JEE Advanced 2022 Question Paper with Solutions

Two spherical stars $A$ and $B$ have densities $\rho_{A}$ and $\rho_{B}$ , respectively. $A$ and $B$ have the same radius, and their masses $M_{A}$ and $M_{B}$ are related by $M_{B}=2 M_{A}$ . Due to an interaction process, star $A$ loses some of its mass, so that its radius is halved, while its spherical shape is retained, and its density remains $\rho_{A}$ . The entire mass lost by $A$ is deposited as a thick spherical shell on $B$ with the density of the shell being $\rho_{A}$ . If $v_{A}$ and $v_{B}$ are the escape velocities from $A$ and $B$ after the interaction process, the ratio $\frac{v_{B}}{v_{A}}=\sqrt{\frac{10 n}{15^{1 / 3}}}$ . The value of $n$ is __________ .

A solid sphere of mass $1 \mathrm{~kg}$ and radius $1 \mathrm{~m}$ rolls without slipping on a fixed inclined plane with an angle of inclination $\theta=30^{\circ}$ from the horizontal. Two forces of magnitude $1 \mathrm{~N}$ each, parallel to the incline, act on the sphere, both at distance $r=0.5 \mathrm{~m}$ from the center of the sphere, as shown in the figure. The acceleration of the sphere down the plane is _________ $m \,s^{-2} .\left(\right.$ Take $g=10\, m s^{-2}$ )

A projectile is fired from horizontal ground with speed $v$ and projection angle $\theta$ . When the acceleration due to gravity is $g$ , the range of the projectile is $d$ . If at the highest point in its trajectory, the projectile enters a different region where the effective acceleration due to gravity is $g^{\prime}=\frac{g}{0.81}$ , then the new range is $d^{\prime}=n d$ . The value of $n$ is ___________ .

List I describes four systems, each with two particles $A$ and $B$ in relative motion as shown in figures. List II gives possible magnitudes of their relative velocities (in $m s^{-1}$ ) at time $t=\frac{\pi}{3} s$ . .tg {border-collapse:collapse;border-spacing:0;} .tg td{border-color:black;border-style:solid;border-width:1px;font-family:Arial, sans-serif;font-size:14px; overflow:hidden;padding:10px 5px;word-break:normal;} .tg th{border-color:black;border-style:solid;border-width:1px;font-family:Arial, sans-serif;font-size:14px; font-weight:normal;overflow:hidden;padding:10px 5px;word-break:normal;} .tg .tg-7btt{border-color:inherit;font-weight:bold;text-align:center;vertical-align:top} .tg .tg-0pky{border-color:inherit;text-align:left;vertical-align:top} List-I List-II (I) $A$ and $B$ are moving on a horizontal circle of radius $1 \mathrm{~m}$ with uniform angular speed $\omega=1 \mathrm{rad} \mathrm{s}^{-1}$ . The initial angular positions of $A$ and $B$ at time $t=0$ are $\theta=0$ and $\theta=\frac{\pi}{2}$ , respectively. (P) $\frac{\sqrt{3}+1}{2}$ (II) Projectiles $A$ and $B$ are fired (in the same vertical plane) at $t=0$ and $t=0.1 \mathrm{~s}$ respectively, with the same speed $v=\frac{5 \pi}{\sqrt{2}} \mathrm{~m} \mathrm{~s}^{-1}$ and at $45^{\circ}$ from the horizontal plane. The initial separation between $A$ and $B$ is large enough so that they do not collide. $\left(g=10 \mathrm{~ms}^{-2}\right)$ . (Q) $\frac{\sqrt{3}-1}{\sqrt{2}}$ (III) Two harmonic oscillators $A$ and $B$ moving in the $x$ direction according to $x_{A}=x_{0} \sin \frac{t}{t_{0}}$ and $x_{B}=x_{0} \sin \left(\frac{t}{t_{0}}+\frac{\pi}{2}\right)$ respectively, starting from $t=0$ . Take $x_{0}=1 \mathrm{~m}, t_{0}=1 \mathrm{~s}$ . (R) $\sqrt{10}$ (IV) Particle $A$ is rotating in a horizontal circular path of radius $1 \mathrm{~m}$ on the $x y$ plane, with constant angular speed $\omega=1 \mathrm{rad} \mathrm{s}^{-1}$ . Particle $B$ is moving up at a constant speed $3 \mathrm{~m} \mathrm{~s}^{-1}$ in the vertical direction as shown in the figure. (Ignore gravity.) (S) $\sqrt{2}$ (T) $\sqrt{25\pi^{2}+1}$ Which one of the following options is correct?

List I describes thermodynamic processes in four different systems. List II gives the magnitudes (either exactly or as a close approximation) of possible changes in the internal energy of the system due to the process. .tg {border-collapse:collapse;border-spacing:0;} .tg td{border-color:black;border-style:solid;border-width:1px;font-family:Arial, sans-serif;font-size:14px; overflow:hidden;padding:10px 5px;word-break:normal;} .tg th{border-color:black;border-style:solid;border-width:1px;font-family:Arial, sans-serif;font-size:14px; font-weight:normal;overflow:hidden;padding:10px 5px;word-break:normal;} .tg .tg-7btt{border-color:inherit;font-weight:bold;text-align:center;vertical-align:top} .tg .tg-0pky{border-color:inherit;text-align:left;vertical-align:top} List-I List-II (I) $10^{-3} \mathrm{~kg}$ of water at $100^{\circ} \mathrm{C}$ is converted to steam at the same temperature, at a pressure of $10^{5} \mathrm{~Pa}$ . The volume of the system changes from $10^{-6} \mathrm{~m}^{3}$ to $10^{-3} \mathrm{~m}^{3}$ in the process. Latent heat of water $=2250\, \mathrm{~kJ} / \mathrm{kg}$ . (P) $2 \mathrm{~kJ}$ (II) $0.2$ moles of a rigid diatomic ideal gas with volume $V$ at temperature $500 \mathrm{~K}$ undergoes an isobaric expansion to volume $3 \mathrm{~V}$ . Assume $R=8.0 \mathrm{Jmol}^{-1} \mathrm{~K}^{-1}$ . (Q) $7 k J$ (III) One mole of a monatomic ideal gas is compressed adiabatically from volume $V=\frac{1}{3} \mathrm{~m}^{3}$ and pressure $2 \mathrm{kPa}$ to volume $\frac{V}{8}$ . (R) $4 \mathrm{~kJ}$ (IV) Three moles of a diatomic ideal gas whose molecules can vibrate, is given $9 \mathrm{~kJ}$ of heat and undergoes isobaric expansion. (S) $5 \mathrm{~kJ}$ (T) $3 \mathrm{~kJ}$ Which one of the following options is correct?

LIST-I contains compounds and LIST-II contains reactions .tg {border-collapse:collapse;border-spacing:0;} .tg td{border-color:black;border-style:solid;border-width:1px;font-family:Arial, sans-serif;font-size:14px; overflow:hidden;padding:10px 5px;word-break:normal;} .tg th{border-color:black;border-style:solid;border-width:1px;font-family:Arial, sans-serif;font-size:14px; font-weight:normal;overflow:hidden;padding:10px 5px;word-break:normal;} .tg .tg-7btt{border-color:inherit;font-weight:bold;text-align:center;vertical-align:top} .tg .tg-0pky{border-color:inherit;text-align:left;vertical-align:top} List-I List-II (I) $\mathrm{H}_{2} \mathrm{O}_{2}$ (P) $\mathrm{Mg}\left(\mathrm{HCO}_{3}\right)_{2}+\mathrm{Ca}(\mathrm{OH})_{2} \rightarrow$ (II) $\mathrm{Mg}(\mathrm{OH})_{2}$ (Q) $\mathrm{BaO}_{2}+\mathrm{H}_{2} \mathrm{SO}_{4} \rightarrow$ (III) $\mathrm{BaCl}_{2}$ (R) $\mathrm{Ca}(\mathrm{OH})_{2}+\mathrm{MgCl}_{2} \rightarrow$ (IV) $\mathrm{CaCO}_{3}$ (S) $\mathrm{BaO}_{2}+\mathrm{HCl} \rightarrow$ (T) $\mathrm{Ca}\left(\mathrm{HCO}_{3}\right)_{2}+\mathrm{Ca}(\mathrm{OH})_{2} \rightarrow$ Match each compound in LIST-I with its formation reaction(s) in LIST-II, and choose the correct option

The treatment of an aqueous solution of $3.74 \mathrm{~g}$ of $\mathrm{Cu}\left(\mathrm{NO}_{3}\right)_{2}$ with excess KI results in a brown solution along with the formation of a precipitate. Passing $\mathrm{H}_{2} \mathrm{~S}$ through this brown solution gives another precipitate $\mathbf{X}$ . The amount of $\mathbf{X}$ (in $g$ ) is ___________. [Given: Atomic mass of $\mathrm{H}=1, \mathrm{~N}=14, \mathrm{O}=16, \mathrm{~S}=32, \mathrm{~K}=39, \mathrm{Cu}=63, \mathrm{I}=127$ ]

A solution is prepared by mixing $0.01 \mathrm{~mol}$ each of $\mathrm{H}_{2} \mathrm{CO}_{3}, \mathrm{NaHCO}_{3}, \mathrm{Na}_{2} \mathrm{CO}_{3}$ , and $\mathrm{NaOH}$ in $100 \mathrm{~mL}$ of water. $p \mathrm{H}$ of the resulting solution is _________. [Given: $p \mathrm{~K}_{\mathrm{a} 1}$ and $p \mathrm{~K}_{\mathrm{a} 2}$ of $\mathrm{H}_{2} \mathrm{CO}_{3}$ are $6.37$ and 10.32, respectively; $\log 2=0.30$ ]

For diatomic molecules, the correct statement(s) about the molecular orbitals formed by the overlap of two $2 p_{z}$ orbitals is(are)

If the reaction sequence given below is carried out with 15 moles of acetylene, the amount of the product $\mathbf{D}$ formed (in $\mathrm{g}$ ) is ___________ . The yields of $\mathbf{A}, \mathbf{B}, \mathbf{C}$ and $\mathbf{D}$ are given in parentheses. [Given: Atomic mass of $\mathrm{H}=1, \mathrm{C}=12, \mathrm{O}=16, \mathrm{Cl}=35$ ]

Let $\bar{z}$ denote the complex conjugate of a complex number $z$ and let $i=\sqrt{-1}$ . In the set of complex numbers, the number of distinct roots of the equation $\bar{z}-z^{2}=i\left(\bar{z}+z^{2}\right)$ is _________.

The number of 4-digit integers in the closed interval [2022, 4482] formed by using the digits 0 , 2 , 3 , 4 , 6 , 7 is _________.

Consider the ellipse $ x 2 4 + y 2 3 = 1 $ Let $H(\alpha, 0), 0<\alpha<2$ , be a point. A straight line drawn through $H$ parallel to the $y$ -axis crosses the ellipse and its auxiliary circle at points $E$ and $F$ respectively, in the first quadrant. The tangent to the ellipse at the point $E$ intersects the positive $x$ -axis at a point $G$ . Suppose the straight line joining $F$ and the origin makes an angle $\phi$ with the positive $x$ -axis. .tg {border-collapse:collapse;border-spacing:0;} .tg td{border-color:black;border-style:solid;border-width:1px;font-family:Arial, sans-serif;font-size:14px; overflow:hidden;padding:10px 5px;word-break:normal;} .tg th{border-color:black;border-style:solid;border-width:1px;font-family:Arial, sans-serif;font-size:14px; font-weight:normal;overflow:hidden;padding:10px 5px;word-break:normal;} .tg .tg-7btt{border-color:inherit;font-weight:bold;text-align:center;vertical-align:top} .tg .tg-0pky{border-color:inherit;text-align:left;vertical-align:top} List-I List-II (I) If $\phi=\frac{\pi}{4}$ , then the area of the triangle $F G H$ is (P) $\frac{(\sqrt{3}-1)^{4}}{8}$ (II) If $\phi=\frac{\pi}{3}$ , then the area of the triangle $F G H$ is (Q) 1 (III) If $\phi=\frac{\pi}{6}$ , then the area of the triangle $F G H$ is (R) $\frac{3}{4}$ (IV) If $\phi=\frac{\pi}{12}$ , then the area of the triangle $F G H$ is (S) $\frac{1}{2 \sqrt{3}}$ (T) $\frac{3 \sqrt{3}}{2}$ The correct option is:

Consider the following lists : .tg {border-collapse:collapse;border-spacing:0;} .tg td{border-color:black;border-style:solid;border-width:1px;font-family:Arial, sans-serif;font-size:14px; overflow:hidden;padding:10px 5px;word-break:normal;} .tg th{border-color:black;border-style:solid;border-width:1px;font-family:Arial, sans-serif;font-size:14px; font-weight:normal;overflow:hidden;padding:10px 5px;word-break:normal;} .tg .tg-amwm{font-weight:bold;text-align:center;vertical-align:top} .tg .tg-0lax{text-align:left;vertical-align:top} List-I List-II (I) $\left\{x \in\left[-\frac{2 \pi}{3}, \frac{2 \pi}{3}\right]: \cos x+\sin x=1\right\}$ (P) has two elements (II) $\left\{x \in\left[-\frac{5 \pi}{18}, \frac{5 \pi}{18}\right]: \sqrt{3} \tan 3 x=1\right\}$ (Q) has three elements (III) $\left\{x \in\left[-\frac{6 \pi}{5}, \frac{6 \pi}{5}\right]: 2 \cos (2 x)=\sqrt{3}\right\}$ (R) has four elements (IV) $\left\{x \in\left[-\frac{7 \pi}{4}, \frac{7 \pi}{4}\right]: \sin x-\cos x=1\right\}$ (S) has five elements (T) has six elements The correct option is:

Let $|M|$ denote the determinant of a square matrix $M$ . Let $g:\left[0, \frac{\pi}{2}\right] \rightarrow \mathbb{R}$ be the function defined by $g(\theta)=\sqrt{f(\theta)-1}+\sqrt{f\left(\frac{\pi}{2}-\theta\right)-1}$ where $f(\theta)=\frac{1}{2}\left|\begin{array}{ccc}1 & \sin \theta & 1 \\-\sin \theta & 1 & \sin \theta \\-1 & -\sin \theta & 1\end{array}\right|+\left|\begin{array}{ccc}\sin \pi & \cos \left(\theta+\frac{\pi}{4}\right) & \tan \left(\theta-\frac{\pi}{4}\right) \\\sin \left(\theta-\frac{\pi}{4}\right) & -\cos \frac{\pi}{2} & \log _{e}\left(\frac{4}{\pi}\right) \\\cot \left(\theta+\frac{\pi}{4}\right) & \log _{e}\left(\frac{\pi}{4}\right) & \tan \pi\end{array}\right| .$ Let $p(x)$ be a quadratic polynomial whose roots are the maximum and minimum values of the function $g(\theta)$ , and $p(2)=2-\sqrt{2}$ . Then, which of the following is/are TRUE ?

Three plane mirrors form an equilateral triangle with each side of length $L$ . There is a small hole at a distance $l>0$ from one of the corners as shown in the figure. A ray of light is passed through the hole at an angle $\theta$ and can only come out through the same hole. The cross section of the mirror configuration and the ray of light lie on the same plane. Which of the following statement(s) is(are) correct?

Consider an LC circuit, with inductance $L=0.1 \,\mathrm{H}$ and capacitance $C=10^{-3} \mathrm{~F}$ , kept on a plane. The area of the circuit is $1 \mathrm{~m}^{2}$ . It is placed in a constant magnetic field of strength $B_{0}$ which is perpendicular to the plane of the circuit. At time $t=0$ , the magnetic field strength starts increasing linearly as $B=B_{0}+\beta t$ with $\beta=0.04 \,\mathrm{T\,s}^{-1}$ . The maximum magnitude of the current in the circuit is _________ $m A$ .

The minimum kinetic energy needed by an alpha particle to cause the nuclear reaction ${ }_{7}^{16} \mathrm{~N}+$ ${ }_{2}^{4} \mathrm{He} \rightarrow{ }_{1}^{1} \mathrm{H}+{ }_{8}^{19} \mathrm{O}$ in a laboratory frame is $n$ (in $M e V$ . Assume that ${ }_{7}^{16} \mathrm{~N}$ is at rest in the laboratory frame. The masses of ${ }_{7}^{16} \mathrm{~N},{ }_{2}^{4} \mathrm{He},{ }_{1}^{1} \mathrm{H}$ and ${ }_{8}^{19} \mathrm{O}$ can be taken to be $16.006 u, 4.003 u, 1.008 u$ and $19.003 u$ , respectively, where $1 u=930 \,\mathrm{MeVc}^{-2}$ . The value of $n$ is ________ .

In the following circuit $C_{1}=12 \mu F, C_{2}=C_{3}=4 \mu F$ and $C_{4}=C_{5}=2 \mu F$ . The charge stored in $C_{3}$ is ____________ $\mu C$ .

A medium having dielectric constant $K>1$ fills the space between the plates of a parallel plate capacitor. The plates have large area, and the distance between them is $d$ . The capacitor is connected to a battery of voltage $V$ , as shown in Figure (a). Now, both the plates are moved by a distance of $\frac{d}{2}$ from their original positions, as shown in Figure (b). In the process of going from the configuration depicted in Figure (a) to that in Figure (b), which of the following statement(s) is(are) correct?

Let $p, q, r$ be nonzero real numbers that are, respectively, the $10^{\text {th }}, 100^{\text {th }}$ and $1000^{\text {th }}$ terms of a harmonic progression. Consider the system of linear equations $ x + y + z = 1 10 x + 100 y + 1000 z = 0 q r x + p r y + p q z = 0 $ .tg {border-collapse:collapse;border-spacing:0;} .tg td{border-color:black;border-style:solid;border-width:1px;font-family:Arial, sans-serif;font-size:14px; overflow:hidden;padding:10px 5px;word-break:normal;} .tg th{border-color:black;border-style:solid;border-width:1px;font-family:Arial, sans-serif;font-size:14px; font-weight:normal;overflow:hidden;padding:10px 5px;word-break:normal;} .tg .tg-7btt{border-color:inherit;font-weight:bold;text-align:center;vertical-align:top} .tg .tg-0pky{border-color:inherit;text-align:left;vertical-align:top} List-I List-II (I) If $\frac{q}{r}=10$ , then the system of linear equations has (P) $x=0, \quad y=\frac{10}{9}, z=-\frac{1}{9}$ as a solution (II) If $\frac{p}{r} \neq 100$ , then the system of linear equations has (Q) $x=\frac{10}{9}, y=-\frac{1}{9}, z=0$ as a solution (III) If $\frac{p}{q} \neq 10$ , then the system of linear equations has (R) infinitely many solutions (IV) If $\frac{p}{q}=10$ , then the system of linear equations has (S) no solution (T) at least one solution The correct option is:

Let $P_{1}$ and $P_{2}$ be two planes given by $\begin{aligned}&P_{1}: 10 x+15 y+12 z-60=0 \\\\&P_{2}:-2 x+5 y+4 z-20=0\end{aligned}$ Which of the following straight lines can be an edge of some tetrahedron whose two faces lie on $P_{1}$ and $P_{2}$ ?

In a study about a pandemic, data of 900 persons was collected. It was found that 190 persons had symptom of fever, 220 persons had symptom of cough, 220 persons had symptom of breathing problem, 330 persons had symptom of fever or cough or both, 350 persons had symptom of cough or breathing problem or both, 340 persons had symptom of fever or breathing problem or both, 30 persons had all three symptoms (fever, cough and breathing problem). If a person is chosen randomly from these 900 persons, then the probability that the person has at most one symptom is ____________.

The greatest integer less than or equal to $\int_{1}^{2} \log _{2}\left(x^{3}+1\right) d x+\int_{1}^{\log _{2} 9}\left(2^{x}-1\right)^{\frac{1}{3}} d x$ is ___________.

If $\beta=\lim \limits_{x \to 0} \frac{e^{x^{3}}-\left(1-x^{3}\right)^{\frac{1}{3}}+\left(\left(1-x^{2}\right)^{\frac{1}{2}}-1\right) \sin x}{x \sin ^{2} x},$ then the value of $6 \beta$ is ___________.

The treatment of galena with $\mathrm{HNO}_{3}$ produces a gas that is

Match the compounds in LIST-I with the observations in LIST-II, and choose the correct option. .tg {border-collapse:collapse;border-spacing:0;} .tg td{border-color:black;border-style:solid;border-width:1px;font-family:Arial, sans-serif;font-size:14px; overflow:hidden;padding:10px 5px;word-break:normal;} .tg th{border-color:black;border-style:solid;border-width:1px;font-family:Arial, sans-serif;font-size:14px; font-weight:normal;overflow:hidden;padding:10px 5px;word-break:normal;} .tg .tg-7btt{border-color:inherit;font-weight:bold;text-align:center;vertical-align:top} .tg .tg-0pky{border-color:inherit;text-align:left;vertical-align:top} List-I List-II (I) Aniline (P) Sodium fusion extract of the compound on boiling with $\mathrm{FeSO}_{4}$ , followed by acidification with conc. $\mathrm{H}_{2} \mathrm{SO}_{4}$ , gives Prussian blue color. (II) $o$ -Cresol (Q) Sodium fusion extract of the compound on treatment with sodium nitroprusside gives blood red color. (III) Cysteine (R) Addition of the compound to a saturated solution of $\mathrm{NaHCO}_{3}$ results in effervescence. (IV) Caprolactam (S) The compound reacts with bromine water to give a white precipitate. (T) Treating the compound with neutral $\mathrm{FeCl}_{3}$ solution produces violet color.

The correct option(s) related to adsorption processes is(are)

Match the rate expressions in LIST-I for the decomposition of $X$ with the corresponding profiles provided in LIST-II. $X_{\mathrm{s}}$ and $\mathrm{k}$ are constants having appropriate units. .tg {border-collapse:collapse;border-spacing:0;} .tg td{border-color:black;border-style:solid;border-width:1px;font-family:Arial, sans-serif;font-size:14px; overflow:hidden;padding:10px 5px;word-break:normal;} .tg th{border-color:black;border-style:solid;border-width:1px;font-family:Arial, sans-serif;font-size:14px; font-weight:normal;overflow:hidden;padding:10px 5px;word-break:normal;} .tg .tg-7btt{border-color:inherit;font-weight:bold;text-align:center;vertical-align:top} .tg .tg-0pky{border-color:inherit;text-align:left;vertical-align:top} List-I List-II (I) rate $=\frac{\mathrm{k}[\mathrm{X}]}{\mathrm{X}_{\mathrm{s}}+[\mathrm{X}]}$ under all possible initial concentrations of $\mathrm{X}$ (P) (II) rate $=\frac{k[X]}{X_{s}+[X]}$ where initial concentrations of $X$ are much less than $X_{s}$ (Q) (III) rate $=\frac{k[X]}{X_{s}+[X]}$ where initial concentrations of $\mathrm{X}$ are much higher than $X_{s}$ (R) (IV) rate $=\frac{k[X]^{2}}{X_{s}+[X]}$ where initial concentration of $X$ is much higher than $\mathrm{X}_{\mathrm{s}}$ (S) (T)

Considering the following reaction sequence, the correct option(s) is(are)