JEE (Advanced) 2025 Paper-1

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 $$y \geq 0$$ 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 $$v_0 \hat{j}$$ m/s, as shown in the figure. Considering the quantity $$K = \frac{B_0^2 L^2}{RM}$$ in appropriate units, ignoring self-inductance of the loop and gravity, which of the following statements is/are correct:

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

Consider a system of three connected strings, $$S_1$$, $$S_2$$ and $$S_3$$ with uniform linear mass densities $$\mu$$ kg/m, $$4\mu$$ kg/m and $$16\mu$$ kg/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 $$y = y_0 \cos(\omega t - kx)$$ cm, where $$y_0$$, $$\omega$$ and $$k$$ are constants of appropriate dimensions. Which of the following statements is/are correct:

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

A cube of unit volume contains $$35 \times 10^7$$ photons of frequency $$10^{15}$$ 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 $$\alpha \times 10^{-9}$$ T. Taking permeability of free space $$\mu_0 = 4\pi \times 10^{-7}$$ Tm/A, Planck's constant $$h = 6 \times 10^{-34}$$ Js and $$\pi = \frac{22}{7}$$, the value of $$\alpha$$ is ________.

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 $$\frac{W_0}{W_S}$$ is ________.

A solid glass sphere of refractive index $$n = \sqrt{3}$$ and radius $$R$$ contains a spherical air cavity of radius $$\frac{R}{2}$$, 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 ________.

A single slit diffraction experiment is performed to determine the slit width using the equation, $$\frac{bd}{D} = m\lambda$$, 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 cm and 1 mm, respectively. The values of $$\lambda$$ and $$m$$ are known precisely to be 600 nm and 3, respectively. The absolute error (in $$\mu$$m) in the value of $$b$$ estimated using the diffraction maximum that occurs for $$m = 3$$ with $$d = 5$$ mm and $$D = 1$$ m is ________.

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 $$T = \frac{Z^2 h^2}{\alpha \pi^2 a_0^2 m_N k_B}$$, (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 ________.

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.