JEE (Advanced) 2020 Paper-1

A football of radius R is kept on a hole of radius r (𝑟 < 𝑅) made on a plank kept horizontally. One end of the plank is now lifted so that it gets tilted making an angle 𝜃 from the horizontal as shown in the figure below. The maximum value of $$\theta$$ so that the football does not start rolling down the plank satisfies (figure is schematic and not drawn to scale)

A light disc made of aluminium (a nonmagnetic material) is kept horizontally and is free to rotate about its axis as shown in the figure. A strong magnet is held vertically at a point above the disc away from its axis. On revolving the magnet about the axis of the disc, the disc will (figure is schematic and not drawn to scale)

A small roller of diameter 20 cm has an axle of diameter 10 cm (see figure below on the left). It is on a horizontal floor and a meter scale is positioned horizontally on its axle with one edge of the scale on top of the axle (see figure on the right). The scale is now pushed slowly on the axle so that it moves without slipping on the axle, and the roller starts rolling without slipping. After the roller has moved 50 cm, the position of the scale will look like (figures are schematic and not drawn to scale)

A circular coil of radius R and N turns has negligible resistance. As shown in the schematic figure, its two ends are connected to two wires and it is hanging by those wires with its plane being vertical. The wires are connected to a capacitor with charge Q through a switch. The coil is in a horizontal uniform magnetic field $$B_{o}$$ parallel to the plane of the coil. When the switch is closed, the capacitor gets discharged through the coil in a very short time. By the time the capacitor is discharged fully, magnitude of the angular momentum gained by the coil will be (assume that the discharge time is so short that the coil has hardly rotated during this time)

A parallel beam of light strikes a piece of transparent glass having cross section as shown in the figure below. Correct shape of the emergent wavefront will be (figures are schematic and not drawn to scale)

An open-ended U-tube of uniform cross-sectional area contains water (density $$10^{3}kg m^{−3}$$). Initially the water level stands at 0.29 m from the bottom in each arm. Kerosene oil (a water-immiscible liquid) of density 800 kg$$ m^{−3}$$ is added to the left arm until its length is 0.1 m, as shown in the schematic figure below. The ratio $$\left(\frac{ℎ1}{ℎ2}\right)$$ of the heights of the liquid in the two arms is 

A particle of mass m moves in circular orbits with potential energy 𝑉(𝑟) = 𝐹𝑟, where F is a positive constant and r is its distance from the origin. Its energies are calculated using the Bohr model. If the radius of the particle’s orbit is denoted by R and its speed and energy are denoted by v and E, respectively, then for the $$n^{th}$$ orbit (here h is the Planck’s constant)

The filament of a light bulb has surface area $$64 mm^{2}$$. The filament can be considered as a black
body at temperature 2500 K emitting radiation like a point source when viewed from far. At night the light bulb is observed from a distance of 100 m. Assume the pupil of the eyes of the observer to be circular with radius 3 mm. Then
(Take Stefan-Boltzmann constant = $$5.67 \times 10−8 Wm^{−2}K^{−4}$$, Wien’s displacement constant = $$2.90 \times 10^{−3} m^{-K}$$, Planck’s constant = $$6.63 \times 10^{−34} $$Js, speed of light in vacuum = $$3.00 \times 10^{8} ms^{−1}$$)

Sometimes it is convenient to construct a system of units so that all quantities can be expressed in
terms of only one physical quantity. In one such system, dimensions of different quantities are given
in terms of a quantity X as follows: $$[position] = [𝑋^{\alpha}]; [speed] = [𝑋^{\beta}]; [acceleration] =[𝑋^{p}]; [linear momentum] = [𝑋^{q}]; [force] = [𝑋^{r}]$$. Then

A uniform electric field, $$\overrightarrow{E} = −400\sqrt{3}\hat{y} NC^{−1}$$ is applied in a region. A charged particle of mass m carrying positive charge q is projected in this region with an initial speed of $$2\sqrt{10} \times 106 ms^{−1}$$. This particle is aimed to hit a target T, which is 5 m away from its entry point into the field as shown schematically in the figure. Take $$\frac{q}{m} = 10^{10} Ckg^{−1}$$. Then