JEE Advanced 2021 · 03 Oct 2021 · Paper 1

Q1.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 sh …

Q2.An ideal gas undergoes a four step cycle as shown in the $P-V$ diagram below. During this cycle, heat is absorbed by the gas in

Q3.An extended object is placed at point O, 10 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 …

Q4.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-decays of Q in the first one hour is

Q5.A projectile is thrown from a point O on the ground at an angle 45° from the vertical and with a speed $5\sqrt{2}$ m/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 …

Q6.A projectile is thrown from a point O on the ground at an angle 45° from the vertical and with a speed $5\sqrt{2}$ m/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 …

Q7.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 ${\mathrm{q}}_1\mu$C. Then S is switched to position Q. After a long time, the charge on the capacitor is ${\mathrm{q}}_2\mu$C.

The magnitude of …

Q8.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 ${\mathrm{q}}_1\mu$C. Then S is switched to position Q. After a long time, the charge on the capacitor is ${\mathrm{q}}_2\mu$C.

The magnitude of …

Q9.Two point charges $-Q$ and $+Q/\sqrt{3}$ 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 lengt …

Q10.Two point charges $-Q$ and $+Q/\sqrt{3}$ 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 lengt …

Q11.A horizontal force $F$ is applied at the center 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 …

Q12.A wide slab consisting of two media of refractive indices ${\mathrm{n}}_1$ and ${\mathrm{n}}_2$ is placed in air as shown in the figure. A ray of light is incident from medium ${\mathrm{n}}_1$ to ${\mathrm{n}}_2$ at an angle $\theta$, where $\sin \theta$ is slightly larger than …

Q13.A particle of mass $M=0.2$ kg is initially at rest in the $xy$-plane at a point ($\mathrm{x}=-l,\mathrm{y}=-h$), where $l=10$ m and $h=1$ m. The particle is accelerated at time $\mathrm{t}=0$ with a constant acceleration $\mathrm{a}=10$ m/s${}^2$ along the positive x-direction. …

Q14.Which of the following statement(s) is(are) correct about the spectrum of hydrogen atom?
(A) The ratio of the longest wavelength to the shortest wavelength in Balmer series is 9/5
(B) There is an overlap between the wavelength ranges of Balmer and Paschen series.
(C) The waveleng …

Q15.A long straight wire carries a current, $I=2$ 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 …

Q16.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, loc …

Q17.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 regio …

Q18.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 fre …

Q19.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 5 …

Q2.Among the following, the conformation that corresponds to the most stable conformation of meso-butane-2,3-diol is -

Q3.For the given close packed structure of a salt made of cation X and anion Y shown below (ions of only one face are shown for clarity), the packing fraction is approximately (packing fraction = $\frac{\text{Packing efficiency}}{100}$)

Q4.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 numbers of Cr and Cu are 24 and 29, respectively)

Q5.For the following reaction scheme, percentage yields are given along the arrow:

$\mathbf{x}$ g and $\mathbf{y}$ g are mass of $\mathbf{R}$ and $\mathbf{U}$, respectively.
(Use: Molar mass (in ${\mathrm{g}\mathrm{mol}}^{-1}$) of H, C and O as 1, 12 and 16, respectively)

The value of …

Q6.For the following reaction scheme, percentage yields are given along the arrow:

$\mathbf{x}$ g and $\mathbf{y}$ g are mass of $\mathbf{R}$ and $\mathbf{U}$, respectively.
(Use: Molar mass (in ${\mathrm{g}\mathrm{mol}}^{-1}$) of H, C and O as 1, 12 and 16, respectively)

The value of …

Q7.For the reaction $\mathbf{X}(\mathrm{s})\rightleftharpoons \mathbf{Y}(\mathrm{s})+\mathbf{Z}(\mathrm{g})$, the plot of $\ln \frac{{\mathrm{p}}_{\mathrm{z}}}{{\mathrm{p}}^{\ominus }}$ versus $\frac{10^4}{T}$ is given below (in solid line), where ${\mathrm{p}}_{\mathrm{z}}$ is the pres …

Q8.For the reaction $\mathbf{X}(\mathrm{s})\rightleftharpoons \mathbf{Y}(\mathrm{s})+\mathbf{Z}(\mathrm{g})$, the plot of $\ln \frac{{\mathrm{p}}_{\mathrm{z}}}{{\mathrm{p}}^{\ominus }}$ versus $\frac{10^4}{T}$ is given below (in solid line), where ${\mathrm{p}}_{\mathrm{z}}$ is the pres …

Q9.The boiling point of water in a 0.1 molal silver nitrate solution (solution A) is ${\mathbf{x}}^{\circ }\mathrm{C}$. To this solution A, an equal volume of 0.1 molal aqueous barium chloride solution is added to make a new solution B. The difference in the boiling points of water in …

Q10.The boiling point of water in a 0.1 molal silver nitrate solution (solution A) is ${\mathbf{x}}^{\circ }\mathrm{C}$. To this solution A, an equal volume of 0.1 molal aqueous barium chloride solution is added to make a new solution B. The difference in the boiling points of water in …

Q11.Given

D-Glucose
The compound(s), which on reaction with ${\mathrm{HNO}}_3$ will give the product having degree of rotation, $[\alpha {]}_{\mathrm{D}}=-52.7^{\circ }$ is (are)

Q12.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).

Q13.The correct statement(s) related to colloids is(are)

Q14.An ideal gas undergoes a reversible isothermal expansion from state I to state II followed by a reversible adiabatic expansion from state II to state III. The correct plot(s) representing the changes from state I to state III is(are)
($p$: pressure, $V$: volume, $T$: temperature, …

Q15.The correct statement(s) related to the metal extraction processes is(are)

Q16.A mixture of two salts is used to prepare a solution $\mathbf{S}$, which gives the following results:
$\underset{\text{only}}{\mathrm{White}}$ precipitate(s) $\underset{\text{Room temperature}}{\overset{\text{Dilute NaOH(aq.)}}{\leftarrow }}$ …

Q17.The maximum number of possible isomers (including stereoisomers) which may be formed on mono-bromination of 1-methylcyclohex-1-ene using ${\mathrm{Br}}_2$ and UV light is ____

Q18.In the reaction given below, the total number of atoms having $s{p}^2$ hybridization in the major product P is ____

Q19.The total number of possible isomers for $\left[\mathrm{Pt}{\left({\mathrm{NH}}_3\right)}_4{\mathrm{Cl}}_2\right]{\mathrm{Br}}_2$ is ____

Q1.Consider a triangle Δ whose two sides lie on the x-axis and the line x+y+1=0. If the orthocenter of Δ is (1, 1), then the equation of the circle passing through the vertices of the triangle Δ is

Q2.The area of the region $\{(x,y):0\le x\le \frac{9}{4},0\le y\le 1,x\ge 3y,x+y\ge 2\}$ is

Q3.Consider three sets $E_1=\{1,2,3\},F_1=\{1,3,4\}$ and $G_1=\{2,3,4,5\}$. Two elements are chosen at random, without replacement, from the set $E_1$, and let $S_1$ denote the set of these chosen elements.
Let $E_2=E_1-S_1$ and $F_2=F_1\cup S_1$. Now two ele …

Q4.Let ${\theta }_1,{\theta }_2,\dots ,{\theta }_{10}$ be positive valued angles (in radian) such that ${\theta }_1+{\theta }_2+\dots +{\theta }_{10}=2\pi$. Define the complex numbers $z_1=e^{i{\theta }_1},z_k=z_{k-1}{e}^{i{\theta }_k}$ for $k=2,3,\dots ,10$, where $i=\sqrt{-1}$. Consid …

Q5.Three numbers are chosen at random, one after another with replacement, from the set $S=\{1,2,3,\dots ,100\}$. 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 …

Q6.Three numbers are chosen at random, one after another with replacement, from the set $S=\{1,2,3,\dots ,100\}$. 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 …

Q7.Let $\alpha ,\beta$ and $\gamma$ be real numbers such that the system of linear equations
$x+2y+3z=\alpha$
$4x+5y+6z=\beta$
$7x+8y+9z=\gamma -1$
is consistent. Let $|M|$ represent the determinant of the matrix …

Q8.Let $\alpha ,\beta$ and $\gamma$ be real numbers such that the system of linear equations
$x+2y+3z=\alpha$
$4x+5y+6z=\beta$
$7x+8y+9z=\gamma -1$
is consistent. Let $|M|$ represent the determinant of the matrix …

Q9.Consider the lines $L_1$ and $L_2$ defined by
$L_1:x\sqrt{2}+y-1=0$ 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 …

Q10.Consider the lines $L_1$ and $L_2$ defined by
$L_1:x\sqrt{2}+y-1=0$ 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 …

Q11.For any 3 × 3 matrix M, let $|M|$ denote the determinant of M. Let
$E=\left[\begin{array}{ccc}1& 2& 3\\ 2& 3& 4\\ 8& 13& 18\end{array}\right],P=\left[\begin{array}{lll}1& 0& 0\\ 0& 0& 1\\ 0& 1& 0\end{array}\right]$ and …

Q12.Let $f:\mathbb{R}\to \mathbb{R}$ be defined by
$f(x)=\frac{x^2-3x-6}{x^2+2x+4}$.
Then which of the following statements is (are) TRUE?

Q13.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 $H^c$ denotes its complement, then which of the following statements is(are) TRUE?

Q14.For any 3 × 3 matrix M, let $|M|$ denote the determinant of M. Let I be the 3 × 3 identity matrix. Let E and F be two 3 × 3 matrices such that (I - EF) is invertible. If $G=(I-EF{)}^{-1}$, then which of the following statements is (are) TRUE?

Q15.For any positive integer n, let $S_n:(0,\infty )\to \mathbb{R}$ 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 $x\in \mathbb{R},{\cot }^{-1}x\in (0,\pi )$ and …

Q16.For any complex number $w=c+id$, let $\arg (w)\in (-\pi ,\pi ]$, where $i=\sqrt{-1}$. Let $\alpha$ and $\beta$ be real numbers such that for all complex numbers $z=x+iy$ satisfying $\arg \left(\frac{z+\alpha }{z+\beta }\right)=\frac{\pi }{4}$, the ordered pair $(x,y)$ lies on the circl …

Q17.For $x\in \mathbb{R}$, the number of real roots of the equation $3x^2-4|x^2-1|+x-1=0$ is ____.

Q18.In a triangle ABC, let $AB=\sqrt{23},BC=3$ and $CA=4$. Then the value of $\frac{\cot A+\cot C}{\cot B}$ is ____.

Q19.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 parallelopiped …