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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 \( T = \dfrac{2\pi}{\omega} \). The correct expression for \( \omega \) is (\( g \) is the acceleration due to gravity):

2.

In a scattering experiment, a particle of mass \( 2m \) collides with another particle of mass \( m \), which is initially at rest. Assuming the collision to be perfectly elastic, the maximum angular deviation \( \theta \) of the heavier particle, as shown in the figure, in radians is:

3.

A conducting square loop initially lies in the XZ plane with its lower edge hinged along the X-axis. Only in the region \( y \ge 0 \), there is a time dependent magnetic field pointing along the Z-direction, \( \vec{B}(t)=B_{0}(\cos \omega t)\,\hat{k} \), where \( B_{0} \) is a constant. The magnetic field is zero everywhere else. At time \( t=0 \), the loop starts rotating with constant angular speed \( \omega \) about the X axis in the clockwise direction as viewed from the \( +X \) axis (as shown in the figure). Ignoring self-inductance of the loop and gravity, which of the following plots correctly represents the induced e.m.f. (\( V \)) in the loop as a function of time:

4.

Figure.\( 1 \) shows the configuration of main scale and Vernier scale before measurement. Figure.\( 2 \) shows the configuration corresponding to the measurement of diameter \( D \) of a tube. The measured value of \( D \) is:

5.

A conducting square loop of side \( L \), mass \( M \) and resistance \( R \) is moving in the \( XY \) plane with its edges parallel to the \( X \) and \( Y \) axes. The region \( y \ge 0 \) has a uniform magnetic field, \( \vec{B} = B_{0}\,\hat{k} \). The magnetic field is zero everywhere else. At time \( t = 0 \), the loop starts to enter the magnetic field with an initial velocity \( v_{0}\hat j \, \text{m/s} \), as shown in the figure. Considering the quantity \( K = \dfrac{B_{0}^{2}L^{2}}{RM} \) in appropriate units, ignoring self-inductance of the loop and gravity, which of the following statements is/are correct:

6.

Length, breadth and thickness of a strip having a uniform cross section are measured to be \( 10.5 \, \text{cm} \), \( 0.05 \, \text{mm} \), and \( 6.0 \, \mu\text{m} \), respectively. Which of the following option(s) give(s) the volume of the strip in \( \text{cm}^3 \) with correct significant figures:

7.

Consider a system of three connected strings, \( S_1, S_2 \) and \( S_3 \) with uniform linear mass densities \( \mu \, \text{kg} / \text{m} \), \( 4\mu \, \text{kg} / \text{m} \) and \( 16\mu \, \text{kg} / \text{m} \), respectively, as shown in the figure. \( S_1 \) and \( S_2 \) are connected at the point \( P \), whereas \( S_2 \) and \( S_3 \) are connected at the point \( Q \), and the other end of \( S_3 \) is connected to a wall. A wave generator \( O \) is connected to the free end of \( S_1 \). The wave from the generator is represented by \( y = y_0 \cos(\omega t - kx) \, \text{cm} \), where \( y_0 \), \( \omega \) and \( k \) are constants of appropriate dimensions.
Which of the following statements is/are correct:

8.

A person sitting inside an elevator performs a weighing experiment with an object of mass \( 50 \, \text{kg} \). Suppose that the variation of the height \( y \) (in \( \text{m} \)) of the elevator, from the ground, with time \( t \) (in \( \text{s} \)) is given by \( y = 8 \left[ 1 + \sin\!\left( \dfrac{2\pi t}{T} \right) \right] \), where \( T = 40\pi \, \text{s} \). Taking acceleration due to gravity, \( g = 10 \, \text{m} / \text{s}^2 \), the maximum variation of the object’s weight (in \( \text{N} \)) as observed in the experiment is ___.

9.

A cube of unit volume contains \( 35 \times 10^{7} \) photons of frequency \( 10^{15} \, \text{Hz} \). If the energy of all the photons is viewed as the average energy being contained in the electromagnetic waves within the same volume, then the amplitude of the magnetic field is \( \alpha \times 10^{-9} \, \text{T} \). Taking permeability of free space \( \mu_{0} = 4\pi \times 10^{-7} \, \text{Tm} / \text{A} \), Planck’s constant \( h = 6 \times 10^{-34} \, \text{J s} \) and \( \pi = \dfrac{22}{7} \), the value of \( \alpha \) is ___

10.

Two identical plates \( P \) and \( Q \), radiating as perfect black bodies, are kept in vacuum at constant absolute temperatures \( T_{P} \) and \( T_{Q} \), respectively, with \( T_{Q} < T_{P} \), as shown in Fig. 1. The radiated power transferred per unit area from \( P \) to \( Q \) is \( W_{0} \). Subsequently, two more plates, identical to \( P \) and \( Q \), are introduced between \( P \) and \( Q \), as shown in Fig. 2. Assume that heat transfer takes place only between adjacent plates. If the power transferred per unit area in the direction from \( P \) to \( Q \) (Fig. 2) in the steady state is \( W_{s} \), then the ratio \( \dfrac{W_{0}}{W_{s}} \) is ___________.

11.

A solid glass sphere of refractive index \( n = \sqrt{3} \) and radius \( R \) contains a spherical air cavity of radius \( \dfrac{R}{2} \), as shown in the figure. A very thin glass layer is present at the point \( O \) so that the air cavity (refractive index \( n = 1 \) ) remains inside the glass sphere. An unpolarized, unidirectional and monochromatic light source \( S \) emits a light ray from a point inside the glass sphere towards the periphery of the glass sphere. If the light is reflected from the point \( O \) and is fully polarized, then the angle of incidence at the inner surface of the glass sphere is \( \theta \). The value of \( \sin \theta \) is__________.

12.

A single slit diffraction experiment is performed to determine the slit width using the equation, \( \dfrac{bd}{D} = m\lambda \), where \( b \) is the slit width, \( D \) the shortest distance between the slit and the screen, \( d \) the distance between the \( m^{\text{th}} \) diffraction maximum and the central maximum, and \( \lambda \) is the wavelength. \( D \) and \( d \) are measured with scales of least count of \( 1 \, \text{cm} \) and \( 1 \, \text{mm} \), respectively. The values of \( \lambda \) and \( m \) are known precisely to be \( 600 \, \text{nm} \) and \( 3 \), respectively. The absolute error (in \( \mu \text{m} \)) in the value of \( b \) estimated using the diffraction maximum that occurs for \( m = 3 \) with \( d = 5 \, \text{mm} \) and \( D = 1 \, \text{m} \) is ___.

13.

Consider an electron in the \( n = 3 \) orbit of a hydrogen-like atom with atomic number \( Z \). At absolute temperature \( T \), a neutron having thermal energy \( k_B T \) has the same de Broglie wavelength as that of this electron. If this temperature is given by \( T = \dfrac{Z^2 h^2}{\alpha \pi^2 a_0^2 m_N k_B} \), (where \( h \) is the Planck’s constant, \( k_B \) is the Boltzmann constant, \( m_N \) is the mass of the neutron and \( a_0 \) is the first Bohr radius of hydrogen atom) then the value of \( \alpha \) is ________.

14.

List-I shows four configurations, each consisting of a pair of ideal electric dipoles. Each dipole has a dipole moment of magnitude \( p \), oriented as marked by arrows in the figures. In all the configurations the dipoles are fixed such that they are at a distance \( 2r \) apart along the \( x \) direction. The midpoint of the line joining the two dipoles is \( X \). The possible resultant electric fields \( \vec{E} \) at \( X \) are given in List-II.
Choose the option that describes the correct match between the entries in List-I to those in List-II.

15.

A circuit with an electrical load having impedance \( Z \) is connected with an AC source as shown in the diagram. The source voltage varies in time as \( V(t) = 300 \sin(400 t) \, \text{V} \), where \( t \) is time in \( \text{s} \). List-I shows various options for the load. The possible currents \( i(t) \) in the circuit as a function of time are given in List-II.

Choose the option that describes the correct match between the entries in List-I to those in List-II.

16.

List-I shows various functional dependencies of energy \( (E) \) on the atomic number \( (Z) \). Energies associated with certain phenomena are given in List-II.
Choose the option that describes the correct match between the entries in List-I to those in List-II.

JEE Advance 2025 Paper 1

Physics Chemistry Mathematics

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A person sitting inside an elevator performs a weighing experiment with an object of mass \( 50 \, \text{kg} \). Suppose that the variation of the height \( y \) (in \( \text{m} \)) of the elevator, from the ground, with time \( t \) (in \( \text{s} \)) is given by \( y = 8 \left[ 1 + \sin\!\left( \dfrac{2\pi t}{T} \right) \right] \), where \( T = 40\pi \, \text{s} \). Taking acceleration due to gravity, \( g = 10 \, \text{m} / \text{s}^2 \), the maximum variation of the object’s weight (in \( \text{N} \)) as observed in the experiment is ___.

JEE MainsJEE AdvanceBITSATJEE Advance | PYQ

JEE Mains JEE Advance BITSAT
JEE Advance | PYQ