A thin prism with angle \( 5^{\circ} \) of refractive index 1.72 is combined with another prism of refractive index 1.9 to produce dispersion without deviation. The angle of second prism is ___________.

Given below are two statements: one is labelled as Assertion \(A\) and the other is labelled as Reason \(R\). Assertion \(A :\) The escape velocities of planet \(A\) and \(B\) are same. But \(A\) and \(B\) are of unequal mass. Reason \(R :\) The product of their mass and radius must be same, \(M _{1} R _{1}= M _{2} R _{2}\) In the light of the above statements, choose the most appropriate answer from the options given below

At \(300\,K\), the rms speed of oxygen molecules is \(\sqrt{\frac{\alpha+5}{\alpha}}\) times to that of its average speed in the gas. Then, the value of \(\alpha\) will be (used \(\pi=\frac{22}{7}\) )

A parallel plate capacitor whose capacitance \(C\) is \(14\, pF\) is charged by a battery to a potential difference \(V =12\, V\) between its plates. The charging battery is now disconnected and a porcelin plate with \(k =7\) is inserted between the plates, then the plate would oscillate back and forth between the plates with a constant mechanical energy of \(..........pJ\). (Assume no friction)

Two circular coils \(P\) and \(Q\)of \(100\) turns each have same radius of \(\pi \mathrm{cm}\). The currents in \(\mathrm{P}\) and \(\mathrm{R}\) are \(1 \mathrm{~A}\) and \(2 \mathrm{~A}\) respectively. \(\mathrm{P}\) and \(\mathrm{Q}\) are placed with their planes mutually perpendicular with their centers coincide. The resultant magnetic field induction at the center of the coils is \(\sqrt{\mathrm{x}} \mathrm{mT}\), where \(\mathrm{x}=\) _______. \(\left[\text { Use } \mu_0=4 \pi \times 10^{-7} \mathrm{TmA}^{-1}\right]\)

A body is dropped on ground from a height \(h_1\) and after hitting the ground, it rebounds to a height \(h _2\) If the ratio of velocities of the body just before and after hitting ground is \(4\), then percentage loss in kinetic energy of the body is \(\frac{x}{4}\). The value of \(x\) is \(.......\)

The energy \(E\) and momentum \(p\) of a moving body of mass \(m\) are related by some equation. Given that c represents the speed of light, identify the correct equation.

An atom absorbs a photon of wavelength \(500\,nm\) and emits another photon of wavelength \(600\,nm\). The net energy absorbed by the atom in this process is \(n \times 10^{-4}\,eV\). The value of \(n\) is ............ [Assume the atom to be stationary during the absorption and emission process] \(\left(\right.\)Take \(h =6.6 \times 10^{-34}\,Js\) and \(\left.c =3 \times 10^8\,m / s \right)\)

For full scale deflection of total \(50\) divisions, \(50\,{mV}\) voltage is required in galvanometer. The resistance of galvanometer if its current sensitivity is \(2\, {div} / {mA}\) will be \(.....\Omega\)

A uniform bar of length 12 cm and mass 20 m lies on a smooth horizontal table. Two point masses m and 2 m are moving in opposite directions with same speed of v and in the same plane as the bar. These masses strike the bar simultaneously and get stuck to it. After collision the entire system is rotating with angular frequency \( \omega \). The ratio of v and \( \omega \) is:

If \(\overrightarrow{\mathrm{L}}\) and \(\overrightarrow{\mathrm{P}}\) represent the angular momentum and linear momentum respectively of a particle of mass ' m ' having position vector \(\overrightarrow{\mathrm{r}}=\mathrm{a}(\hat{\mathrm{i}} \cos \omega \mathrm{t}+\hat{\mathrm{j}} \sin \omega \mathrm{t})\). The direction of force is

The pressure \(P_{1}\) and density \(d_{1}\) of diatomic gas \(\left(\gamma=\frac{7}{5}\right)\) changes suddenly to \(P _{2}\left(> P _{1}\right)\) and \(d _{2}\) respectively during an adiabatic process. The temperature of the gas increases and becomes \(......\) times of its initial temperature.\(\left(\right.\) given \(\left.\frac{ d _{2}}{ d _{1}}=32\right)\)

A simple pendulum made of a bob of mass \(m\) and a metallic wire of negligible mass has time period \(2s\) at  \(T = 0\,^oC\) . If the temeprature of the wire is increased and the corresponding change in its time peirod is plotted against its temperature, the resulting graph is a line of slope \(S\). If the coefficient of linear expansion of metal is \(\alpha \) then the value of  \(S\) is