JEE Main 2024 Jan 27 Shift 1 — Question Paper

Let $S = \{1, 2, 3, \ldots, 10\}$. Suppose $M$ is the set of all the subsets of $S$, then the relation $R = \{(A, B) : A \cap B \neq \phi;\ A, B \in M\}$ is :

The function $f : \mathbf{N} - \{1\} \to \mathbf{N}$; defined by $f(n) =$ the highest prime factor of $n$, is :

If $S = \{z \in \mathbf{C} : |z - i| = |z + i| = |z - 1|\}$, then, $n(S)$ is :

Consider the matrix $f(x) = \begin{bmatrix} \cos x & -\sin x & 0 \\ \sin x & \cos x & 0 \\ 0 & 0 & 1 \end{bmatrix}$.

Given below are two statements :

Statement I : $f(-x)$ is the inverse of the matrix $f(x)$.

Statement II : $f(x)\, f(y) = f(x + y)$.

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

$^{n-1}C_r = (k^2 - 8)\ ^nC_{r+1}$ if and only if :

If A denotes the sum of all the coefficients in the expansion of $(1 - 3x + 10x^2)^n$ and B denotes the sum of the coefficients in the expansion of $(1 + x^2)^n$, then :

The number of common terms in the progressions $4, 9, 14, 19, \ldots$, up to $25^{\text{th}}$ term and $3, 6, 9, 12, \ldots$, up to $37^{\text{th}}$ term is :

Consider the function.

$$f(x) = \begin{cases} \dfrac{a(7x - 12 - x^2)}{b|x^2 - 7x + 12|} & , \ x < 3 \\ 2^{\frac{\sin(x-3)}{x - [x]}} & , \ x > 3 \\ b & , \ x = 3 , \end{cases}$$

where $[x]$ denotes the greatest integer less than or equal to $x$. If S denotes the set of all ordered pairs (a, b) such that $f(x)$ is continuous at $x = 3$, then the number of elements in S is :

If $a = \lim\limits_{x \to 0} \dfrac{\sqrt{1 + \sqrt{1 + x^4}} - \sqrt{2}}{x^4}$ and $b = \lim\limits_{x \to 0} \dfrac{\sin^2 x}{\sqrt{2} - \sqrt{1 + \cos x}}$, then the value of $ab^3$ is :

If the shortest distance of the parabola $y^2 = 4x$ from the centre of the circle $x^2 + y^2 - 4x - 16y + 64 = 0$ is $d$, then $d^2$ is equal to :

If $\displaystyle\int_0^1 \frac{1}{\sqrt{3 + x} + \sqrt{1 + x}}\, dx = a + b\sqrt{2} + c\sqrt{3}$, where a, b, c are rational numbers, then $2a + 3b - 4c$ is equal to :

If (a, b) be the orthocentre of the triangle whose vertices are (1, 2), (2, 3) and (3, 1), and $I_1 = \displaystyle\int_a^b x \sin(4x - x^2)\, dx$, $I_2 = \displaystyle\int_a^b \sin(4x - x^2)\, dx$, then $36 \dfrac{I_1}{I_2}$ is equal to :

Let $x = x(t)$ and $y = y(t)$ be solutions of the differential equations $\dfrac{dx}{dt} + ax = 0$ and $\dfrac{dy}{dt} + by = 0$ respectively, $a, b \in \mathbf{R}$. Given that $x(0) = 2$; $y(0) = 1$ and $3y(1) = 2x(1)$, the value of $t$, for which $x(t) = y(t)$, is :

Four distinct points $(2k, 3k)$, $(1, 0)$, $(0, 1)$ and $(0, 0)$ lie on a circle for $k$ equal to :

The portion of the line $4x + 5y = 20$ in the first quadrant is trisected by the lines $L_1$ and $L_2$ passing through the origin. The tangent of an angle between the lines $L_1$ and $L_2$ is :

The length of the chord of the ellipse $\dfrac{x^2}{25} + \dfrac{y^2}{16} = 1$, whose mid point is $\left(1, \dfrac{2}{5}\right)$, is equal to :

The distance, of the point $(7, -2, 11)$ from the line $\dfrac{x - 6}{1} = \dfrac{y - 4}{0} = \dfrac{z - 8}{3}$ along the line $\dfrac{x - 5}{2} = \dfrac{y - 1}{-3} = \dfrac{z - 5}{6}$, is :

If the shortest distance between the lines $\dfrac{x - 4}{1} = \dfrac{y + 1}{2} = \dfrac{z}{-3}$ and $\dfrac{x - \lambda}{2} = \dfrac{y + 1}{4} = \dfrac{z - 2}{-5}$ is $\dfrac{6}{\sqrt{5}}$, then the sum of all possible values of $\lambda$ is :

Let $\vec{a} = \hat{i} + 2\hat{j} + \hat{k}$, $\vec{b} = 3(\hat{i} - \hat{j} + \hat{k})$. Let $\vec{c}$ be the vector such that $\vec{a} \times \vec{c} = \vec{b}$ and $\vec{a} \cdot \vec{c} = 3$. Then $\vec{a} \cdot \left( (\vec{c} \times \vec{b}) - \vec{b} - \vec{c} \right)$ is equal to :

Let $a_1, a_2, \ldots, a_{10}$ be 10 observations such that $\sum\limits_{k=1}^{10} a_k = 50$ and $\sum\limits_{\forall\, k < j} a_k \cdot a_j = 1100$. Then the standard deviation of $a_1, a_2, \ldots, a_{10}$ is equal to :

If $\alpha$ satisfies the equation $x^2 + x + 1 = 0$ and $(1 + \alpha)^7 = A + B\alpha + C\alpha^2$, $A, B, C \geq 0$, then $5(3A - 2B - C)$ is equal to ________.

Let $A = \begin{bmatrix} 2 & 0 & 1 \\ 1 & 1 & 0 \\ 1 & 0 & 1 \end{bmatrix}$, $B = [B_1, B_2, B_3]$, where $B_1, B_2, B_3$ are column matrics, and

$$AB_1 = \begin{bmatrix} 1 \\ 0 \\ 0 \end{bmatrix},\ AB_2 = \begin{bmatrix} 2 \\ 3 \\ 0 \end{bmatrix},\ AB_3 = \begin{bmatrix} 3 \\ 2 \\ 1 \end{bmatrix}$$

If $\alpha = |B|$ and $\beta$ is the sum of all the diagonal elements of $B$, then $\alpha^3 + \beta^3$ is equal to ________.

Let for a differentiable function $f : (0, \infty) \to \mathbf{R}$, $f(x) - f(y) \geq \log_e\left(\dfrac{x}{y}\right) + x - y,\ \forall\ x, y \in (0, \infty)$.

Then $\sum\limits_{n=1}^{20} f'\left(\dfrac{1}{n^2}\right)$ is equal to ________.

If $8 = 3 + \dfrac{1}{4}(3 + p) + \dfrac{1}{4^2}(3 + 2p) + \dfrac{1}{4^3}(3 + 3p) + \cdots \infty$, then the value of $p$ is ________.

Let $f(x) = x^3 + x^2 f'(1) + x f''(2) + f'''(3)$, $x \in \mathbf{R}$. Then $f'(10)$ is equal to ________.

Let the area of the region $\{(x, y) : x - 2y + 4 \geq 0,\ x + 2y^2 \geq 0,\ x + 4y^2 \leq 8,\ y \geq 0\}$ be $\dfrac{m}{n}$, where $m$ and $n$ are coprime numbers. Then $m + n$ is equal to ________.

If the solution of the differential equation $(2x + 3y - 2)\, dx + (4x + 6y - 7)\, dy = 0$, $y(0) = 3$, is $\alpha x + \beta y + 3 \log_e |2x + 3y - \gamma| = 6$, then $\alpha + 2\beta + 3\gamma$ is equal to ________.

The least positive integral value of $\alpha$, for which the angle between the vectors $\alpha\hat{i} - 2\hat{j} + 2\hat{k}$ and $\alpha\hat{i} + 2\alpha\hat{j} - 2\hat{k}$ is acute, is ________.

A fair die is tossed repeatedly until a six is obtained. Let X denote the number of tosses required and let $a = P(X = 3)$, $b = P(X \geq 3)$ and $c = P(X \geq 6 \mid X > 3)$. Then $\dfrac{b + c}{a}$ is equal to ________.

Let the set of all $a \in \mathbf{R}$ such that the equation $\cos 2x + a \sin x = 2a - 7$ has a solution be $[p, q]$ and $r = \tan 9° - \tan 27° - \dfrac{1}{\cot 63°} + \tan 81°$, then $pqr$ is equal to ________.

Given below are two statements :

Statement (I) : Planck's constant and angular momentum have same dimensions.

Statement (II) : Linear momentum and moment of force have same dimensions.

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

Position of an ant (S in metres) moving in Y-Z plane is given by $S = 2t^2\hat{j} + 5\hat{k}$ (where t is in second). The magnitude and direction of velocity of the ant at $t = 1$ s will be :

A body of mass 1000 kg is moving horizontally with a velocity 6 m/s. If 200 kg extra mass is added, the final velocity (in m/s) is :

A train is moving with a speed of 12 m/s on rails which are 1.5 m apart. To negotiate a curve of radius 400 m, the height by which the outer rail should be raised with respect to the inner rail is (Given, $g = 10\ \mathrm{m/s^2}$) :

Two bodies of mass 4 g and 25 g are moving with equal kinetic energies. The ratio of magnitude of their linear momentum is :

The acceleration due to gravity on the surface of earth is g. If the diameter of earth reduces to half of its original value and mass remains constant, then acceleration due to gravity on the surface of earth would be :

Given below are two statements :

Statement (I) : Viscosity of gases is greater than that of liquids.

Statement (II) : Surface tension of a liquid decreases due to the presence of insoluble impurities.

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

0.08 kg air is heated at constant volume through $5^{\circ}\mathrm{C}$. The specific heat of air at constant volume is $0.17\ \mathrm{kcal/kg^{\circ}C}$ and $J = 4.18$ joule/cal. The change in its internal energy is approximately.

The average kinetic energy of a monoatomic molecule is 0.414 eV at temperature :

(Use $\mathrm{K_B} = 1.38 \times 10^{-23}$ J/mol-K)

An electric charge $10^{-6}\ \mu\mathrm{C}$ is placed at origin (0, 0) m of X–Y co-ordinate system. Two points P and Q are situated at $(\sqrt{3}, \sqrt{3})$ m and $(\sqrt{6}, 0)$ m respectively. The potential difference between the points P and Q will be :

A wire of resistance R and length L is cut into 5 equal parts. If these parts are joined parallely, then resultant resistance will be :

A proton moving with a constant velocity passes through a region of space without any change in its velocity. If $\vec{E}$ and $\vec{B}$ represent the electric and magnetic fields respectively, then the region of space may have :

(A) $E = 0,\ B = 0$

(B) $E = 0,\ B \neq 0$

(C) $E \neq 0,\ B = 0$

(D) $E \neq 0,\ B \neq 0$

Choose the most appropriate answer from the options given below :

A rectangular loop of length 2.5 m and width 2 m is placed at $60^{\circ}$ to a magnetic field of 4 T. The loop is removed from the field in 10 s. The average emf induced in the loop during this time is :

A plane electromagnetic wave propagating in $x$-direction is described by

$E_y = (200\ \mathrm{Vm^{-1}})\ \sin[1.5 \times 10^7 t - 0.05\ x]$; The intensity of the wave is :

(Use $\epsilon_0 = 8.85 \times 10^{-12}\ \mathrm{C^2 N^{-1} m^{-2}}$)

If the refractive index of the material of a prism is $\cot\left(\frac{A}{2}\right)$, where A is the angle of prism then the angle of minimum deviation will be :

A convex lens of focal length 40 cm forms an image of an extended source of light on a photo-electric cell. A current I is produced. The lens is replaced by another convex lens having the same diameter but focal length 20 cm. The photoelectric current now is :

The radius of third stationary orbit of electron for Bohr's atom is R. The radius of fourth stationary orbit will be :

Which of the following circuits is reverse - biased ?

Identify the physical quantity that cannot be measured using spherometer :

A wire of length 10 cm and radius $\sqrt{7} \times 10^{-4}$ m is connected across the right gap of a meter bridge. When a resistance of $4.5\ \Omega$ is connected on the left gap by using a resistance box, the balance length is found to be at 60 cm from the left end. If the resistivity of the wire is $\mathrm{R} \times 10^{-7}\ \Omega$m, then value of R is :

A particle starts from origin at $t = 0$ with a velocity $5\hat{i}$ m/s and moves in $x$-$y$ plane under action of a force which produces a constant acceleration of $(3\hat{i} + 2\hat{j})\ \mathrm{m/s^2}$. If the $x$-coordinate of the particle at that instant is 84 m, then the speed of the particle at this time is $\sqrt{\alpha}$ m/s. The value of $\alpha$ is ________.

Four particles each of mass 1 kg are placed at four corners of a square of side 2 m. Moment of inertia of system about an axis perpendicular to its plane and passing through one of its vertex is ________ $\mathrm{kgm^2}$.

If average depth of an ocean is 4000 m and the bulk modulus of water is $2 \times 10^9\ \mathrm{Nm^{-2}}$, then fractional compression $\frac{\Delta V}{V}$ of water at the bottom of ocean is $\alpha \times 10^{-2}$. The value of $\alpha$ is ________. (Given, $g = 10\ \mathrm{ms^{-2}}$, $\rho = 1000\ \mathrm{kg\ m^{-3}}$)

A particle executes simple harmonic motion with an amplitude of 4 cm. At the mean position, velocity of the particle is 10 cm/s. The distance of the particle from the mean position when its speed becomes 5 cm/s is $\sqrt{\alpha}$ cm, where $\alpha =$ ________.

A thin metallic wire having cross sectional area of $10^{-4}\ \mathrm{m^2}$ is used to make a ring of radius 30 cm. A positive charge of $2\pi$ C is uniformly distributed over the ring, while another positive charge of 30 pC is kept at the centre of the ring. The tension in the ring is ________ N; provided that the ring does not get deformed (neglect the influence of gravity).

$\left(\text{given, } \frac{1}{4\pi\epsilon_0} = 9 \times 10^9\ \text{SI units}\right)$

The charge accumulated on the capacitor connected in the following circuit is ________ $\mu$C.

(Given $C = 150\ \mu$F)

[Figure: Wheatstone-bridge circuit driven by a 10 V battery; upper branch has $R_1 = 1\ \Omega$ then $R_3 = 6\ \Omega$, lower branch has $R_2 = 2\ \Omega$ then $R_4 = 4\ \Omega$; capacitor C is connected between the midpoints A (between $R_1$ and $R_2$) and B (between $R_3$ and $R_4$)]

Two long, straight wires carry equal currents in opposite directions as shown in figure. The separation between the wires is 5.0 cm. The magnitude of the magnetic field at a point P midway between the wires is ________ $\mu$T. (Given : $\mu_0 = 4\pi \times 10^{-7}\ \mathrm{TmA^{-1}}$)

[Figure: two vertical parallel wires each carrying 10 A, the left wire's current directed upward and the right wire's current directed downward, with point P midway between them]

Two coils have mutual inductance 0.002 H. The current changes in the first coil according to the relation $i = i_0 \sin \omega t$, where $i_0 = 5$A and $\omega = 50\pi$ rad/s. The maximum value of emf in the second coil is $\frac{\pi}{\alpha}$ V. The value of $\alpha$ is ________.

Two immiscible liquids of refractive indices $\frac{8}{5}$ and $\frac{3}{2}$ respectively are put in a beaker as shown in the figure. The height of each column is 6 cm. A coin is placed at the bottom of the beaker. For near normal vision, the apparent depth of the coin is $\frac{\alpha}{4}$ cm. The value of $\alpha$ is ________.

[Figure: a beaker with two liquid layers, each 6 cm high — upper layer $\mu_2 = 3/2$, lower layer $\mu_1 = 8/5$ — with a coin at the bottom]

In a nuclear fission process, a high mass nuclide $(A \approx 236)$ with binding energy 7.6 MeV/Nucleon dissociated into middle mass nuclides $(A \approx 118)$, having binding energy of 8.6 MeV/Nucleon. The energy released in the process would be ________ MeV.

Choose the polar molecule from the following :

A solution of two miscible liquids showing negative deviation from Raoult's law will have :

Given below are two statements :

Statement (I) : Aqueous solution of ammonium carbonate is basic.

Statement (II) : Acidic/basic nature of salt solution of a salt of weak acid and weak base depends on $\mathrm{K_a}$ and $\mathrm{K_b}$ value of acid and the base forming it.

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

Given below are two statements :

Statement (I) : The $4f$ and $5f$ - series of elements are placed separately in the Periodic table to preserve the principle of classification.

Statement (II) : s-block elements can be found in pure form in nature.

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

Element not showing variable oxidation state is :

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

Assertion (A) : Melting point of Boron (2453 K) is unusually high in group 13 elements.

Reason (R) : Solid Boron has very strong crystalline lattice.

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

Which of the following electronic configuration would be associated with the highest magnetic moment ?

The electronic configuration for Neodymium is :

[Atomic Number for Neodymium 60]

Consider the following complex ions

$\mathrm{P = [FeF_6]^{3-}}$

$\mathrm{Q = [V(H_2O)_6]^{2+}}$

$\mathrm{R = [Fe(H_2O)_6]^{2+}}$

The correct order of the complex ions, according to their spin only magnetic moment values (in B.M.) is :

IUPAC name of following compound (P) is :

[Figure: cyclohexane ring bearing two methyl groups on the same carbon (gem-dimethyl) and an ethyl group on the carbon two positions away]

Cyclohexene [Figure: cyclohexene ring structure] is ________ type of an organic compound.

Which of the following has highly acidic hydrogen ?

The correct statement regarding nucleophilic substitution reaction in a chiral alkyl halide is :

Given below are two statements :

Statement (I) : p-nitrophenol is more acidic than m-nitrophenol and o-nitrophenol.

Statement (II) : Ethanol will give immediate turbidity with Lucas reagent.

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

The ascending order of acidity of $\mathrm{-OH}$ group in the following compounds is :

(A) $\mathrm{Bu{-}OH}$

(B) [Structure: 4-nitrophenol, $\mathrm{O_2N{-}C_6H_4{-}OH}$ (para)]

(C) [Structure: 4-methoxyphenol, $\mathrm{MeO{-}C_6H_4{-}OH}$ (para)]

(D) [Structure: phenol, $\mathrm{C_6H_5{-}OH}$]

(E) [Structure: 2,4-dinitrophenol, phenol with $\mathrm{NO_2}$ groups at ortho and para positions]

Choose the correct answer from the options given below :

Highest enol content will be shown by :

Which of the following is strongest Bronsted base ?

Two nucleotides are joined together by a linkage known as :

Yellow compound of lead chromate gets dissolved on treatment with hot NaOH solution. The product of lead formed is a :

NaCl reacts with conc. $\mathrm{H_2SO_4}$ and $\mathrm{K_2Cr_2O_7}$ to give reddish fumes (B), which react with NaOH to give yellow solution (C). (B) and (C) respectively are :

Mass of methane required to produce 22 g of $\mathrm{CO_2}$ after complete combustion is ________ g.

(Given Molar mass in $\mathrm{g\ mol^{-1}}$   C = 12.0   H = 1.0   O = 16.0)

The number of electrons present in all the completely filled subshells having $n = 4$ and $s = +\frac{1}{2}$ is ________.

(Where $n$ = principal quantum number and $s$ = spin quantum number)

Sum of bond order of CO and $\mathrm{NO^+}$ is ________.

If three moles of an ideal gas at 300 K expand isothermally from $30\ \mathrm{dm^3}$ to $45\ \mathrm{dm^3}$ against a constant opposing pressure of 80 kPa, then the amount of heat transferred is ________ J.

From the given list, the number of compounds with +4 oxidation state of Sulphur is ________.

$\mathrm{SO_3,\ H_2SO_3,\ SOCl_2,\ SF_4,\ BaSO_4,\ H_2S_2O_7}$

The mass of silver (Molar mass of Ag : $108\ \mathrm{g\,mol^{-1}}$) displaced by a quantity of electricity which displaces 5600 mL of $\mathrm{O_2}$ at S.T.P. will be ________ g.

Consider the following data for the given reaction

$$\mathrm{2\,HI_{(g)} \rightarrow H_{2(g)} + I_{2(g)}}$$

	1	2	3
$\mathrm{HI\ (mol\ L^{-1})}$	0.005	0.01	0.02
$\mathrm{Rate\ (mol\ L^{-1}s^{-1})}$	$7.5 \times 10^{-4}$	$3.0 \times 10^{-3}$	$1.2 \times 10^{-2}$

The order of the reaction is ________.

Among the following, total number of meta directing functional groups is ________. (Integer based)

$\mathrm{-OCH_3,\ -NO_2,\ -CN,\ -CH_3,\ -NHCOCH_3,\ -COR,\ -OH,\ -COOH,\ -Cl}$

Among the given organic compounds, the total number of aromatic compounds is ________.

(A) [Structure: two fused six-membered rings, each ring containing two C=C double bonds with sp3 ring-fusion carbons (4a,8a-dihydronaphthalene)]

(B) [Structure: benzene ring connected through a $\mathrm{CH_2}$ group to a cyclohexene ring]

(C) [Structure: benzene ring fused to a five-membered ring whose apex carbon bears a lone pair (carbanion-type centre)]

(D) [Structure: two benzene rings fused to a central five-membered ring whose carbon bears a negative charge (fluorenyl anion)]

3-Methylhex-2-ene on reaction with HBr in presence of peroxide forms an addition product (A). The number of possible stereoisomers for 'A' is ________.