JEE Advanced 2024 Paper 1 Question Paper

Question 1

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

Accepted Answer

(B) $\frac{9}{11x} + \frac{13}{11} x^{10}$

Answer

(A) $\frac{31}{11x} - \frac{9}{11} x^{10}$

Answer

(C) $\frac{-9}{11x} + \frac{31}{11} x^{10}$

Answer

(D) $\frac{13}{11x} + \frac{9}{11} x^{10}$

Question 2

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 answer. Assume that the probability of the student giving the correct answer for a question, given that he has guess it, is $\frac{1}{2}$. Also assume that the probability of the answer for a question being guessed, given that the student's answer is correct, is $\frac{1}{6}$. Then the probability that the student knows the answer of a randomly chosen question is

Accepted Answer

(C) $\frac{5}{7}$

Answer

(A) $\frac{1}{12}$

Answer

(B) $\frac{1}{7}$

Answer

(D) $\frac{5}{12}$

Question 3

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

Accepted Answer

(B) $\frac{\sqrt{11} + 1}{2\sqrt{3}}$

Answer

(A) $\frac{\sqrt{11} - 1}{2\sqrt{3}}$

Answer

(C) $\frac{\sqrt{11} + 1}{3\sqrt{2}}$

Answer

(D) $\frac{\sqrt{11} - 1}{3\sqrt{2}}$

Question 4

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 centre of the ellipse. If the area of the triangle $	riangle ORT$ is $\frac{3}{2}$, then which of the following options is correct ?

Accepted Answer

(A) $q = 2, p = 3\sqrt{3}$

Answer

(B) $q = 2, p = 4\sqrt{3}$

Answer

(C) $q = 1, p = 5\sqrt{3}$

Answer

(D) $q = 1, p = 6\sqrt{3}$

Question 5

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

Accepted Answer

(A) $Z \cup T_{1} \cup T_{2} \subset S$

Accepted Answer

(C) $T_{2} \cap (2024, \infty) 
eq \phi$

Accepted Answer

(D) For any given $a, b \in Z$, $\cos\left(\pi\left(a + b\sqrt{2}\right)\right) + i \sin\left(\pi\left(a + b\sqrt{2}\right)\right) \in Z$ if and only if $b = 0$, where $i = \sqrt{-1}$

Answer

(B) $T_{1} \cap \left(0, \frac{1}{2024}\right) = \phi$, where $\phi$ denotes the empty set

Question 6

Let $R^{2}$ denote $R 	imes R$.
Let $S = \left\{(a, b, c) : a, b, c \in R 	ext{ and } ax^{2} + 2bxy + cy^{2} > 0 	ext{ for all } (x, y) \in R^{2} - \{(0,0)\}ight\}$
Then which of the following statements is (are) TRUE?

Accepted Answer

(B) If $\left(3, b, \frac{1}{12}\right) \in S$, then $|2b| < 1$

Accepted Answer

(C) For any given $(a, b, c) \in S$, then the system of linear equations
$ax + by = 1$
$bx + cy = -1$
has a unique solution.

Accepted Answer

(D) For any given $(a, b, c) \in S$, then the system of linear equations
$(a + 1)x + by = 0$
$bx + (c + 1)y = 0$
has a unique solution.

Answer

(A) $\left(2, \frac{7}{2}, 6\right) \in S$

Question 7

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

Accepted Answer

(A) There is a triangle whose area is 1 and all of whose vertices are from $S$.

Accepted Answer

(B) There are two distinct points $L$ and $M$ in $T$ such that each point on the line segments $LM$ is also in $T$.

Accepted Answer

(C) There are infinitely many rectangles of perimeter 48, two of whose vertices are from $S$ and the other two vertices are from $T$.

Accepted Answer

(D) There is a square of perimeter 48, two of whose vertices are from $S$ and the other two vertices are from $T$.

Question 8

Let $a = 3\sqrt{2}$ and $b = \frac{1}{5^{1/6}\sqrt{6}}$. If $x, y \in R$ are such that
$3x + 2y = \log_{a}(18)^{5/4}$ and $2x - y = \log_{b}\left(\sqrt{1080}\right)$,
then $4x + 5y$ is equal to ______ .

Accepted Answer

8

Question 9

Let $f(x) = x^{4} + ax^{3} + bx^{2} + c$ be a polynomial with real coefficients such that $f(1) = -9$. Suppose that $i\sqrt{3}$ is a root of the equation $4x^{3} + 3ax^{2} + 2bx = 0$, where $i = \sqrt{-1}$. If $\alpha_{1}, \alpha_{2}, \alpha_{3},$ and $\alpha_{4}$ are all the roots of the equation $f(x) = 0$, then $\left|\alpha_{1}\right|^{2} + \left|\alpha_{2}\right|^{2} + \left|\alpha_{3}\right|^{2} + \left|\alpha_{4}\right|^{2}$ is equal to ______ .

Accepted Answer

20

Question 10

Let $S = \left\{A\begin{pmatrix} 0 & 1 & c \ 1 & a & d \ 1 & b & e \end{pmatrix} : a, b, c, d, e \in \{0, 1\} 	ext{ and } |A| \in \{-1, 1\}ight\}$, where $|A|$ denotes the determinant of $A$. Then the number of elements in $S$ is ______ .

Accepted Answer

16

Question 11

A group of 9 students $s_{1}, s_{2}, \ldots\ldots s_{9}$ 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 team $Y$. Then the number of ways to form such teams, is ______ .

Accepted Answer

665

Question 12

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

Accepted Answer

5

Question 13

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

Accepted Answer

42

Question 14

Let $\alpha$ and $\beta$ be the distinct roots of the equation $x^{2} + x - 1 = 0$. Consider the set $T = \{1, \alpha, \beta\}$. For a $3 	imes 3$ matrix $M = (a_{ij})_{3	imes3}$, define $R_{i} = a_{i1} + a_{i2} + a_{i3}$ and $C_{j} = a_{1j} + a_{2j} + a_{3j}$ for $i = 1, 2, 3$ and $j = 1, 2, 3$.
Match each entry in List-I to the correct entries in List-II.
The correct option is:

Accepted Answer

(C) (P) $	o$ (2)   (Q) $	o$ (4)   (R) $	o$ (3)   (S) $	o$ (5)

Answer

(A) (P) $	o$ (4)   (Q) $	o$ (2)   (R) $	o$ (5)   (S) $	o$ (1)

Answer

(B) (P) $	o$ (2)   (Q) $	o$ (4)   (R) $	o$ (1)   (S) $	o$ (5)

Answer

(D) (P) $	o$ (1)   (Q) $	o$ (5)   (R) $	o$ (3)   (S) $	o$ (4)

Question 15

Let the straight line $y = 2x$ touch a circle with centre $(0, \alpha)$, $\alpha > 0$, and radius $r$ at a point $A_{1}$. Let $B_{1}$ be the point on the circle such the line segment $A_{1}B_{1}$ is a diameter of the circle. Let $\alpha + r = 5 + \sqrt{5}$.
Match each entry in List-I to the correct entries in List-II.
The correct option is:

Accepted Answer

(C) (P) $	o$ (4)   (Q) $	o$ (2)   (R) $	o$ (5)   (S) $	o$ (3)

Answer

(A) (P) $	o$ (4)   (Q) $	o$ (2)   (R) $	o$ (1)   (S) $	o$ (3)

Answer

(B) (P) $	o$ (2)   (Q) $	o$ (4)   (R) $	o$ (1)   (S) $	o$ (3)

Answer

(D) (P) $	o$ (2)   (Q) $	o$ (4)   (R) $	o$ (3)   (S) $	o$ (5)

Question 16

Let $\gamma \in R$ be such that the lines $L_{1} : \frac{x + 11}{1} = \frac{y + 21}{2} = \frac{z + 29}{3}$ and $L_{2} : \frac{x + 16}{3} = \frac{y + 11}{2} = \frac{z + 4}{\gamma}$ intersect. Let $R_{1}$ be the point of intersection of $L_{1}$ and $L_{2}$. Let $O = (0, 0, 0)$, and $\hat{n}$ denote a unit normal vector to the plane containing both the lines $L_{1}$ and $L_{2}$.
Match each entry in List-I to the correct entries in List-II.
The correct option is:

Accepted Answer

(C) (P) $	o$ (3)   (Q) $	o$ (4)   (R) $	o$ (1)   (S) $	o$ (5)

Answer

(A) (P) $	o$ (3)   (Q) $	o$ (4)   (R) $	o$ (1)   (S) $	o$ (2)

Answer

(B) (P) $	o$ (5)   (Q) $	o$ (4)   (R) $	o$ (1)   (S) $	o$ (2)

Answer

(D) (P) $	o$ (3)   (Q) $	o$ (1)   (R) $	o$ (4)   (S) $	o$ (5)

Question 17

Let $f : R 	o R$ and $g : R 	o R$ be functions defined by
$f(x) = \begin{cases} x|x|\sin\left(\frac{1}{x}ight), & x 
eq 0, \ 0, & x = 0, \end{cases}$ and $g(x) = \begin{cases} 1 - 2x, & 0 \leq x \leq \frac{1}{2}, \ 0, & 	ext{otherwise.} \end{cases}$
Let $a, b, c, d \in R$. Define the function $h : R 	o R$ by
$h(x) = af(x) + b\left(g(x) + g\left(\frac{1}{2} - xight)ight) + c(x - g(x)) + d\,g(x), x \in R$
Match each entry in List-I to the correct entries in List-II.
The correct option is:

Accepted Answer

(C) (P) $	o$ (5)   (Q) $	o$ (3)   (R) $	o$ (2)   (S) $	o$ (4)

Answer

(A) (P) $	o$ (4)   (Q) $	o$ (3)   (R) $	o$ (1)   (S) $	o$ (2)

Answer

(B) (P) $	o$ (5)   (Q) $	o$ (2)   (R) $	o$ (4)   (S) $	o$ (3)

Answer

(D) (P) $	o$ (4)   (Q) $	o$ (2)   (R) $	o$ (1)   (S) $	o$ (3)

Question 18

A dimensionless quantity is constructed in terms of electronic charge $e$, permittivity of free space $\varepsilon_{0}$, Planck's constant $h$, and speed of light $c$. If the dimensionless quantity is written as $e^{\alpha}\varepsilon_{0}^{\beta}h^{\gamma}c^{\delta}$ and $n$ is a non-zero integer, then $(\alpha, \beta, \gamma, \delta)$ is given by

Accepted Answer

(A) $(2n, -n, -n, -n)$

Answer

(B) $(n, -n, -2n, -n)$

Answer

(C) $(n, -n, -n, -2n)$

Answer

(D) $(2n, -n, -2n, -2n)$

Question 19

An infinitely long wire, located on the z-axis, carries a current $I$ along the +z-direction and produces the magnetic field $\vec{B}$. The magnitude of the line integral $\int \vec{B} \cdot \overrightarrow{dl}$ along a straight line from the point $\left(-\sqrt{3}a, a, 0\right)$ to $(a, a, 0)$ is given by
[$\mu_{0}$ is the magnetic permeability of free space.]

Accepted Answer

(A) $7\mu_{0}I / 24$

Answer

(B) $7\mu_{0}I / 12$

Answer

(C) $\mu_{0}I / 8$

Answer

(D) $\mu_{0}I / 6$

Question 20

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
[$\varepsilon_{0}$ is the permittivity of free space.]

Accepted Answer

(B) $q^{2} / \left(32\pi\varepsilon_{0}R^{3}m\right)$

Answer

(A) $q^{2} / \left(4\pi\varepsilon_{0}R^{3}m\right)$

Answer

(C) $q^{2} / \left(8\pi\varepsilon_{0}R^{3}m\right)$

Answer

(D) $q^{2} / \left(16\pi\varepsilon_{0}R^{3}m\right)$

Question 21

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 $t = (\pi/4)$ s are

Accepted Answer

(C) 0.5 m, $-5$ kg m/s

Answer

(A) $-0.5$ m, 5 kg m/s

Answer

(B) 0.5 m, 0 kg m/s

Answer

(D) $-1$ m, 5 kg m/s

Question 22

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 $V(r) = kr^{2}/2$, 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 $L = n\hbar$, where $\hbar = h/(2\pi)$, $h$ 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?

Accepted Answer

(A) $r^{2} = n\hbar\sqrt{\frac{1}{mk}}$

Accepted Answer

(B) $v^{2} = n\hbar\sqrt{\frac{k}{m^{3}}}$

Accepted Answer

(C) $\frac{L}{mr^{2}} = \sqrt{\frac{k}{m}}$

Answer

(D) $E = \frac{n\hbar}{2}\sqrt{\frac{k}{m}}$

Question 23

Two uniform strings 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 $v_{0} = \frac{1}{2L}\sqrt{\frac{T}{\mu}}$, which of the following statement(s) is(are) correct?

Accepted Answer

(A) With a node at $O$, the minimum frequency of vibration of the composite string is $v_{0}$.

Accepted Answer

(C) When the composite string vibrates at the minimum frequency with a node at $O$, it has 6 nodes, including the end nodes.

Accepted Answer

(D) No vibrational mode with an antinode at $O$ is possible for the composite string.

Answer

(B) With an antinode at $O$, the minimum frequency of vibration of the composite string is $2v_{0}$.

Question 24

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 level QPR. If the image of a point object $O$ at a height of $h$ (OT in the figure) is formed onto itself, then, which of the following option(s) is(are) correct?

Accepted Answer

(A) For $n = 1.42$, $h = 50$ cm.

Accepted Answer

(B) For $n = 1.35$, $h = 36$ cm.

Answer

(C) For $n = 1.45$, $h = 65$ cm.

Answer

(D) For $n = 1.48$, $h = 85$ cm.

Question 25

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 from $-73^{\circ}$C to $27^{\circ}$C is $nk$, the value of $n$ is ______.
[Given: 0 K $= -273^{\circ}$C]

Accepted Answer

25000

Question 26

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 motor's thin shaft. Initially, both the discs are at rest. The motor is switched on so that the smaller disc rotates at a uniform angular speed $\omega$. If the angular speed at which the large disc rotates is $\omega/n$, then the value of $n$ is ______.

Accepted Answer

12

Question 27

A point source $S$ emits unpolarized light uniformly in all directions. At two points $A$ and $B$, the ratio $r = I_{A}/I_{B}$ of the intensities of light is 2. If a set of two polaroids having $45^{\circ}$ angle between their pass-axes is placed just before point $B$, then the new value of $r$ will be ______.

Accepted Answer

8

Question 28

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 the source move away from each other at the same speed $v$ with respect to the ground (as shown in Case 2 in the figure), the observer measures the frequency of sound to be $n$ Hz. The value of $n$ is ______.

Accepted Answer

200

Question 29

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 that of the hole is connected to tank 2, and the pipe ends at the ground level. When the water flows from the tanks 1 and 2 through the holes, the times taken to empty the tanks are $t_{1}$ and $t_{2}$, respectively. If $H = \left(\frac{16}{9}\right)h$, then the ratio $t_{1}/t_{2}$ is ______.

Accepted Answer

3

Question 30

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 imparted to the rod at a distance $x = L/n$ from the mid-point of the rod (see figure), then the rod and string revolve together around the point $O$, with the rod remaining aligned with the string. In such a case, the value of $n$ is ______.

Accepted Answer

18

Question 31

One mole of a monatomic ideal gas undergoes the cyclic process $J 	o K 	o L 	o M 	o J$, as shown in the P-T diagram.
Match the quantities mentioned in List-I with their values in List-II and choose the correct option.
[R is the gas constant.]

Accepted Answer

(B) P $	o$ 4;  Q $	o$ 3;  R $	o$ 5;  S $	o$ 2

Answer

(A) P $	o$ 1;  Q $	o$ 3;  R $	o$ 5;  S $	o$ 4

Answer

(C) P $	o$ 4;  Q $	o$ 1;  R $	o$ 2;  S $	o$ 2

Answer

(D) P $	o$ 2;  Q $	o$ 5;  R $	o$ 3;  S $	o$ 4

Question 32

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 $C_{0} = \varepsilon_{0}a^{2}/d$, 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.

Accepted Answer

(C) P $	o$ 3;  Q $	o$ 2;  R $	o$ 4;  S $	o$ 1

Answer

(A) P $	o$ 3;  Q $	o$ 2;  R $	o$ 4;  S $	o$ 5

Answer

(B) P $	o$ 2;  Q $	o$ 3;  R $	o$ 2;  S $	o$ 1

Answer

(D) P $	o$ 3;  Q $	o$ 2;  R $	o$ 2;  S $	o$ 5

Question 33

A light ray is incident on the surface of a sphere of refractive index $n$ at an angle of incidence $	heta_{0}$. The ray partially refracts into the sphere with angle of refraction $\phi_{0}$ and then partly reflects from the back surface. The reflected ray then emerges out of the sphere after a partial refraction. The total angle of deviation of the emergent ray with respect to the incident ray is $\alpha$. Match the quantities mentioned in List-I with their values in List-II and choose the correct option.

Accepted Answer

(A) P $	o$ 5;  Q $	o$ 2;  R $	o$ 1;  S $	o$ 4

Answer

(B) P $	o$ 5;  Q $	o$ 1;  R $	o$ 2;  S $	o$ 4

Answer

(C) P $	o$ 3;  Q $	o$ 2;  R $	o$ 1;  S $	o$ 4

Answer

(D) P $	o$ 3;  Q $	o$ 1;  R $	o$ 2;  S $	o$ 5

Question 34

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 $K_{1}$ and $K_{2}$. Initially, both the keys are open and there is no charge on the capacitor. At an instant, key $K_{1}$ is closed and immediately after this the current in $R_{0}$ is found to be $I_{1}$. After a long time, the current attains a steady state value $I_{2}$. Thereafter, $K_{2}$ is closed and simultaneously $K_{1}$ is opened and the voltage across $C_{0}$ oscillates with amplitude $V_{0}$ and angular frequency $\omega_{0}$.
Match the quantities mentioned in List-I with their values in List-II and choose the correct option.

Accepted Answer

(A) P $	o$ 1; Q $	o$ 3; R $	o$ 2; S $	o$ 5

Answer

(B) P $	o$ 1; Q $	o$ 2; R $	o$ 3; S $	o$ 5

Answer

(C) P $	o$ 1; Q $	o$ 3; R $	o$ 2  S $	o$ 4

Answer

(D) P $	o$ 2  Q $	o$ 5  R $	o$ 3  S $	o$ 4

Question 35

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

Accepted Answer

(D) $2 : 1$

Answer

(A) $2\sqrt{2} : \sqrt{3}$

Answer

(B) $2\sqrt{2} : 1$

Answer

(C) $1 : 2$

Question 36

At room temperature, disproportionation of an aqueous solution of $	extit{in situ}$ generated nitrous acid (HNO$_{2}$) gives the species

Accepted Answer

(A) H$_{3}$O$^{+}$, NO$_{3}^{-}$ and NO

Answer

(B) H$_{3}$O$^{+}$, NO$_{3}^{-}$ and NO$_{2}$

Answer

(C) H$_{3}$O$^{+}$, NO$^{-}$ and NO$_{2}$

Answer

(D) H$_{3}$O$^{+}$, NO$_{3}^{-}$ and N$_{2}$O

Question 37

Aspartame, an artificial sweetener, is a dipeptide aspartyl phenylalanine methyl ester. The structure of aspartame is
Structure of phenylalanine and aspartic acid are given below.

Accepted Answer

(B) (B)

Answer

(A) (A)

Answer

(C) (C)

Answer

(D) (D)

Question 38

Among the following options, select the option in which each complex in $\mathbf{Set	ext{-}I}$ shows geometrical isomerism and the two complexes in $\mathbf{Set	ext{-}II}$ are ionization isomers of each other.
[en = H$_{2}$NCH$_{2}$CH$_{2}$NH$_{2}$]

Accepted Answer

(C) $\mathbf{Set - I}$: [Co(NH$_{3}$)$_{3}$(NO$_{2}$)$_{3}$] and [Co(en)$_{2}$Cl$_{2}$]
$\mathbf{Set - II}$: [Co(NH$_{3}$)$_{5}$Cl]SO$_{4}$ and [Co(NH$_{3}$)$_{5}$(SO$_{4}$)]Cl

Answer

(A) $\mathbf{Set - I}$: [Ni(CO)$_{4}$] and [PdCl$_{2}$(PPh$_{3}$)$_{2}$]
$\mathbf{Set - II}$: [Co(NH$_{3}$)$_{5}$Cl]SO$_{4}$ and [Co(NH$_{3}$)$_{5}$(SO$_{4}$)]Cl

Answer

(B) $\mathbf{Set - I}$: [Co(en)(NH$_{3}$)$_{2}$Cl$_{2}$] and [PdCl$_{2}$(PPh$_{3}$)$_{2}$]
$\mathbf{Set - II}$: [Co(NH$_{3}$)$_{6}$][Cr(CN)$_{6}$] and [Cr(NH$_{3}$)$_{6}$[Co(CN)$_{6}$]

Answer

(D) $\mathbf{Set - I}$: [Cr(NH$_{3}$)$_{5}$Cl]Cl$_{2}$ and [Co(en)(NH$_{3}$)$_{2}$Cl$_{2}$]
$\mathbf{Set - II}$: [Cr(H$_{2}$O)$_{6}$]Cl$_{3}$ and [Cr(H$_{2}$O)$_{5}$Cl]Cl$_{2}$.H$_{2}$O

Question 39

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

Accepted Answer

(A) Uncertainty principle rules out the existence of definite paths for electrons.

Accepted Answer

(B) The energy of an electrons in 2s orbital of an atom is lower than the energy of an electron that is infinitely far away from the nucleus.

Accepted Answer

(C) According to Bohr's model, the most negative energy value for an electron is given by n = 1, which corresponds to the most stable orbit.

Answer

(D) According to Bohr's model, the magnitude of velocity of electrons increases with increase in values of n.

Question 40

Reaction of $	extit{iso}$-propylbenzene with O$_{2}$ followed by the treatment with H$_{3}$O$^{+}$ forms phenol and a by product $\mathbf{P}$. Reaction of $\mathbf{P}$ with 3 equivalents of Cl$_{2}$ gives compound $\mathbf{Q}$. Treatement of $\mathbf{Q}$ with Ca(OH)$_{2}$ produces compound $\mathbf{R}$ and calcium salt $\mathbf{S}$.
The correct statement(s) regarding $\mathbf{P, Q, R}$ and $\mathbf{S}$ is(are)

Accepted Answer

(A) Reaction of $\mathbf{P}$ with $\mathbf{R}$ in the presence of KOH followed by acidification gives

Accepted Answer

(B) Reaction of $\mathbf{R}$ with O$_{2}$ in the presence of light gives phosgene gas

Accepted Answer

(D) $\mathbf{S}$ on heating gives $\mathbf{P}$

Answer

(C) $\mathbf{Q}$ reacts with aqueous NaOH to produce Cl$_{3}$CCH$_{2}$OH and Cl$_{3}$CCOONa

Question 41

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

Accepted Answer

(A) CO$_{2}$, C$_{2}$H$_{4}$, NO and HCl

Accepted Answer

(D) CO$_{2}$, BCl$_{3}$, O$_{3}$ and C$_{2}$H$_{4}$

Answer

(B) NO$_{2}$, O$_{3}$, HCl and H$_{2}$SO$_{4}$

Answer

(C) BCl$_{3}$, NO, NO$_{2}$ and H$_{2}$SO$_{4}$

Question 42

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

Accepted Answer

8120

Question 43

Consider the following reaction,
$2\mathrm{H}_{2}(g) + 2\mathrm{NO}(g) \longrightarrow \mathrm{N}_{2}(g) + 2\mathrm{H}_{2}\mathrm{O}(g)$
Which follows the mechanism given below?
$2\mathrm{NO}(g) \underset{\mathrm{K}_{-1}}{\overset{\mathrm{K}_{1}}{\rightleftharpoons}} \mathrm{N}_{2}\mathrm{O}_{2}(g)$ (fast equilibrium)
$\mathrm{N}_{2}\mathrm{O}_{2}(g) + \mathrm{H}_{2}(g) \xrightarrow{\mathrm{k}_{2}} \mathrm{N}_{2}\mathrm{O}(g) + \mathrm{H}_{2}\mathrm{O}(g)$ (slow reaction)
$\mathrm{N}_{2}\mathrm{O}(g) + \mathrm{H}_{2}(g) \xrightarrow{\mathrm{k}_{3}} \mathrm{N}_{2}(g) + \mathrm{H}_{2}\mathrm{O}(g)$ (fast reaction)
The order of the reaction is _________.

Accepted Answer

3

Question 44

Complete reaction of acetaldehyde with excess formaldehyde, upon heating with conc. NaOH solution, gives $\mathbf{P}$ and $\mathbf{Q}$. Compound $\mathbf{P}$ does not give Tollen's test, whereas $\mathbf{Q}$ on acidification gives positive Tollen's test. Treatment of $\mathbf{P}$ with excess cyclohexanone in the presence of catalytic amount of p-tolunesulfonic acid (PTSA) gives product $\mathbf{R}$.
Sum of the number of methylene groups ($-$CH$_{2}-$) and oxygen atoms in $\mathbf{R}$ is __________.

Accepted Answer

18

Question 45

Among V(CO)$_{6}$, Cr(CO)$_{5}$, Mn(CO)$_{5}$, Fe(CO)$_{5}$, [Co(CO)$_{3}$]$^{3-}$, [Cr(CO)$_{4}$]$^{4-}$ and Ir(CO)$_{3}$, the total number of species isoelectronic with Ni(CO)$_{4}$ is ________________.
[Given, atomic number: V = 23, Cr = 24, Mn = 25, Fe = 26, Co = 27, Ni = 28, Cu = 29, Ir = 77]

Accepted Answer

1

Question 46

In the following reaction sequence, major product $\mathbf{P}$ is formed.
Glycerol reacts completely with excess $\mathbf{P}$ in the presence of an acid catalyst to form $\mathbf{Q}$. Reaction of $\mathbf{Q}$ with excess NaOH followed by the treatment with CaCl$_{2}$ yields Ca – soap $\mathbf{R}$, quantitatively. Starting with one mole of $\mathbf{Q}$, the amount of $\mathbf{R}$ produced in gram is ___________.
[Given, atomic weight: H = 1, C = 12, N = 14, O = 16, Na = 23, Cl = 35, Ca = 40]

Accepted Answer

909

Question 47

Among the following complexes, the total number of diamagnetic species is ________.
[Mn(NH$_{3}$)$_{6}$]$^{3+}$, [MnCl$_{6}$]$^{3-}$, [FeF$_{6}$]$^{3-}$, [CoF$_{6}$]$^{3-}$, [Fe(NH$_{3}$)$_{6}$]$^{3+}$ and [Co(en)$_{3}$]$^{3+}$
[Given, atomic number: Mn = 25, Fe = 26, Co = 27; en = H$_{2}$NCH$_{2}$CH$_{2}$NH$_{2}$]

Accepted Answer

1

Question 48

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 $\left(\Lambda_{0}\right)$ values (in mS m$^{2}$ mol$^{-1}$) for different ions in aqueous solutions are given below:
Ag$^{+}$ 6.2; K$^{+}$ 7.4; Na$^{+}$ 5.0; H$^{+}$ 35.0; NO$_{3}^{-}$ 7.2; Cl$^{-}$ 7.6; SO$_{4}^{2-}$ 16.0; OH$^{-}$ 19.9; CH$_{3}$COO$^{-}$ 4.1
For different combinations of titrates and titrants given in List-I, the graphs of 'conductance' versus 'volume of titrant' are given in List-II.
Match each entry List-I with the appropriate entry in List-II and choose the correct option.

Accepted Answer

(C) (P) $	o$ (3), (Q) $	o$ (4), (R) $	o$ (2), (S) $	o$ (5)

Answer

(A) (P) $	o$ (4), (Q) $	o$ (3), (R) $	o$ (2), (S) $	o$ (5)

Answer

(B) (P) $	o$ (2), (Q) $	o$ (4), (R) $	o$ (3), (S) $	o$ (1)

Answer

(D) (P) $	o$ (4), (Q) $	o$ (3), (R) $	o$ (2), (S) $	o$ (1)

Question 49

Based on $\mathbf{VSEPR}$ 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.

Accepted Answer

(B) (P) $	o$ (5), (Q) $	o$ (3), (R) $	o$ (2), (S) $	o$ (4)

Answer

(A) (P) $	o$ (5), (Q) $	o$ (2), (R) $	o$ (3), (S) $	o$ (1)

Answer

(C) (P) $	o$ (4), (Q) $	o$ (3), (R) $	o$ (2), (S) $	o$ (1)

Answer

(D) (P) $	o$ (4), (Q) $	o$ (2), (R) $	o$ (5), (S) $	o$ (3)

Question 50

List-I contains various reaction sequence 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.

Accepted Answer

(A) (P) $	o$ (3), (Q) $	o$ (5), (R) $	o$ (4), (S) $	o$ (1)

Answer

(B) (P) $	o$ (3), (Q) $	o$ (2), (R) $	o$ (4), (S) $	o$ (1)

Answer

(C) (P) $	o$ (3), (Q) $	o$ (5), (R) $	o$ (1), (S) $	o$ (4)

Answer

(D) (P) $	o$ (5), (Q) $	o$ (2), (R) $	o$ (4), (S) $	o$ (1)

Question 51

List-I contains various reaction sequence 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.

Accepted Answer

(C) (P) $	o$ (3), (Q) $	o$ (5), (R) $	o$ (4), (S) $	o$ (1)

Answer

(A) (P) $	o$ (2), (Q) $	o$ (3), (R) $	o$ (4), (S) $	o$ (5)

Answer

(B) (P) $	o$ (2), (Q) $	o$ (3), (R) $	o$ (5), (S) $	o$ (1)

Answer

(D) (P) $	o$ (3), (Q) $	o$ (2), (R) $	o$ (5), (S) $	o$ (4)

List-I	List-II
P.Work done in the complete cyclic process	1. $RT_{0} - 4RT_{0}\ln 2$
Q.Change in the internal energy of the gas in the process JK	2.0
R.Heat given to the gas in the process KL	3. $3RT_{0}$
S.Change in the internal energy of the gas in the process MJ	4. $-2RT_{0}\ln 2$
	5. $-3RT_{0}\ln 2$

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. $C_{0}/2$
R.The capacitance between $S_{1}$ and $S_{3}$ , with $S_{2}$ shorted to $S_{4}$ , is	3. $C_{0}/3$
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. $2C_{0}/3$
	5. $2C_{0}$

List-I	List-II
P.If $n = 2$ and $\alpha = 180^{\circ}$ , then all the possible values of $\theta_{0}$ will be	1. $30^{\circ}$ and $0^{\circ}$
Q.If $n = \sqrt{3}$ and $\alpha = 180^{\circ}$ , then all the possible values of $\theta_{0}$ will be	2. $60^{\circ}$ and $0^{\circ}$
R.If $n = \sqrt{3}$ and $\alpha = 180^{\circ}$ , then all the possible values of $\phi_{0}$ will be	3. $45^{\circ}$ and $0^{\circ}$
S.If $n = \sqrt{2}$ and $\theta_{0} = 45^{\circ}$ , then all the possible values of $\alpha$ will be	4. $150^{\circ}$
	5. $0^{\circ}$

List-I	List-II
P.The value of $I_{1}$ in Ampere is	1.0
Q.The value of $I_{2}$ in Ampere is	2.2
R.The value of $\omega_{0}$ in kilo-radians/s	3.4
S.The value of $V_{0}$ in Volt is	4.20
	5.200

List-I	List-II
P.Titrate: KCl Titrant: AgNO $_{3}$	1.see figure
Q.Titrate: AgNO $_{3}$ Titrant: KCl	2.see figure
R.Titrate: NaOH Titrant: HCl	3.see figure
S.Titrate: NaOH Titrant: CH $_{3}$ COOH	4.see figure
	5.see figure

JEE Advanced 2024 Paper 1 Question Paper with Answers

Physics — JEE Advanced 2024 Paper 1

Chemistry — JEE Advanced 2024 Paper 1

Mathematics — JEE Advanced 2024 Paper 1

Chapters tested in this paper

Attempt JEE Advanced 2024 Paper 1 under exam timing.

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

List-I	List-II
P.see figure	1.see figure
Q.see figure	2.see figure
R.see figure	3.see figure
S.see figure	4.see figure
	5.see figure

List-I	List-II
P.The number of matrices $M = (a_{ij})_{3\times3}$ with all entries in $T$ such that $R_{i} = C_{j} = 0$ for all $i, j$ , is	1.1
Q.The number of symmetric matrices $M = (a_{ij})_{3\times3}$ with all entries in $T$ such that $C_{j} = 0$ for all $j$ , is	2.12
R.Let $M = (a_{ij})_{3\times3}$ be a skew symmetric matrix such that $a_{ij} \in T$ for $i > j$ . Then the number of elements in the set $\left\{\begin{pmatrix} x \\ y \\ z \end{pmatrix} : x, y, z \in R, M\begin{pmatrix} x \\ y \\ z \end{pmatrix} = \begin{pmatrix} a_{12} \\ 0 \\ -a_{23} \end{pmatrix}\right\}$ is	3.infinite
S.Let $M = (a_{ij})_{3\times3}$ be a matrix with all entries in $T$ such that $R_{i} = 0$ for all $i$ . Then the absolute value of determinant of $M$ is	4.6
	5.0

List-I	List-II
P. $\alpha$ equals	1. $(-2, 4)$
Q. $r$ equals	2. $\sqrt{5}$
R. $A_{1}$ equals	3. $(-2, 6)$
S. $B_{1}$ equals	4.5
	5. $(2, 4)$

List-I	List-II
P. $\gamma$ equals	1. $-\hat{i} - \hat{j} + \hat{k}$
Q.A possible choice for $\hat{n}$ is	2. $\sqrt{\frac{3}{2}}$
R. $\overrightarrow{OR_{1}}$ equals	3.1
S.A possible value of $\overrightarrow{OR_{1}} \cdot \hat{n}$ is	4. $\frac{1}{\sqrt{6}}\hat{i} - \frac{2}{\sqrt{6}}\hat{j} + \frac{1}{\sqrt{6}}\hat{k}$
	5. $\sqrt{\frac{2}{3}}$

List-I	List-II
P.If $a = 0$ , $b = 1$ , $c = 0$ and $d = 0$ , then	1. $h$ is one-one.
Q.If $a = 1$ , $b = 0$ , $c = 0$ and $d = 0$ , then	2. $h$ is onto.
R.If $a = 0$ , $b = 0$ , $c = 1$ and $d = 0$ , then	3. $h$ is differentiable on $R$ .
S.If $a = 0$ , $b = 0$ , $c = 0$ and $d = 1$ , then	4.the range of $h$ is $[0, 1]$ .
	5.the range of $h$ is $\{0, 1\}$ .