\(\frac{x}{x+4}\) is the ratio of energies of photons produced due to transition of an electron of hydrogen atom from its \((i)\) third permitted energy level to the second level and \((ii)\) the highest permitted energy level to the second permitted level. The value of \(x\) will be.

A standing wave is formed by the superposition of two waves travelling in opposite directions. The transverse displacement is given by \(y\left( {x,t} \right) = 0.5\sin\, \left( {\frac{{5\pi }}{4}x} \right)\,\cos\, \left( {200\,\pi t} \right)\). What is the speed of the travelling wave moving in the positive \(x\) direction .... \(m/s\) ? (\(x\) and \(t\) are in meter and second, respectively.)

Region \(I\) and \(II\) are separated by a spherical surface of radius \(25\, {cm}\). An object is kept in region \(I\) at a distance of \(40\, {cm}\) from the surface. The distance of the image from the surface is \(.....\,cm\)

Let the moment of inertia of a hollow cylinder of length \(30\, cm\) (inner radius \(10\, cm\) and outer radius \(20\, cm\)), about its axis be \(I\). The radius of a thin cylinder of the same mass such that its moment of inertia about its axis is also \(I\), is ......... \(cm\).

Given below are two statements: one is labelled as Assertion A and the other is labelled as Reason R
Statement I: Change in internal energy of a system containing \(n\) mole of ideal gas can be written as \(\Delta U = n C_v (T_f - T_i) = \dfrac{nR}{\gamma - 1}(T_f - T_i)\), where \(\gamma = \dfrac{C_p}{C_v}\), \(T_i =\) initial temperature, \(T_f =\) final temperature.
Statement II: Relation between degree of freedom \(f\) and \(\gamma (= C_p/C_v)\) is \(\left(\gamma = 1 + \dfrac{2}{f}\right)\)
Choose the correct answer from the options given below

The dimension of stopping potential \(\mathrm{V}_{0}\) in photoelectric effect in units of Planck's constant \(h\), speed of light \(c\) and Gravitational constant \(G\) and ampere \(A\) is

A metal exposed to light of wavelength \(800\,nm\) and emits photoelectrons with a certain kinetic energy. The maximum kinetic energy of photo-electron doubles when light of wavelength \(500\, nm\) is used. The work function of the metal is \(...eV\) (Take \(hc =1230 \,eV - nm\) ).

The angular speed of truck wheel is increased from \(900\, rpm\) to \(2460\, rpm\) in \(26\) seconds. The number of revolutions by the truck engine during this time is (Assuming the acceleration to be uniform).

A particle is moving in a circular path of radius \(a,\) with a constant velocity \(v\) as shown in the figure.The centre of circle is marked by \('C'\). The angular momentum from the origin \(O\) can be written as

Two forces \(\vec{F}_1\) and \(\vec{F}_2\) are acting on a body. One force has magnitude thrice that of the other force and the resultant of the two forces is equal to the force of larger magnitude. The angle between \(\vec{F}_1\) and \(\overrightarrow{\mathrm{F}}_2\) is \(\cos ^{-1}\left(\frac{1}{\mathrm{n}}\right)\). The value of \(|\mathrm{n}|\) is __________.

A bi convex lens of focal length \(10\,cm\) is cut in two identical parts along a plane perpendicular to the principal axis. The power of each lens after cut is \(...........\,D\).

A gas based geyser heats water flowing at the rate of 5.0 litres per minute from \(27^{\circ} C\) to \(87^{\circ} C\).
The rate of consumption of the gas is ___________ \(g / s\).
(Take heat of combustion of gas \(=5.0 \times 10^4 J / g\) ) specific heat capacity of water \(=4200 J / kg.~{ }^{\circ} C\)