JEE Advanced 2021 — Paper 1

The smallest division on the main scale of a Vernier calipers is 0.1 cm . Ten divisions of the Vernier scale correspond to nine divisions of the main scale. The figure below on the left shows the reading of this calipers with no gap between its two jaws. The figure on the right shows the reading with a solid sphere held between the jaws. The correct diameter of the sphere is

An extended object is placed at point $\mathrm{O}, 10 \mathrm{~cm}$ in front of a convex lens $\mathrm{L}_{1}$ and a concave lens $\mathrm{L}_{2}$

is placed 10 cm behind it, as shown in the figure. The radii of curvature of all the curved surfaces in both the lenses are 20 cm .

The refractive index of both the lenses is 1.5 . The total magnification of this lens system is

A heavy nucleus Q of half-life 20 minutes undergoes alpha-decay with probability of $60 \%$ and beta-decay with probability of $40 \%$.

Initially, the number of Q nuclei is 1000 . The number of alpha-decay of Q in the first one hour is

A horizontal force F is applied at the centre of mass of a cylindrical object of mass m and radius R , perpendicular to its axis as shown in the figure. The coefficient of friction between the object and the ground is $\mu$. The center of mass of the object has an acceleration $a$. The acceleration due to gravity is g . Given that the object rolls without slipping, which of the following statement(s) is(are) correct?

A particle of mass $\mathrm{M}=0.2 \mathrm{~kg}$ is initially at rest in the xy -plane at a point $(\mathrm{x}=-\ell, \mathrm{y}=-\mathrm{h})$, where $\ell=10 \mathrm{~m}$ and $\mathrm{h}=1 \mathrm{~m}$. The particle is accelerated at time $\mathrm{t}=0$ with a constant acceleration $\mathrm{a}=10 \mathrm{~m} / \mathrm{s}^{2}$ along the positive x-direction. Its angular momentum and torque with respect to the origin, in SI units, are represented by $\overrightarrow{\mathrm{L}}$ and $\vec{\tau}$ respectively. $\hat{\mathrm{i}}, \hat{\mathrm{j}}$ and $\hat{\mathrm{k}}$ are unit vectors along the positive x , y and z-directions, respectively. If $\hat{k}=\hat{i} \times \hat{j}$ then which of the following statement(s) is(are) correct?

Which of the following statement(s) is(are) correct about the spectrum of hydrogen atom?

A long straight wire carries a current, $\mathrm{I}=2$ ampere. A semi-circular conducting rod is placed beside it on two conducting parallel rails of negligible resistance. Both the rails are parallel to the wire. The wire, the rod and the rails lie in the same horizontal plane, as shown in the figure.

Two ends of the semi-circular rod are at distances 1 cm and 4 cm from the wire. At time $t=0$, the rod starts moving on the rails with a speed

$\mathrm{v}=3.0 \mathrm{~m} / \mathrm{s}$ (see the figure) A resistor $\mathrm{R}=1.4 \Omega$ and a capacitor $\mathrm{C}_{0}=5.0 \mu \mathrm{~F}$

are connected in series between the rails. At time $\mathrm{t}=0, \mathrm{C}_{0}$ is uncharged.

Which of the following statement(s) is(are) correct? [ $\mu_{0}=4 \pi \times 10^{-7}$ SI units. Take $\ell \mathrm{n} 2=0.7$ ]

A cylindrical tube, with its base as shown in the figure, is filled with water. It is moving down with a constant acceleration $a$

along a fixed inclined plane with angle $\theta=45^{\circ} . \mathrm{P}_{1}$ and $\mathrm{P}_{2}$ are pressures at points 1 and 2 ,

respectively located at the base of the tube. Let $\beta=\left(P_{1}-P_{2}\right) /(\rho g d)$, where $\rho$ is density of water, $d$

is the inner diameter of the tube and $g$ is the acceleration due to gravity. Which of the following statement(s) is(are) correct?

An $\alpha$-particle (mass 4 amu ) and a singly charged sulfur ion (mass 32 amu ) are initially at rest. They are accelerated through a potential V and then

allowed to pass into a region of uniform magnetic field which is normal to the velocities of the particles. Within this region,

the $\alpha$-particle and the sulfur ion move in circular orbits of radii $r_{\alpha}$ and $r_{s}$, respectively.

The ratio $\left(r_{s} / r_{\alpha}\right)$ is $\qquad$ .

A thin rod of mass M and length $a$ is free to rotate in horizontal plane about a fixed vertical axis passing through point O . A thin circular disc of mass M and of radius

$a / 4$ is pivoted on this rod with its center at a distance $a / 4$ from the free end so that it can rotate freely about its vertical axis, as shown in the figure.

Assume that both the rod and the disc have uniform density and they remain horizontal during the motion.

An outside stationary observer finds the rod rotating with an angular velocity $\Omega$ and the disc rotating about its vertical axis with angular velocity

$4 \Omega$. The total angular momentum of the system about the point O is $\left(\frac{\mathrm{Ma}^{2} \Omega}{48}\right) \mathrm{n}$.

The value of n is $\qquad$ .

A small object is placed at the center of a large evacuated hollow spherical container. Assume that the container is maintained at 0 K . At time $\mathrm{t}=0$, the temperature of the object is 200 K . The temperature of the object becomes 100 K at $\mathrm{t}=\mathrm{t}_{1}$ and 50 K at $\mathrm{t}=\mathrm{t}_{2}$. Assume the object and the container to be ideal black bodies. The heat capacity of the object does not depend on temperature. The ratio $\left(t_{2} / t_{1}\right)$ is $\qquad$ ,

Among the following, the conformation that corresponds to the most stable conformation of meso-butane- 2,3-diol is

For the given close packed structure of a salt made of cation $\mathbf{X}$ and anion $\mathbf{Y}$ shown below (ions of only one face are shown for clarity), the packing fraction is approximately (packing fraction $=\frac{\text { packing\; efficiency }}{100}$ )

The calculated spin only magnetic moments of $\left[\mathrm{Cr}\left(\mathrm{NH}_{3}\right)_{6}\right]^{3+}$ and

$\left[\mathrm{CuF}_{6}\right]^{3-}$ in BM , respectively, are (Atomic number of Cr and Cu are 24 and 29, respectively)

The reaction of $\mathbf{Q}$ with PhSNa yields an organic compound (major product) that gives positive Carius test on treatment with

$\mathrm{Na}_{2} \mathrm{O}_{2}$ followed by addition of $\mathrm{BaCl}_{2}$. The correct option(s) for $\mathbf{Q}$ is(are)

The correct statement(s) related to colloids is(are)

The correct statement(s) related to the metal extraction processes is(are)

A mixture of two salts is used to prepare a solution S , which gives the following results: The correct option(s) for the salt mixture is(are)

Consider a triangle $\Delta$ whose two sides lie on the x -axis and the line $\mathrm{x}+\mathrm{y}+1=0$.

If the orthocentre of $\Delta$ is $(1,1)$, then the equation of the circle passing through the vertices of the triangle $\Delta$ is

The area of the region $\left\{(x, y) ; \quad 0 \leq x \leq \frac{9}{4}, \quad 0 \leq y \leq 1, \quad x \geq 3 y, \quad x+y \geq 2\right\}$ is

Consider three sets $\mathrm{E}_{1}=\{1,2,3\}, \mathrm{F}_{1}=\{1,3,4\}$ and $\mathrm{G}_{1}=\{2,3,4,5\}$.

Two elements are chosen at random, without replacement, from the set $E_{1}$ and let $S_{1}$ denote the set of these chose elements.

Let $E_{2}=E_{1}-S_{1}$ and $F_{2}=F_{1} \cup S_{1}$. Now two elements are chosen at random, without replacement, from the set

$F_{2}$ and let $S_{2}$ denote the set of these chosen elements. Let $G_{2}=F_{1} \cup S_{2}$. Finally, two elements are chosen at random,

without replacement, from the set $G_{2}$ and let $\mathrm{S}_{3}$ denote the set of these chosen elements. Let $E_{3}=E_{2} \cup S_{3}$.

Given that $E_{1}=E_{3}$, let $p$ be the conditional probability of the event $S_{1}=\{1,2\}$. Then the value of $p$ is

Let $\theta_{1}, \theta_{2}, \ldots . ., \theta_{10}$ be positive valued angles (in radian) such that $\theta_{1}+\theta_{2}+\ldots . .+\theta_{10}=2 \pi$. Define the complex numbers $z_{1}=e^{i \theta_{1}}, z_{k}=z_{k-1} e^{i \theta_{k}}$ for $k=2,3, \ldots . ., 10$, where $i=\sqrt{-1}$. Consider the statements $P$ and Q given below:

For any $3 \times 3$ matrix $M$, let $|M|$ denote the determinant of $M$.

Let

If Q is a non-singular matrix of order

$3 \times 3$, then which of the following statements is(are) TRUE?

Let $f: R \rightarrow R$ be defined by $f(x)=\frac{x^{2}-3 x-6}{x^{2}+2 x+4}$. Then which of the following statements is(are) TRUE?

Let $E, F$ and $G$ be three events having probabilities $P(E)=\frac{1}{8}, P(F)=\frac{1}{6}$ and $P(G)=\frac{1}{4}$, and

let $P(E \cap F \cap G)=\frac{1}{10}$. For any event H , if $\mathrm{H}^{\mathrm{C}}$ denotes its complement,

then which of the following statements is(are) TRUE?

For any $3 \times 3$ matrix M , let $|\mathrm{M}|$ denote the determinant of M . Let I be the $3 \times 3$ identity matrix. Let E and F be two $3 \times 3$ matrices such that ( $I-E F$ ) is invertible. If $G=(I-E F)^{-1}$, then which of the following statements is(are) TRUE?

For any positive integer $n$, let $S_{n}:(0, \infty) \rightarrow R$ be defined by $S_{n}(x)=\sum_{k=1}^{n} \cot ^{-1}\left(\frac{1+k(k+1) x^{2}}{x}\right)$

where for any $x \in R, \cot ^{-1}(x) \in(0, \pi)$ and $\tan ^{-1}(x) \in\left(-\frac{\pi}{2}, \frac{\pi}{2}\right)$.

Then which of the following statements is(are) TRUE?

For any complex number $\mathrm{w}=\mathrm{c}+$ id, let $\arg (\mathrm{w}) \in(-\pi, \pi]$, where $\mathrm{i}=\sqrt{-1}$. Let $\alpha$ and $\beta$ be real numbers such that for all complex numbers Let $z = x + iy$ satisfy

Then the ordered pair $(x, y)$ lies on the circle

Which of the following statements is(are) TRUE?

For $x \in R$, the number of real roots of the equation $3 x^{2}-4\left|x^{2}-1\right|+x-1=0$ is $\qquad$

In a triangle $A B C$, let $A B=\sqrt{23}, B C=3$ and $C A=4$. Then the value of $\frac{\cot A+\cot C}{\cot B}$ is $\qquad$

Let $\vec{u}, \vec{v}$ and $\vec{w}$ be vectors in three-dimensional space, where $\vec{u}$ and $\vec{v}$ are unit vectors which are not perpendicular to each other and $\vec{u} \cdot \vec{w}=1, \vec{v} \cdot \vec{w}=1, \vec{w} \cdot \vec{w}=4$. If the volume of the parallelepiped, whose adjacent sides are represented by the vectors $\vec{u}, \vec{v}$ and $\vec{w}$ is $\sqrt{2}$ , then the value of $|3 \vec{u}+5 \vec{v}|$ is $\qquad$

A projectile is thrown from a point $O$ on the ground at an angle $45^{\circ}$ from the vertical and with a speed $5 \sqrt{2} \mathrm{~m} / \mathrm{s}$. The projectile at the highest point of its trajectory splits into two equal parts. One part falls vertically down to the ground, 0.5 s after the splitting. The other part, t seconds after the splitting, falls to the ground at a distance x meters from the point O . The acceleration due to gravity $\mathrm{g}=10 \mathrm{~m} / \mathrm{s}^{2}$.

The value of $t$ is $\text{ }\!\!~\!\!\text{ }$

A projectile is thrown from a point $O$ on the ground at an angle $45^{\circ}$ from the vertical and with a speed $5 \sqrt{2} \mathrm{~m} / \mathrm{s}$. The projectile at the highest point of its trajectory splits into two equal parts. One part falls vertically down to the ground, 0.5 s after the splitting. The other part, t seconds after the splitting, falls to the ground at a distance x meters from the point O . The acceleration due to gravity $\mathrm{g}=10 \mathrm{~m} / \mathrm{s}^{2}$.

The value of $x$ is $\text{ }\!\!~\!\!\text{ }$ .

The magnitude of ${{\text{q}}_{1}}$ is $\text{ }\!\!~\!\!\text{ }$ .

In the circuit shown below, the switch $S$ is connected to position $P$ for a long time so that the charge on the capacitor becomes

$\mathrm{q}_{1} \mu \mathrm{C}$. Then S is switched to position Q . After a long time, the charge on the capacitor is

$\mathrm{q}_{2} \mu \mathrm{C}$.

The magnitude of ${{\text{q}}_{2}}$ is $\text{ }\!\!~\!\!\text{ }$ .

Two point charges -Q and $+\mathrm{Q} / \sqrt{3}$ are placed in the xy-plane at the origin $(0,0)$ and a point $(2,0)$,respectively, as shown in the figure.This results in an equipotential circle of radius R and potential $\mathrm{V}=0$ in the xy -plane with its center at (b, 0). All lengths are measured in meters.

The value of R is _______ meter.

Two point charges -Q and $+\mathrm{Q} / \sqrt{3}$ are placed in the xy-plane at the origin $(0,0)$ and a point $(2,0)$,respectively, as shown in the figure.This results in an equipotential circle of radius R and potential $\mathrm{V}=0$ in the xy -plane with its center at (b, 0). All lengths are measured in meters.

The value of b is _______ meter.

The value of $\mathbf{x}$ is $\text{ }\!\!~\!\!\text{ }$ .

The value of $\mathbf{y}$ is $\text{ }\!\!~\!\!\text{ }$. .

The value of standard enthalpy, $\text{ }\!\!\Delta\!\!\text{ }{{H}^{\ominus }}\left( \text{inkJmo}{{\text{l}}^{-1}} \right)$ for the given reaction is $\text{ }\!\!~\!\!\text{ }$ .

For the reaction, $\mathbf{X}(s) \rightleftharpoons \mathbf{Y}(s)+\mathbf{Z}(g)$, the plot of $\ln \frac{p_{\mathbf{Z}}}{p^{\theta}}$

versus $\frac{10^{4}}{T}$ is given below (in solid line), where $p_{\mathbf{Z}}$ is the pressure (in bar) of the gas $\mathbf{Z}$ at temperature

$T$ and $p^{\theta}=1$ bar. $\ln \frac{p_{Z}}{p^{\theta}}$ (Given, $\frac{\mathrm{d}(\ln K)}{\mathrm{d}\left(\frac{1}{T}\right)}=-\frac{\Delta H^{\theta}}{R}$,

where the equilibrium constant, $K=\frac{p_{z}}{p^{\alpha}}$ and the gas constant, $\mathrm{R}=8.314 \mathrm{~J} \mathrm{~K}^{-1} \mathrm{~mol}^{-1}$ )

The value of $\text{ }\!\!\Delta\!\!\text{ }{{S}^{\ominus }}$ (in $\text{J}{{\text{K}}^{-1}}\text{ }\!\!~\!\!\text{ mo}{{\text{l}}^{-1}}$ ) for the given reaction, at 1000 K is

The value of $\mathbf{x}$ is $\text{ }\!\!~\!\!\text{ }$ .

The boiling point of water in a 0.1 molal silver nitrate solution (solution $\mathbf{A}$ ) is $\mathbf{x}^{\circ} \mathrm{C}$. To this solution $\mathbf{A}$,

an equal volume of 0.1 molal aqueous barium chloride solution is added to make a new solution $\mathbf{B}$.

The difference in the boiling points of water in the two solutions $\mathbf{A}$ and $\mathbf{B}$ is $\mathbf{y} \times 10^{-20} \mathrm{C}$.

(Assume: Densities of the solutions $\mathbf{A}$ and $\mathbf{B}$ are the same as that of water and the soluble salts dissociate completely.

Use: Molal elevation constant (Ebullioscopic Constant), $\mathrm{K}_{\mathrm{b}}=0.5 \mathrm{~K} \mathrm{~kg} \mathrm{~mol}^{-1}$;

Boiling point of pure water as $100^{\circ} \mathrm{C}$.)

The value of $\left| \mathbf{y} \right|$ is $\text{ }\!\!~\!\!\text{ }$ .

Three numbers are chosen at random, one after another with replacement, from the set $S=\{1,2,3, \ldots \ldots, 100\}$. Let $p_{1}$ be the probability that the maximum of chosen numbers is at least 81 and $p_{2}$ be the probability that the minimum of chosen numbers is at most 40 .

The value of $\frac{125}{4}{{p}_{2}}$ is $\text{ }\!\!~\!\!\text{ }$

Let $\alpha, \beta$ and $\gamma$ be real numbers such that the system of linear equation $\begin{gathered}x+2 y+3 z=\alpha ,4 x+5 y+6 z=\beta ,7 x+8 y+9 z=\gamma-1\end{gathered}$ is consistent. Let $|\mathrm{M}|$ represent the determinant of the matrix $M=\left[\begin{array}{ccc}\alpha & 2 & \gamma \\\beta & 1 & 0 \\-1 & 0 & 1\end{array}\right]$ Let P be the plane containing all those $(\alpha, \beta, \gamma)$ for which the above system of linear equations is consistent, and $D$ be the square of the distance of the point $(0,1,0)$ from the plane $P$.

The value of $\left| \text{M} \right|$ is

Consider the lines $L_{1}$ and $L_{2}$ defined by $L_{1}: x \sqrt{2}+y-1=0 \text { and } L_{2}: x \sqrt{2}-y+1=0$ For a fixed constant $\lambda$, let $C$ be the locus of a point $P$ such that the product of the distance of $P$ from $L_{1}$ and the distance of $P$ from $L_{2}$ is $\lambda^{2}$. The line $y=2 x+1$ meets $C$ at two points $R$ and $S$, where the distance between R and S is $\sqrt{270}$. Let the perpendicular bisector of RS meet $C$ at two distinct points $R^{\prime}$ and $S^{\prime}$. Let $D$ be the square of the distance between $\mathrm{R}^{\prime}$ and $\mathrm{S}^{\prime}$

The value of ${{\lambda }^{2}}$ is $\text{ }\!\!~\!\!\text{ }$