JEE Advanced 2020 — Paper 1

A football of radius R is kept on a hole of radius $\mathrm{r}(\mathrm{r}<\mathrm{R})$ 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 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 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)

An open-ended U-tube of uniform cross-sectional area contains water (density $10^{3} \mathrm{~kg} \mathrm{~m}^{-3}$ ). Initially the water level stands at 0.29 m from the bottom in each arm. Kerosene oil (a water-immiscible liquid) of density $800 \mathrm{~kg} \mathrm{~m}^{-3}$ is added to the left arm until its length is 0.1 m , as shown in the schematic figure below. The ratio $\left(\frac{h_{1}}{h_{2}}\right)$ of the heights of the liquid in the two arms is

A particle of mass $m$ moves in circular orbits with potential energy $V(r)=F r$, 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 $\mathrm{n}^{\text {th }}$ orbit (here h is the Planck's constant)

The filament of a light bulb has surface area $64 \mathrm{~mm}^{2}$. 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 $=5.67 \times 10^{-8} \mathrm{Wm}^{-2} \mathrm{~K}^{-4}$, Wien's displacement constant

$=2.90 \times 10^{-3} \mathrm{~m}-\mathrm{K}$, Planck's constant $=6.63 \times 10^{-34} \mathrm{Js}$, speed of light in vacuum

$=3.00 \times 108 \mathrm{~ms}^{-1}$ )

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 $]=\left[\mathrm{X}^{\alpha}\right] ;[$ Speed $]=\left[\mathrm{X}^{\beta}\right]$; [acceleration $]=\left[\mathrm{X}^{\mathrm{p}}\right]$; [Linear momentum $]=\left[\mathrm{X}^{q}\right]$; [force] $=\left[X^{\prime}\right]$. Then

A uniform electric field, $\overrightarrow{\mathrm{E}}=-400 \sqrt{3} \hat{\mathrm{y}} \mathrm{NC}^{-1}$ 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 $2 \sqrt{10} \times 10^{6} \mathrm{~ms}^{-1}$. 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 $\frac{\mathrm{q}}{\mathrm{m}}=10^{10} \mathrm{Ckg}^{-1}$. Then

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 $\Delta \mathrm{V}$ 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)

As shown schematically in the figure, two vessels contain water solutions (at temperature T) of potassium permanganate $\left(\mathrm{KMnO}_{4}\right)$ of different concentrations $\mathrm{n}_{1}$ and $\mathrm{n}_{2}\left(\mathrm{n}_{1}>\mathrm{n}_{2}\right)$ molecules per unit volume with $\Delta \mathrm{n}=\left(\mathrm{n}_{1}-\mathrm{n}_{2}\right) \ll \mathrm{n}_{1}$. When they are connected by a tube of small length $\ell$ and cross-sectional area $\mathrm{S}, \mathrm{KMnO}_{4}$ starts to diffuse from the left to the right vessel through the tube. Consider the collection of molecules to behave as dilute ideal gases and the difference in their partial pressure in the two vessels causing the diffusion. The speed $v$ of the molecules is limited by the viscous force $-\beta v$ on each molecule, where $\beta$ is a constant. Neglecting all terms of the order $(\Delta \mathrm{n})^{2}$, which of the following is/are correct? ( $k_{B}$ is the Boltzmann constant)

Put a uniform meter scale horizontally on your extended index fingers with the left one at 0.00 cm and the right one at 90.00 cm . When you attempt to move both the fingers slowly towards the center, initially only the left finger slips with respect to the scale and the right finger does not. After some distance, the left finger stops and the right one starts slipping. Then the right finger stops at a distance $x_{R}$ from the center ( 50.00 cm ) of the scale and the left one starts slipping again. This happens because of the difference in the frictional forces on the two fingers. If the coefficients of static and dynamic friction between the fingers and the scale are 0.40 and 0.32 , respectively, the value of $x_{R}(\mathrm{in} \mathrm{cm})$ is $\qquad$ —.

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 $10^{3} \mathrm{~kg} \mathrm{~m}^{-3}, 0.07 \mathrm{Nm}^{-1}$ and $10 \mathrm{~ms}^{-2}$, respectively, the value of h (in mm ) is $\_\_\_\_$ .

One end of a spring of negligible unstretched length and spring constant k is fixed at the origin ( 0,0 ). A point particle of mass $m$ carrying a positive charge $q$ is attached at its other end. The entire system is kept on a smooth horizontal surface. When a point dipole $\overrightarrow{\mathrm{p}}$ pointing towards the charge q is fixed at the origin, the spring gets stretched to a length $l$ and attains a new equilibrium position (see figure below). If the point mass is now displaced slightly by $\Delta l \ll l$ from its equilibrium position and released, it is found to oscillate at frequency $\frac{1}{\delta} \sqrt{\frac{\mathrm{k}}{\mathrm{m}}}$. The value of $\delta$ is $\_\_\_\_$ .

Consider one mole of helium gas enclosed in a container at initial pressure $P_{1}$ and volume $V_{1}$. It expands isothermally to volume $4 V_{1}$. After this, the gas expands adiabatically and its volume becomes $32 V_{1}$. The work done by the gas during isothermal and adiabatic expansion processes are $W_{\text {iso }}$ and $W_{\text {adia }}$, respectively. If the ratio $\frac{W_{\text {iso }}}{W_{\text {adia }}}=f \ln 2$, then $f$ is $\_\_\_\_$ .

A stationary tuning fork is in resonance with an air column in a pipe. If the tuning fork is moved with a speed of $2 \mathrm{~ms}^{-1}$ in front of the open end of the pipe and parallel to it, the length of the pipe should be changed for the resonance to occur with the moving tuning fork. If the speed of sound in air is $320 \mathrm{~ms}^{-1}$, the smallest value of the percentage change required in the length of the pipe is $\_\_\_\_\_\_\_\_$ .

A circular disc of radius $R$ carries surface charge density $\sigma_{0}(r)=\sigma_{0}\left(1-\frac{r}{R}\right)$, where $\sigma_{0}$ is a constant and r is the distance from the center of the disc. Electric flux through a large spherical surface that encloses the charged disc completely is $\phi_{0}$. Electric flux through another spherical surface of radius $\frac{R}{4}$ and concentric with the disc is $\phi$. Then the ratio $\frac{\phi_{0}}{\phi}$ 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, $\mathrm{Q}, \mathrm{R}$ and S with D-erythrose from Column-II.

Column-I	Column-II
P.	1. Diastereomer
Q.	2. Identical
R.	3. Enantiomer
S.

In thermodynamics, the $\mathrm{P}-\mathrm{V}$ work done is given by $\mathrm{w}=-\int \mathrm{dV} \mathrm{P}_{\mathrm{ext}}$ For a system undergoing a

particular process, the work done is $\mathrm{w}=-\int \mathrm{dV}\left(\frac{\mathrm{RT}}{\mathrm{~V}-\mathrm{b}}-\frac{\mathrm{a}}{\mathrm{~V}^{2}}\right)$ This equation is applicable to a

With respect to the compounds $\mathbf{I}-\mathbf{V}$, choose the correct statement(s).

Choose the correct statement(s) among the following:

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

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).

$5.00 \mathrm{~mL}$ 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?

Consider the reaction $A \rightleftharpoons B$ at 1000 K . At time $\mathrm{t}^{\prime}$, the temperature of the system was increased to 2000 K and the system was allowed to reach equilibrium. Throughout this experiment the partial pressure of $A$ was maintained at 1 bar. Given below is the plot of the partial pressure of $B$ with time. What is the ratio of standard Gibbs energy of the reaction at 1000 K to that at 2000 K ?

Consider a $70 \%$ efficient hydrogen-oxygen fuel cell working under standard conditions at 1 bar and 298 K . Its cell

reaction is

$\mathrm{H}_{2}(\mathrm{~g})+\frac{1}{2} \mathrm{O}_{2}(\mathrm{~g}) \longrightarrow \mathrm{H}_{2} \mathrm{O}(\ell)$

The work derived from the cell on the consumption of $10 \times 10^{-3} \mathrm{~mol}$ of $\mathrm{H}_{2}(\mathrm{~g})$ is used to compress 1.00

mol of a monoatomic ideal gas in a thermally insulated container. What is the change in the temperature (in K ) of

the ideal gas?

The standard reduction potentials for the two half - cells are given below.

$\mathrm{O}_{2}(\mathrm{~g})+4 \mathrm{H}^{+}(\mathrm{aq})+4 \mathrm{e}^{-} \longrightarrow 2 \mathrm{H}_{2} \mathrm{O}(\ell), \mathrm{E}^{0}=1.23 \mathrm{~V}$,

$2 \mathrm{H}^{+}(\mathrm{aq})+2 \mathrm{e}^{-} \longrightarrow \mathrm{H}_{2}(\mathrm{~g}), \mathrm{E}^{0}=0.00 \mathrm{~V}$.

Use $F=96500 \mathrm{C} \mathrm{mol}^{-1}, \mathrm{R}=8.314 \mathrm{~J} \mathrm{~mol}^{-1} \mathrm{~K}^{-1}$.

Aluminium reacts with sulfuric acid to form aluminium sulfate and hydrogen. What is the volume of hydrogen gas

in litre (L) produced at 300 K and 1.0 atm pressure, when 5.4 g of aluminium and 50.0 mL of 5.0 M sulfuric acid are

combined for the reaction? (Use molar mass of aluminium as $27.0 \mathrm{~g} \mathrm{~mol}^{-1}, \mathrm{R}=0.082 \mathrm{~atm} \mathrm{~L} \mathrm{~mol}^{-1}$

$\mathrm{~K}^{-1}$ )

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

Suppose $a$, $b$ denote the distinct real roots of the quadratic polynomial $x^{2}+20 x-2020$ and suppose $c, d$

denote the distinct complex roots of the quadratic polynomial $x^{2}-20 x+2020$. Then the value of

If the function $f: R \rightarrow R$ is defined by $f(x)=|x|(x-\sin x)$, then which of the following statements is TRUE?

Let the function $\mathrm{f}: \mathrm{R} \rightarrow \mathrm{R}$ and $\mathrm{g}: \mathrm{R} \rightarrow \mathrm{R}$ be defined by

Then the area of the region in the first quadrant bounded by the curves $y=f(x), y=g(x)$ and $x=0$ is

Let $\mathrm{a}, \mathrm{b}$ and $\lambda$ be positive real numbers. Suppose P is an end point of the latus rectum of the parabola $\mathrm{y}^{2}=$ $4 \lambda x$, and suppose the ellipse $\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1$ passes through the point $P$. If the tangents to the parabola and the ellipse at the point $P$ are perpendicular to each other, then the eccentricity of the ellipse is

Let $C_{1}$ and $C_{2}$ be two biased coins such that the probabilities of getting head in a single toss are $\frac{2}{3}$ and $\frac{1}{3}$, respectively. Suppose $\alpha$ is the number of heads that appear when $C_{1}$ is tossed twice, independently, and suppose $\beta$ is the number of heads that appear when $\mathrm{C}_{2}$ is tossed twice, independently. Then the probability that the roots of the quadratic polynomial $x^{2}-a x+\beta$ are real and equal, is

Consider all rectangles lying in the region

$\left\{(x, y) \in R \times R: 0 \leq x \leq \frac{\pi}{2} \text { and } 0 \leq y \leq 2 \sin (2 x)\right\}$ 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 function $f: R \rightarrow R$ be defined by $f(x)=x^{3}-x^{2}+(x-1)$ sin $x$ and let $g: R \rightarrow R$ be an

arbitrary function. Let $\mathrm{fg}: \mathrm{R} \rightarrow \mathrm{R}$ be the product function defined by $(\mathrm{fg})(\mathrm{x})=\mathrm{f}(\mathrm{x}) \mathrm{g}(\mathrm{x})$. Then which of

the following statements is/are TRUE?

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

Let $S$ be the set of all complex numbers $z$ satisfying $\left|z^{2}+z+1\right|=1$. Then which of the following statements is/are TRUE?

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

Let $L_{1}$ and $L_{2}$ be the following straight lines.

Suppose the straight line

lies in the plane containing $L_{1}$ and $L_{2}$, and passes through the point of intersection of $L_{1}$ and $L_{2}$. If the line $L$

bisects the acute angle between the lines $L_{1}$ and $L_{2}$, then which of the following statements is/are TRUE?

Let m be the minimum possible value of $\log _{3}\left(3^{y_{1}}+3^{y_{2}}+3^{y_{3}}\right)$, where $y_{1}, y_{2}, y_{3}$ are real numbers for

which $y_{1}+y_{2}+y_{3}=9$. Let $M$ be the maximum possible value of $\left(\log _{3} x_{1}+\log _{3} x_{2}+\log _{3} x_{3}\right)$,

where $x_{1}$, $x_{2}, x_{3}$ are positive real numbers for which $x_{1}+x_{2}+x_{3}=9$. Then the value of

$\log _{2}\left(m^{3}\right)+\log _{3}\left(M^{2}\right)$ is

Let $a_{1}, a_{2}, a_{3}, \ldots .$. be a sequence of positive integers in arithmetic progression with common difference 2 .

Also, let $b_{1}, b_{2}, b_{3}, \ldots .$. be a sequence of positive integers in geometric progression with common ratio 2. If

$\mathrm{a}_{1}=\mathrm{b}_{1}=\mathrm{c}$, then the number of all possible values of c , for which the equality

$2\left(a_{1}+a_{2}+\ldots . .+a_{n}\right)=b_{1}+b_{2}+\ldots . .+b_{n}$ holds for some positive integer $n$, is $\_\_\_\_$

Let $\mathrm{f}:[0,2] \rightarrow \mathrm{R}$ be the function defined by

If $\alpha, \beta \in[0,2]$ are such that $\{x \in[0,2]: f(x) \geq 0\}=[\alpha, \beta]$, then the value of $\beta-\alpha$ is $\_\_\_\_$

In a triangle $P Q R$, let $\vec{a}=\overrightarrow{Q R}, \vec{b}=\overrightarrow{R P}$ and $\vec{c}=\overrightarrow{P Q}$. If

$|\overrightarrow{\mathrm{a}}|=3,|\overrightarrow{\mathrm{~b}}|=4 \text { and } \frac{\overrightarrow{\mathrm{a}} \cdot(\overrightarrow{\mathrm{c}}-\overrightarrow{\mathrm{b}})}{\overrightarrow{\mathrm{c}} \cdot(\overrightarrow{\mathrm{a}}-\overrightarrow{\mathrm{b}})}=\frac{|\overrightarrow{\mathrm{a}}|}{|\overrightarrow{\mathrm{a}}|+|\overrightarrow{\mathrm{b}}|},$ then the value of $|\vec{a} \times \vec{b}|^{2}$ is $\_\_\_\_$

For a polynomial $g(x)$ with real coefficients, let $\mathrm{m}_{\mathrm{g}}$ denote the number of distinct real roots of $\mathrm{g}(\mathrm{x})$. Suppose S in the set of polynomials with real coefficients defined by $S=\left\{\left(x^{2}-1\right)^{2}\left(a_{0}+a_{1} x+a_{2} x^{2}+a_{3} x^{3}\right): a_{0}, a_{1}, a_{2}, a_{3} \in R\right\}$ For a polynomial $f$, let $f^{\prime}$ and $f^{\prime \prime}$ denote its first and second order derivatives, respectively. Then the minimum possible value of $\left(\mathrm{m}_{\mathrm{f}^{\prime}}+\mathrm{m}_{\mathrm{f}^{\prime \prime}}\right)$, where $\mathrm{f} \in \mathrm{S}$, is $\_\_\_\_$

Let e denote the base of the natural logarithm. The value of the real number a for which the right hand limit

$\lim _{x \rightarrow 0^{+}} \frac{(1-x)^{1 / x}-e^{-1}}{x^{a}}$ is equal to a nonzero real number, is $\_\_\_\_$