JEE Advanced 2020 — Paper 2

A train with cross-sectional area $S_{t}$ is moving with speed $v_{t}$ inside a long tunnel of cross-sectional area $\mathrm{S}_{0}\left(\mathrm{~S}_{0}=4 \mathrm{~S}_{\mathrm{t}}\right)$. 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 $\mathrm{p}_{0}$. If the pressure in the region between the sides of the train and the tunnel walls is $p$, then $p_{0}-p=\frac{7}{2 N} \rho v_{t}^{2}$. 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 $900 \mathrm{~kg} \mathrm{~m}^{-3}$ 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 $8 \times 10^{-7} \mathrm{~m}$ 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 $=10 \mathrm{~ms}^{-2}$ and charge on an electron (e) $=1.6 \times 10^{-19} \mathrm{C}$ )

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 $\rho_{0}=1.25 \mathrm{~kg} \mathrm{~m}^{-3}$ and $h_{0}=6000 \mathrm{~m}$, 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 $\vec{p}$ 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 $\mathrm{N} \times(\mathrm{mgh})$, 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, $\frac{\mathrm{F}}{\sin \theta}$ 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 $=10 \mathrm{~ms}^{-2}$ and the universal gas constant $=8.3 \mathrm{~J} \mathrm{~mol}-1 \mathrm{~K}^{-1}$ ).

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 $\mathrm{v}_{0}$ 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 $2 V$, and the separation distance $d$ is reduced to $\frac{d}{2}$, then

Two identical non-conducting solid spheres of same mass and charge are suspended in air from a common point by two non-conducting, massless strings of same length. At equilibrium, the angle between the strings is $\alpha$. The spheres are now immersed in a dielectric liquid of density $800 \mathrm{~kg} \mathrm{~m}^{-3}$ and dielectric constant 21. If the angle between the strings remains the same after the immersion, then

Starting at time $\mathrm{t}=0$ from the origin with speed $1 \mathrm{~ms}^{-1}$, a particle follows a two-dimensional trajectory in the $x-y$ plane so that its coordinates are related by the equation $y=\frac{x^{2}}{2}$. The $x$ and $y$ components of its acceleration are denoted by $\mathrm{a}_{\mathrm{x}}$ and $\mathrm{a}_{\mathrm{y}}$, respectively. Then

A spherical bubble inside water has radius R. Take the pressure inside the bubble and the water pressure to be $\mathrm{p}_{0}$. The bubble now gets compressed radially in an adiabatic manner so that its radius becomes ( $\mathrm{R}-\mathrm{a}$ ). For $\mathrm{a} \ll \mathrm{R}$ the magnitude of the work done in the process is given by $\left(4 \pi \mathrm{p}_{0} \mathrm{Ra}^{2}\right) \mathrm{X}$, where X is a constant and $\gamma=C_{p} / C_{V}=41 / 30$. The value of $X$ is $\_\_\_\_$

In the balanced condition, the values of the resistances of the four arms of a Wheatstone bridge are shown in the figure below. The resistance $\mathrm{R}_{3}$ has temperature coefficient $0.0004{ }^{\circ} \mathrm{C}^{-1}$. If the temperature of $\mathrm{R}_{3}$ is increased by $100{ }^{\circ} \mathrm{C}$, the voltage developed between S and T will be $\_\_\_\_$ volt.

Two capacitors with capacitance values $\mathrm{C}_{1}=2000 \pm 10 \mathrm{pF}$ and $\mathrm{C}_{2}=3000 \pm 15 \mathrm{pF}$ are connected in series. The voltage applied across this combination is $\mathrm{V}=5.00 \pm 0.02 \mathrm{~V}$. The percentage error in the calculation of the energy stored in this combination of capacitors is $\_\_\_\_$

A cubical solid aluminium (bulk modulus $=-V \frac{\mathrm{dP}}{\mathrm{dV}}=70 \mathrm{GPa}$ ) block has an edge length of 1 m on the surface of the earth. It is kept on the floor of a 5 km deep ocean. Taking the average density of water and the acceleration due to gravity to be $10^{3} \mathrm{~kg} \mathrm{~m}^{-3}$ and $10 \mathrm{~ms}^{-2}$, respectively, the change in the edge length of the block in mm is $\_\_\_\_$

The inductors of two LR circuits are placed next to each other, as shown in the figure. The values of the self-inductance of the inductors, resistances, mutual-inductance and applied voltages are specified in the given circuit. After both the switches are closed simultaneously, the total work done by the batteries against the induced EMF in the inductors by the time the currents reach their steady state values is $\_\_\_\_$ mJ .

A container with 1 kg of water in it is kept in sunlight, which causes the water to get warmer than the surroundings. The average energy per unit time per unit area received due to the sunlight is $700 \mathrm{Wm}^{-2}$ and it is absorbed by the water over an effective area of $0.05 \mathrm{~m}^{2}$. Assuming that the heat loss from the water to the surroundings is governed by Newton's law of cooling, the difference (in ${ }^{\circ} \mathrm{C}$ ) in the temperature of water and the surroundings after a long time will be $\_\_\_\_$ . (Ignore effect of the container, and take constant for Newton's law of cooling $=0.001 \mathrm{~s}^{-1}$, Heat capacity of water $=4200 \mathrm{~J} \mathrm{~kg}^{-1} \mathrm{~K}^{-1}$ )

The $1^{\text {st }}, 2^{\text {nd }}$, and the $3^{\text {rd }}$ ionization enthalpies, $I_{1}, I_{2}$, and $I_{3}$, of four atoms with atomic numbers $n, n+1$, $n+2$, and $n+3$, where $n<10$, are tabulated below. What is the value of $n$ ?

Consider the following compounds in the liquid form:

$\mathrm{O}_{2}$, $\mathrm{HF}, \mathrm{H}_{2} \mathrm{O}, \mathrm{NH}_{3}, \mathrm{H}_{2} \mathrm{O}_{2}, \mathrm{CCl}_{4}, \mathrm{CHCl}_{3}, \mathrm{C}_{6} \mathrm{H}_{6}, \mathrm{C}_{6} \mathrm{H}_{5} \mathrm{Cl}$.

cWhen a charged comb is brought near their flowing stream, how many of them show deflection as per the

following figure?

In the chemical reaction between stoichiometric quantities of $\mathrm{KMnO}_{4}$ and KI in weakly basic solution, what is the number of moles of $\mathrm{I}_{2}$ released for 4 moles of $\mathrm{KMnO}_{4}$ consumed?

An acidified solution of potassium chromate was layered with an equal volume of amyl alcohol. When it was shaken after the addition of 1 mL of $3 \% \mathrm{H}_{2} \mathrm{O}_{2}$, a blue alcohol layer was obtained. The blue color is due to the formation of a chromium (VI) compound ' $\mathbf{X}$ '. What is the number of oxygen atoms bonded to chromium through only single bonds in a molecule of $\mathbf{X}$ ?

The structure of a peptide is given below:

If the absolute values of the net charge of the peptide at $\mathrm{pH}=2, \mathrm{pH}=6$, and $\mathrm{pH}=11$ are $\left|\mathrm{z}_{1}\right|,\left|\mathrm{z}_{2}\right|$, and $\left|\mathrm{z}_{3}\right|$, respectively, then what is $\left|\mathrm{z}_{1}\right|+\left|\mathrm{z}_{2}\right|+\left|\mathrm{z}_{3}\right|$ ?

An organic compound $\left(\mathrm{C}_{8} \mathrm{H}_{10} \mathrm{O}_{2}\right)$ rotates plane-polarized light. It produces pink color with neutral $\mathrm{FeCl}_{3}$ solution. What is the total number of all the possible isomers for this compound?

In an experiment, $m$ grams of a compound $\mathbf{X}$ (gas/liquid/solid) taken in a container is loaded in a balance as shown in figure $\mathbf{I}$ below. In the presence of a magnetic filed, the pan with $\mathbf{X}$ is either deflected upwards (figure II), or deflected downwards (figure III), depending on the compound $\mathbf{X}$. Identify the correct statements(s).

Which of the following plots is (are) correct for the given reaction? $\left([\mathrm{P}]_{0}\right.$ is the initial concentration of $\mathbf{P}$ )

Which among the following statement(s) is(are) true for the extraction of aluminium from bauxite?

Choose the correct statement(s) among the following

Consider the following four compounds I, II, III and IV. Choose the correct statement(s).

A solution of 0.1 M weak base (B) is titrated with 0.1 M of a strong acid (HA). The variation of pH of the solution with the volume of HA added is shown in the figure below. What is the $\mathrm{p} K_{\mathrm{b}}$ of the base? The neutralization reaction is given by $\mathrm{B}+\mathrm{HA} \rightarrow \mathrm{BH}^{+}+\mathrm{A}^{-}$.

Liquid $\mathbf{A}$ and $\mathbf{B}$ form ideal solution for all compositions of $\mathbf{A}$ and $\mathbf{B}$ at $25^{\circ} \mathrm{C}$. Two such solutions with 0.25 and 0.50 mole fractions of $\mathbf{A}$ have the total vapour pressures of 0.3 and 0.4 bar, respectively. What is the vapour pressure of pure liquid $\mathbf{B}$ in bar?

The figure below is the plot of potential energy versus internuclear distance (d) of $\mathrm{H}_{2}$ molecule in the electronic ground state. What is the value of the net potential energy $E_{0}$ (as indicated in figure) in $\mathrm{kJ} \mathrm{mol}^{-1}$, for $d=d_{0}$ at which the electron-electron repulsion and the nucleus-nucleus repulsion energies are absent? As reference, the potential energy of H atom is taken as zero when its electron and the nucleus are infinitely far apart.

Use Avogadro constant as $6.023 \times 10^{23} \mathrm{~mol}^{-1}$.

Consider the reaction sequence from $\mathbf{P}$ to $\mathbf{Q}$ shown below. The overall yield of the major product $\mathbf{Q}$ from $\mathbf{P}$ is

$75 \%$. What is the amount in grams of $\mathbf{Q}$ obtained from 9.3 mL of $\mathbf{P}$ ? (Use density of $\mathbf{P}=1.00 \mathrm{~g} \mathrm{~mL}^{-1}$;

Molar mass of $\mathrm{C}=12.0, \mathrm{H}=1.0, \mathrm{O}=16.0$ and $\mathrm{N}=14.0 \mathrm{~g} \mathrm{~mol}^{-1}$ )

Tin is obtained from cassiterite by reduction with coke. Use the data given below to determine the minimum

temperature (in K ) at which the reduction of cassiterite by coke would take place.

At $298 \mathrm{~K} ; \Delta_{f} H^{0}\left(\mathrm{SnO}_{2}(\mathrm{~s})\right)=-581.0 \mathrm{~kJ} \mathrm{~mol}^{-1}, \Delta_{f} H^{0}\left(\mathrm{CO}_{2}(g)\right)=-394.0 \mathrm{~kJ} \mathrm{~mol}^{-1}$,

$S^{0}\left(\mathrm{SnO}_{2}(s)\right)=56.0 \mathrm{~J} \mathrm{~K}^{-1} \mathrm{~mol}^{-1}, S^{0}(\mathrm{Sn}(\mathrm{s}))=52.0 \mathrm{~J} \mathrm{~K}^{-1} \mathrm{~mol}^{-1}$,

$S^{0}(\mathrm{C}(s))=6.0 \mathrm{~J} \mathrm{~K}^{-1} \mathrm{~mol}^{-1}, S^{0}\left(\mathrm{CO}_{2}(\mathrm{~g})\right)=210.0 \mathrm{~J} \mathrm{~K}^{-1} \mathrm{~mol}^{-1}$,

Assume that the enthalpies and the entropies are temperature independent.

An acidified solution of $0.05 \mathrm{M} \mathrm{Zn}^{2+}$ is saturated with $0.1 \mathrm{M} \mathrm{H}_{2} \mathrm{~S}$. What is the minimum molar concentration

(M) of $\mathrm{H}^{+}$required to prevent the precipitation of ZnS ?

Use $K_{s p}(\mathrm{ZnS})=1.25 \times 10^{-22}$ and overall dissociation constant of

$\mathrm{H}_{2} \mathrm{~S} . \mathrm{K}_{\mathrm{NET}}=K_{1} K_{2}=1 \times 10^{-21}$

For a complex number z , let $\operatorname{Re}(\mathrm{z})$ denote the real part of z . Let S be the set of all complex numbers z satisfying $z^{4}-|z|^{4}=4 i z^{2}$, where $i=\sqrt{-1}$. Then the minimum possible value of $\left|z_{1}-z_{2}\right|^{2}$, where $z_{1}, z_{2} \in S$ with $\operatorname{Re}\left(\mathrm{z}_{1}\right)>0$ and $\operatorname{Re}\left(\mathrm{z}_{2}\right)<0$, is $\_\_\_\_$

The probability that a missile hits a target successfully is 0.75 . In order to destroy the target completely, at least three successful hits are required. Then the minimum number of missiles that have to be fired so that the probability of completely destroying the target is NOT less than 0.95 , is $\_\_\_\_$

Let $O$ be the centre of the circle $x^{2}+y^{2}=r^{2}$, where $r>\frac{\sqrt{5}}{2}$. Suppose PQ is a chord of this circle and the equation of the line passing through $P$ and $Q$ is $2 x+4 y=5$. If the centre of the circumcircle of the triangle OPQ lies on the line $x+2 y=4$, then the value of $r$ is

The trace of a square matrix is defined to be the sum of its diagonal entries. If A is a $2 \times 2$ matrix such that the trace of $A$ is 3 and the trace of $A^{3}$ is -18 , then the value of the determinant of $A$ is $\_\_\_\_$

Let the functions $\mathrm{f}:(-1,1) \rightarrow \mathbb{R}$ and $\mathrm{g}:(-1,1) \rightarrow(-1,1)$ be defined by $f(x)=|2 x-1|+|2 x+1| \text { and } g(x)=x-[x],$ where $[\mathrm{x}$ ] denotes the greatest integer less than or equal to x . Let fog : $(-1,1) \rightarrow \mathbb{R}$ be the composite function defined by $(f 0 g)(x)=f(g(x))$. Suppose c is the number of points in the interval $(-1,1)$ at which fog is NOT continuous, and suppose $d$ is the number of points in the interval $(-1,1)$ at which fog is NOT differentiable. Then the value of $\mathrm{c}+\mathrm{d}$ is $\_\_\_\_$

The value of the limit $\lim _{x \rightarrow \frac{\pi}{2}} \frac{4 \sqrt{2}(\sin 3 x+\sin x)}{\left(2 \sin 2 x \sin \frac{3 x}{2}+\cos \frac{5 x}{2}\right)-\left(\sqrt{2}+\sqrt{2} \cos 2 x+\cos \frac{3 x}{2}\right)}$ is

$\_\_\_\_$

Let b be a nonzero real number. Suppose $f : \mathbb{R} \rightarrow \mathbb{R}$ is a differentiable function such that $f(0)=1$. If the derivative $f^{\prime}$ of $f$ satisfies the equation $f^{\prime}(x)=\frac{f(x)}{b^{2}+x^{2}}$ for all $x \in \mathbb{R}$, then which of the following statements is/are TRUE?

Let a and b be positive real numbers such that $\mathrm{a}>1$ and $\mathrm{b}<\mathrm{a}$. Let P be a point in the first quadrant that lies on the hyperbola $\frac{x^{2}}{a^{2}}-\frac{y^{2}}{b^{2}}=1$. Suppose the tangent to the hyperbola at $P$ passes through the point $(1,0)$, and suppose the normal to the hyperbola at P cuts off equal intercepts on the coordinate axes. Let $\Delta$ denote the area of the triangle formed by the tangent at P , the normal at P and the x -axis. If e denotes the eccentricity of the hyperbola, then which of the following statements is/are TRUE?

Let $\mathrm{f}: \mathbb{R} \rightarrow \mathbb{R}$ and $\mathrm{g}: \mathbb{R} \rightarrow \mathbb{R}$ be functions satisfying $f(x+y)=f(x)+f(y)+f(x) f(y) \text { and } f(x)=x g(x)$ for all $x, y \in \mathbb{R}$. If $\lim _{x \rightarrow 0} g(x)=1$, then which of the following statements is/are TRUE?

Let $\alpha, \beta, \gamma, \delta$ be real numbers such that $\alpha^{2}+\beta^{2}+\gamma^{2} \neq 0$ and $\alpha+\gamma=1$. Suppose the point (3, 2, -1 ) is the mirror image of the point $(1,0,-1)$ with respect to the plane $\alpha \mathrm{x}+\beta \mathrm{y}+\gamma \mathrm{z}=\delta$. Then which of the following statements is/are TRUE?

Let $a$ and $b$ be positive real numbers. Suppose $\overrightarrow{P Q}=a \hat{i}+b \hat{j}$ and $\overrightarrow{P S}=a \hat{i}-b \hat{j}$ are adjacent sides of $a$ parallelogram PQRS. Let $\vec{u}$ and $\vec{v}$ be the projection vectors of $\vec{w}=\hat{i}+\hat{j}$ along $\overrightarrow{P Q}$ and $\overrightarrow{P S}$, respectively. If $|\overrightarrow{\mathrm{u}}|+|\overrightarrow{\mathrm{v}}|=|\overrightarrow{\mathrm{w}}|$ and if the area of the parallelogram PQRS is 8 , then which of the following statements is/are TRUE?

For nonnegative integers s and r, let $\binom{s}{r} = \begin{cases} \frac{s!}{r!(s-r)!} & \text{if } 0 \leq r \leq s, \\ 0 & \text{if } r > s. \end{cases}$ For positive integers m and n, let $g(m, n) = \sum_{p=0}^{m+n} \frac{f(m, n, p)}{\binom{n+p}{p}}$ where for any nonnegative integer p, $f(m, n, p) = \sum_{i=0}^{p} \binom{m}{i} \binom{n+i}{p} \binom{p+n}{p-i}$

An engineer is required to visit a factory for exactly four days during the first 15 days of every month and it is mandatory that no two visits take place on consecutive days. Then the number of all possible ways in which such visits to the factory can be made by the engineer during 1-15 June 2021 is $\_\_\_\_$

In a hotel, four rooms are available. Six persons are to be accommodated in these four rooms in such a way that each of these rooms contains at least one person and at most two persons. Then the number of all possible ways in which this can be done is $\_\_\_\_$

Two fair dice, each with faces numbered $1,2,3,4,5$ and 6 , are rolled together and the sum of the numbers on the faces is observed. This process is repeated till the sum is either a prime number or a perfect square. Suppose the sum turns out to be a perfect square before it turns out to be a prime number. If $p$ is the probability that this perfect square is an odd number, then the value of 14 p is $\_\_\_\_$

Let the function $\mathrm{f}:[0,1] \rightarrow \mathbb{R}$ be defined by

$f(x)=\frac{4^{x}}{4^{x}+2}$ Then the value of $f\left(\frac{1}{40}\right)+f\left(\frac{2}{40}\right)+f\left(\frac{3}{40}\right)+\ldots+f\left(\frac{39}{40}\right)-f\left(\frac{1}{2}\right)$

is $\_\_\_\_$

Let $f: \mathbb{R} \rightarrow \mathbb{R}$ be a differentiable function such that its derivative $f^{\prime}$ is continuous and $f(\pi)=-6$. If

$F$ : $[0$, $\pi] \rightarrow \mathbb{R}$ is defined by $F(x)=\int_{0}^{x} f(t) d t$, and if $\int_{0}^{\pi}\left(f^{\prime}(x)+F(x)\right) \cos x d x=2$ then the

value of $f(0)$ is $\_\_\_\_$

Let the function $\mathrm{f}:(0, \pi) \rightarrow \mathbb{R}$ 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 $\theta \in\left\{\lambda_{1} \pi, \ldots, \lambda_{\mathrm{r}} \pi\right\}$, where $0<\lambda_{1}<\ldots<\lambda_{\mathrm{r}}$ $<1$. Then the value of $\lambda_{1}+\ldots+\lambda_{\mathrm{r}}$ is $\_\_\_\_$

A beaker of radius r is filled with water $\text{Refractive index of water} = \frac{4}{3}$ 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 . This makes the water surface curved so that the difference in the height of water level at the centre and at the circumference of the beaker is h $h \ll H,\quad h \ll r$ 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)