JEE (Advanced) 2020 Paper-2

A large square container with thin transparent vertical walls and filled with water (refractive index $$\frac{4}{3}$$) is kept on a horizontal table. A student holds a thin straight wire vertically inside the water 12 cm from one of its corners, as shown schematically in the figure. Looking at the wire from this corner, another student sees two images of the wire, located symmetrically on each side of the line of sight as shown. The separation (in cm) between these images is ____________.

A train with cross-sectional area $$S_t$$ is moving with speed $$v_t$$ inside a long tunnel of cross-sectional area $$S_0(S_0 = 4S_t)$$. Assume that almost all the air (density $$\rho$$)in front of the train flows back between its sides and the walls of the tunnel. Also, the air flow with respect to the train is steady and laminar. Take the ambient pressure and that inside the train to be $$p_0$$. If the pressure in the region between the sides of the train and the tunnel walls is 𝑝, then $$p_0 - p = \frac{7}{2N}\rho v_t^2$$. The value of N is _____________.

Two large circular discs separated by a distance of 0.01 m are connected to a battery via a switch as shown in the figure. Charged oil drops of density 900 kg m$$^{−3}$$ are released through a tiny hole at the center of the top disc. Once some oil drops achieve terminal velocity, the switch is closed to apply a voltage of 200 V across the discs. As a result, an oil drop of radius $$8 \times 10^{−7}$$ m stops moving vertically and floats between the discs. The number of electrons present in this oil drop is ________. (neglect the buoyancy force, take acceleration due to gravity = 10 ms$$^{−2}$$ and charge on an electron (e) = $$1.6 \times 10^{-19}$$ C)

A hot air balloon is carrying some passengers, and a few sandbags of mass 1 kg each so that its total mass is 480 kg. Its effective volume giving the balloon its buoyancy is 𝑉. The balloon is floating at an equilibrium height of 100 m. When 𝑁 number of sandbags are thrown out, the balloon rises to a new equilibrium height close to 150 m with its volume 𝑉 remaining unchanged. If the variation of the density of air with height ℎ from the ground is $$\rho(h) = \rho_0 e^{-\frac{h}{h_0}}$$, where $$\rho_0 = 1.25$$ kg m$$^{-3}$$ and $$h_0 = 6000$$ m, the value of N is ________________.

A point charge q of mass 𝑚 is suspended vertically by a string of length 𝑙. A point dipole of dipole moment $$\overrightarrow{p}$$ is now brought towards q from infinity so that the charge moves away. The final equilibrium position of the system including the direction of the dipole, the angles and distances is shown in the figure below. If the work done in bringing the dipole to this position is $$N \times (mgh)$$, where g is the acceleration due to gravity, then the value of 𝑁 is _________ . (Note that for three coplanar forces keeping a point mass in equilibrium, $$\frac{F}{\sin \theta}$$ is the same for all forces, where 𝐹 is any one of the forces and $$\theta$$ is the angle between the other two forces)

A thermally isolated cylindrical closed vessel of height 8 m is kept vertically. It is divided into two equal parts by a diathermic (perfect thermal conductor) frictionless partition of mass 8.3 kg. Thus the partition is held initially at a distance of 4 m from the top, as shown in the schematic figure below. Each of the two parts of the vessel contains 0.1 mole of an ideal gas at temperature 300 K. The partition is now released and moves without any gas leaking from one part of the vessel to the other. When equilibrium is reached, the distance of the partition from the top (in m) will be _______ (take the acceleration due to gravity = 10 ms$$^{-2}$$ and the universal gas constant = 8.3 J mol$$^{-1}$$ K$$^{-1}$$).

A beaker of radius 𝑟 is filled with water (refractive index $$\frac{4}{3}$$) up to a height 𝐻 as shown in the figure on the left. The beaker is kept on a horizontal table rotating with angular speed $$\omega$$. This makes the water surface curved so that the difference in the height of water level at the center and at the circumference of the beaker is $$h(h \ll H, h \ll r)$$, as shown in the figure on the right. Take this surface to be approximately spherical with a radius of curvature 𝑅. Which of the following is/are correct? (g is the acceleration due to gravity)

A student skates up a ramp that makes an angle $$30^\circ$$ with the horizontal. He/she starts (as shown in the figure) at the bottom of the ramp with speed $$𝑣_0$$ and wants to turn around over a semicircular path xyz of radius 𝑅 during which he/she reaches a maximum height ℎ (at point y) from the ground as shown in the figure. Assume that the energy loss is negligible and the force required for this turn at the highest point is provided by his/her weight only. Then (g is the acceleration due to gravity)

A rod of mass 𝑚 and length 𝐿, pivoted at one of its ends, is hanging vertically. A bullet of the same mass moving at speed 𝑣 strikes the rod horizontally at a distance 𝑥 from its pivoted end and gets embedded in it. The combined system now rotates with angular speed $$\omega$$ about the pivot. The maximum angular speed $$\omega_M$$ is achieved for $$x = x_M$$. Then

In an X-ray tube, electrons emitted from a filament (cathode) carrying current I hit a target (anode) at a distance 𝑑 from the cathode. The target is kept at a potential 𝑉 higher than the cathode resulting in emission of continuous and characteristic X-rays. If the filament current 𝐼 is decreased to $$\frac{I}{2}$$, the potential difference 𝑉 is increased to 2𝑉, and the separation distance 𝑑 is reduced to $$\frac{d}{2}$$, then