JEE Advanced 2024 · 26 May 2024 · Paper 1

Q1.A dimensionless quantity is constructed in terms of electronic charge $e$, permittivity of free space ${\epsilon }_0$, Planck's constant h, and speed of light $c$. If the dimensionless quantity is written as …

Q2.An infinitely long wire, located on the $z$-axis, carries a current $I$ along the +z-direction and produces the magnetic field $\overrightarrow{\mathrm{B}}$. The magnitude of the line integral $\int \overrightarrow{\mathrm{B}}.\overrightarrow{dl}$ along a straight line …

Q3.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. …

Q4.A block of mass 5 kg moves along the x-direction subject to the force $F=(-20x+10)\mathrm{N}$, with the value of $x$ in metre. At time $t=0\mathrm{s}$, it is at rest at position $x=1\mathrm{m}$. The position and momentum of the block at $t=(\pi /4)\mathrm{s}$ ar …

Q5.A particle of mass $m$ is moving in a circular orbit under the influence of the central force $\mathrm{F}(r)=-kr$, corresponding to the potential energy $V(r)=k{r}^2/2$, where $k$ is a positive force constant and $r$ is the radial distance from the origin. According …

Q6.Two uniform string of mass per unit length $\mu$ and $4\mu$, 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 …

Q7.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 cm, while the planar surface (STU) acts as a mirror. This beaker is filled with a liquid of refractive index $n$ up to the le …

Q8.The specific heat capacity of a substance is temperature dependent and is given by the formula $C=kT$, where $k$ is a constant of suitable dimensions in SI units, and $T$ is the absolute temperature. If the heat required to raise the temperature of 1 kg of the substance fr …

Q9.A disc of mass $M$ and radius $R$ is free to rotate about its vertical axis as shown in the figure. A battery operated motor of negligible mass is fixed to this disc at a point on its circumference. Another disc of the same mass $M$ and radius $R/2$ is fixed to the moto …

Q10.A point source S emits unpolarized light uniformly in all directions. At two points A and B, the ratio $\mathrm{r}={\mathrm{I}}_{\mathrm{A}}/{\mathrm{I}}_{\mathrm{B}}$ of the intensities of light is 2. If a set of two polaroids having 45° angle between their pass-axes is placed j …

Q11.A source (S) of sound has frequency 240 Hz. When the observer (O) and the source move towards each other at a speed $v$ with respect to the ground (as shown in Case 1 in the figure), the observer measures the frequency of the sound to be 288 Hz. However, when the observer and t …

Q12.Two large, identical water tanks, 1 and 2, kept on the top of a building of height $H$, are filled with water up to height $h$ in each tank. Both the tanks contain an identical hole of small radius on their sides, close to their bottom. A pipe of the same internal radius as t …

Q13.A thin uniform rod of length $L$ and certain mass is kept on a frictionless horizontal table with a massless string of length $L$ fixed to one end (top view is shown in the figure). The other end of the string is pivoted to a point O . If a horizontal impulse $P$ is imparte …

Q14.One mole of a monatomic ideal gas undergoes the cyclic process $\mathrm{J}\to \mathrm{K}\to \mathrm{L}\to \mathrm{M}\to \mathrm{J}$, as shown in the P-T diagram.

Match the quantities mentioned in List-I with their values in List-II and choos …

Q15.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 $C_0={\epsilon }_0{\mathrm{a}}^2/\mathrm{d}$, where …

Q16.A light ray is incident on the surface of a sphere of refractive index $n$ at an angle of incidence ${\theta }_0$. The ray partially refracts into the sphere with angle of refraction ${\varphi }_0$ and then partly reflects from the back surface. The reflected ray then emerges ou …

Q17.The circuit shown in the figure contains an inductor $L$, a capacitor $C_0$, a resistor $R_0$ and an ideal battery. The circuit also contains two keys ${\mathrm{K}}_1$ and ${\mathrm{K}}_2$. Initially, both the keys are open and there is no charge on the capacitor. A …

Q1.A closed vessel contains 10 g of an ideal gas X at 300 K, which exerts 2 atm pressure. At the same temperature, 80 g of another ideal gas Y is added to it and the pressure becomes 6 atm. The ratio of root mean square velocities of X and Y at 300 K is

Q2.At room temperature, disproportionation of an aqueous solution of in situ generated nitrous acid (${\mathrm{HNO}}_2$) gives the species

Q3.Aspartame, an artificial sweetener, is a dipeptide aspartyl phenylalanine methyl ester. The structure of aspartame is

Q4.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.
[$\mathrm{en}={\mathrm{H}}_2{\mathrm{NCH}}_2{\mathrm{CH}}_2{\mathrm{NH}}_2$]

Q5.Among the following, the correct statement(s) for electrons in an atom is(are)

Q6.Reaction of iso-propylbenzene with ${\mathrm{O}}_2$ followed by the treatment with ${\mathrm{H}}_3{\mathrm{O}}^{+}$ forms phenol and a by-product P. Reaction of P with 3 equivalents of ${\mathrm{Cl}}_2$ gives compound Q. Treatment of Q with $\mathrm{Ca}(\mathrm{OH}{)}_2$ produces c …

Q7.The option(s) in which at least three molecules follow Octet Rule is(are)

Q8.Consider the following volume-temperature (V - T) diagram for the expansion of 5 moles of an ideal monoatomic gas.

(see diagram)

Considering only P-V work is involved, the total change in enthalpy (in Joule) for the transformation of state in the sequence …

Q9.Consider the following reaction,

which follows the mechanism given below: …

Q10.Complete reaction of acetaldehyde with excess formaldehyde, upon heating with conc. NaOH solution, gives P and Q. Compound P does not give Tollens' test, whereas Q on acidification gives positive Tollens' test. Treatment of P with excess cyclohexanone in the presence of catalytic …

Q11.Among $\mathrm{V}(\mathrm{CO}{)}_6$, $\mathrm{Cr}(\mathrm{CO}{)}_5$, $\mathrm{Cu}(\mathrm{CO}{)}_3$, $\mathrm{Mn}(\mathrm{CO}{)}_5$, $\mathrm{Fe}(\mathrm{CO}{)}_5$, $[\mathrm{Co}(\mathrm{CO}{)}_3{]}^{3-}$, $[\mathrm{Cr}(\mathrm{CO}{)}_4{]}^{4-}$, and $\mathrm{Ir}(\mathrm{CO}{)}_3$, t …

Q12.In the following reaction sequence, the major product P is formed.

(see reaction scheme)

Glycerol reacts completely with excess P in the presence of an acid catalyst to form Q. Reaction of Q with excess NaOH followed by the treatment with ${\mathrm{CaCl}}_2$ yields Ca-soap R, q …

Q13.Among the following complexes, the total number of diamagnetic species is ____.
$[\mathrm{Mn}({\mathrm{NH}}_3{)}_6{]}^{3+}$, $[{\mathrm{MnCl}}_6{]}^{3-}$, $[{\mathrm{FeF}}_6{]}^{3-}$, $[{\mathrm{CoF}}_6{]}^{3-}$, $[\mathrm{Fe}({\mathrm{NH}}_3{)}_6{]}^{3+}$, and …

Q14.In a conductometric titration, small volume of titrant of higher concentration is added stepwise to a larger volume of titrate of much lower concentration, and the conductance is measured after each addition.
The limiting ionic conductivity (${\Lambda }_0$) values (in …

Q15.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.

List-I
(P) ${\mathrm{XeF}}_2$
(Q) ${\mathrm{XeF}}_4$
(R) ${\mathrm{XeO}}_3$
(S) …

Q16.List-I contains various reaction sequences and List-II contains the possible products. Match each entry in List-I with the appropriate entry in List-II and choose the correct option.

List-I
(P) (see reaction scheme)
(Q) (see reaction scheme)
(R) (see reaction scheme)
(S) (see re …

Q17.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.

List-I
(P) (see reaction scheme)
(Q) (see reaction scheme)
(R) (see reaction scheme)
(S) …

Q1.Let $f(\mathrm{x})$ be a continuously differentiable function on the interval $(0,\infty )$ such that $f(1)=2$ and ${\lim }_{\mathrm{t}\to \mathrm{x}}\frac{{\mathrm{t}}^{10}f(\mathrm{x})-{\mathrm{x}}^{10}f(\mathrm{t})}{{\mathrm{t}}^9-{\mathrm{x}}^9}=1$ for each …

Q2.A student appears for a quiz consisting of only true-false type questions and answers all the questions. The student knows the answers of some questions and guesses the answers for the remaining questions. Whenever the student knows the answer of a question, he gives the correct …

Q3.Let $\frac{\pi }{2}<\mathrm{x}<\pi$ be such that $\cot \mathrm{x}=\frac{-5}{\sqrt{11}}$. Then
$\left(\sin \frac{11x}{2}\right)(\sin 6x-\cos 6x)+\left(\cos \frac{11x}{2}\right)(\sin 6x+\cos 6x)$ is equal to :

Q4.Consider the ellipse $\frac{{\mathrm{x}}^2}{9}+\frac{{\mathrm{y}}^2}{4}=1$. Let $\mathrm{S}(\mathrm{p},\mathrm{q})$ be a point in the first quadrant such that $\frac{{\mathrm{p}}^2}{9}+\frac{{\mathrm{q}}^2}{4}>1$. Two tangents are drawn from S to the ellipse, of which one …

Q5.Let $\mathrm{S}=\{\mathrm{a}+\mathrm{b}\sqrt{2}:\mathrm{a},\mathrm{b}\in \mathbb{Z}\},{\mathrm{T}}_1=\left\{(-1+\sqrt{2}{)}^{\mathrm{n}}:\mathrm{n}\in \mathbb{N}\right\}$ and ${\mathrm{T}}_2=\left\{(1+\sqrt{2}{)}^{\mathrm{n}}:\mathrm{n}\in \mathbb{N}\right\}$. Then which …

Q6.Let ${\mathbb{R}}^2$ denote $\mathbb{R}\times \mathbb{R}$. Let
$\mathrm{S}=\{(\mathrm{a},\mathrm{b},\mathrm{c}):\mathrm{a},\mathrm{b},\mathrm{c}\in \mathbb{R}$ and ${\mathrm{ax}}^2+2\mathrm{bxy}+{\mathrm{cy}}^2>0$ for all …

Q7.Let ${\mathbb{R}}^3$ denote the three-dimensional space. Take two points $\mathrm{P}=(1,2,3)$ and $\mathrm{Q}=(4,2,7)$.
Let dist (X, Y) denote the distance between two points X and Y in ${\mathbb{R}}^3$. Let …

Q8.Let $\mathrm{a}=3\sqrt{2}$ and $\mathrm{b}=\frac{1}{5^{1/6}\sqrt{6}}$. If $\mathrm{x},\mathrm{y}\in \mathbb{R}$ are such that

then $4x+5y$ equ …

Q9.Let $f(\mathrm{x})={\mathrm{x}}^4+{\mathrm{ax}}^3+{\mathrm{bx}}^2+\mathrm{c}$ be a polynomial with real coefficients such that $f(1)=-9$. Suppose that $\mathrm{i}\sqrt{3}$ is a root of the equation $4{\mathrm{x}}^3+3{\mathrm{ax}}^2+2\mathrm{bx}=0$, where …

Q10.Let $\mathrm{S}=\{\mathrm{A}=\left(\begin{array}{lll}0& 1& \mathrm{c}\\ 1& \mathrm{a}& \mathrm{d}\\ 1& \mathrm{b}& \mathrm{e}\end{array}\right):\mathrm{a},\mathrm{b},\mathrm{c},\mathrm{d},\mathrm{e}\in \{0,1\}$ and …

Q11.A group of 9 students, ${\mathrm{s}}_1,{\mathrm{s}}_2,\dots ..,{\mathrm{s}}_9$, is to be divided to from three teams X, Y, and Z of sizes 2, 3, and 4, respectively. Suppose that ${\mathrm{s}}_1$ cannot be selected for the team X, and ${\mathrm{s}}_2$ cannot be selecte …

Q12.Let $\overrightarrow{\mathrm{OP}}=\frac{\alpha -1}{\alpha }\hat{\mathrm{i}}+\hat{\mathrm{j}}+\hat{\mathrm{k}},\overrightarrow{\mathrm{OQ}}=\hat{\mathrm{i}}+\frac{\beta -1}{\beta }\hat{\mathrm{j}}+\hat{\mathrm{k}}$ and …

Q13.Let X be a random variable, and let $\mathrm{P}(\mathrm{X}=\mathrm{x})$ denote the probability that X takes the value x. Suppose that the points ( $\mathrm{x},\mathrm{P}(\mathrm{X}=\mathrm{x})$ ), $\mathrm{x}=0,1,2,3,4$, lie on a fixed straight line in the xy-plane, and …

Q14.Let $\alpha$ and $\beta$ be the distinct roots of the equation ${\mathrm{x}}^2+\mathrm{x}-1=0$. Consider the set $\mathrm{T}=\{1,\alpha ,\beta \}$. For a $3\times 3$ matrix $\mathrm{M}={\left({\mathrm{a}}_{\mathrm{ij}}\right)}_{3\times 3}$, define …

Q15.Let the straight line $\mathrm{y}=2\mathrm{x}$ touch a circle with center $(0,\alpha ),\alpha >0$, and radius r at a point ${\mathrm{A}}_1$. Let ${\mathrm{B}}_1$ be the point on the circle such that the line segment ${\mathrm{A}}_1{\mathrm{B}}_1$ is a diameter of the …

Q16.Let $\gamma \in \mathbb{R}$ be such that the lines ${\mathrm{L}}_1:\frac{\mathrm{x}+11}{1}=\frac{\mathrm{y}+21}{2}=\frac{\mathrm{z}+29}{3}$ and ${\mathrm{L}}_2:\frac{\mathrm{x}+16}{3}=\frac{\mathrm{y}+11}{2}=\frac{\mathrm{z}+4}{\gamma }$ intersect. Let ${\mathrm{R}}_1$ b …

Q17.Let $f:\mathbb{R}\to \mathbb{R}$ and $\mathrm{g}:\mathbb{R}\to \mathbb{R}$ be functions defined by …