JEE (Advanced) 2023 Paper-1

A bar of mass $$M = 1.00$$ kg and length $$L = 0.20$$ m is lying on a horizontal frictionless surface. One end of the bar is pivoted at a point about which it is free to rotate. A small mass $$m = 0.10$$ kg is moving on the same horizontal surface with 5.00 m s$$^{-1}$$ speed on a path perpendicular to the bar. It hits the bar at a distance $$L/2$$ from the pivoted end and returns back on the same path with speed v. After this elastic collision, the bar rotates with an angular velocity $$\omega$$. Which of the following statement is correct?

A container has a base of 50 cm $$\times$$ 5 cm and height 50 cm, as shown in the figure. It has two parallel electrically conducting walls each of area 50 cm $$\times$$ 50 cm. The remaining walls of the container are thin and non-conducting. The container is being filled with a liquid of dielectric constant 3 at a uniform rate of 250 cm$$^3$$ s$$^{-1}$$. What is the value of the capacitance of the container after 10 seconds?

[Given: Permittivity of free space $$\epsilon_0 = 9 \times 10^{-12}$$ C$$^2$$ N$$^{-1}$$m$$^{-2}$$, the effects of the non-conducting walls on the capacitance are negligible]

One mole of an ideal gas expands adiabatically from an initial state $$(T_A, V_0)$$ to final state $$(T_f, 5V_0)$$. Another mole of the same gas expands isothermally from a different initial state $$(T_B, V_0)$$ to the same final state $$(T_f, 5V_0)$$. The ratio of the specific heats at constant pressure and constant volume of this ideal gas is $$\gamma$$. What is the ratio $$T_A/T_B$$?

Two satellites P and Q are moving in different circular orbits around the Earth (radius $$R$$). The heights of P and Q from the Earth surface are $$h_P$$ and $$h_Q$$, respectively, where $$h_P = R/3$$. The accelerations of P and Q due to Earth's gravity are $$g_P$$ and $$g_Q$$, respectively. If $$g_P/g_Q = 36/25$$, what is the value of $$h_Q$$?

A Hydrogen-like atom has atomic number $$Z$$. Photons emitted in the electronic transitions from level $$n = 4$$ to level $$n = 3$$ in these atoms are used to perform photoelectric effect experiment on a target metal. The maximum kinetic energy of the photoelectrons generated is 1.95 eV. If the photoelectric threshold wavelength for the target metal is 310 nm, the value of $$Z$$ is ______.

[Given: $$hc = 1240$$ eV-nm and $$Rhc = 13.6$$ eV, where $$R$$ is the Rydberg constant, $$h$$ is the Planck's constant and $$c$$ is the speed of light in vacuum]

An optical arrangement consists of two concave mirrors M$$_1$$ and M$$_2$$, and a convex lens L with a common principal axis, as shown in the figure. The focal length of L is 10 cm. The radii of curvature of M$$_1$$ and M$$_2$$ are 20 cm and 24 cm, respectively. The distance between L and M$$_2$$ is 20 cm. A point object S is placed at the mid-point between L and M$$_2$$ on the axis. When the distance between L and M$$_1$$ is $$n/7$$ cm, one of the images coincides with S. The value of $$n$$ is ______.

In an experiment for determination of the focal length of a thin convex lens, the distance of the object from the lens is $$10 \pm 0.1$$ cm and the distance of its real image from the lens is $$20 \pm 0.2$$ cm. The error in the determination of focal length of the lens is $$n$$ %. The value of $$n$$ is ______.

A closed container contains a homogeneous mixture of two moles of an ideal monatomic gas ($$\gamma = 5/3$$) and one mole of an ideal diatomic gas ($$\gamma = 7/5$$). Here, $$\gamma$$ is the ratio of the specific heats at constant pressure and constant volume of an ideal gas. The gas mixture does a work of 66 Joule when heated at constant pressure. The change in its internal energy is ______ Joule.

A person of height 1.6 m is walking away from a lamp post of height 4 m along a straight path on the flat ground. The lamp post and the person are always perpendicular to the ground. If the speed of the person is 60 cm s$$^{-1}$$, the speed of the tip of the person's shadow on the ground with respect to the person is ______ cm s$$^{-1}$$.

Two point-like objects of masses 20 gm and 30 gm are fixed at the two ends of a rigid massless rod of length 10 cm. This system is suspended vertically from a rigid ceiling using a thin wire attached to its center of mass, as shown in the figure. The resulting torsional pendulum undergoes small oscillations. The torsional constant of the wire is $$1.2 \times 10^{-8}$$ N m rad$$^{-1}$$. The angular frequency of the oscillations in $$n \times 10^{-3}$$ rad s$$^{-1}$$. The value of $$n$$ is ______.