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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:
Light travels as a

A planet of radius $\mathrm{R}=\frac{1}{10}\times$ (radius of Earth) has the same mass density as Earth. Scientists dig a well of depth $\frac{\mathrm{R}}{5}$ on it and lower a wire of the same length and of linear mass density ${10}^{-3}\mathrm{ }\mathrm{k}\mathrm{g}{\mathrm{m}}^{-1}$ into it. If the wire is not touching anywhere, the force applied at the top of the wire by a person holding it in place is (take the radius of Earth $=6\times {10}^6\mathrm{ }\mathrm{m}$ and the acceleration due to gravity of Earth is $10\mathrm{ }\mathrm{m}{\mathrm{s}}^{-2}$ )

A circular disc of radius $R$ carries surface charge density $\sigma \left(r\right)={\sigma }_0\left(1-\frac{r}{R}\right)$, where ${\sigma }_0$ is a constant and $r$ is the distance from the center of the disc. Electric flux through a large spherical surface that encloses the charged disc completely is ${\varphi }_0$. Electric flux through another spherical surface of radius $\frac{R}{4}$ and concentric with the disc is $\varphi$. Then the ratio $\frac{{\varphi }_0}{\varphi }$ is

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A thermally insulating cylinder has a thermally insulating and frictionless movable partition in the middle, as shown in the figure below. On each side of the partition, there is one mole of an ideal gas, with specific heat at constant volume, \(C_{V}=2 R\). Here, \(R\) is the gas constant. Initially, each side has a volume \(V_{0}\) and temperature \(T_{0}\). The left side has an electric heater, which is turned on at very low power to transfer heat \(Q\) to the gas on the left side. As a result the partition moves slowly towards the right reducing the right side volume to \(V_{0} / 2\). Consequently, the gas temperatures on the left and the right sides become \(T_{L}\) and \(T_{R}\), respectively. Ignore the changes in the temperatures of the cylinder, heater and the partition.

Question :
The value of $\frac{T_R}{T_0}$ is:

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______

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 $\left[E\right]$ and $\left[B\right]$ is

The figures below depict two situations in which two infinitely long static line charges of constant positive line charge density $\mathrm{\lambda }$ are kept parallel to each other. In their resulting electric field, point charges $\mathrm{q}$ and $-\mathrm{q}$ are kept in equilibrium between them. The point charges are confined to move in the $\mathrm{x}$ direction only. If they are given a small displacement about their equilibrium positions, then the correct statements(s) is (are)

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The capacitor of capacitance \(C\) can be charged (with the help of a resistance \(R\) ) by a voltage source \(V\), by closing switch \(S_1\) while keeping switch \(S_2\) open. The capacitor can be connected in series with an inductor \(L\) by closing switch \(S_2\) and opening \(S_1\).

Question :
After the capacitor gets fully charged, \(S_1\) is opened and \(S_2\) is closed so that the inductor is connected in series with the capacitor. Then,

Function \(x=A \sin ^2 \omega t+B \cos ^2 \omega t+C\) \(\sin \omega t \cos \omega t\) represents SHM.

Let \(f:\left[0, \frac{\pi}{2}\right] \rightarrow[0,1]\) be the function defined by \(f(x)=\sin ^2 x\) and let \(g:\left[0, \frac{\pi}{2}\right] \rightarrow[0, \infty)\) be the function defined by \(g(x)=\sqrt{\frac{\pi x}{2}-x^2}\).
The value of \(\frac{16}{\pi^3} \int_0^{\frac{\pi}{2}} f(x) g(x) d x\) is

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Consider the polynomial \(f(x)=1+2 x+3 x^2+4 x^3\). Let \(s\) be the sum of all distinct real roots of \(f(x)\) and let \(t=|s|\).Question:
The area bounded by the curve \(y=f(x)\) and the lines \(x=0, y=0\) and \(x=t\), lies in the interval

A small particle of mass $\mathrm{m}$ moving inside a heavy, hollow and straight tube along the tube axis, undergoes elastic collision at two ends. The tube has no friction and it is closed at one end by a flat surface while the other end is fitted with a heavy movable flat piston as shown in figure. When the distance of the piston from closed end is $\mathrm{L}={\mathrm{L}}_0$ the particle speed is $\mathrm{v}={\mathrm{v}}_0.$ The piston is moved inward at a very low speed $\mathrm{V}$ such that $\mathrm{V}<<\frac{\mathrm{d}\mathrm{L}}{\mathrm{L}}{\mathrm{v}}_0,$ where $\mathrm{d}\mathrm{L}$ is the infinitesimal displacement of the piston. Which of the following statement(s) is/are correct?

The minimum number of times a fair coin needs to be tossed so that the probability of getting at least two heads is at least $0.96$, is