JEE (Advanced) 2022 Paper-2

Two resistances $$R_1 = X$$ $$\Omega$$ and $$R_2 = 1$$ $$\Omega$$ are connected to a wire AB of uniform resistivity, as shown in the figure. The radius of the wire varies linearly along its axis from 0.2 mm at A to 1 mm at B. A galvanometer (G) connected to the center of the wire, 50 cm from each end along its axis, shows zero deflection when A and B are connected to a battery. The value of X is _______.

In a particular system of units, a physical quantity can be expressed in terms of the electric charge $$e$$, electron mass $$m_e$$, Planck's constant $$h$$, and Coulomb's constant $$k = \dfrac{1}{4\pi \epsilon_0}$$, where $$\epsilon_0$$ is the permittivity of vacuum. In terms of these physical constants, the dimension of the magnetic field is [B] = [e]$$^\alpha$$ [m$$_e$$]$$^\beta$$ [h]$$^\gamma$$ [k]$$^\delta$$. The value of $$\alpha + \beta + \gamma + \delta$$ is _______.

Consider a configuration of $$n$$ identical units, each consisting of three layers. The first layer is a column of air of height $$h = \dfrac{1}{3}$$ cm, and the second and third layers are of equal thickness $$d = \dfrac{\sqrt{3}-1}{2}$$ cm, and refractive indices $$\mu_1 = \dfrac{\sqrt{3}}{2}$$ and $$\mu_2 = \sqrt{3}$$, respectively. A light source O is placed on the top of the first unit as shown in the figure. A Ray of light from $$O$$ is incident on the second layer of the first unit at an angle of $$\theta = 60^\circ$$ to the normal. For a specific value of $$n$$, the ray of light emerges from the bottom of the configuration at a distance $$l = \dfrac{8}{\sqrt{3}}$$ cm, as shown in the figure. The value of $$n$$ is _______.

A charge $$q$$ is surrounded by a closed surface consisting of an inverted cone of height $$h$$ and base radius $$R$$, and a hemisphere of radius $$R$$ as shown in the figure. The electric flux through the conical surface is $$\dfrac{nq}{6\epsilon_0}$$ (in SI units). The value of $$n$$ is _______.

On a frictionless horizontal plane, a bob of mass $$m = 0.1$$ kg is attached to a spring with natural length $$l_0 = 0.1$$ m. The spring constant is $$k_1 = 0.009$$ N m$$^{-1}$$ when the length of the spring $$l > l_0$$ and is $$k_2 = 0.016$$ N m$$^{-1}$$ when $$l < l_0$$. Initially the bob is released from $$l = 0.15$$ m. Assume that Hooke's law remains valid throughout the motion. If the time period of the full oscillation is T = $$(n\pi)$$ s, then the integer closest to $$n$$ is _______.

An object and a concave mirror of focal length $$f = 10$$ cm both move along the principal axis of the mirror with constant speeds. The object moves with speed $$V_0 = 15$$ cm s$$^{-1}$$ towards the mirror with respect to a laboratory frame. The distance between the object and the mirror at a given moment is denoted by $$u$$. When $$u = 30$$ cm, the speed of the mirror $$V_m$$ is such that the image is instantaneously at rest with respect to the laboratory frame, and the object forms a real image. The magnitude of $$V_m$$ is _______ cm s$$^{-1}$$.

In the figure, the inner (shaded) region A represents a sphere of radius $$r_A = 1$$, within which the electrostatic charge density varies with the radial distance $$r$$ from the center as $$\rho_A = kr$$, where $$k$$ is positive. In the spherical shell B of outer radius $$r_B$$, the electrostatic charge density varies as $$\rho_B = \dfrac{2k}{r}$$. Assume that dimensions are taken care of. All physical quantities are in their SI units.

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

In Circuit-1 and Circuit-2 shown in the figures, $$R_1 = 1$$ $$\Omega$$, $$R_2 = 2$$ $$\Omega$$ and $$R_3 = 3$$ $$\Omega$$. $$P_1$$ and $$P_2$$ are the power dissipations in Circuit-1 and Circuit-2 when the switches $$S_1$$ and $$S_2$$ are in open conditions, respectively.

$$Q_1$$ and $$Q_2$$ are the power dissipations in Circuit-1 and Circuit-2 when the switches $$S_1$$ and $$S_2$$ are in closed conditions, respectively.

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

A bubble has surface tension $$S$$. The ideal gas inside the bubble has ratio of specific heats $$\gamma = \dfrac{5}{3}$$. The bubble is exposed to the atmosphere and it always retains its spherical shape. When the atmospheric pressure is $$P_{a1}$$, the radius of the bubble is found to be $$r_1$$ and the temperature of the enclosed gas is $$T_1$$. When the atmospheric pressure is $$P_{a2}$$, the radius of the bubble and the temperature of the enclosed gas are $$r_2$$ and $$T_2$$, respectively.

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

A disk of radius $$R$$ with uniform positive charge density $$\sigma$$ is placed on the $$xy$$ plane with its center at the origin. The Coulomb potential along the z-axis is

$$V(z) = \dfrac{\sigma}{2\epsilon_0}\left(\sqrt{R^2 + z^2} - z\right)$$

A particle of positive charge $$q$$ is placed initially at rest at a point on the z axis with $$z = z_0$$ and $$z_0 > 0$$. In addition to the Coulomb force, the particle experiences a vertical force $$\vec{F} = -c\hat{k}$$ with $$c > 0$$. Let $$\beta = \dfrac{2c\epsilon_0}{q\sigma}$$. Which of the following statement(s) is(are) correct?