JEE (Advanced) 2025 Paper-2

A positive point charge of $$10^{-8}$$ C is kept at a distance of 20 cm from the center of a neutral conducting sphere of radius 10 cm. The sphere is then grounded and the charge on the sphere is measured. The grounding is then removed and subsequently the point charge is moved a distance of 10 cm further away from the center of the sphere along the radial direction. Taking $$\frac{1}{4\pi\epsilon_0} = 9 \times 10^9$$ Nm$$^2$$/C$$^2$$ (where $$\epsilon_0$$ is the permittivity of free space), which of the following statements is/are correct:

Two identical concave mirrors each of focal length $$f$$ are facing each other as shown in the schematic diagram. The focal length $$f$$ is much larger than the size of the mirrors. A glass slab of thickness $$t$$ and refractive index $$n_0$$ is kept equidistant from the mirrors and perpendicular to their common principal axis. A monochromatic point light source $$S$$ is embedded at the center of the slab on the principal axis, as shown in the schematic diagram. For the image to be formed on $$S$$ itself, which of the following distances between the two mirrors is/are correct:

Six infinitely large and thin non-conducting sheets are fixed in configurations I and II. As shown in the figure, the sheets carry uniform surface charge densities which are indicated in terms of $$\sigma_0$$. The separation between any two consecutive sheets is $$1 \, \mu$$m. The various regions between the sheets are denoted as 1, 2, 3, 4 and 5. If $$\sigma_0 = 9 \, \mu$$C/m$$^2$$, then which of the following statements is/are correct:

(Take permittivity of free space $$\epsilon_0 = 9 \times 10^{-12}$$ F/m)

The efficiency of a Carnot engine operating with a hot reservoir kept at a temperature of 1000 K is 0.4. It extracts 150 J of heat per cycle from the hot reservoir. The work extracted from this engine is being fully used to run a heat pump which has a coefficient of performance 10. The hot reservoir of the heat pump is at a temperature of 300 K. Which of the following statements is/are correct:

A conducting solid sphere of radius $$R$$ and mass $$M$$ carries a charge $$Q$$. The sphere is rotating about an axis passing through its center with a uniform angular speed $$\omega$$. The ratio of the magnitudes of the magnetic dipole moment to the angular momentum about the same axis is given as $$\alpha \frac{Q}{2M}$$. The value of $$\alpha$$ is ______.

A hydrogen atom, initially at rest in its ground state, absorbs a photon of frequency $$\nu_1$$ and ejects the electron with a kinetic energy of 10 eV. The electron then combines with a positron at rest to form a positronium atom in its ground state and simultaneously emits a photon of frequency $$\nu_2$$. The center of mass of the resulting positronium atom moves with a kinetic energy of 5 eV. It is given that positron has the same mass as that of electron and the positronium atom can be considered as a Bohr atom, in which the electron and the positron orbit around their center of mass. Considering no other energy loss during the whole process, the difference between the two photon energies (in eV) is ______.

An ideal monatomic gas of $$n$$ moles is taken through a cycle $$WXYZW$$ consisting of consecutive adiabatic and isobaric quasi-static processes, as shown in the schematic $$V$$-$$T$$ diagram. The volume of the gas at $$W$$, $$X$$ and $$Y$$ points are, 64 cm$$^3$$, 125 cm$$^3$$ and 250 cm$$^3$$, respectively. If the absolute temperature of the gas $$T_W$$ at the point $$W$$ is such that $$nRT_W = 1$$ J ($$R$$ is the universal gas constant), then the amount of heat absorbed (in J) by the gas along the path $$XY$$ is ______.

A geostationary satellite above the equator is orbiting around the earth at a fixed distance $$r_1$$ from the center of the earth. A second satellite is orbiting in the equatorial plane in the opposite direction to the earth's rotation, at a distance $$r_2$$ from the center of the earth, such that $$r_1 = 1.21 \, r_2$$. The time period of the second satellite as measured from the geostationary satellite is $$\frac{24}{p}$$ hours. The value of $$p$$ is ______.

The left and right compartments of a thermally isolated container of length $$L$$ are separated by a thermally conducting, movable piston of area $$A$$. The left and right compartments are filled with $$\frac{3}{2}$$ and 1 moles of an ideal gas, respectively. In the left compartment the piston is attached to a spring with spring constant $$k$$ and natural length $$\frac{2L}{5}$$. In thermodynamic equilibrium, the piston is at a distance $$\frac{L}{2}$$ from the left and right edges of the container as shown in the figure. Under the above conditions, if the pressure in the right compartment is $$P = \frac{kL}{A} \alpha$$, then the value of $$\alpha$$ is ______.

In a Young's double slit experiment, a combination of two glass wedges $$A$$ and $$B$$, having refractive indices 1.7 and 1.5, respectively, are placed in front of the slits, as shown in the figure. The separation between the slits is $$d = 2$$ mm and the shortest distance between the slits and the screen is $$D = 2$$ m. Thickness of the combination of the wedges is $$t = 12 \, \mu$$m. The value of $$l$$ as shown in the figure is 1 mm. Neglect any refraction effect at the slanted interface of the wedges. Due to the combination of the wedges, the central maximum shifts (in mm) with respect to O by ______.