Parallel rays of light of intensity $\mathrm{I}=912\mathrm{ }\mathrm{W}{\mathrm{m}}^{-2}$ are incident on a spherical black body kept in surroundings of temperature 300 K. Take Stefan-Boltzmann constant $\mathrm{\sigma }=5.7\times {10}^{-8}\mathrm{ }\mathrm{W}{\mathrm{m}}^{-2}\mathrm{ }{\mathrm{K}}^{-4}$ and assume that the energy exchange with the surroundings is only through radiation. The final steady state temperature of the black body is close to

Statement I The total translational kinetic energy of all the molecules of a given mass of an ideal gas is \(1.5\) times the product of its pressure and its volume.
Statement II The molecules of a gas collide with each other and the velocities of the molecules change due to the collision.

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When a particle is restricted to move along \(x\)-axis between \(x=0\) and \(x=a\), where \(a\) is of nanometer dimension, its energy can take only certain specific values. The allowed energies of the particle moving in such a restricted region, correspond to the formation of standing waves with nodes at its ends \(x=0\) and \(x=a\). The wavelength of this standing wave is related to the liner momentum \(p\) of the particle according to the de Broglie relation. The energy of the particle of mass \(m\) is related to its linear momentum as \(E=\frac{p^2}{2 m}\). Thus, the energy of the particle can be denoted by a quantum number \(n\) taking values \(1,2,3, \ldots(n=1\), called the ground state) corresponding to the number of loops in the standing wave.
Use the model described above to answer the following three questions for a particle moving in the line \(x=0\) to \(x=a\) [Take \(h=6.6 \times 10^{-34} \mathrm{Js}\) and \(e=1.6 \times 10^{-19} \mathrm{C}\) ]
Question:
The speed of the particle that can take discrete values is proportional to

In the given circuit, a charge of \(+80 \mu C\) is given to the upper plate of the \(4 \mu F\) capacitor. Then in the steady state, the charge on the upper plate of the \(3 \mu F\) capacitor is

A ball of mass \(0.2 \mathrm{~kg}\) rests on a vertical post of height \(5 \mathrm{~m}\). A bullet of mass \(0.01 \mathrm{~kg}\), travelling with a velocity \(v \mathrm{~m} / \mathrm{s}\) in a horizontal direction, hits the centre of the ball. After the collision, the ball and bullet travel independently. The ball hits the ground at a distance of \(20 \mathrm{~m}\) and the bullet at a distance of \(100 \mathrm{~m}\) from the foot of the post. The initial velocity \(v\) of the bullet is

For an isosceles prism of angle $A$ and refractive index $\mu$ it is found that, the angle of minimum deviation is ${ \delta }_m=A$. Which of the following options is/are correct?

In an experiment to determine the focal length \((f)\) of a concave mirror by the \(u-v\) method, a student places the object pin \(A\) on the principal axis at a distance \(x\) from the pole \(P\). The student looks at the pin and its inverted image from a distance keeping his/her eye in line with PA. When the student shifts his/her eye towards left, the image appears to the right of the object pin. Then

A solution of colourless salt \(\mathrm{H}\) on boiling with excess \(\mathrm{NaOH}\) produces a non-flammable gas. The gas evolution ceases after sometime. Upon addition of \(\mathrm{Zn}\) dust to the same solution, the gas evolution restarts. The colourless salt (s) \(\mathrm{H}\) is (are)

Two particles of mass \(m\) each are tied at the ends of a light string of length \(2 a\). The whole system is kept on a frictionless horizontal surface with the string held tight so that each mass is at a distance \(a\) from the centre \(P\) (as shown in the figure). Now, the mid-point of the string is pulled vertically upwards with a small but constant force \(F\). As a result, the particles move towards each other on the surface.
The magnitude of acceleration, when the separation between them becomes \(2 x\), is

A transparent slab of thickness d has a refractive index n (z) that increases with z. Here z is the vertical distance inside the slab, measured from the top. The slab is placed between two media with uniform refractive indices $n_{1}and{n}_2\left(>n_1\right),$ as shown in the figure. A ray of light is incident with angle ${\theta }_i$ from medium 1 and emerges in medium 2 with refraction angle ${\theta }_f$ with a lateral displacement $\mathcal{l}$ . Which of the following statement(s) is (are) true?

Match the conversions in Column I with the type(s) given in Column II. Indicate your answer by darkening the appropriate bubbles of the \(4 \times 4\) matrix given in the ORS.

One mole of an ideal gas expands adiabatically from an initial state $(T_A, V_0)$ to final state $(T_f, 5V_0)$. Another mole of the same gas expands isothermally from a different initial state $(T_B, V_0)$ to the same final state $(T_f, 5V_0)$. The ratio of the specific heats at constant pressure and constant volume of this ideal gas is $\gamma$. What is the ratio $\frac{T_A}{T_B}$?

In the experiment to determine the speed of sound using a resonance column: