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 ${\mu }_0=4\pi \times {10}^{-7} \mathrm{N} {\mathrm{A}}^{-2}$]

Two beams of red and violet colours are made to pass separately through a prism (angle of the prism is \(60^{\circ}\) ). In the position of minimum deviation, the angle of refraction will be

A thin circular coin of mass $5 \mathrm{gm}$ and radius $\frac{4}{3} \mathrm{cm}$ is initially in a horizontal $xy$-plane. The coin is tossed vertically up ($+z$ direction) by applying an impulse of $\sqrt{\frac{\mathrm{\pi }}{2}}\times {10}^{-2} \mathrm{N}-\mathrm{s}$ at a distance $\frac{2}{3} \mathrm{cm}$ from its center. The coin spins about its diameter and moves along the $+z$ direction. By the time the coin reaches back to its initial position, it completes $n$ rotations. The value of $n$ is _____. [Given: The acceleration due to gravity $g=10 \mathrm{m} {\mathrm{s}}^{-2}$]

A hydrogen atom, initially at rest in its ground state, absorbs a photon of frequency \(v_1\) and ejects the electron with a kinetic energy of 10 eV . The electron then combines with a positron at rest to form a positronium atom in its ground state and simultaneously emits a photon of frequency \(v_2\). The center of mass of the resulting positronium atom moves with a kinetic energy of 5 eV . It is given that positron has the same mass as that of electron and the positronium atom can be considered as a Bohr atom, in which the electron and the positron orbit around their center of mass. Considering no other energy loss during the whole process, the difference between the two photon energies (in eV ) is \(\qquad\)

Match List I with List II and select the correct answer using the codes given below the lists:

	\(\text{List I}\)		\(\text{List II}\)
a.	Boltzmann constant	p.	\(\left[M L^2 T^{-1}\right]\)
b.	Coefficient of viscosity	q.	\(\left[M L^{-1} T^{-1}\right]\)
c.	Planck constant	r.	\(\left[M L T^{-3} K^{-1}\right]\)
d.	Thermal conductivity	s.	\(\left[M L^2 T^{-2} K^{-1}\right]\)

A human body has a surface area of approximately $1 m^2$ . The normal body temperature is $10 K$ above the surrounding room temperature $T_0$ . Take the room temperature to be $T_0=300 K$ . For $T_0=300 K$ , the value of $\sigma T^4=460\frac{W}{m^2}$ (where $\sigma$ is the Stefan-Boltzmann constant). Which of the following options is/are correct?

A ball is projected from the ground at an angle of ${45}^o$ with the horizontal surface. It reaches a maximum height of $120\mathrm{ m}$ and returns to the ground. Upon hitting the ground for the first time, it loses half of its kinetic energy. Immediately after the bounce, the velocity of the ball makes an angle of ${30}^o$ with the horizontal surface. The maximum height it reaches after the bounce (in meters) is _____________.

The major product(s) of the following reaction is/are:

Match the following.

Match the chemical substance in Column I with type of polymers/type of bond in Column II. Indicate your answer by darkening the appropriate bubbles of \(4 \times 4\) matrix given in the ORS.

Column I	Column II
A. Cellulose	p. Natural polymer
B. Nylon-6,6	q. Synthetic polymer
C. Protein	r. Amide linkage
D. Sucrose	s. Glycoside linkage

Incandescent bulbs are designed by keeping in mind that the resistance of their filament increases with the increase in temperature. If at room temperature, \(100 \mathrm{~W}, 60 \mathrm{~W}\) and \(40 \mathrm{~W}\) bulbs have filament resistances \(R_{100}, R_{60}\) and \(R_{40}\), respectively, the relation between these resistances is

A particle of mass $m$ is initially at rest at the origin. It is subjected to a force and starts moving along the x - axis. Its kinetic energy changes with time as $dK/dt=\gamma t,$ where $\gamma$ is a positive constant of appropriate dimensions. Which of the following statements is (are) true?

The total number of alkenes possible by dehydrobromination of 3-bromo-3-cyclopentylhexane using alcoholic \(\mathrm{KOH}\) is