JEE Main 2024 Apr 4 Shift 2 — Question Paper

Let a relation R on $\mathbb{N} \times \mathbb{N}$ be defined as: $(x_1, y_1)$ R $(x_2, y_2)$ if and only if $x_1 \le x_2$ or $y_1 \le y_2$. Consider the two statements: (I) R is reflexive but not symmetric. (II) R is transitive Then which one of the following is true?

The area (in sq. units) of the region $S = \{z \in \mathbb{C} : |z-1| \le 2; (z+\bar{z}) + i(z-\bar{z}) \le 2, \operatorname{Im}(z) \ge 0\}$ is

Let $A = \begin{bmatrix} 1 & 2 \\ 0 & 1 \end{bmatrix}$ and $B = I + \operatorname{adj}(A) + (\operatorname{adj} A)^2 + \ldots + (\operatorname{adj} A)^{10}$. Then, the sum of all the elements of the matrix $B$ is:

If the coefficients of $x^4$, $x^5$ and $x^6$ in the expansion of $(1+x)^n$ are in the arithmetic progression, then the maximum value of $n$ is:

The value of $\frac{1 \times 2^2 + 2 \times 3^2 + \ldots + 100 \times (101)^2}{1^2 \times 2 + 2^2 \times 3 + \ldots + 100^2 \times 101}$ is

Let three real numbers $a, b, c$ be in arithmetic progression and $a+1, b, c+3$ be in geometric progression. If $a > 10$ and the arithmetic mean of $a$, $b$ and $c$ is $8$, then the cube of the geometric mean of $a$, $b$ and $c$ is

Let $f(x) = 3\sqrt{x-2} + \sqrt{4-x}$ be a real valued function. If $\alpha$ and $\beta$ are respectively the minimum and the maximum values of $f$, then $\alpha^2 + 2\beta^2$ is equal to

If the function $$f(x) = \begin{cases} \dfrac{72^x - 9^x - 8^x + 1}{\sqrt{2} - \sqrt{1 + \cos x}} & , \ x \ne 0 \\ a \log_e 2 \log_e 3 & , \ x = 0 \end{cases}$$ is continuous at $x = 0$, then the value of $a^2$ is equal to

Let $f(x) = \int_0^x \left(t + \sin\left(1 - e^t\right)\right) dt, x \in \mathbb{R}$. Then, $\lim\limits_{x \to 0} \dfrac{f(x)}{x^3}$ is equal to

If the value of the integral $\int_{-1}^{1} \dfrac{\cos \alpha x}{1 + 3^x} dx$ is $\dfrac{2}{\pi}$. Then, a value of $\alpha$ is

The area (in sq. units) of the region described by $\{(x, y) : y^2 \le 2x, \text{ and } y \ge 4x - 1\}$ is

Let $y = y(x)$ be the solution of the differential equation $(x^2 + 4)^2 dy + (2x^3y + 8xy - 2)dx = 0$. If $y(0) = 0$, then $y(2)$ is equal to

Let C be a circle with radius $\sqrt{10}$ units and centre at the origin. Let the line $x + y = 2$ intersects the circle C at the points P and Q. Let MN be a chord of C of length 2 unit and slope $-1$. Then, a distance (in units) between the chord PQ and the chord MN is

Consider a hyperbola H having centre at the origin and foci on the x-axis. Let $C_1$ be the circle touching the hyperbola H and having the centre at the origin. Let $C_2$ be the circle touching the hyperbola H at its vertex and having the centre at one of its foci. If areas (in sq units) of $C_1$ and $C_2$ are $36\pi$ and $4\pi$, respectively, then the length (in units) of latus rectum of H is

Let PQ be a chord of the parabola $y^2 = 12x$ and the midpoint of PQ be at $(4, 1)$. Then, which of the following point lies on the line passing through the points P and Q?

Let P be the point of intersection of the lines $\dfrac{x-2}{1} = \dfrac{y-4}{5} = \dfrac{z-2}{1}$ and $\dfrac{x-3}{2} = \dfrac{y-2}{3} = \dfrac{z-3}{2}$. Then, the shortest distance of P from the line $4x = 2y = z$ is

Let $\vec{a} = \hat{i} + \hat{j} + \hat{k}$, $\vec{b} = 2\hat{i} + 4\hat{j} - 5\hat{k}$ and $\vec{c} = x\hat{i} + 2\hat{j} + 3\hat{k}, x \in \mathbb{R}$. If $\vec{d}$ is the unit vector in the direction of $\vec{b} + \vec{c}$ such that $\vec{a} \cdot \vec{d} = 1$, then $(\vec{a} \times \vec{b}) \cdot \vec{c}$ is equal to

For $\lambda > 0$, let $\theta$ be the angle between the vectors $\vec{a} = \hat{i} + \lambda\hat{j} - 3\hat{k}$ and $\vec{b} = 3\hat{i} - \hat{j} + 2\hat{k}$. If the vectors $\vec{a} + \vec{b}$ and $\vec{a} - \vec{b}$ are mutually perpendicular, then the value of $(14 \cos\theta)^2$ is equal to

If the mean of the following probability distribution of a radam variable X:

X	0	2	4	6	8
P(X)	$a$	$2a$	$a+b$	$2b$	$3b$

is $\dfrac{46}{9}$, then the variance of the distribution is

Given that the inverse trigonometric function assumes principal values only. Let $x$, $y$ be any two real numbers in $[-1, 1]$ such that $\cos^{-1} x - \sin^{-1} y = \alpha$, $\dfrac{-\pi}{2} \le \alpha \le \pi$. Then, the minimum value of $x^2 + y^2 + 2xy \sin\alpha$ is

Consider the function $f : \mathbb{R} \to \mathbb{R}$ defined by $f(x) = \dfrac{2x}{\sqrt{1 + 9x^2}}$. If the composition of $f$, $\underbrace{(f \circ f \circ f \circ \cdots \circ f)}_{10 \text{ times}}(x) = \dfrac{2^{10} x}{\sqrt{1 + 9\alpha x^2}}$, then the value of $\sqrt{3\alpha + 1}$ is equal to ______

Let $S = \{\sin^2 2\theta : (\sin^4\theta + \cos^4\theta)x^2 + (\sin 2\theta)x + (\sin^6\theta + \cos^6\theta) = 0$ has real roots$\}$. If $\alpha$ and $\beta$ be the smallest and largest elements of the set S, respectively, then $3((\alpha - 2)^2 + (\beta - 1)^2)$ equals _________

Let $A$ be a $2 \times 2$ symmetric matrix such that $A \begin{bmatrix} 1 \\ 1 \end{bmatrix} = \begin{bmatrix} 3 \\ 7 \end{bmatrix}$ and the determinant of $A$ be 1. If $A^{-1} = \alpha A + \beta I$, where $I$ is an identity matrix of order $2 \times 2$, then $\alpha + \beta$ equals _________

There are 4 men and 5 women in Group A, and 5 men and 4 women in Group B. If 4 persons are selected from each group, then the number of ways of selecting 4 men and 4 women is ___________

Let $f : \mathbb{R} \to \mathbb{R}$ be a thrice differentiable function such that $f(0) = 0, f(1) = 1, f(2) = -1, f(3) = 2$ and $f(4) = -2$. Then, the minimum number of zeros of $(3f'f'' + ff''')(x)$ is ___________

If $\int \operatorname{cosec}^5 x \, dx = \alpha \cot x \operatorname{cosec} x \left(\operatorname{coesc}^2 x + \dfrac{3}{2}\right) + \beta \log_e \left|\tan \dfrac{x}{2}\right| + C$ where $\alpha, \beta \in \mathbb{R}$ and C is the constant of integration, then the value of $8(\alpha + \beta)$ equals ________

Let $y = y(x)$ be the solution of the differential equation $(x + y + 2)^2 dx = dy$, $y(0) = -2$. Let the maximum and minimum values of the function $y = y(x)$ in $\left[0, \dfrac{\pi}{3}\right]$ be $\alpha$ and $\beta$, respectively. If $(3\alpha + \pi)^2 + \beta^2 = \gamma + \delta\sqrt{3}, \gamma, \delta \in \mathbb{Z}$, then $\gamma + \delta$ equals ________

Consider a triangle ABC having the vertices $A(1, 2)$, $B(\alpha, \beta)$ and $C(\gamma, \delta)$ and angles $\angle ABC = \dfrac{\pi}{6}$ and $\angle BAC = \dfrac{2\pi}{3}$. If the points B and C lie on the line $y = x + 4$, then $\alpha^2 + \gamma^2$ is equal to _____.

Consider a line L passing through the points $P(1, 2, 1)$ and $Q(2, 1, -1)$. If the mirror image of the point $A(2, 2, 2)$ in the line L is $(\alpha, \beta, \gamma)$, then $\alpha + \beta + 6\gamma$ is equal to _______.

In a tournament, a team plays 10 matches with probabilities of winning and losing each match as $\dfrac{1}{3}$ and $\dfrac{2}{3}$ respectively. Let $x$ be the number of matches that the team wins, and $y$ be the number of matches that team loses. If the probability $\mathrm{P}(|x - y| \le 2)$ is $p$, then $3^9 p$ equals ___________.

Which of the diode circuit shows correct biasing used for the measurement of dynamic resistance of p-n junction diode :

Applying the principle of homogeneity of dimensions, determine which one is correct,

where $T$ is time period, $G$ is gravitational constant, $M$ is mass, $r$ is radius of orbit.

A cyclist starts from the point $P$ of a circular ground of radius 2 km and travels along its circumference to the point S. The displacement of a cyclist is:

[Figure: a circle with centre O; point P at the top, Q at the right, R at the bottom, S at the left; arrows along the circumference show travel from P through Q and R to S]

A 2 kg brick begins to slide over a surface which is inclined at an angle of $45°$ with respect to horizontal axis. The co-efficient of static friction between their surfaces is:

A 90 kg body placed at 2R distance from surface of earth experiences gravitational pull of :

($\mathrm{R}$ = Radius of earth, $\mathrm{g} = 10~\mathrm{m\,s^{-2}}$)

A body of $m$ kg slides from rest along the curve of vertical circle from point $A$ to $B$ in friction less path. The velocity of the body at $B$ is:

[Figure: a vertical circle with centre marked; point A on the upper right of the circle at $45°$ above the horizontal radius, point B at the bottom of the circle; the vertical diameter is shown dashed]

(given, $R = 14~m$, $g = 10~m/s^2$ and $\sqrt{2} = 1.4$ )

Correct formula for height of a satellite from earths surface is:

Given below are two statements :

Statement I : The contact angle between a solid and a liquid is a property of the material of the solid and liquid as well.

Statement II : The rise of a liquid in a capillary tube does not depend on the inner radius of the tube.

In the light of the above statements, choose the correct answer from the options given below :

A sample of gas at temperature $T$ is adiabatically expanded to double its volume. Adiabatic constant for the gas is $\gamma = 3/2$. The work done by the gas in the process is:

($\mu = 1$ mole)

The translational degrees of freedom ($f_t$) and rotational degrees of freedom ($f_r$) of $CH_4$ molecule are:

A charge $q$ is placed at the center of one of the surface of a cube. The flux linked with the cube is:

An electric bulb rated $50~W$ - $200~V$ is connected across a $100~V$ supply. The power dissipation of the bulb is:

The magnetic moment of a bar magnet is $0.5~\mathrm{Am^2}$. It is suspended in a uniform magnetic field of $8\times 10^{-2}~\mathrm{T}$. The work done in rotating it from its most stable to most unstable position is:

Match List I with List II

	LIST I		LIST II
A.	Purely capacitive circuit	I.	[Phasor diagram: current $I$ along the vertical axis, voltage $V$ along the horizontal axis, with a $90°$ angle between them ($I$ leads $V$ by $90°$)]
B.	Purely inductive circuit	II.	[Phasor diagram: $V$ and $I$ both along the same horizontal direction (in phase)]
C.	LCR series at resonance	III.	[Phasor diagram: $V$ at an angle $\theta$ above the horizontal $I$ axis ($V$ leads $I$ by $\theta$)]
D.	LCR series circuit	IV.	[Phasor diagram: voltage $V$ along the vertical axis, current $I$ along the horizontal axis, with a $90°$ angle between them ($V$ leads $I$ by $90°$)]

Choose the correct answer from the options given below:

Arrange the following in the ascending order of wavelength:

A. Gamma rays ($\lambda_1$) B. x - rays ($\lambda_2$) C. Infrared waves ($\lambda_3$) D. Microwaves ($\lambda_4$)

Choose the most appropriate answer from the options given below

The width of one of the two slits in a Young's double slit experiment is 4 times that of the other slit. The ratio of the maximum of the minimum intensity in the interference pattern is:

Given below are two statements: one is labelled as Assertion A and the other is labelled as Reason R.

Assertion A: Number of photons increases with increase in frequency of light.

Reason R: Maximum kinetic energy of emitted electrons increases with the frequency of incident radiation.

In the light of the above statements, choose the most appropriate answer from the options given below:

According to Bohr's theory, the moment of momentum of an electron revolving in $4^{th}$ orbit of hydrogen atom is:

Identify the logic gate given in the circuit:

[Figure: inputs A and B each pass through a NOT gate (inverter); the two inverted outputs feed a two-input OR-shaped gate whose output carries a bubble (inversion), giving output Y]

In simple harmonic motion, the total mechanical energy of given system is $E$. If mass of oscillating particle $P$ is doubled then the new energy of the system for same amplitude is:

[Figure: a spring of stiffness $k$ hanging from a ceiling with a block of mass $m$ (particle $P$) attached at its lower end]

The disintegration energy $Q$ for the nuclear fission of $^{235}U \rightarrow\ ^{140}Ce +\ ^{94}Zr + n$ is ______ MeV.

Given atomic masses of $^{235}U$: $235.0439\,u$; $^{140}Ce$ : $139.9054\,u$,

$^{94}Zr$ : $93.9063\,u$ ; $n$ : $1.0086\,u$,

Value of $c^2 = 931~MeV/u$.

A light ray is incident on a glass slab of thickness $4\sqrt{3}$ cm and refractive index $\sqrt{2}$. The angle of incidence is equal to the critical angle for the glass slab with air. The lateral displacement of ray after passing through glass slab is ______ cm.

(Given $\sin 15° = 0.25$)

A rod of length 60 $cm$ rotates with a uniform angular velocity 20 rad $s^{-1}$ about its perpendicular bisector, in a uniform magnetic filed $0.5T$. The direction of magnetic field is parallel to the axis of rotation. The potential difference between the two ends of the rod is ______ V.

Two parallel long current carrying wire separated by a distance $2r$ are shown in the figure. The ratio of magnetic field at $A$ to the magnetic field produced at $C$ is $\frac{x}{7}$. The value of $x$ is ______.

[Figure: two vertical parallel wires carrying currents $I$ (left wire) and $2I$ (right wire), separated by distance $2r$ with midpoint $O$; point $A$ lies at distance $r$ to the left of the left wire, and point $C$ lies at distance $r$ to the right of the right wire]

Two wires $A$ and $B$ are made up of the same material and have the same mass. Wire $A$ has radius of $2.0~mm$ and wire $B$ has radius of $4.0~mm$. The resistance of wire $B$ is $2\Omega$. The resistance of wire $A$ is ______ $\Omega$.

The displacement of a particle executing SHM is given by $x = 10\sin\left(wt + \frac{\pi}{3}\right) m$. The time period of motion is 3.14 s. The velocity of the particle at $t = 0$ is ______ $m/s$.

Mercury is filled in a tube of radius 2 $cm$ up to a height of 30 $cm$. The force exerted by mercury on the bottom of the tube is ______ N.

(Given, atmospheric pressure $= 10^5~Nm^{-2}$, density of mercury $= 1.36 \times 10^4~kg\,m^{-3}$, $g = 10~m\,s^{-2}$, $\pi = \frac{22}{7}$)

In a system two particles of masses $m_1 = 3~kg$ and $m_2 = 2~kg$ are placed at certain distance from each other. The particle of mass $m_1$ is moved towards the center of mass of the system through a distance 2 cm. In order to keep the center of mass of the system at the original position, the particle of mass $m_2$ should move towards the center of mass by the distance ______ $cm$.

A bus moving along a straight highway with speed of $72~km/h$ is brought to halt within 4 $s$ after applying the brakes. The distance travelled by the bus during this time (Assume the retardation is uniform) is ______ $m$.

A parallel plate capacitor of capacitance $12.5~pF$ is charged by a battery connected between its plates to potential difference of 12.0 V. The battery is now disconnected and a dielectric slab ($\epsilon_r = 6$) is inserted between the plates. The change in its potential energy after inserting the dielectric slab is ______ $\times 10^{-12}$ J.

Choose the Incorrect Statement about Dalton's Atomic Theory

The correct statement/s about Hydrogen bonding is/are

A. Hydrogen bonding exists when H is covalently bonded to the highly electro negative atom. B. Intermolecular H bonding is present in o-nitro phenol C. Intramolecular H bonding is present in HF. D. The magnitude of H bonding depends on the physical state of the compound. E. H-bonding has powerful effect on the structure and properties of compounds

Choose the correct answer from the options given below:

The equilibrium constant for the reaction

$$\mathrm{SO_3(g) \rightleftharpoons SO_2(g) + \frac{1}{2}\,O_2(g)}$$

is $\mathrm{K_c} = 4.9 \times 10^{-2}$. The value of $\mathrm{K_c}$ for the reaction given below is

$$\mathrm{2\,SO_2(g) + O_2(g) \rightleftharpoons 2\,SO_3(g)}$$

is :

Fuel cell, using hydrogen and oxygen as fuels,

A. has been used in spaceship B. has as efficiency of 40% to produce electricity C. uses aluminum as catalysts D. is eco-friendry E. is actually a type of Galvanic cell only

Choose the correct answer from the options given below:

For a strong electrolyte, a plot of molar conductivity against $(\text{concentration})^{1/2}$ is a straight line, with a negative slope, the correct unit for the slope is

Given below are two statements :

Statement I : The correct order of first ionization enthalpy values of Li, Na, F and Cl is Na < Li < Cl < F.

Statement II : The correct order of negative electron gain enthalpy values of Li, Na, F and Cl is Na < Li < F < Cl

In the light of the above statements, choose the correct answer from the options given below :

The correct order of the first ionization enthalpy is

The number of species from the following that have pyramidal geometry around the central atom is ______

$$\mathrm{S_2O_3^{2-},\ SO_4^{2-},\ SO_3^{2-},\ S_2O_7^{2-}}$$

A first row transition metal in its +2 oxidation state has a spin-only magnetic moment value of 3.86 BM. The atomic number of the metal is

The number of unpaired d-electrons in $\mathrm{[Co(H_2O)_6]^{3+}}$ is ________.

If an iron (III) complex with the formula $\mathrm{\left[Fe(NH_3)_x(CN)_y\right]^-}$ has no electron in its $\mathrm{e_g}$ orbital, then the value of $x + y$ is

When $\mathrm{MnO_2}$ and $\mathrm{H_2SO_4}$ is added to a salt (A), the greenish yellow gas liberated as salt (A) is :

The adsorbent used in adsorption chromatography is/are -

A. silica gel B. alumina C. quick lime D. magnesia

Choose the most appropriate answer from the options given below :

Correct order of stability of carbanion is -

[Figure: four cyclic carbanions — (a) cycloprop-2-en-1-yl anion (three-membered ring with one C=C, negative charge on the sp3 carbon); (b) cyclobutyl anion (four-membered saturated ring bearing a negative charge); (c) cyclopentyl anion (five-membered saturated ring bearing a negative charge); (d) cyclopentadienyl anion (five-membered ring with two C=C and a lone pair/negative charge on the fifth carbon)]

[Figure: reaction sequence — 1-methylcyclohex-1-ene $\xrightarrow{\text{"A"}}$ open-chain keto aldehyde $\mathrm{CH_3CO(CH_2)_4CHO}$ (6-oxoheptanal) $\xrightarrow{\text{"B"}}$ sodium carboxylate $\mathrm{NaOOC(CH_2)_4CHO}$ (sodium 6-oxohexanoate)]

In the above chemical reaction sequence "A" and "B" respectively are

Find out the major product formed from the following reaction. [Me : $-\mathrm{CH_3}$]

[Figure: 3,5-dibromocyclopent-1-ene $\xrightarrow{\mathrm{Me_2NH\ (2\ equiv)}}$ ?]

[Figure: 2-bromo-3-methyl-1-phenylbutane, $\mathrm{C_6H_5CH_2\text{-}CH(Br)\text{-}CH(CH_3)_2}$ $\xrightarrow[\Delta]{\mathrm{KOH(alc)}}$ major product "P"]

Product P is

Common name of Benzene - 1, 2 - diol is -

$$\mathrm{CH_3-CH_2-CH_2-Br + NaOH \xrightarrow{C_2H_5OH} Product\ 'A'}$$

Product A $\xrightarrow[\mathrm{H^+}]{\mathrm{H_2O}}$ Product "B"

Product A $\xrightarrow[\mathrm{H_2O/H_2O_2/\overline{O}H}]{\text{Diborane}}$ Product "C"

Consider the above reactions, identify product B and product C.

Match List I with List II

	LIST I		LIST II
A.	α - Glucose and α - Galactose	I.	Functional isomers
B.	α - Glucose and β - Glucose	II.	Homologous
C.	α - Glucose and α - Fructose	III.	Anomers
D.	α - Glucose and α - Ribose	IV.	Epimers

Choose the correct answer from the options given below:

The maximum number of orbitals which can be identified with $n = 4$ and $m_l = 0$ is ________.

Number of compounds / species from the following with non-zero dipole moment is ________.

$$\mathrm{BeCl_2,\ BCl_3,\ NF_3,\ XeF_4,\ CCl_4,\ H_2O,\ H_2S,\ HBr,\ CO_2,\ H_2,\ HCl}$$

Three moles of an ideal gas are compressed isothermally from 60 L to 20 L using constant pressure of 5 atm. Heat exchange Q for the compression is – ________ Lit. atm.

2.7 kg of each of water and acetic acid are mixed. The freezing point of the solution will be $-x\ {}^\circ\mathrm{C}$. Consider the acetic acid does not dimerise in water, nor dissociates in water. $x =$ ________ (nearest integer)

[Given: Molar mass of water = $18\ \mathrm{g\ mol^{-1}}$, acetic acid = $60\ \mathrm{g\ mol^{-1}}$

$\mathrm{K_f}$ $\mathrm{H_2O}$ : $1.86\ \mathrm{K\ kg\ mol^{-1}}$

$\mathrm{K_f}$ acetic acid : $3.90\ \mathrm{K\ kg\ mol^{-1}}$

freezing point : $\mathrm{H_2O} = 273$ K, acetic acid $= 290$ K]