A spherical bubble inside water has radius $R$. Take the pressure inside the bubble and the water pressure to be $p_0.$ The bubble now gets compressed radially in an adiabatic manner so that its radius becomes $(R-a)$. For $a\ll R$ the magnitude of the work done in the process is given by $\left(4\pi P_{0}R{a}^2\right)X,$ where $X$ is a constant and $\gamma =C_p/C_V=41/30$. The value of $X$ is______

Two transparent media of refractive indices \(\mu_1\) and \(\mu_3\) have a solid lens shaped transparent material of refractive index \(\mu_2\) between them as shown in figures in Column II. A ray traversing these media is also shown in the figures. In Column I different relationships between \(\mu_1, \mu_2\) and \(\mu_3\) are given. Match them to the ray diagram shown in Column II.

An ideal gas is expanding such that \(p T^2=\) constant. The coefficient of volume expansion of the gas is

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The mass of a nucleus \({ }_{Z}^{A} X\) is less than the sum of the masses of \((A-Z)\) number of neutrons and \(Z\) number of protons in the nucleus. The energy equivalent to the corresponding mass difference is known as the binding energy of the nucleus. A heavy nucleus of mass \(M\) can break into two light nuclei of masses \(m_{1}\) and \(m_{2}\) only if \(\left(m_{1}+m_{2}\right) \lt M\). Also two light nuclei of masses \(m_{3}\) and \(m_{4}\) can undergo complete fusion and form a heavy nucleus of mass \(M^{\prime}\) only if \(\left(m_{3}+m_{4}\right) \gt M^{\prime}\). The masses of some neutral atoms are given in the table below:


\(\left(1 u=932 M e V / c^{2}\right)\)

Question:
The kinetic energy (in \(k e V\) ) of the alpha particle, when the nucleus \({ }_{84}^{210} P o\) at rest undergoes alpha decay, is

A charged particle is introduced at the origin $x=0, y=0, z=0$ with a given initial velocity $\overrightarrow{v}$ . A uniform electric field $\overrightarrow{E}$ and a uniform magnetic field $\overrightarrow{B}$ exist everywhere. The velocity $\overrightarrow{v}$ , electric field $\overrightarrow{E}$ and a uniform magnetic field $\overrightarrow{B}$ are given in the columns below

Column 1	Column 2	Column 3
1. Electron with $\overrightarrow{v}=2\frac{E_0}{B_0}\widehat{x}$	(i) $\overrightarrow{E}=E_0\widehat{z}$	(P) $\overrightarrow{B}=-B_0\widehat{x}$
2. Electron with $\overrightarrow{v}=2\frac{E_0}{B_0}\widehat{y}$	(ii) $\overrightarrow{E}=-E_0\widehat{y}$	(Q) $\overrightarrow{B}=B_0\widehat{x}$
3.Proton with $\overrightarrow{v}$ =0	(iii) $\overrightarrow{E}={-E}_0\widehat{x}$	(R) $\overrightarrow{B}=-B_0\widehat{y}$
4. Proton with $\overrightarrow{v}=2\frac{E_0}{B_0}\widehat{x}$	(iv) $\overrightarrow{E}=E_0\widehat{x}$	(S) $\overrightarrow{B}=-B_0\widehat{z}$

In which case would the particle move in a straight line along the negative direction of y axis

A glass beaker has a solid, plano-convex base of refractive index 1.60, as shown in the figure. The radius of curvature of the convex surface (SPU) is \(9 \mathrm{~cm}\), while the planar surface (STU) acts as a mirror. This beaker is filled with a liquid of refractive index \(n\) up to the level QPR. If the image of a point object \(\mathrm{O}\) at a height of \(h\) (OT in the figure) is formed onto itself, then, which of the following option(s) is(are) correct?

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The figure shows a surface \(X Y\) separating two transparent media, medium-1 and medium-2. The lines \(a b\) and \(c d\) represent wavefronts of a light wave travelling in medium-1 and incident on \(X Y\). The lines ef and \(g h\) represent wavefronts of the light wave in medium-2 after refraction.
Question:
The phases of the light wave at \(c, d, e\) and \(f\) are \(\phi_c, \phi_d, \phi_e\) and \(\phi_f\), respectively. It is given that \(\phi_c \neq \phi_f\).

A container of fixed volume has a mixture of one mole of hydrogen and one mole of helium in equilibrium at temperature $\mathrm{T}$ . Assuming the gases are ideal, the correct statement(s) is (are)

A particle of unit mass is moving along the x-axis under the influence of a force and its total energy is conserved. Four possible forms of the potential energy of the particle are given in column I (a and ${\mathrm{U}}_0$ are constants). Match the potential energies in column I to the corresponding statement(s) in column II.

	Column I		Column II
A.	${\mathrm{U}}_1\left(\mathrm{x}\right)=\frac{{\mathrm{U}}_0}{2}{\left[1-{\left(\frac{\mathrm{x}}{\mathrm{a}}\right)}^2\right]}^2$	P.	The force acting on the particle is zero at $\mathrm{x}\mathrm{}=\mathrm{}\mathrm{a}$
B.	${\mathrm{U}}_2\left(\mathrm{x}\right)=\frac{{\mathrm{U}}_0}{2}{\left(\frac{\mathrm{x}}{\mathrm{a}}\right)}^2$	Q.	The force acting on the particle is zero at $\mathrm{x}\mathrm{}=\mathrm{}0$ .
C.	${\mathrm{U}}_3\left(\mathrm{x}\right)=\frac{{\mathrm{U}}_0}{2}{\left(\frac{\mathrm{x}}{\mathrm{a}}\right)}^2\exp \left[-{\left(\frac{\mathrm{x}}{\mathrm{a}}\right)}^2\right]$	R.	The force acting on the particle is zero at $\mathrm{x}=\mathrm{}-\mathrm{a}$
D.	${\mathrm{U}}_4\left(\mathrm{x}\right)=\frac{{\mathrm{U}}_0}{2}\left[\frac{\mathrm{x}}{\mathrm{a}}-\frac{1}{3}{\left(\frac{\mathrm{x}}{\mathrm{a}}\right)}^3\right]$	S.	The particle experiences an attractive force towards $\mathrm{x}\mathrm{}=\mathrm{}0$ in the region $\left|\mathrm{x}\right|<\mathrm{a}$
		T.	The particle with total energy $\frac{{\mathrm{U}}_0}{4}$ can oscillate about the point $\mathrm{x}=\mathrm{}-\mathrm{a}$

One Indian and four American men and their wives are to be seated randomly around a circular table. Then, the conditional probability that the Indian man is seated adjacent to his wife given that each American man is seated adjacent to his wife, is

The number of hexagonal faces that are present in a truncated octahedron is

Two solid spheres \(A\) and \(B\) of equal volumes but of different densities \(d_A\) and \(d_B\) are connected by a string. They are fully immersed in a fluid of density \(d_F\). They get arranged into an equilibrium state as shown in the figure with a tension in the string. The arrangement is possible only if

Paragraph I: An organic compound \(\mathbf{P}\) with molecular formula \(\mathrm{C}_9 \mathrm{H}_{18} \mathrm{O}_2\) decolorizes bromine water and also shows positive iodoform test. \(\mathbf{P}\) on ozonolysis followed by treatment with \(\mathrm{H}_2 \mathrm{O}_2\) gives \(\mathbf{Q}\) and \(\mathbf{R}\). While compound \(\mathbf{Q}\) shows positive iodoform test, compound \(\mathbf{R}\) does not give positive iodoform test. \(\mathbf{Q}\) and \(\mathbf{R}\) on oxidation with pyridinium chlorochromate (PCC) followed by heating give \(\mathbf{S}\) and \(\mathbf{T}\), respectively. Both \(\mathbf{S}\) and \(\mathbf{T}\) show positive iodoform test.
Complete copolymerization of 500 moles of \(\mathbf{Q}\) and 500 moles of \(\mathbf{R}\) gives one mole of a single acyclic copolymer \(\mathbf{U}\).
[Given, atomic mass: \(\mathrm{H}=1, \mathrm{C}=12, \mathrm{O}=16\)]
Question: Sum of number of oxygen atoms in \(\mathbf{S}\) and \(\mathbf{T}\) is