JEE Advanced 2020 · 27 Sep 2020 · Paper 2

Q1.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 …

Q2.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 …

Q3.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 $900{\text{ kg m}}^{-3}$ are released through a tiny hole at the center of the top disc. Once some oil drops achieve terminal vel …

Q4.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 …

Q5.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 an …

Q6.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 …

Q7.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 hei …

Q8.A student skates up a ramp that makes an angle 30° 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 …

Q9.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 …

Q10.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 c …

Q11.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 de …

Q12.Starting at time $t=0$ from the origin with speed $1{\text{ ms}}^{-1}$, 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 $a_x$ and …

Q13.A spherical bubble inside water has radius $R$. Take the pressure inside the bubble and the water pressure to be $p_0$. The bubble now gets compressed radially in an adiabatic manner so that its radius becomes $(R-a)$. For $a\ll R$ the magnitude of the work done in the process i …

Q14.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 $R_3$ has temperature coefficient $0.0004°{\text{C}}^{-1}$. If the temperature of $R_3$ is increased by $100°\text{C}$, the voltage develo …

Q15.Two capacitors with capacitance values $C_1=2000\pm 10$ pF and $C_2=3000\pm 15$ pF are connected in series. The voltage applied across this combination is $V=5.00\pm 0.02$ V. The percentage error in the calculation of the energy stored in this combination of capacitors i …

Q16.A cubical solid aluminium (bulk modulus $=-V\frac{dP}{dV}=70$ 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 $10^3{\text{ kg m}}^{-3}$ …

Q17.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, …

Q18.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 $700{\text{ Wm}}^{-2}$ and it is absorbed by the water over an effective area of …

Q1.The $1^{\text{st }},2^{\text{nd }}$ and the $3^{\text{rd }}$ ionization enthalpies ${\mathrm{I}}_1,{\mathrm{I}}_2$ and ${\mathrm{I}}_3$, of four atoms with atomic numbers $\mathrm{n},\mathrm{n}+1$, $\mathrm{n}+2$ and $\mathrm{n}+3$, where $\mathrm{n}<10$, ar …

Q2.Consider the following compounds in the liquid form :
${\mathrm{O}}_2,\mathrm{HF},{\mathrm{H}}_2\mathrm{O},{\mathrm{NH}}_3,{\mathrm{H}}_2{\mathrm{O}}_2,{\mathrm{CCl}}_4,{\mathrm{CHCl}}_3,{\mathrm{C}}_6{\mathrm{H}}_6,{\mathrm{C}}_6{\mathrm{H}}_5\mathrm{Cl}$.
When a …

Q3.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?

Q4.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) comp …

Q5.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 $\left|z_3\right|$, respectively, then what is …

Q6.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?

Q7.In an experiment, $m$ grams of a compound X (gas/liquid/solid) taken in a container is loaded in a balance as shown in figure I below. In the presence of a magnetic field, the pan with X is either deflected upwards (figure II), or deflected downwards (figure III), depending on …

Q8.Which of the following plots is(are) correct for the given reaction?
( $[\mathrm{P}{]}_0$ is the initial concentration of $\mathbf{P}$ )

Q9.Which among the following statement(s) is(are) true for the extraction of aluminium from bauxite?

Q10.Choose the correct statement(s) among the following.

Q11.Consider the following four compounds I, II, III, and IV.

Choose the correct statement(s).

Q12.Consider the following transformations of a compound $\mathbf{P}$.

Choose the correct option(s).

Q13.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 …

Q14.Liquids A and B form ideal solution for all compositions of A and B at 25°C. Two such solutions with 0.25 and 0.50 mole fractions of A have the total vapor pressures of 0.3 and 0.4 bar, respectively. What is the vapor pressure of pure liquid B in bar?

Q15.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 the figure) in $\mathrm{kJ}{\mathrm{mol}}^{-1}$, for …

Q16.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 …

Q17.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 …

Q18.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 …

Q1.For a complex number $z$, let $\mathrm{Re}(z)$ denote the real part of $z$. Let $S$ be the set of all complex numbers $z$ satisfying $z^4-|z{|}^4=4i{z}^2$, where $i=\sqrt{-1}$. Then the minimum possible value of ${\left|z_1-z_2\right|}^2$, where …

Q2.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 …

Q3.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 $2x+4y=5$. If the centre of the circumcircle of the triangle $OPQ$ lies on the …

Q4.The trace of a square matrix is defined to be the sum of its diagonal entries. If $A$ is a 2 × 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 ____

Q5.Let the functions $f:(-1,1)\to \mathbb{R}$ and $g:(-1,1)\to (-1,1)$ be defined by

where $[x]$ denotes the greatest integer less than or equal to $x$. Let $f\circ g:(-1,1)\to \mathbb{R}$ be the co …

Q6.The value of the limit

is ____

Q7.Let $b$ be a nonzero real number. Suppose $f:\mathbb{R}\to \mathbb{R}$ 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 w …

Q8.Let $a$ and $b$ be positive real numbers such that $a>1$ and $b<a$. 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 suppo …

Q9.Let $f:\mathbb{R}\to \mathbb{R}$ and $g:\mathbb{R}\to \mathbb{R}$ be functions satisfying $f(x+y)=f(x)+f(y)+f(x)f(y)$ and $f(x)=xg(x)$ for all $x,y\in \mathbb{R}$. If ${\lim }_{x\to 0}g(x)=1$, then which of the following statements is/are …

Q10.Let $\alpha ,\beta ,\gamma ,\delta$ be real numbers such that ${\alpha }^2+{\beta }^2+{\gamma }^2\ne 0$ 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 $\alpha x+\beta y+\gamma z=\delta$. Then wh …

Q11.Let $a$ and $b$ be positive real numbers. Suppose $\overrightarrow{PQ}=a\hat{i}+b\hat{j}$ and $\overrightarrow{PS}=a\hat{i}-b\hat{j}$ are adjacent sides of a parallelogram $PQRS$. Let $\overrightarrow{u}$ and $\overrightarrow{v}$ be the projection vectors of …

Q12.For non-negative integers $s$ and $r$, let

For positive integers $m$ and $n$, let

where for any non …

Q13.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 dur …

Q14.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 ____

Q15.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 tur …

Q16.Let the function $f:[0,1]\to \mathbb{R}$ be defined by $f(x)=\frac{4^x}{4^x+2}$
Then the value of $f\left(\frac{1}{40}\right)+f\left(\frac{2}{40}\right)+f\left(\frac{3}{40}\right)+\dots +f\left(\frac{39}{40}\right)-f\left(\frac{1}{2}\right)$ is ____

Q17.Let $f:\mathbb{R}\to \mathbb{R}$ be a differentiable function such that its derivative $f^{\prime }$ is continuous and $f(\pi )=-6$. If $F:[0,\pi ]\to \mathbb{R}$ is defined by $F(x)={\int }_0^{x}f(t)dt$, and if …

Q18.Let the function $f:(0,\pi )\to \mathbb{R}$ be defined by

Suppose the function $f$ has a local minimum at $\theta$ precisely when $\theta \in \{{\lambda }_1\pi ,\dots ,{\lambda }_r\pi \}$, w …