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Physics

A region in the form of an equilateral triangle (in $ x\text{-}y $ plane) of height $ L $ has a uniform magnetic field $ \\vec{B} $ pointing in the $ +z $-direction. A conducting loop $ PQR $, in the form of an equilateral triangle of the same height $ L $, is placed in the $ x\text{-}y $ plane with its vertex $ P $ at $ x = 0 $ in the orientation shown in the figure. At $ t = 0 $, the loop starts entering the region of the magnetic field with a uniform velocity $ \\vec{v} $ along the $ +x $-direction. The plane of the loop and its orientation remain unchanged throughout its motion.

Which of the following graph best depicts the variation of the induced emf $ (E) $ in the loop as a function of the distance $ (x) $ starting from $ x = 0 $?

2.

A particle of mass $ m $ is under the influence of the gravitational field of a body of mass $ M $ $ ( \gg m ) $. The particle is moving in a circular orbit of radius $ r_0 $ with time period $ T_0 $ around the mass $ M $. Then, the particle is subjected to an additional central force, corresponding to the potential energy $ V_c(r) = \dfrac{m\alpha}{r^3} $, where $ \alpha $ is a positive constant of suitable dimensions and $ r $ is the distance from the centre of the orbit. If the particle moves in the same circular orbit of radius $ r_0 $ in the combined gravitational potential due to $ M $ and $ V_c(r) $, but with a new time period $ T_1 $, then $ \dfrac{(T_1^2 - T_0^2)}{T_1^2} $ is given by: $[G \text{ is the gravitational constant.}]$

3.

A metal target with atomic number $ Z = 46 $ is bombarded with a high energy electron beam. The emission of X-rays from the target is analyzed. The ratio $ r $ of the wavelengths of the $ K_{\alpha} $-line and the cut-off is found to be $ r = 2 $. If the same electron beam bombards another metal target with $ Z = 41 $, the value of $ r $ will be

4.

A thin stiff insulated metal wire is bent into a circular loop with its two ends extending tangentially from the same point of the loop. The wire loop has mass $ m $ and radius $ r $ and it is in a uniform vertical magnetic field $ B_0 $, as shown in the figure. Initially, it hangs vertically downwards, because of acceleration due to gravity $ g $, on two conducting supports at $ P $ and $ Q $. When a current $ I $ is passed through the loop, the loop turns about the line $ PQ $ by an angle $ \theta $ given by

5.

A small electric dipole $ \vec{p}_0 $, having a moment of inertia $ I $ about its center, is kept at a distance $ r $ from the center of a spherical shell of radius $ R $. The surface charge density $ \sigma $ is uniformly distributed on the spherical shell. The dipole is initially oriented at a small angle $ \theta $ as shown in the figure. While staying at a distance $ r $, the dipole is free to rotate about its center.

If released from rest, then which of the following statement(s) is(are) correct? $[ \varepsilon_0 \text{ is the permittivity of free space}]$

6.

A table tennis ball has radius $ \left(\frac{3}{2}\right)\times 10^{-2} \, \text{m} $ and mass $ \left(\frac{22}{7}\right)\times 10^{-3} \, \text{kg} $. It is slowly pushed down into a swimming pool to a depth of $ d = 0.7 \, \text{m} $ below the water surface and then released from rest. It emerges from the water surface at speed $ v $, without getting wet, and rises up to a height $ H $. Which of the following option(s) is(are) correct?
[given: $ \pi = \frac{22}{7} $, $ g = 10 \, \text{m s}^{-2} $, density of water $ = 1 \times 10^{3} \, \text{kg m}^{-3} $, viscosity of water $ = 1 \times 10^{3} \, \text{Pa−s} $.]

7.

A positive, singly ionized atom of mass number $A_M$ is accelerated from rest by the voltage $192\text{ V}$. Thereafter, it enters a rectangular region of width $w$ with magnetic field $\vec{B}_0 = 0.1\hat{k}\text{ Tesla}$, as shown in the figure. The ion finally hits a detector at the distance $x$ below its starting trajectory.
$\text{[Given : Mass of neutron/proton} = (5/3) \times 10^{-27}\text{ kg , charge of the electron} = 1.6 \times 10^{-19}\text{C]}$

Which of the following option(s) is(are) correct?

8.

The dimensions of a cone are measured using a scale with a least count of $ 2 \, \text{mm} $. The diameter of the base and the height are both measured to be $ 20.0 \, \text{cm} $. The maximum percentage error in the determination of the volume is ______.

9.

A ball is thrown from the location $ (x_{0}, y_{0}) = (0, 0) $ of a horizontal playground with an initial speed $ v_{0} $ at an angle $ \theta_{0} $ from the $ +x $-direction. The ball is to be hit by a stone, which is thrown at the same time from the location $ (x_{1}, y_{1}) = (L, 0) $. The stone is thrown at an angle $ (180^{\circ} - \theta_{1}) $ from the $ +x $-direction with a suitable initial speed. For a fixed $ v_{0} $, when $ (\theta_{0}, \theta_{1}) = (45^{\circ}, 45^{\circ}) $, the stone hits the ball after time $ T_{1} $, and when $ (\theta_{0}, \theta_{1}) = (60^{\circ}, 30^{\circ}) $, it hits the ball after time $ T_{2} $. In such a case, $ \left(\frac{T_{1}}{T_{2}}\right)^{2} $ is ______.

10.

A charge is kept at the central point $ P $ of a cylindrical region. The two edges subtend a half-angle $ \theta $ at $ P $, as shown in the figure. When $ \theta = 30^{\circ} $, then the electric flux through the curved surface of the cylinder is $ \Phi $. If $ \theta = 60^{\circ} $, then the electric flux through the curved surface becomes $ \Phi / \sqrt{n} $, where the value of $ n $ is ______.

11.

Two equilateral-triangular prisms $ P_1 $ and $ P_2 $ are kept with their sides parallel to each other, in vacuum, as shown in the figure. A light ray enters prism $ P_1 $ at an angle of incidence $ \theta $ such that the outgoing ray undergoes minimum deviation in prism $ P_2 $. If the respective refractive indices of $ P_1 $ and $ P_2 $ are $ \sqrt{\frac{3}{2}} $ and $ \sqrt{3} $, then $ \theta = \sin^{-1}\!\left[ \sqrt{\frac{3}{2}} \sin\!\left( \frac{\pi}{\beta} \right) \right] $, where the value of $ \beta $ is _______.

12.

An infinitely long thin wire, having a uniform charge density per unit length of $ 5 \,\text{nC/m} $, is passing through a spherical shell of radius $ 1 \,\text{m} $, as shown in the figure. A $ 10 \,\text{nC} $ charge is distributed uniformly over the spherical shell. If the configuration of the charges remains static, the magnitude of the potential difference between points $ P $ and $ R $, in Volt, is ____.
[Given: In SI units $ \frac{1}{4\pi \epsilon_0} = 9 \times 10^{9} $, $ \ln 2 = 0.7 $. Ignore the area pierced by the wire.]

13.

A spherical soap bubble inside an air chamber at pressure $ P_0 = 10^{5} \,\text{Pa} $ has a certain radius so that the excess pressure inside the bubble is $ \Delta P = 144 \,\text{Pa} $. Now, the chamber pressure is reduced to $ \frac{8 P_0}{27} $ so that the bubble radius and its excess pressure change. In this process, all the temperatures remain unchanged. Assume air to be an ideal gas and the excess pressure $ \Delta P $ in both the cases to be much smaller than the chamber pressure. The new excess pressure $ \Delta P $ in $ \text{Pa} $ is: ___.

14.

PARAGRAPH – 1
In a young’s double slit experiment, each of the two slits $ A $ and $ B $, as shown in the figure, are oscillating about their fixed center and with a mean separation of $ 0.8 \,\text{mm} $. The distance between the slits at time $ t $ is given by $ d = (0.8 + 0.04 \sin \omega t)\,\text{mm} $, where $ \omega = 0.08 \,\text{rad s}^{-1} $. The distance of the screen from the slits is $ 1 \,\text{m} $ and the wavelength of the light used to illuminate the slits is 6000Å. The interference pattern on the screen changes with time, while the central bright fringe (zeroth fringe) remains fixed at point $ O $.

The $ 8^{\text{th}} $ bright fringe above the point $ O $ oscillates with time between two extreme positions. The separation between these two extreme positions, in micrometer $ (\mu m) $, is _______.

15.

PARAGRAPH – 1
In a young’s double slit experiment, each of the two slits $ A $ and $ B $, as shown in the figure, are oscillating about their fixed center and with a mean separation of $ 0.8 \,\text{mm} $. The distance between the slits at time $ t $ is given by $ d = (0.8 + 0.04 \sin \omega t)\,\text{mm} $, where $ \omega = 0.08 \,\text{rad s}^{-1} $. The distance of the screen from the slits is $ 1 \,\text{m} $ and the wavelength of the light used to illuminate the slits is 6000Å. The interference pattern on the screen changes with time, while the central bright fringe (zeroth fringe) remains fixed at point $ O $.

The maximum speed in $ \mu m / s $ at which the $ 8^{\text{th}} $ bright fringe will move is ___________.

16.

PARAGRAPH – 2
Two particles, $ 1 $ and $ 2 $, each of mass $ m $, are connected by a massless spring, and are on a horizontal frictionless plane, as shown in the figure. Initially, the two particles, with their center of mass at $ x_0 $, are oscillating with amplitude $ a $ and angular frequency $ \omega $. Thus, their positions at time $ t $ are given by $ x_1(t) = (x_0 + d) + a \sin \omega t $ and $ x_2(t) = (x_0 - d) - a \sin \omega t $, respectively, where $ d > 2a $. Particle $ 3 $ of mass $ m $ moves towards this system with speed $ u_0 = a\omega/2 $, and undergoes instantaneous elastic collision with particle $ 2 $, at time $ t_0 $. Finally, particles $ 1 $ and $ 2 $ acquire a center of mass speed $ v_{cm} $ and oscillate with amplitude $ b $ and the same angular frequency $ \omega $.

if the collision occurs at time $ t_0 = 0 $, the value of $ v_{cm}/(a\omega) $ will be ___.

17.

PARAGRAPH – $ 2 $
Two particles, $ 1 $ and $ 2 $, each of mass $ m $, are connected by a massless spring, and are on a horizontal frictionless plane, as shown in the figure. Initially, the two particles, with their center of mass at $ x_0 $, are oscillating with amplitude $ a $ and angular frequency $ \omega $. Thus, their positions at time $ t $ are given by $ x_1(t) = (x_0 + d) + a \sin \omega t $ and $ x_2(t) = (x_0 - d) - a \sin \omega t $, respectively, where $ d > 2a $. Particle $ 3 $ of mass $ m $ moves towards this system with speed $ u_0 = \frac{a\omega}{2} $, and undergoes instantaneous elastic collision with particle $ 2 $, at time $ t_0 $. Finally, particles $ 1 $ and $ 2 $ acquire a center of mass speed $ v_{\mathrm{cm}} $ and oscillate with amplitude $ b $ and the same angular frequency $ \omega $.

If the collision occurs at time $ t_0 = \frac{\pi}{2\omega} $, then the value of $ \frac{4b^2}{a^2} $ will be _____.

JEE Advanced 2024 Paper - 2

Physics Chemistry Mathematics

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An infinitely long thin wire, having a uniform charge density per unit length of $ 5 \,\text{nC/m} $, is passing through a spherical shell of radius $ 1 \,\text{m} $, as shown in the figure. A $ 10 \,\text{nC} $ charge is distributed uniformly over the spherical shell. If the configuration of the charges remains static, the magnitude of the potential difference between points $ P $ and $ R $, in Volt, is ____.
[Given: In SI units $ \frac{1}{4\pi \epsilon_0} = 9 \times 10^{9} $, $ \ln 2 = 0.7 $. Ignore the area pierced by the wire.]

JEE MainsJEE AdvanceBITSATJEE Advance | PYQ

JEE Mains JEE Advance BITSAT
JEE Advance | PYQ