A block of mass ${\mathrm{m}}_1=1\mathrm{}\mathrm{k}\mathrm{g}$ another mass ${\mathrm{m}}_2=2\mathrm{}\mathrm{k}\mathrm{g}$ , are placed together (see figure) on an inclined plane with angle of inclination $\mathrm{\theta }$ . Various values of $\mathrm{\theta }$ are given in List I. The coefficient of friction between the block ${\mathrm{m}}_1$ and the plane is always zero. The coefficient of static and dynamic friction between the block ${\mathrm{m}}_2$ and the plane are equal to $\mathrm{\mu }=0.3$ . In List II expressions for the friction on the block ${\mathrm{m}}_2$ are given. Match the correct expression of the friction in List II with the angles given in List I, and choose the correct option. The acceleration due to gravity is denoted by g.
[Useful information: $\tan \left({5.5}^{\mathrm{o}}\right)\approx 0.1;\tan \left({11.5}^{\mathrm{o}}\right)\approx 0.2;\tan \left({16.5}^{\mathrm{o}}\right)\approx 0.3$ ]

	List I		List II
A.	$\mathrm{\theta }=5^{\mathrm{o}}$	$\mathrm{P}.$	${\mathrm{m}}_2\mathrm{g}\sin \mathrm{\theta }$
B.	$\mathrm{\theta }={10}^{\mathrm{o}}$	$\mathrm{Q}.$	$\left({\mathrm{m}}_1+{\mathrm{m}}_2\right)\mathrm{g}\sin \mathrm{\theta }$
C.	$\mathrm{\theta }={15}^{\mathrm{o}}$	$\mathrm{R}.$	$\mathrm{\mu }{\mathrm{m}}_2\mathrm{}\mathrm{g}\cos \mathrm{\theta }$
D.	$\mathrm{\theta }={20}^{\mathrm{o}}$	$\mathrm{S}.$	$\mathrm{\mu }\left({\mathrm{m}}_1+{\mathrm{m}}_2\right)\mathrm{g}\cos \mathrm{\theta }$

A thermally isolated cylindrical closed vessel of height $8\mathrm{m}$ is kept vertically. It is divided into two equal parts by a diathermic (perfect thermal conductor) frictionless partition of mass $8.3\mathrm{kg}$. Thus the partition is held initially at a distance of $4\mathrm{m}$ from the top, as shown in the schematic figure below. Each of the two parts of the vessel contains $0.1$ mole of an ideal gas at temperature $300\mathrm{K}$. The partition is now released and moves without any gas leaking from one part of the vessel to the other. When equilibrium is reached, the distance of the partition from the top (in $\mathrm{m}$) will be ____________ (take the acceleration due to gravity $=10\mathrm{m}{\mathrm{s}}^{-2}$ and the universal gas constant$=8.3\mathrm{J}{\mathrm{mol}}^{-1}{\mathrm{K}}^{-1}$).

A loop carrying current \(I\) lies in the \(x-y\) plane as shown in the figure. The unit vector \(\hat{k}\) is coming out of the plane of the paper. The magnetic moment of the current loop is

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A special metal \(S\) conducts electricity without any resistance. A closed wire loop, made of \(S\), does not allow any change in flux through itself by inducing a suitable current to generate a compensating flux. The induced current in the loop cannot decay due to its zero resistance. This current gives rise to a magnetic moment which in turn repels the source of magnetic field or flux. Consider such a loop, of radius \(a\), with its center at the origin. A magnetic dipole of moment \(m\) is brought along the axis of this loop from infinity to a point at distance \(r(\gg a)\) from the center of the loop with its north pole always facing the loop, as shown in the figure below.

The magnitude of magnetic field of a dipole \(m\), at a point on its axis at distance \(r\), is \(\frac{\mu_{0}}{2 \pi} \frac{m}{r^{3}}\), where \(\mu_{0}\) is the permeability of free space. The magnitude of the force between two magnetic dipoles with moments, \(m_{1}\) and \(m_{2}\), separated by a distance \(r\) on the common axis, with their north poles facing each other, is \(\frac{k m_{1} m_{2}}{r^{4}}\), where \(k\) is a constant of appropriate dimensions. The direction of this force is along the line joining the two dipoles.

Question :
When the dipole m is placed at a distance r from the centre of the loop (as shown in the figure), the current induced in the loop will be proportional to:

A series LCR circuit is connected to a $45 \sin \left(\omega t\right)$ Volt source. The resonant angular frequency of the circuit is ${10}^5 \mathrm{rad} {\mathrm{s}}^{-1}$ and current amplitude at resonance is $I_0$. When the angular frequency of the source is $\omega =8\times {10}^4 \mathrm{rad} {\mathrm{s}}^{-1}$, the current amplitude in the circuit is $0.05 I_0$. If $L=50 \mathrm{mH}$, match each entry in List-$\mathrm{I}$ with an appropriate value from List-$\mathrm{II}$ and choose the correct option.

	List-$\mathrm{I}$		List-$\mathrm{II}$
$\mathrm{P}$	$I_0$ in $\mathrm{mA}$	$1$	$44.4$
$\mathrm{Q}$	The quality factor of the circuit	$2$	$18$
$\mathrm{R}$	The bandwidth of the circuit in $\mathrm{rad} {\mathrm{s}}^{-1}$	$3$	$400$
$\mathrm{S}$	The peak power dissipated at resonance in Watt	$4$	$2250$
		$5$	$500$

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A point charge \(Q\) is moving in a circular orbit of radius \(R\) in the \(x-y\) plane with an angular velocity \(\omega .\) This can be considered as equivalent to a loop carrying a steady current \(\frac{Q \omega}{2 \pi}\). A uniform magnetic field along the positive \(z\)-axis is now switched on, which increases at a constant rate from 0 to \(B\) in one second. Assume that the radius of the orbit remains constant. The application of the magnetic field induces an emf in the orbit. The induced emf is defined as the work done by an induced electric field in moving a unit positive charge around a closed loop. It is known that, for an orbiting charge, the magnetic dipole moment is proportional to the angular momentum with a proportionality constant \(\gamma\).

Question:

The magnitude of the induced electric field in the orbit at any instant of time during the time interval of the magnetic field change is

A plane polarized blue light ray is incident on a prism such that there is no reflection from the surface of the prism. The angle of deviation of the emergent ray is $\delta ={60}^{\mathrm{o}}$ (see Figure-$1$). The angle of minimum deviation for red light from the same prism is ${\delta }_{\min }={30}^{\mathrm{o}}$ (see Figure-$2$). The refractive index of the prism material for blue light is $\sqrt{3}$. Which of the following statement(s) is(are) correct?

In a photoemission experiment, the maximum kinetic energies of photoelectrons from metals $P,Q$ and $R$ are $E_P, E_Q$ and $E_R$, respectively, and they are related by $E_P=2E_Q=2E_R$. In this experiment, the same source of monochromatic light is used for metals $P$ and $Q$ while a different source of monochromatic light is used for the metal $R$. The work functions for metals $P,Q$ and $R$ are $4.0 \mathrm{eV}, 4.5 \mathrm{eV}$ and $5.5 \mathrm{eV}$, respectively. The energy of the incident photon used for metal $R$, in $eV$ is ___.

A pulse of light of duration 100 ns is absorbed completely by a small object initially at rest. Power of the pulse is 30 mW and the speed of light is 3 ×10⁸ ms^-1. The final momentum of the object is

Ozonolysis of ${\mathrm{ClO}}_2$ produces an oxide of chlorine. The average oxidation state of chlorine in this oxide is ______ .

The heating of \(\mathrm{NH}_4 \mathrm{NO}_2\) at \(60-70^{\circ} \mathrm{C}\) and \(\mathrm{NH}_4 \mathrm{NO}_3\) at \(200-250^{\circ} \mathrm{C}\) is associated with the formation of nitrogen containing compounds \(\mathbf{X}\) and \(\mathbf{Y}\), respectively. \(\mathbf{X}\) and \(\mathbf{Y}\), respectively, are

Among the triatomic molecules/ions, $\mathrm{B}\mathrm{e}\mathrm{C}{\mathrm{l}}_2,\mathrm{ }{\mathrm{N}}_3^{-},\mathrm{ }{\mathrm{N}}_2\mathrm{O},\mathrm{ }{\mathrm{N}\mathrm{O}}_2^{+},\mathrm{ }{\mathrm{O}}_3,\mathrm{S}\mathrm{C}{\mathrm{l}}_2,\mathrm{ }{\mathrm{I}\mathrm{C}\mathrm{l}}_2^{-},\mathrm{ }{\mathrm{I}}_3^{-}$ and $\mathrm{X}\mathrm{e}{\mathrm{F}}_2$ , the total number of linear molecule(s)/ion(s) where the hybridization of the central atom does not have contribution from the d-orbital(s) is
[Atomic number: S = 16, Cl = 17, I = 53 and Xe = 54]

Consider a uniform spherical charge distribution of radius $R_1$ centred at the origin O. In this distribution, a spherical cavity of radius $R_2$ , centred at $P$ with distance $OP=a=R_1-R_2$ (see figure) is made. If the electric field inside the cavity at position $\overrightarrow{r}$ is $\overrightarrow{E}\left(\overrightarrow{r}\right)$ , then the correct statement(s) is (are)