JEE Advanced 2024 Question Paper with Solutions

Let $\overrightarrow{O P}=\frac{\alpha-1}{\alpha} \hat{i}+\hat{j}+\hat{k}, \overrightarrow{O Q}=\hat{i}+\frac{\beta-1}{\beta} \hat{j}+\hat{k}$ and $\overrightarrow{O R}=\hat{i}+\hat{j}+\frac{1}{2} \hat{k}$ be three vectors, where $\alpha, \beta \in \mathbb{R}-\{0\}$ and $O$ denotes the origin. If $(\overrightarrow{O P} \times \overrightarrow{O Q}) \cdot \overrightarrow{O R}=0$ and the point $(\alpha, \beta, 2)$ lies on the plane $3 x+3 y-z+l=0$ , then the value of $l$ is ____________.

Let $\vec{p}=2 \hat{i}+\hat{j}+3 \hat{k}$ and $\vec{q}=\hat{i}-\hat{j}+\hat{k}$ . If for some real numbers $\alpha, \beta$ , and $\gamma$ , we have $15 \hat{i}+10 \hat{j}+6 \hat{k}=\alpha(2 \vec{p}+\vec{q})+\beta(\vec{p}-2 \vec{q})+\gamma(\vec{p} \times \vec{q}),$ then the value of $\gamma$ is ________.

Consider the following volume-temperature $(\mathrm{V}-\mathrm{T})$ diagram for the expansion of 5 moles of an ideal monoatomic gas. Considering only $\mathrm{P}-\mathrm{V}$ work is involved, the total change in enthalpy (in Joule) for the transformation of state in the sequence $\mathbf{X} \rightarrow \mathbf{Y} \rightarrow \mathbf{Z}$ is ____________. [Use the given data: Molar heat capacity of the gas for the given temperature range, $\mathrm{C}_{\mathrm{V}, \mathrm{m}}=12 \mathrm{~J} \mathrm{~K}^{-1}$ $\mathrm{mol}^{-1}$ and gas constant, $\left.\mathrm{R}=8.3 \mathrm{~J} \mathrm{~K}^{-1} \mathrm{~mol}^{-1}\right]$

Based on VSEPR model, match the xenon compounds given in List-I with the corresponding geometries and the number of lone pairs on xenon given in List-II and choose the correct option. table { width: 80%; border-collapse: collapse; margin: 20px 0; } th { background-color: blue; color: white; border: 1px solid lightgrey; padding: 10px; } td { border: 1px solid lightgrey; padding: 10px; text-align: left; } List-I List-II (P) XeF $_2$ (1) Trigonal bipyramidal and two lone pair of electrons (Q) XeF $_4$ (2) Tetrahedral and one lone pair of electrons (R) XeO $_3$ (3) Octahedral and two lone pair of electrons (S) XeO $_3$ F $_2$ (4) Trigonal bipyramidal and no lone pair of electrons (5) Trigonal bipyramidal and three lone pair of electrons

The option(s) in which at least three molecules follow Octet Rule is(are)

A closed vessel contains $10 \mathrm{~g}$ of an ideal gas $\mathbf{X}$ at $300 \mathrm{~K}$ , which exerts $2 \mathrm{~atm}$ pressure. At the same temperature, $80 \mathrm{~g}$ of another ideal gas $\mathbf{Y}$ is added to it and the pressure becomes $6 \mathrm{~atm}$ . The ratio of root mean square velocities of $\mathbf{X}$ and $\mathbf{Y}$ at $300 \mathrm{~K}$ is

Among the following, the correct statement(s) for electrons in an atom is(are)

A group of 9 students, $s_1, s_2, \ldots, s_9$ , is to be divided to form three teams $X, Y$ , and $Z$ of sizes 2,3 , and 4 , respectively. Suppose that $s_1$ cannot be selected for the team $X$ , and $s_2$ cannot be selected for the team $Y$ . Then the number of ways to form such teams, is ____________.

Let $a=3 \sqrt{2}$ and $b=\frac{1}{5^{1 / 6} \sqrt{6}}$ . If $x, y \in \mathbb{R}$ are such that 3 x + 2 y = log a ⁡ ( 18 ) 5 4  and  2 x − y = log b ⁡ ( 1080 ) , then $4 x+5 y$ is equal to __________.

Let $S=\{a+b \sqrt{2}: a, b \in \mathbb{Z}\}, T_1=\left\{(-1+\sqrt{2})^n: n \in \mathbb{N}\right\}$ , and $T_2=\left\{(1+\sqrt{2})^n: n \in \mathbb{N}\right\}$ . Then which of the following statements is (are) TRUE?

Consider the ellipse $\frac{x^2}{9}+\frac{y^2}{4}=1$ . Let $S(p, q)$ be a point in the first quadrant such that $\frac{p^2}{9}+\frac{q^2}{4}>1$ . Two tangents are drawn from $S$ to the ellipse, of which one meets the ellipse at one end point of the minor axis and the other meets the ellipse at a point $T$ in the fourth quadrant. Let $R$ be the vertex of the ellipse with positive $x$ -coordinate and $O$ be the center of the ellipse. If the area of the triangle $\triangle O R T$ is $\frac{3}{2}$ , then which of the following options is correct?

Let the function $f: \mathbb{R} \rightarrow \mathbb{R}$ be defined by $f(x)=\frac{\sin x}{e^{\pi x}} \frac{\left(x^{2023}+2024 x+2025\right)}{\left(x^2-x+3\right)}+\frac{2}{e^{\pi x}} \frac{\left(x^{2023}+2024 x+2025\right)}{\left(x^2-x+3\right)} .$ Then the number of solutions of $f(x)=0$ in $\mathbb{R}$ is _________.

A glass beaker has a solid, plano-convex base of refractive index 1.60, as shown in the figure. The radius of curvature of the convex surface (SPU) is $9 \mathrm{~cm}$ , while the planar surface (STU) acts as a mirror. This beaker is filled with a liquid of refractive index $n$ up to the level QPR. If the image of a point object $\mathrm{O}$ at a height of $h$ (OT in the figure) is formed onto itself, then, which of the following option(s) is(are) correct?

Four identical thin, square metal sheets, $S_1, S_2, S_3$ and $S_4$ , each of side $a$ are kept parallel to each other with equal distance $d(\ll a)$ between them, as shown in the figure. Let ${C_0} = {{{\varepsilon _0}{a^2}} \over d}$ , where $\varepsilon_0$ is the permittivity of free space. Match the quantities mentioned in List-I with their values in List-II and choose the correct option. table { width: 100%; border-collapse: collapse; } th { background-color: #1E90FF; color: white; } th, td { border: 1px solid lightgrey; padding: 8px; text-align: left; } body[data-theme='dark'] th, body[data-theme='dark'] td { border: 1px solid white; } List-I List-II (P) The capacitance between $S_1$ and $S_4$ , with $S_2$ and $S_3$ not connected, is (1) $3C_0$ (Q) The capacitance between $S_1$ and $S_4$ , with $S_2$ shorted to $S_3$ , is (2) $\frac{C_0}{2}$ (R) The capacitance between $S_1$ and $S_3$ , with $S_2$ shorted to $S_4$ , is (3) $\frac{C_0}{3}$ (S) The capacitance between $S_1$ and $S_2$ , with $S_3$ shorted to $S_1$ , and $S_2$ shorted to $S_4$ , is (4) $\frac{2C_0}{3}$ (5) $2C_0$

Two equilateral-triangular prisms $\mathrm{P}_1$ and $\mathrm{P}_2$ are kept with their sides parallel to each other, in vacuum, as shown in the figure. A light ray enters prism $\mathrm{P}_1$ at an angle of incidence $\theta$ such that the outgoing ray undergoes minimum deviation in prism $\mathrm{P}_2$ . If the respective refractive indices of $\mathrm{P}_1$ and $\mathrm{P}_2$ are $\sqrt{\frac{3}{2}}$ and $\sqrt{3}$ , then $\theta=\sin ^{-1}\left[\sqrt{\frac{3}{2}} \sin \left(\frac{\pi}{\beta}\right)\right]$ , where the value of $\beta$ is ____.

A thin stiff insulated metal wire is bent into a circular loop with its two ends extending tangentially from the same point of the loop. The wire loop has mass $m$ and radius $r$ and it is in a uniform vertical magnetic field $B_0$ , as shown in the figure. Initially, it hangs vertically downwards, because of acceleration due to gravity $g$ , on two conducting supports at $\mathrm{P}$ and $\mathrm{Q}$ . When a current $I$ is passed through the loop, the loop turns about the line $\mathrm{PQ}$ by an angle $\theta$ given by

A positive, singly ionized atom of mass number $A_{\mathrm{M}}$ is accelerated from rest by the voltage $192 \mathrm{~V}$ . Thereafter, it enters a rectangular region of width $w$ with magnetic field $\vec{B}_0=0.1 \hat{k}$ Tesla, as shown in the figure. The ion finally hits a detector at the distance $x$ below its starting trajectory. [Given: Mass of neutron/proton $=(5 / 3) \times 10^{-27} \mathrm{~kg}$ , charge of the electron $=1.6 \times 10^{-19} \mathrm{C}$ .] Which of the following option(s) is(are) correct?

Let $S=\left\{A=\left( 0 1 c 1 a d 1 b e \right): a, b, c, d, e \in\{0,1\}\right. $and$ \left.|A| \in\{-1,1\}\right\} $, where$ |A| $denotes the determinant of$ A $. Then the number of elements in$ S$ is __________.

Let $\mathbb{R}^3$ denote the three-dimensional space. Take two points $P=(1,2,3)$ and $Q=(4,2,7)$ . Let $\operatorname{dist}(X, Y)$ denote the distance between two points $X$ and $Y$ in $\mathbb{R}^3$ . Let $\begin{gathered} S=\left\{X \in \mathbb{R}^3:(\operatorname{dist}(X, P))^2-(\operatorname{dist}(X, Q))^2=50\right\} \text { and } \\ T=\left\{Y \in \mathbb{R}^3:(\operatorname{dist}(Y, Q))^2-(\operatorname{dist}(Y, P))^2=50\right\} . \end{gathered}$ Then which of the following statements is (are) TRUE?

Let $f(x)$ be a continuously differentiable function on the interval $(0, \infty)$ such that $f(1)=2$ and $\lim\limits_{t \rightarrow x} \frac{t^{10} f(x)-x^{10} f(t)}{t^9-x^9}=1$ for each $x>0$ . Then, for all $x>0, f(x)$ is equal to :

Let $X$ be a random variable, and let $P(X=x)$ denote the probability that $X$ takes the value $x$ . Suppose that the points $(x, P(X=x)), x=0,1,2,3,4$ , lie on a fixed straight line in the $x y$ -plane, and $P(X=x)=0$ for all $x \in \mathbb{R}-\{0,1,2,3,4\}$ . If the mean of $X$ is $\frac{5}{2}$ , and the variance of $X$ is $\alpha$ , then the value of $24 \alpha$ is _____________.

Considering only the principal values of the inverse trigonometric functions, the value of tan ⁡ ( sin − 1 ⁡ ( 3 5 ) − 2 cos − 1 ⁡ ( 2 5 ) ) is

List-I contains various reaction sequences and List-II contains different phenolic compounds. Match each entry in List-I with the appropriate entry in List-II and choose the correct option. table { width: 100%; border-collapse: collapse; margin: 20px 0; } th { background-color: blue; color: white; border: 1px solid lightgrey; padding: 10px; } td { border: 1px solid lightgrey; padding: 10px; text-align: left; vertical-align: top; } img { max-width: 100px; height: auto; } List-I List-II

Complete reaction of acetaldehyde with excess formaldehyde, upon heating with conc. $\mathrm{NaOH}$ solution, gives $\mathbf{P}$ and $\mathbf{Q}$ . Compound $\mathbf{P}$ does not give Tollens' test, whereas $\mathbf{Q}$ on acidification gives positive Tollens' test. Treatment of $\mathbf{P}$ with excess cyclohexanone in the presence of catalytic amount of $p$ -toluenesulfonic acid (PTSA) gives product $\mathbf{R}$ . Sum of the number of methylene groups $\left(-\mathrm{CH}_2-\right)$ and oxygen atoms in $\mathbf{R}$ is __________.

Aspartame, an artificial sweetener, is a dipeptide aspartyl phenylalanine methyl ester. The structure of aspartame is

Reaction of iso-propylbenzene with $\mathrm{O}_2$ followed by the treatment with $\mathrm{H}_3 \mathrm{O}^{+}$ forms phenol and a by-product $\mathbf{P}$ . Reaction of $\mathbf{P}$ with 3 equivalents of $\mathrm{Cl}_2$ gives compound $\mathbf{Q}$ . Treatment of $\mathbf{Q}$ with $\mathrm{Ca}(\mathrm{OH})_2$ produces compound $\mathbf{R}$ and calcium salt $\mathbf{S}$ . The correct statement(s) regarding $\mathbf{P}, \mathbf{Q}, \mathbf{R}$ and $\mathbf{S}$ is(are)

Among the following options, select the option in which each complex in Set-I shows geometrical isomerism and the two complexes in Set-II are ionization isomers of each other. $\text { [en }=\mathrm{H}_2 \mathrm{NCH}_2 \mathrm{CH}_2 \mathrm{NH}_2 \text { ] }$

Q.3 Two beads, each with charge 𝑞 and mass 𝑚, are on a horizontal, frictionless, non-conducting, circular hoop of radius 𝑅. One of the beads is glued to the hoop at some point, while the other one performs small oscillations about its equilibrium position along the hoop. The square of the angular frequency of the small oscillations is given by  [𝜀0 is the permittivity of free space.]

Q.4 A block of mass 5 kg moves along the x -direction subject to the force 𝐹 = (−20𝑥 + 10) N, with the value of 𝑥 in metre. At time 𝑡 = 0 s, it is at rest at position 𝑥 = 1 m. The position and momentum of the block at 𝑡 = (𝜋/4) s are

Q.1  A dimensionless quantity is constructed in terms of electronic charge 𝑒, permittivity of free space 𝜀0, Planck’s constant ℎ, and speed of light 𝑐. If the dimensionless quantity is written as 𝑒𝛼𝜀0𝛽ℎ𝛾𝑐𝛿 and 𝑛 is a non-zero integer, then (𝛼, 𝛽, 𝛾, 𝛿) is given by

Q.5  A particle of mass 𝑚 is moving in a circular orbit under the influence of the central force 𝐹(𝑟) = −𝑘𝑟, corresponding to the potential energy 𝑉(𝑟) = 𝑘𝑟2/2, where 𝑘 is a positive force constant and 𝑟 is the radial distance from the origin. According to the Bohr’s quantization rule, the angular momentum of the particle is given by 𝐿 = 𝑛ℏ, where ℏ = ℎ/(2𝜋), ℎ is the Planck’s constant, and 𝑛 a positive integer. If 𝑣 and 𝐸 are the speed and total energy of the particle, respectively, then which of the following expression(s) is(are) correct?

Q.6 Two uniform strings of mass per unit length 𝜇 and 4𝜇, and length 𝐿 and 2𝐿, respectively, are joined at point O, and tied at two fixed ends P and Q, as shown in the figure. The strings are under a uniform tension 𝑇. If we define the frequency   ν₀ = (1/2L)√(T/μ), which of the following statement(s) is(are) correct?