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A small spherical monoatomic ideal gas bubble \(\left(\gamma=\frac{5}{3}\right)\) is trapped inside a liquid of density \(\rho_1\) (see figure). Assume that the bubble does not exchange any heat with the liquid. The bubble contains \(n\) moles of gas. The temperature of the gas when the bubble is at the bottom is \(T_0\), the height of the liquid is \(H\) and the atmospheric pressure is \(p_0\) (Neglect surface tension).

Question:
When the gas bubble is at height \(y\) from the bottom, its temperature is

Most materials have the refractive index, \(n>1\). So, when a light ray from air enters a naturally occurring material, then by Snell's

law, \(\frac{\sin \theta_{1}}{\sin \theta_{2}}=\frac{n_{2}}{n_{1}}\), it is understood that the refracted ray bends towards the normal. But it never emerges on the same side of the normal as the incident ray. According to electromagnetism, the refractive index of the medium is given by the relation, \(\mathrm{n}=c / v=\pm \sqrt{\varepsilon_{r} \mu_{r}}\), where \(c\) is the speed of electromagnetic waves in vacuum, \(v\) its speed in the medium, \(\varepsilon_{r}\) and \(\mu_{r}\) are the relative permittivity and permeability of the medium respectively. In normal materials, both \(\varepsilon_{r}\) and \(\mu_{r}\), are positive, implying positive \(n\) for the medium. When both \(\varepsilon_{r}\) and \(\mu_{r}\) are negative, one must choose the negative root of \(n\). Such negative refractive index materials can now be artificially prepared and are called metamaterials. They exhibit significantly different optical behavior, without violating any physical laws. Since \(n\) is negative, it results in a change in the direction of propagation of the refracted light. However, similar to normal materials, the frequency of light remains unchanged upon refraction even in meta-materials.
Question : Choose the correct statement.

The image of an object, formed by a plano - convex lens at a distance of 8 m behind the lens, is real and is one-third the size of the object. The wavelength of light inside the lens is  $\frac{2}{3}$ times the wavelength in free space. The radius of the curved surface of the lens is

Two conducting cylinders of equal length but different radii are connected in series between two heat baths kept at temperatures $T_1=300\mathrm{K}\mathrm{a}\mathrm{n}\mathrm{d}{T}_2=100\mathrm{K}$as shown in the figure. The radius of the bigger cylinder is twice that of the smaller one and the thermal conductivities of the materials of the smaller and the larger cylinders are $K_1\mathrm{a}\mathrm{n}\mathrm{d}{K}_2$, respectively. If the temperature at the junction of the two cylinders in the steady state is $200\mathrm{}\mathrm{K}$, then $K_1/K_2$is

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A small block of mass \(1 \mathrm{~kg}\) is released from rest at the top of a rough track. The track is a circular arc of radius \(40 \mathrm{~m}\). The block slides along the track without toppling and a frictional force acts on it in the direction opposite to the instantaneous velocity. The work done in overcoming the friction up to the point \(Q\), as shown in the figure below, is \(150 \mathrm{~J}\). (Take the acceleration due to gravity, \(g=10 \mathrm{~m} \mathrm{~s}^{-2}\) ).


Question :
The magnitude of the normal reaction that acts on the block at the point \(Q\) is

Which of the following statement(s) is (are) correct about the spectrum of hydrogen atom?

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A thermal power plant produces electric power of \(600 \mathrm{~kW}\) at \(4000 \mathrm{~V}\), which is to be transported to a place \(20 \mathrm{~km}\) away from the power plant for consumers' usage. It can be transported either directly with a cable of large current carrying capacity or by using a combination of step-up and step-down transformers at the two ends. The drawback of the direct transmission is the large energy dissipation. In the method using transformers, the dissipation is much smaller. In this method, a step-up transformer is used at the plant side so that the current is reduced to a smaller value. At the consumers' end, a step-down transformer is used to supply power to the consumers at the specified lower voltage. It is reasonable to assume that the power cable is purely resistive and the transformers are ideal with a power factor unity. All the currents and voltages mentioned are rms values.


Question:

If the direct transmission method with a cable of resistance \(0.4 \Omega \mathrm{km}^{-1}\) is used, the power dissipation (in %) during transmission is

A planar structure of length $\mathrm{L}$ and width $\mathrm{W}$ is made of two different optical media of refractive indices ${\mathrm{n}}_1=1.5$ and ${\mathrm{n}}_2=1.44$ as shown in figure. If $\mathrm{L}>>\mathrm{W},$ a ray entering from end $\mathrm{A}\mathrm{B}$ will emerge from end $\mathrm{C}\mathrm{D}$ only if the total internal reflection condition is met inside the structure. For $\mathrm{L}=9.6\mathrm{}\mathrm{m},$ if the incident angle $\mathrm{\theta }$ is varied, the maximum time taken by a ray to exit the plane $\mathrm{C}\mathrm{D}$ is $\mathrm{t}\times {10}^{-9}\mathrm{s},$ where $\mathrm{t}$ is ____.
[Speed of light $\mathrm{c}=3\times {10}^8\mathrm{m}/\mathrm{s}]$

A rectangular conducting loop of length $4 \mathrm{cm}$ and width $2 \mathrm{cm}$ is in the $xy$-plane, as shown in the figure. It is being moved away from a thin and long conducting wire along the direction $\frac{\sqrt{3}}{2}\stackrel{\wedge }{x}+\frac{{\displaystyle 1}}{{\displaystyle 2}}\stackrel{\wedge }{y}$ with a constant speed $v$. The wire is carrying a steady current $I=10 \mathrm{A}$ in the positive $x$-direction. A current of $10 \mu \mathrm{A}$ flows through the loop when it is at a distance $d=4 \mathrm{cm}$ from the wire. If the resistance of the loop is $0.1 \Omega$, then the value of $v$ is _____ $\mathrm{m} {\mathrm{s}}^{-1}$.
[Given: The permeability of free space ${\mu }_0=4\pi \times {10}^{-7} \mathrm{N} {\mathrm{A}}^{-2}$]

A circuit with an electrical load having impedance \(Z\) is connected with an AC source as shown in the diagram. The source voltage varies in time as \(V(t)=300 \sin (400 t) \mathrm{V}\), where \(t\) is time in s. List-I shows various options for the load. The possible currents \(i(t)\) in the circuit as a function of time are given in List-II.

Choose the option that describes the correct match between the entries in List-I to those in List-II.

	List-I		List-II
\((P)\)		\((1)\)
\((Q)\)		\((2)\)
\((R)\)		\((3)\)
\((S)\)		\((4)\)
		\((5)\)

Let \(f_{1}:(0, \infty) \rightarrow \mathbb{R}\) and \(f_{2}:(0, \infty) \rightarrow \mathbb{R}\) be defined by

\[

f_{1}(x)=\int_{0}^{x} \prod_{j=1}^{21}(t-j)^{j} d t, \quad x>0

\]

and

\(

f_{2}(x)=98(x-1)^{50}-600(x-1)^{49}+2450, \quad x>0

\)

where, for any positive integer \(n\) and real numbers \(a_{1}, a_{2}, \ldots, a_{n}, \prod_{i=1}^{n} a_{i}\) denotes the product of \(a_{1}, a_{2}, \ldots, a_{n} .\) Let \(m_{i}\) and \(n_{i}\), respectively, denote the number of points of local minima and the number of points of local maxima of function \(f_{i}, i=1,2\), in the interval \((0, \infty)\).

The value of $2m_1+3n_1+m_1{n}_1$ is _____.

An electron in an excited state of $\mathrm{L}{\mathrm{i}}^{2+}$ ion has angular momentum $\frac{3\mathrm{h}}{2\mathrm{\pi }}$ . The de Broglie wavelength of the electron in this state is $\mathrm{p}\mathrm{\pi }{\mathrm{\alpha }}_0$ (where ${\mathrm{a}}_0$ is the Bohr radius). The value of $\mathrm{p}$ is

Two independent harmonic oscillators of equal mass are oscillating about the origin with angular frequencies ${\mathrm{\omega }}_1$ and ${\mathrm{\omega }}_2$ and have total energies ${\mathrm{E}}_1$ and ${\mathrm{E}}_2$ , respectively. The variations of their momenta $\mathrm{p}$ with positions $\mathrm{x}$ are shown in the figures. If $\frac{\mathrm{a}}{\mathrm{b}}={\mathrm{n}}^2$ and $\frac{\mathrm{a}}{\mathrm{R}}=\mathrm{n}$ , then the correct equation(s) is (are)