A block of mass M has a circular cut with a frictionless surface as shown. The block rests on the horizontal frictionless surface of a fixed table. Initially the right edge of the block is at $x=0$ , in a coordinate system fixed to the table. A point mass m is released from rest at the topmost point of the path as shown and it slides down. When the mass loses contact with the block its position is x and velocity is v. At that instant which of the following options is correct?

The focal length of a thin biconvex lens is \(20 \mathrm{~cm}\). When an object is moved from a distance of \(25 \mathrm{~cm}\) in front of it to \(50 \mathrm{~cm}\), the magnification of its image changes from \(m_{25}\) to \(m_{50}\). The ratio \(\frac{m_{25}}{m_{50}}\) is

A thin and uniform rod of mass $\mathrm{M}$ and length $\mathrm{L}$ is held vertical on a floor with large friction. The rod is released from rest so that it falls by rotating about its contact-point with the floor without slipping. Which of the following statement(s) is/are correct, when the rod makes an angle ${60}^{\mathrm{o}}$ with vertical?
[ $\mathrm{g}$ is the acceleration due to gravity]

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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:
The buoyancy force acting on the gas bubble is (Assume \(R\) is the universal gas constant)

A person in a lift is holding a water jar, which has a small hole at the lower end of its side. When the lift is at rest, the water jet coming out of the hole hits the floor of the lift at a distance d of 1.2 m from the person. In the following, state of the lift's motion is given in List I and the distance where the water jet hits the floor of the lift is given in List II. Match the statements from List I with those in List II and select the correct answer using the code given below the lists.
 

	List I		List II
A.	Lift is accelerating vertically up.	$P.$	$\mathrm{d}=1.2\mathrm{ }\mathrm{m}$
B.	Lift is accelerating vertically down with an acceleration less than the gravitational acceleration.	$Q.$	$\mathrm{d}>1.2\mathrm{ }\mathrm{m}$
C.	Lift is moving vertically up with constant speed.	$R.$	$\mathrm{d}<1.2\mathrm{ }\mathrm{m}$
D.	Lift is falling freely.	$S.$	No water leaks out of the jar

One mole of an ideal gas undergoes two different cyclic processes $\mathrm{I}$ and $\mathrm{II}$, as shown in the $P-V$ diagrams below. In cycle $\mathrm{I}$, processes $a, b, c$ and $d$ are isobaric, isothermal, isobaric and isochoric, respectively. In cycle $\mathrm{II}$, processes $a\text{'}, b\text{'}, c\text{'}$ and $d\text{'}$ are isothermal, isochoric, isobaric and isochoric, respectively. The total work done during cycle $\mathrm{I}$ is $W_{\mathrm{I}}$ and that during cycle $\mathrm{II}$ is $W_{\mathrm{II}}$. The ratio $\frac{W_{\mathrm{I}}}{W_{\mathrm{II}}}$ is _____.

Column I shows four situations of standard Young's double slit arrangement with the screen placed far away from the slits \(S_1\) and \(S_2\). In each of these cases \(S_1 P_0=S_2 P_0\), \(S_1 P_1-S_2 P_1=\frac{\lambda}{4}\) and \(S_1 P_2-S_2 P_2=\frac{\lambda}{3}\), where \(\lambda\) is the wavelength of the light used. In the cases \(\mathrm{B}, \mathrm{C}\) and \(\mathrm{D}\), a transparent sheet of refractive index \(\mu\) and thickness \(t\) is pasted on slit \(S_2\). The thickness of the sheets are different in different cases. The phase difference between the light waves reaching a point \(P\) on the screen from the two slits is denoted by \(\delta(P)\) and the intensity by \(I(P)\). Match each situation given in Column-I with the statement(s) in Column-II valid for that situation.

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Scientists are working hard to develop nuclear fusion reactor. Nuclei of heavy hydrogen, \({ }_1^2 \mathrm{H}\) known as deuteron and denoted by \(D\) can be thought of as a candidate for fusion reactor. The \(D\) - \(D\) reaction is \({ }_1^2 \mathrm{H}+{ }_1^2 \mathrm{H} \rightarrow{ }_2^3 \mathrm{He}+n+\) energy. In the core of fusion reactor. A gas of heavy hydrogen is fully ionized into deuteron nuclei and electrons. This collection of \({ }_1^4 \mathrm{H}\) nuclei and electrons is known as plasma. The nuclei move randomly in the reactor core and occasionally come close enough for nuclear fusion to take place. Usually, the temperatures in the reactor core are too high and no material wall can be used to confine the plasma. Special techniques are used which confine the plasma for a time \(t_0\) before the particles fly away from the core. If n is the density (number/volume) of deutrons, the product t \(_0\) is called Lawson number. In one of the criteria, \(a\) reactor is termed successful if Lawson number is greater than \(5 \times 10^{14} \mathrm{scm}^{-3}\).
It may be helpful to use the following : Boltzmann constant \(k=8.6 \times 10^{-5} \mathrm{eV} / \mathrm{K}\); \(\frac{e^2}{4 \pi \varepsilon_0}=1.44 \times 10^{-9} \mathrm{eVm}\)
Question:
Results of calculations for four different designs of a fusion reactor using D-D reaction are given below. Which of these is most promising based on Lawson criterion?

The compound(s) with \(\mathrm{P}-\mathrm{H}\) bond(s) is(are)

Consider a configuration of $n$ identical units, each consisting of three layers. The first layer is a column of air of height $h=\frac{1}{3} \mathrm{cm}$, and the second and third layers are of equal thickness $d=\frac{\sqrt{3}-1}{2} \mathrm{cm}$, and refractive indices ${\mu }_1=\sqrt{\frac{3}{2}}$ and ${\mu }_2=\sqrt{3}$, respectively. A light source $O$ is placed on the top of the first unit, as shown in the figure. A ray of light from $O$ is incident on the second layer of the first unit at an angle of $\theta =60°$ to the normal. For a specific value of $n$, the ray of light emerges from the bottom of the configuration at a distance $l=\frac{8}{\sqrt{3}} \mathrm{cm}$, as shown in the figure. The value of $n$ is _______.

Consider the kinetic data given in the following table for the reaction $\mathrm{A}+\mathrm{B}+\mathrm{C}\to$ Product.

Ex peri ment No.	\([\text A]\) \((\text{mol}\) \(\text{dm} ^{-3})\)	\([\text B]\) \((\text{mol}\) \(\text{dm} ^{-3})\)	\([\text C]\) \((\text{mol}\) \(\text{dm} ^{-3})\)	Rate of reaction \((\text{mol}\) \(\text{dm} ^{-3}\) \(\text s^{-1})\)
1	0.2	0.1	0.1	\(6.0 \times\) \(10^{-5}\)
2	0.2	0.2	0.1	\(6.0 \times \) \(10^{-5}\)
3	0.2	0.1	0.2	\(1.2 \times \) \(10^{-4}\)
4	0.3	0.1	0.1	\(9.0 \times \) \(10^{-5}\)

The rate of the reaction for $\left[\mathrm{A}\right]=0.15\mathrm{ }\mathrm{m}\mathrm{o}\mathrm{l}\mathrm{ }\mathrm{d}{\mathrm{m}}^{-3},\mathrm{ }\left[\mathrm{B}\right]=0.25\mathrm{ }\mathrm{m}\mathrm{o}\mathrm{l}\mathrm{ }\mathrm{d}{\mathrm{m}}^{-3}$ and $\left[\mathrm{C}\right]=0.15\mathrm{ }\mathrm{m}\mathrm{o}\mathrm{l}\mathrm{ }\mathrm{d}{\mathrm{m}}^{-3}$ is found to be $\mathrm{Y}\times {10}^{-5}\mathrm{ }\mathrm{m}\mathrm{o}\mathrm{l}\mathrm{ }\mathrm{d}{\mathrm{m}}^{-3}{\mathrm{s}}^{-1}.$ The value of $\mathrm{Y}$ is __________

A series \(R-C\) combination is connected to an \(\mathrm{AC}\) voltage of angular frequency \(\omega=500 \mathrm{rad} / \mathrm{s}\). If the impedance of the \(R-C\) circuit is \(R \sqrt{1.25}\), the time constant (in millisecond) of the circuit is

Match the following columns