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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:
The phases of the light wave at \(c, d, e\) and \(f\) are \(\phi_c, \phi_d, \phi_e\) and \(\phi_f\), respectively. It is given that \(\phi_c \neq \phi_f\).

Under the influence of the coulomb field of charge \(+Q\), a charge \(-q\) is moving around it in an elliptical orbit. Find out the correct statement(s).

Consider a thin square plate floating on a viscous liquid in a large tank. The height $h$ of the liquid in the tank is much less than the width of the tank. The floating plate is pulled horizontally with a constant velocity $u_0$ . Which of the following statements is (are) true?

Airplanes A and B are flying with constant velocity in the same vertical plane at angles ${30}^{\mathrm{o}}$ and ${60}^{\mathrm{o}}$ with respect to the horizontal respectively as shown in the figure. The speed of A is $100\sqrt{3}\mathrm{}\mathrm{m}{\mathrm{s}}^{-1}$ . At time t = 0 s, an observer in A finds B at a distance of 500 m. This observer sees B moving with a constant velocity perpendicular to the line of motion of A. If at $\mathrm{t}={\mathrm{t}}_0$ , A just escapes being hit by B, t₀ in seconds is

A ray of light travelling in water is incident on its surface open to air. The angle of incidence is \(\theta\), which is less than the critical angle. Then there will be

The center of a disk of radius \(r\) and mass \(m\) is attached to a spring of spring constant \(k\), inside a ring of radius \(R>r\) as shown in the figure. The other end of the spring is attached on the periphery of the ring. Both the ring and the disk are in the same vertical plane. The disk can only roll along the inside periphery of the ring, without slipping. The spring can only be stretched or compressed along the periphery of the ring, following the Hooke's law. In equilibrium, the disk is at the bottom of the ring. Assuming small displacement of the disc, the time period of oscillation of center of mass of the disk is written as \(T=\frac{2 \pi}{\omega}\). The correct expression for \(\omega\) is ( \(g\) is the acceleration due to gravity):

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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

Let \((x, y)\) be any point on the parabola \(y^2=4 x\). Let \(P\) be the point that divides the line segment from \((0,0)\) to \((x, y)\) in the ratio \(1: 3\). Then, the locus of \(P\) is

Match the transformation in Column I with appropriate options in Column II.

Column I	Column II
A. \(CO _2(s) \rightarrow CO _2(g)\)	p. Phase transition
B. \(CaCO _3(s) \rightarrow CaO (s)\) \(+~ CO _2(g)\)	q. Allotropic change
C. \(2 H \rightarrow H _2(g)\)	r. \(\Delta\text{H}\) is positive
D. \(P _{\text {(white, solid) }} \rightarrow P _{\text {(red, solid) }}\)	s. \(\Delta\text{S}\) is positive
	t. \(\Delta\text{S}\) is negative

${ }^{131}I is$ an isotope of Iodine that $\beta$ decays to an isotope of Xenon with a half-life of $8 days$ . A small amount of a serum labelled with ${ }^{131}I$ is injected into the blood of a person. The activity of the amount of ${ }^{131}I$ injected was $2.4\times {10}^5$ Becquerel (Bq). It is known that the injected serum will get distributed uniformly in the blood stream in less than half an hour. After 11.5 hours, 2.5 ml of blood is drawn from the person's body, and gives an activity of $115 Bq$ . The total volume of blood in the person's body, in litres is approximately (you may use \(\mathrm{e}^{\mathrm{x}} \approx 1+\mathrm{x}\) for \(|\mathrm{x}|<<1\) and \(\ln 2 \approx 0.7\)).

One Indian and four American men and their wives are to be seated randomly around a circular table. Then, the conditional probability that the Indian man is seated adjacent to his wife given that each American man is seated adjacent to his wife, is

You are given many resistances, capacitors and inductors. These are connected to a variable DC voltage source (the first two circuits) or an AC voltage source of \(50 \mathrm{~Hz}\) frequency (the next three circuits) in different ways as shown in Column II. When a current \(I\) (steady state for DC or rms for \(A C\) ) flows through the circuit, the corresponding voltage \(V_1\) and \(V_2\) (indicated in circuits) are related as shown in Column I. Match the two

An incompressible liquid is kept in a container having a weightless piston with a hole. A capillary tube of inner radius $0.1\mathrm{mm}$ is dipped vertically into the liquid through the airtight piston hole, as shown in the figure. The air in the container is isothermally compressed from its original volume $V_0$to$\frac{100}{101}{V}_0$with the movable piston.Considering air as an ideal gas, the height $\left(h\right)$ of the liquid column in the capillary above the liquid level in $\mathrm{cm}$is _____.
[Given: Surface tension of the liquid is $0.075\mathrm{N}{\mathrm{m}}^{-1}$, atmospheric pressure is ${10}^5\mathrm{N}{\mathrm{m}}^{-2}$, acceleration due to gravity $\left(g\right)$ is $10\mathrm{m}{\mathrm{s}}^{-2}$, density of the liquid is ${10}^3\mathrm{kg}{\mathrm{m}}^{-3}$and contact angle of capillary surface with the liquid is zero]