A fission reaction is given by ${}_{92}{}^{236}U\to {}_{54}{}^{140}Xe+{}_{38}{}^{94}Sr+x+y$ , where $x$ and $y$ are two particles. Considering ${}_{92}{}^{236}U$ to be at rest, the kinetic energies of the products are denoted by $K_{Xe},\mathrm{ }{K}_{Sr},\mathrm{ }{K}_x\left(2\mathrm{ }MeV\right)$ and $K_y\left(2\mathrm{ }MeV\right)$ , respectively. Let the binding energies per nucleon of ${}_{92}{}^{236}U,\mathrm{ }{}_{54}{}^{140}Xe\mathrm{ }$ and ${}_{38}{}^{94}Sr$ be 7.5 MeV, 8.5 MeV, and 8.5 MeV respectively. Considering different conservation laws, the correct option(s) is (are)

A heavy nucleus $Q$ of half-life $20$ minutes undergoes alpha-decay with probability of $60\%$  and beta-decay with probability of $40\%$. Initially, the number of $Q$ nuclei is $1000$. The number of alpha-decays of $Q$ in the first one hour is:

Three identical capacitors $C_1,C_2\mathrm{a}\mathrm{n}\mathrm{d}{C}_3$ have a capacitance of $1.0\mu F$ each and they are uncharged initially. They are connected in a circuit as shown in the figure and $C_1$ is then filled completely with a dielectric material of relative permittivity ${ϵ}_r$ . The cell electromotive force (emf) $V_0=8V$ . First the switch $S_1$ is closed while the switch $S_2$ is kept open. When the capacitor $C_3$ is fully charged, $S_1$ is opened and $S_2$ is closed simultaneously. When all the capacitors reach equilibrium, the charge on $C_3$ is found to be $5\mu C$ . The value of ${ϵ}_r$ =____________.

Consider a Vernier callipers in which each 1 cm on the main scale is divided into 8 equal divisions and a screw gauge with 100 divisions on its circular scale. In the Vernier callipers, 5 divisions of the Vernier scale coincide with 4 divisions on the main scale and in the screw gauge, one complete rotation of the circular scale moves it by two divisions on the linear scale. Then:

A bi-convex lens is formed with two thin plano-convex lenses as shown in the figure. Refractive index \(n\) of the first lens is \(1.5\) and that of the second lens is \(1.2\). Both the curved surface are of the same radius of curvature \(R=14\) \(\mathrm{cm}\). For this bi-convex lens, for an object distance of 40 \(\mathrm{cm}\), the image distance will be

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Light guidance in an optical fiber can be understood by considering a structure comprising of thin solid glass cylinder of refractive index \(n_{1}\) surrounded by a medium of lower refractive index \(n_{2}\). The light guidance in the structure takes place due to successive total internal reflections at the interface of the media \(n_{1}\) and \(n_{2}\) as shown in the figure. All rays with the angle of incidence \(i\) less than a particular value \(i_{m}\) are confined in the medium of refractive index \(n_{1}\). The numerical aperture (NA) of the structure is defined as \(\sin i_{m}\).



Question:

If two structures of same cross-sectional area, but different numerical apertures \(N A_{1}\) and \(N A_{2}\left(N A_{2} < N A_{1}\right)\) are joined longitudinally, the numerical aperture of the combined structure is

In plotting stress versus strain curves for two materials \(P\) and \(Q\), a student by mistake puts strain on the \(y\)-axis and stress on the \(x\)-axis as shown in the figure. Then the correct statement(s) is(are)

Statement 1 For practical purposes, the earth is used as a reference at zero potential in electrical circuits.
and Statement 2 The electrical potential of a sphere of radius \(R\) with charge \(Q\) uniformly distributed on the surface is given by \(\frac{Q}{4 \pi \varepsilon_0 R}\).

For a radioactive material, its activity $\mathrm{A}$ and rate of change of its activity $\mathrm{R}$ are defined as $\mathrm{A}=-\frac{\mathrm{d}\mathrm{N}}{\mathrm{d}\mathrm{t}}$ and $\mathrm{R}=-\frac{\mathrm{d}\mathrm{A}}{\mathrm{d}\mathrm{t}}$ , where $\mathrm{N}\left(\mathrm{t}\right)$ is the number of nuclei at time $\mathrm{t}$ . Two radioactive sources $\mathrm{P}$ (mean life $\mathrm{\tau }$ ) and $\mathrm{Q}$ (mean life $2\mathrm{\tau }$ ) have the same activity at $\mathrm{t}=0$ . Their rates of change of activities at $\mathrm{t}=2\mathrm{\tau }$ are ${\mathrm{R}}_{\mathrm{P}}$ and ${\mathrm{R}}_{\mathrm{Q}}$ , respectively. If $\frac{{\mathrm{R}}_{\mathrm{P}}}{{\mathrm{R}}_{\mathrm{Q}}}=\frac{\mathrm{n}}{\mathrm{e}}$ , then the value of $\mathrm{n}$ is

A cubical solid aluminum (bulk modulus \(=-\text V \frac{\text{dP}}{\text{dV}}=70\text{ GPa}\)) block has an edge length of 1 m on the surface of the earth. It is kept on the floor of a 5 km deep ocean. Taking the average density of water and the acceleration due to gravity to be \(10^3\text{ kg m} ^{-3}\) and \(10\text{ ms} ^{-2}\), respectively, the change in the edge length of the block in mm is _____________ .

A solid sphere of radius R and density ρ is attached to one end of a mass-less spring of force constant k. The other end of the spring is connected to another solid sphere of radius R and density 3 ρ The complete arrangement is placed in a liquid of density 2ρ and is allowed to reach equilibrium. The correct statement(s) is (are)

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The reaction of \(\mathrm{K}_{3}\left[\mathrm{Fe}(\mathrm{CN})_{6}\right]\) with freshly prepared \(\mathrm{FeSO}_{4}\) solution produces a dark blue precipitate called Turnbull's blue. Reaction of \(\mathrm{K}_{4}\left[\mathrm{Fe}(\mathrm{CN})_{6}\right]\) with the \(\mathrm{FeSO}_{4}\) solution in complete absence of air produces a white precipitate \(\mathbf{X}\), which turns blue in air. Mixing the \(\mathrm{FeSO}_{4}\) solution with \(\mathrm{NaNO}_{3}\), followed by a slow addition of concentrated \(\mathrm{H}_{2} \mathrm{SO}_{4}\) through the side of the test tube produces a brown ring.
Question:
Among the following, the brown ring is due to the formation of

The value of $\sum _{k=1}^{13}\frac{1}{\sin \left(\frac{\pi }{4}+\frac{\left(k-1\right)\pi }{6}\right)\sin \left(\frac{\pi }{4}+\frac{k\pi }{6}\right)}$ is equal to