JEE (Advanced) 2022 Paper-1

In the following circuit $$C_1 = 12$$ $$\mu F$$, $$C_2 = C_3 = 4$$ $$\mu F$$ and $$C_4 = C_5 = 2$$ $$\mu F$$. The charge stored in $$C_3$$ is ______ $$\mu C$$.

A rod of length 2 cm makes an angle $$\frac{2\pi}{3}$$ rad with the principal axis of a thin convex lens. The lens has a focal length of 10 cm and is placed at a distance of $$\frac{40}{3}$$ cm from the object as shown in the figure. The height of the image is $$\frac{30\sqrt{3}}{13}$$ cm and the angle made by it with respect to the principal axis is $$\alpha$$ rad. The value of $$\alpha$$ is $$\frac{\pi}{n}$$ rad, where $$n$$ is ______.

At time $$t = 0$$, a disk of radius 1 m starts to roll without slipping on a horizontal plane with an angular acceleration of $$\alpha = \frac{2}{3}$$ rad s$$^{-2}$$. A small stone is stuck to the disk. At $$t = 0$$, it is at the contact point of the disk and the plane. Later, at time $$t = \sqrt{\pi}$$ s, the stone detaches itself and flies off tangentially from the disk. The maximum height (in m) reached by the stone measured from the plane is $$\frac{1}{2} + \frac{x}{10}$$. The value of $$x$$ is ______. [Take $$g = 10$$ m s$$^{-2}$$.]

A solid sphere of mass 1 kg and radius 1 m rolls without slipping on a fixed inclined plane with an angle of inclination $$\theta = 30^\circ$$ from the horizontal. Two forces of magnitude 1 N each, parallel to the incline, act on the sphere, both at distance $$r = 0.5$$ m from the center of the sphere, as shown in the figure. The acceleration of the sphere down the plane is ______ ms$$^{-2}$$. (Take $$g = 10$$ m s$$^{-2}$$.)

Consider an LC circuit, with inductance $$L = 0.1$$ H and capacitance $$C = 10^{-3}$$ F, kept on a plane. The area of the circuit is 1 m$$^2$$. It is placed in a constant magnetic field of strength $$B_0$$ which is perpendicular to the plane of the circuit. At time $$t = 0$$, the magnetic field strength starts increasing linearly as $$B = B_0 + \beta t$$ with $$\beta = 0.04$$ Ts$$^{-1}$$. The maximum magnitude of the current in the circuit is ______ mA.

A projectile is fired from horizontal ground with speed $$v$$ and projection angle $$\theta$$. When the acceleration due to gravity is $$g$$, the range of the projectile is $$d$$. If at the highest point in its trajectory, the projectile enters a different region where the effective acceleration due to gravity is $$g' = \frac{g}{0.81}$$, then the new range is $$d' = nd$$. The value of $$n$$ is ______.

A medium having dielectric constant $$K > 1$$ fills the space between the plates of a parallel plate capacitor. The plates have large area, and the distance between them is $$d$$. The capacitor is connected to a battery of voltage $$V$$, as shown in Figure (a). Now, both the plates are moved by a distance of $$\frac{d}{2}$$ from their original positions, as shown in Figure (b).

In the process of going from the configuration depicted in Figure (a) to that in Figure (b), which of the following statement(s) is(are) correct?

The figure shows a circuit having eight resistances of 1 $$\Omega$$ each, labelled $$R_1$$ to $$R_8$$, and two ideal batteries with voltages $$\varepsilon_1 = 12$$ V and $$\varepsilon_2 = 6$$ V.

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

An ideal gas of density $$\rho = 0.2$$ kg m$$^{-3}$$ enters a chimney of height h at the rate of $$\alpha = 0.8$$ kg s$$^{-1}$$ from its lower end, and escapes through the upper end as shown in the figure. The cross-sectional area of the lower end is $$A_1 = 0.1$$ m$$^2$$ and the upper end is $$A_2 = 0.4$$ m$$^2$$. The pressure and the temperature of the gas at the lower end are 600 Pa and 300 K, respectively, while its temperature at the upper end is 150 K. The chimney is heat insulated so that the gas undergoes adiabatic expansion. Take $$g = 10$$ ms$$^{-2}$$ and the ratio of specific heats of the gas $$\gamma = 2$$. Ignore atmospheric pressure.

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

Three plane mirrors form an equilateral triangle with each side of length $$L$$. There is a small hole at a distance $$l > 0$$ from one of the corners as shown in the figure. A ray of light is passed through the hole at an angle $$\theta$$ and can only come out through the same hole. The cross section of the mirror configuration and the ray of light lie on the same plane.

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