In a scattering experiment, a particle of mass \(2 m\) collides with another particle of mass \(m\), which is initially at rest. Assuming the collision to be perfectly elastic, the maximum angular deviation \(\theta\) of the heavier particle, as shown in the figure, in radians is:

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.

One mole of an ideal gas undergoes two different cyclic processes $\mathrm{I}$ and $\mathrm{II}$, as shown in the $P-V$ diagrams below. In cycle $\mathrm{I}$, processes $a, b, c$ and $d$ are isobaric, isothermal, isobaric and isochoric, respectively. In cycle $\mathrm{II}$, processes $a\text{'}, b\text{'}, c\text{'}$ and $d\text{'}$ are isothermal, isochoric, isobaric and isochoric, respectively. The total work done during cycle $\mathrm{I}$ is $W_{\mathrm{I}}$ and that during cycle $\mathrm{II}$ is $W_{\mathrm{II}}$. The ratio $\frac{W_{\mathrm{I}}}{W_{\mathrm{II}}}$ is _____.

Gravitational acceleration on the surface of a planet is \(\frac{\sqrt{6}}{11} \mathrm{~g}\), where \(g\) is the gravitational acceleration on the surface of the earth. The average mass density of the planet is \(\frac{2}{3}\) times that of the earth. If the escape speed on the surface of the earth is taken to be 11 \(\mathrm{kms}^{-1}\), the escape speed on the surface of the planet in \(\mathrm{kms}^{-1}\) will be

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?

Two beams of red and violet colours are made to pass separately through a prism (angle of the prism is \(60^{\circ}\) ). In the position of minimum deviation, the angle of refraction will be

Look at the drawing given in the figure, which has been drawn with ink of uniform line-thickness.

The mass of ink used to draw each of the two inner circles, and each of the two line segments is \(m\). The mass of the ink used to draw the outer circle is \(6 \mathrm{~m}\). The coordinates of the centres of the different parts are, outer circle 10 , \(0)\), left inner circle \((0,0)\), right inner circle \((a, a)\), vertical line \((0,0)\) and horizontal line \((0,-a)\). The \(y\)-coordinate of the centre of mass of the ink in this drawing is

Consider a disc rotating in the horizontal plane with a constant angular speed \(\omega\) about its centre \(\mathrm{O}\). The disc has a shaded region on one side of the diameter and an unshaded region on the other side as shown in the figure. When the disc is in the orientation as shown, two pebbles \(P\) and \(Q\) are simultaneously projected at an angle towards \(R\). The velocity of projection is in the \(y\) - \(z\) plane and is same for both pebbles with respect to the disc. Assume that (i) they land back on the disc before the disc has completed \(1 / 8\) rotation, (ii) their range is less than half the disc radius, and (iii) \(\omega\) remains constant throughout. Then

A hydrogen atom, initially at rest in its ground state, absorbs a photon of frequency \(v_1\) and ejects the electron with a kinetic energy of 10 eV . The electron then combines with a positron at rest to form a positronium atom in its ground state and simultaneously emits a photon of frequency \(v_2\). The center of mass of the resulting positronium atom moves with a kinetic energy of 5 eV . It is given that positron has the same mass as that of electron and the positronium atom can be considered as a Bohr atom, in which the electron and the positron orbit around their center of mass. Considering no other energy loss during the whole process, the difference between the two photon energies (in eV ) is \(\qquad\)

A small object is placed at the centre of a large evacuated hollow spherical container. Assume the container is maintained at $0 \mathrm{K}$. At time $t=0$, the temperature of the object is $200 \mathrm{K}$. The temperature of the object becomes $100 \mathrm{K}$ at $t=t_1$ and $50 \mathrm{K}$ at $t=t_2$. Assume the object and the container to be ideal black bodies. The heat capacity of the object does not depend on temperature. The ratio $\left(\frac{t_2}{t_1}\right)$ _________ .

Tangents are drawn from the point \((17,7)\) to the circle \(x^2+y^2=169\).
Statement I The tangents are mutually perpendicular.
Statement II The locus of the points from which mutually perpendicular tangents can be drawn to the given circle is \(x^2+y^2=338\)

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The key feature of Bohr's theory of spectrum of hydrogen atom is the quantization of angular momentum when an electron is revolving around a proton. We will extend this to a general rotational motion to find quantized rotational energy of a diatomic molecule assuming it to be rigid. The rule to be applied is Bohr's quantization condition.
Question:
It is found that the excitation frequency from ground to the first excited state of rotation for the \(\mathrm{CO}\) molecule is close to \(\frac{4}{\pi} \times 10^{11} \mathrm{~Hz}\). Then the moment of inertia of CO molecule about its centre of mass is close to (Take \(h=2 \pi \times 10^{-34} J-s\) )

Paragraph : In electromagnetic theory, the electric and magnetic phenomena are related to each other. Therefore, the dimensions of electric and magnetic quantities must also be related to each other. In the questions below, $\left[E\right] \mathrm{a}\mathrm{n}\mathrm{d} \left[B\right]$ stand for dimensions of electric and magnetic fields respectively, while $\left[{\epsilon }_0\right]$ and $\left[{\mu }_0\right]$ stand for dimensions of the permittivity and permeability of free space respectively. $\left[L\right]\mathrm{a}\mathrm{n}\mathrm{d} \left[T\right]$ are dimensions of length and time respectively. All the quantities are given in $SI$ units

Question : The relation between $\left[{ϵ}_0\right] \mathrm{a}\mathrm{n}\mathrm{d} \left[{\mu }_0\right]$ is