JEE Advanced 2020 Question Paper with Solutions

When water is filled carefully in a glass, one can fill it to a height h above the rim of the glass due to the surface tension of water. To calculate h just before water starts flowing, model the shape of the water above the rim as a disc of thickness h having semicircular edges, as shown schematically in the figure. When the pressure of water at the bottom of this disc exceeds what can be withstood due to the surface tension, the water surface breaks near the rim and water starts flowing from there. If the density of water, its surface tension and the acceleration due to gravity are 10 3 kg m −3 , 0.07 Nm −1 and 10 ms −2 , respectively, the value of h (in mm) is _________.

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 )

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 $\left( {{{{h_1}} \over {{h_2}}}} \right)$ of the heights of the liquid in the two arms is :

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 $\theta$ 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 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)

Newman projections P, Q, R and S are shown below : Which one of the following options represents identical molecules?

5.00 mL of 0.10 M oxalic acid solution taken in a conical flask is titrated against NaOH from a burette using phenolphthalein indicator. The volume of NaOH required for the appearance of permanent faint pink color is tabulated below for five experiments. What is the concentration, in molarity, of the NaOH solution? Exp. No. Vol. of NaOH (mL) 1 12.5 2 10.5 3 9.0 4 9.0 5 9.0

If the distribution of molecular speeds of a gas is as per the figure shown below, then the ratio of the most probable, the average, and the root mean square speeds, respectively, is

The Fischer projection of D-erythrose is shown below : D-erythrose and its isomers are listed as P, Q, R, and S in Column - I. Choose the correct relationship of P, Q, R, and S with D-erythrose from Column - II.

With respect to the compounds I-V, choose the correct statement(s).

Let x, y and z be positive real numbers. Suppose x, y and z are the lengths of the sides of a triangle opposite to its angles X, Y, and Z, respectively. If $\tan {X \over 2} + \tan {Z \over 2} = {{2y} \over {x + y + z}}$ , then which of the following statements is/are TRUE?

For a polynomial g(x) with real coefficients, let m g denote the number of distinct real roots of g(x). Suppose S is the set of polynomials with real coefficients defined by S = { ( x 2 − 1 ) 2 ( a 0 + a 1 x + a 2 x 2 + a 3 x 3 ) : a 0 , a 1 , a 2 , a 3 ∈ R } ; For a polynomial f, let f' and f'' denote its first and second order derivatives, respectively. Then the minimum possible value of (m f' + m f'' ), where f ∈ S, is ..............

In a triangle PQR, let a = QR, b = RP, and c = PQ. If |a| = 3, |b| = 4 and ${{a\,.(\,c - \,b)} \over {c\,.\,(a - \,b)}} = {{|a|} \over {|a| + |b|}}$ , then the value of |a $\times$ b| 2 is ......

Let a 1 , a 2 , a 3 , .... be a sequence of positive integers in arithmetic progression with common difference 2. Also, let b 1 , b 2 , b 3 , .... be a sequence of positive integers in geometric progression with common ratio 2. If a 1 = b 1 = c, then the number of all possible values of c, for which the equality 2(a 1 + a 2 + ... + a n ) = b 1 + b 2 + ... + b n holds for some positive integer n, is ...........

Let the function $f:(0,\pi ) \to R$ be defined by $f(\theta ) = {(\sin \theta + \cos \theta )^2} + {(\sin \theta - \cos \theta )^4}$ Suppose the function f has a local minimum at $\theta$ precisely when $\theta \in \{ {\lambda _1}\pi ,....,{\lambda _r}\pi \}$ , where $0 < {\lambda _1} < ...{\lambda _r} < 1$ . Then the value of ${\lambda _1} + ... + {\lambda _r}$ 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)

Shown in the figure is a semicircular metallic strip that has thickness t and resistivity $\rho$ . Its inner radius is R 1 and outer radius is R 2 . If a voltage V 0 is applied between its two ends, a current I flows in it. In addition, it is observed that a transverse voltage $\Delta$ V develops between its inner and outer surfaces due to purely kinetic effects of moving electrons (ignore any role of the magnetic field due to the current). Then (figure is schematic and not drawn to scale)

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)

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 Bo 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 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)

Which of the following inequalities is/are TRUE?

Let C 1 and C 2 be two biased coins such that the probabilities of getting head in a single toss are 2 3 and 1 3 , respectively. Suppose α is the number of heads that appear when C 1 is tossed twice, independently, and suppose β is the number of heads that appear when C 2 is tossed twice, independently. Then the probability that the roots of the quadratic polynomial x 2 − ax + β are real and equal, is

Consider the rectangles lying the region $\left\{ {(x,y) \in R \times R:0\, \le \,x\, \le \,{\pi \over 2}} \right.$ and $\left. {0\, \le \,y\, \le \,2\sin (2x)} \right\}$ and having one side on the X-axis. The area of the rectangle which has the maximum perimeter among all such rectangles, is

Let the functions f : R $\to$ R and g : R $\to$ R be defined by f(x) = e x $-$ 1 $-$ e $-$ |x $-$ 1| and g(x) = ${1 \over 2}$ (e x $-$ 1 + e 1 $-$ x ). The the area of the region in the first quadrant bounded by the curves y = f(x), y = g(x) and x = 0 is

Let M be a 3 $\times$ 3 invertible matrix with real entries and let I denote the 3 $\times$ 3 identity matrix. If M $-$ 1 = adj(adj M), then which of the following statements is/are ALWAYS TRUE?

$_{92}^{238}U$ is known to undergo radioactive decay to form $_{82}^{206}Pb$ by emitting alpha and beta particles. A rock initially contained 68 $\times$ 10 $-$ 6 g of $_{92}^{238}U$ . If the number of alpha particles that it would emit during its radioactive decay of $_{92}^{238}U$ to $_{82}^{206}Pb$ in three half-lives is Z $\times$ 10 18 , then what is the value of Z?

Choose the correct statement(s) among the following :

With respect to hypochlorite, chlorate and perchlorate ions, choose the correct statement(s).

A colourless aqueous solution contains nitrates of two metals, X and Y. When it was added to an aqueous solution of NaCl, a white precipitate was formed. This precipitate was found to be partly soluble in hot water to give a residue P and a solution Q. The residue P was soluble in aqueous NH 3 and also in excess sodium thiosulphate. The hot solution Q gave a yellow precipitate with KI. The metals X and Y, respectively, are

The cubic unit cell structure of a compound containing cation M and anion X is shown below. When compared to the anion, the cation has smaller ionic radius. Choose the correct statement(s).