{-31} \mathrm{~kg}\) and \(\mathrm{m}_{\mathrm{p}}=1836\) times \(\mathrm{m}_{\mathrm{e}}\) )

Match the LIST-I with LIST-II

LIST-I	LIST-II
(A) Gravitational constant	(I) \(\left[\mathrm{LT}^{-2}\right]\)
(B) Gravitational potential energy	(II) \(\left[\mathrm{L}^2 \mathrm{~T}^{-2}\right]\)
(C) Gravitational potential	(III) \(\left[\mathrm{ML}^2 \mathrm{~T}^{-2}\right]\)
(D) Acceleration due to gravity	(IV) \(\left[\mathrm{M}^{-1} \mathrm{~L}^3 \mathrm{~T}^{-2}\right]\)

Choose the correct answer from the options given below :

A container contains a liquid with refractive index of 1.2 up to a height of 60 cm and another liquid having refractive index 1.6 is added to height H above first liquid. If viewed from above, the apparent shift in the position of bottom of container is 40 cm. The value of H is ______ cm.
(Consider liquids are immisible)

The internal energy of air in \(4 \mathrm{~m} \times 4 \mathrm{~m} \times 3 \mathrm{~m}\) sized room at 1 atmospheric pressure will be _______ \(\times 10^6 \mathrm{~J}\). (Consider air as diatomic molecule)

The linear mass density of a thin rod \(A B\) of length \(L\) varies from \(A\) to \(B\) as \(\lambda( x )=\lambda_{0}\left(1+\frac{ x }{ L }\right),\) where \(x\) is the distance from A. If \(M\) is the mass of the rod then its moment of inertia about an axis passing through \(A\) and perpendicular to the rod is\(......ML^{2}\)

A particle moves such that its position vector \(\overrightarrow{\mathrm{r}}(\mathrm{t})=\cos \omega \mathrm{t} \hat{\mathrm{i}}+\sin \omega \mathrm{t} \hat{\mathrm{j}}\) where \(\omega\) is a constant and \(t\) is time. Then which of the following statements is true for the velocity \(\overrightarrow{\mathrm{v}}(\mathrm{t})\) and acceleration \(\overrightarrow{\mathrm{a}}(\mathrm{t})\) of the particle

A particle of mass \(m\) moves in a circular orbit in a central potential field \(U ( r )= U _{0} r ^{4}\). If Bohr's quantization conditions are applied, radii of possible orbitals \(r _{ n }\) vary with \(n ^{1 / \alpha}\), where \(\alpha\) is ....... .

A small point of mass \(m\) is placed at a distance \(2 R\) from the centre ' \(O^{\prime}\) of a big uniform solid sphere of mass M and radius R . The gravitational force on ' m ' due to M is \(\mathrm{F}_1\). A spherical part of radius \(\mathrm{R} / 3\) is removed from the big sphere as shown in the figure and the gravitational force on m due to remaining part of M is found to be \(\mathrm{F}_2\). The value of ratio \(\mathrm{F}_1: \mathrm{F}_2\) is _______.

\(5\, beats/ second\) are heard when a turning fork is sounded with a sonometer wire under tension, when the length of the sonometer wire is either \(0.95\,m\) or \(1\,m\) . The frequency of the fork will be ... \(Hz\)