JEE (Advanced) 2021 Paper-1

A wide slab consisting of two media of refractive indices $$n_1$$ and $$n_2$$ is placed in air as shown in the figure. A ray of light is incident from medium $$n_1$$ to $$n_2$$ at an angle $$\theta$$, where $$\sin \theta$$ is slightly larger than $$\frac{1}{n_1}$$. Take refractive index of air as 1. Which of the following statement(s) is(are) correct?

A particle of mass M = 0.2 kg is initially at rest in the xy-plane at a point (x = −l, y = −h), where l = 10 m and h = 1 m. The particle is accelerated at time t = 0 with a constant acceleration 𝑎 = 10 m/s$$^2$$ along the positive x-direction. Its angular momentum and torque with respect to the origin, in SI units, are represented by $$\overrightarrow{L}$$ and $$\overrightarrow{\tau}$$, respectively. $$\hat{i}, \hat{j}$$ and $$\hat{k}$$ are unit vectors along the positive x, y and z-directions, respectively. If $$\hat{k} = \hat{i} \times \hat{j}$$ then which of the following statement(s) is(are) correct?

Which of the following statement(s) is(are) correct about the spectrum of hydrogen atom?

A long straight wire carries a current, I = 2 ampere. A semi-circular conducting rod is placed beside it on two conducting parallel rails of negligible resistance. Both the rails are parallel to the wire. The wire, the rod and the rails lie in the same horizontal plane, as shown in the figure. Two ends of the semi-circular rod are at distances 1 cm and 4 cm from the wire. At time t = 0, the rod starts moving on the rails with a speed v = 3.0 m/s (see the figure).
A resistor R = 1.4 Ω and a capacitor $$C_o = 5.0 \mu F$$ are connected in series between the rails. At time t = 0, $$C_o$$ is uncharged. Which of the following statement(s) is(are) correct?
$$[\mu_0 = 4 \pi \times 10^{−7}$$ SI units. Take $$\ln 2 = 0.7]$$

A cylindrical tube, with its base as shown in the figure, is filled with water. It is moving down with a constant acceleration 𝑎 along a fixed inclined plane with angle $$\theta = 45^\circ. P_1$$ and $$P_2$$ are pressures at points 1 and 2, respectively, located at the base of the tube. Let $$\beta = \frac{(P_1 − P_2)}{(\rho gd)}$$, where $$\rho$$ is density of water, 𝑑 is the inner diameter of the tube and 𝑔 is the acceleration due to gravity. Which of the following statement(s) is(are) correct?

An $$\alpha$$-particle (mass 4 amu) and a singly charged sulfur ion (mass 32 amu) are initially at rest. They are accelerated through a potential V and then allowed to pass into a region of uniform magnetic field which is normal to the velocities of the particles. Within this region, the $$\alpha$$-particle and the sulfur ion move in circular orbits of radii $$r_{\alpha}$$ and $$r_{S}$$, respectively. The ratio $$\left(\frac{r_S}{r_{\alpha}} \right)$$ is ___.

A thin rod of mass M and length 𝑎 is free to rotate in horizontal plane about a fixed vertical axis passing through point O. A thin circular disc of mass M and of radius 𝑎/4 is pivoted on this rod with its center at a distance 𝑎/4 from the free end so that it can rotate freely about its vertical axis, as shown in the figure. Assume that both the rod and the disc have uniform density and they remain horizontal during the motion. An outside stationary observer finds the rod rotating with an angular velocity Ω and the disc rotating about its vertical axis with angular velocity 4Ω. The total angular momentum of the system about the point O is $$\left(\frac{M a^2 Ω}{48}\right)n$$.
The value of n is .........

A small object is placed at the center of a large evacuated hollow spherical container. Assume that the container is maintained at 0 K. At time 𝑡=0, the temperature of the object is 200 K. The temperature of the object becomes 100 K at $$t = t_1$$ and 50 K at $$t = t_2$$. Assume the object and the container to be ideal black bodies. The heat capacity of the object does not depend on temperature. The ratio $$\left(\frac{t_2}{t_1} \right)$$ is ___.

The major product formed in the following reaction is

Among the following, the conformation that corresponds to the most stable conformation of meso-butane-2,3-diol is