JEE Main 2024 Jan 30 Shift 1 — Question Paper

If the domain of the function $f(x)=\cos^{-1}\left(\frac{2-|x|}{4}\right)+\{\log_e(3-x)\}^{-1}$ is $[-\alpha, \beta)-\{\gamma\}$, then $\alpha+\beta+\gamma$ is equal to :

If $z=x+\mathrm{i}y$, $xy\neq 0$, satisfies the equation $z^2+\mathrm{i}\bar{z}=0$, then $|z^2|$ is equal to :

Consider the system of linear equations $x+y+z=4\mu$, $x+2y+2\lambda z=10\mu$, $x+3y+4\lambda^2 z=\mu^2+15$, where $\lambda, \mu \in \mathbf{R}$. Which one of the following statements is NOT correct ?

If $f(x)=\begin{vmatrix} 2\cos^4 x & 2\sin^4 x & 3+\sin^2 2x \\ 3+2\cos^4 x & 2\sin^4 x & \sin^2 2x \\ 2\cos^4 x & 3+2\sin^4 x & \sin^2 2x \end{vmatrix}$,

then $\frac{1}{5}f'(0)=$ is equal to :

Two integers $x$ and $y$ are chosen with replacement from the set $\{0, 1, 2, 3, ..., 10\}$. Then the probability that $|x-y|>5$, is :

Let $S_n$ denote the sum of first $n$ terms of an arithmetic progression. If $S_{20}=790$ and $S_{10}=145$, then $S_{15}-S_5$ is :

The maximum area of a triangle whose one vertex is at $(0, 0)$ and the other two vertices lie on the curve $y=-2x^2+54$ at points $(x, y)$ and $(-x, y)$, where $y > 0$, is :

Let $f:\left[-\frac{\pi}{2}, \frac{\pi}{2}\right]\to \mathbf{R}$ be a differentiable function such that $f(0)=\frac{1}{2}$. If the $\lim\limits_{x\to 0} \frac{x\int_0^x f(t)\,\mathrm{d}t}{e^{x^2}-1}=\alpha$, then $8\alpha^2$ is equal to :

Let $g:\mathbf{R}\to\mathbf{R}$ be a non constant twice differentiable function such that $g'\left(\frac{1}{2}\right)=g'\left(\frac{3}{2}\right)$. If a real valued function $f$ is defined as $f(x)=\frac{1}{2}[g(x)+g(2-x)]$, then

The value of $\lim\limits_{n\to\infty} \sum\limits_{k=1}^{n} \frac{n^3}{(n^2+k^2)(n^2+3k^2)}$ is :

The area (in square units) of the region bounded by the parabola $y^2=4(x-2)$ and the line $y=2x-8$, is :

Let $y=y(x)$ be the solution of the differential equation $\sec x\,\mathrm{d}y+\{2(1-x)\tan x+x(2-x)\}\,\mathrm{d}x=0$ such that $y(0)=2$. Then $y(2)$ is equal to :

A line passing through the point $\mathrm{A}(9, 0)$ makes an angle of $30°$ with the positive direction of $x$-axis. If this line is rotated about A through an angle of $15°$ in the clockwise direction, then its equation in the new position is :

If the circles $(x+1)^2+(y+2)^2=r^2$ and $x^2+y^2-4x-4y+4=0$ intersect at exactly two distinct points, then

If the length of the minor axis of an ellipse is equal to half of the distance between the foci, then the eccentricity of the ellipse is :

Let $(\alpha, \beta, \gamma)$ be the foot of perpendicular from the point $(1, 2, 3)$ on the line $\frac{x+3}{5}=\frac{y-1}{2}=\frac{z+4}{3}$. Then $19(\alpha+\beta+\gamma)$ is equal to :

Let $\mathrm{A}(2, 3, 5)$ and $\mathrm{C}(-3, 4, -2)$ be opposite vertices of a parallelogram ABCD. If the diagonal $\overrightarrow{\mathrm{BD}}=\hat{i}+2\hat{j}+3\hat{k}$, then the area of the parallelogram is equal to :

Let $\vec{a}=a_1\hat{i}+a_2\hat{j}+a_3\hat{k}$ and $\vec{b}=b_1\hat{i}+b_2\hat{j}+b_3\hat{k}$ be two vectors such that $|\vec{a}|=1$, $\vec{a}\cdot\vec{b}=2$ and $|\vec{b}|=4$. If $\vec{c}=2(\vec{a}\times\vec{b})-3\vec{b}$, then the angle between $\vec{b}$ and $\vec{c}$ is equal to :

Let M denote the median of the following frequency distribution

Class	0 – 4	4 – 8	8 – 12	12 – 16	16 – 20
Frequency	3	9	10	8	6

Then $20\mathrm{M}$ is equal to :

If $2\sin^3 x+\sin 2x\cos x+4\sin x-4=0$ has exactly 3 solutions in the interval $\left[0, \frac{n\pi}{2}\right]$, $n\in\mathbf{N}$, then the roots of the equation $x^2+nx+(n-3)=0$ belong to :

Let $\mathrm{A}=\{1, 2, 3, ...., 7\}$ and let $\mathrm{P(A)}$ denote the power set of A. If the number of functions $f:\mathrm{A}\to\mathrm{P(A)}$ such that $a\in f(a)$, $\forall\ a\in\mathrm{A}$ is $m^n$, $m$ and $n\in\mathbf{N}$ and $m$ is least, then $m+n$ is equal to ________.

Let $\alpha, \beta \in\mathbf{N}$ be roots of the equation $x^2-70x+\lambda=0$, where $\frac{\lambda}{2}, \frac{\lambda}{3}\notin\mathbf{N}$. If $\lambda$ assumes the minimum possible value, then $\frac{(\sqrt{\alpha-1}+\sqrt{\beta-1})(\lambda+35)}{|\alpha-\beta|}$ is equal to :

Number of integral terms in the expansion of $\left\{7^{\left(\frac{1}{2}\right)}+11^{\left(\frac{1}{6}\right)}\right\}^{824}$ is equal to ________.

Let $\alpha=1^2+4^2+8^2+13^2+19^2+26^2+...$ upto 10 terms and $\beta=\sum\limits_{n=1}^{10} n^4$. If $4\alpha-\beta=55k+40$, then $k$ is equal to ________.

The value of $9\int_0^9 \left[\sqrt{\frac{10x}{x+1}}\right]\mathrm{d}x$, where $[t]$ denotes the greatest integer less than or equal to $t$, is ________.

Let $y=y(x)$ be the solution of the differential equation $(1-x^2)\,\mathrm{d}y=\left[xy+(x^3+2)\sqrt{3(1-x^2)}\right]\mathrm{d}x$, $-1<x<1$, $y(0)=0$. If $y\left(\frac{1}{2}\right)=\frac{m}{n}$, $m$ and $n$ are co-prime numbers, then $m+n$ is equal to ________.

Let the latus ractum of the hyperbola $\frac{x^2}{9}-\frac{y^2}{b^2}=1$ subtend an angle of $\frac{\pi}{3}$ at the centre of the hyperbola. If $b^2$ is equal to $\frac{l}{m}(1+\sqrt{n})$, where $l$ and $m$ are co-prime numbers, then $l^2+m^2+n^2$ is equal to ________.

If $\mathrm{d}_1$ is the shortest distance between the lines $x+1=2y=-12z$, $x=y+2=6z-6$ and $\mathrm{d}_2$ is the shortest distance between the lines $\frac{x-1}{2}=\frac{y+8}{-7}=\frac{z-4}{5}$, $\frac{x-1}{2}=\frac{y-2}{1}=\frac{z-6}{-3}$, then the value of $\frac{32\sqrt{3}\,\mathrm{d}_1}{\mathrm{d}_2}$ is :

A group of 40 students appeared in an examination of 3 subjects - Mathematics, Physics and Chemistry. It was found that all students passed in atleast one of the subjects, 20 students passed in Mathematics, 25 students passed in Physics, 16 students passed in Chemistry, atmost 11 students passed in both Mathematics and Physics, atmost 15 students passed in both Physics and Chemistry, atmost 15 students passed in both Mathematics and Chemistry. The maximum number of students passed in all the three subjects is ________.

If the function $$f(x)=\begin{cases} \frac{1}{|x|}, & |x|\geq 2 \\ ax^2+2b, & |x|<2 \end{cases}$$ is differentiable on $\mathbf{R}$, then $48(a+b)$ is equal to ________.

Match List - I with List - II.

	List - I		List - II
(A)	Coefficient of viscosity	(I)	$[\mathrm{M\,L^2\,T^{-2}}]$
(B)	Surface tension	(II)	$[\mathrm{M\,L^2\,T^{-1}}]$
(C)	Angular momentum	(III)	$[\mathrm{M\,L^{-1}\,T^{-1}}]$
(D)	Rotational kinetic energy	(IV)	$[\mathrm{M\,L^0\,T^{-2}}]$

Choose the correct answer from the options given below :

A particle of mass m is projected with a velocity 'u' making an angle of $30°$ with the horizontal. The magnitude of angular momentum of the projectile about the point of projection when the particle is at its maximum height h is :

A spherical body of mass 100 g is dropped from a height of 10 m from the ground. After hitting the ground, the body rebounds to a height of 5 m. The impulse of force imparted by the ground to the body is given by : (given, $\mathrm{g} = 9.8~\mathrm{m/s^2}$)

All surfaces shown in figure are assumed to be frictionless and the pulleys and the string are light. The acceleration of the block of mass 2 kg is :

[Figure: a 2 kg block on a $30°$ incline connected by a string over pulleys to a hanging 4 kg block; a movable pulley hangs from the ceiling]

A particle is placed at the point A of a frictionless track ABC as shown in figure. It is gently pushed towards right. The speed of the particle when it reaches the point B is : (Take $\mathrm{g} = 10~\mathrm{m/s^2}$).

[Figure: a curved track with point A at height 1 m, point B at height 0.5 m inside a valley, and point C at the top on the right]

The gravitational potential at a point above the surface of earth is $-5.12 \times 10^7~\mathrm{J/kg}$ and the acceleration due to gravity at that point is $6.4~\mathrm{m/s^2}$. Assume that the mean radius of earth to be 6400 km. The height of this point above the earth's surface is :

Young's modules of material of a wire of length 'L' and cross-sectional area A is Y. If the length of the wire is doubled and cross-sectional area is halved then Young's modules will be :

Two thermodynamical processes are shown in the figure. The molar heat capacity for process A and B are $\mathrm{C_A}$ and $\mathrm{C_B}$. The molar heat capacity at constant pressure and constant volume are represented by $\mathrm{C_P}$ and $\mathrm{C_V}$, respectively. Choose the correct statement.

[Figure: log P vs log V graph; process A is a straight line through origin at angle $\tan^{-1}\gamma$, process B is a straight line at $45°$]

At which temperature the r.m.s. velocity of a hydrogen molecule equal to that of an oxygen molecule at $47~°\mathrm{C}$ ?

The electrostatic potential due to an electric dipole at a distance 'r' varies as :

An electric toaster has resistance of $60~\Omega$ at room temperature ($27~°\mathrm{C}$). The toaster is connected to a 220 V supply. If the current flowing through it reaches 2.75 A, the temperature attained by toaster is around : ( if $\alpha = 2 \times 10^{-4}/°\mathrm{C}$)

Two insulated circular loop A and B of radius 'a' carrying a current of 'I' in the anti clockwise direction as shown in the figure. The magnitude of the magnetic induction at the centre will be :

[Figure: two circular loops of equal radius with a common centre O, loop A horizontal and loop B vertical (mutually perpendicular planes), each carrying current I anticlockwise]

A series L.R circuit connected with an ac source $\mathrm{E} = (25 \sin 1000\,\mathrm{t})~\mathrm{V}$ has a power factor of $\frac{1}{\sqrt{2}}$. If the source of emf is changed to $\mathrm{E} = (20 \sin 2000\,\mathrm{t})~\mathrm{V}$, the new power factor of the circuit will be :

The electric field of an electromagnetic wave in free space is represented as $\vec{\mathrm{E}} = \mathrm{E}_0 \cos(\omega t - kz)\,\hat{i}$. The corresponding magnetic induction vector will be :

The diffraction pattern of a light of wavelength 400 nm diffracting from a slit of width 0.2 mm is focused on the focal plane of a convex lens of focal length 100 cm. The width of the $1^{\text{st}}$ secondary maxima will be :

The work function of a substance is 3.0 eV. The longest wavelength of light that can cause the emission of photoelectrons from this substance is approximately;

The ratio of the magnitude of the kinetic energy to the potential energy of an electron in the $5^{\text{th}}$ excited state of a hydrogen atom is :

A Zener diode of breakdown voltage 10 V is used as a voltage regulator as shown in the figure. The current through the Zener diode is :

[Figure: 20 V battery in series with a $200~\Omega$ resistor; a Zener diode in parallel with a branch containing a $500~\Omega$ resistor]

A potential divider circuit is shown in figure. The output voltage $\mathrm{V}_0$ is :

[Figure: a 4 V battery across a series chain of a $3.3~\mathrm{k}\Omega$ resistor followed by seven $100~\Omega$ resistors; $\mathrm{V}_0$ is taken across the last six $100~\Omega$ resistors (tap after the first two resistors)]

Primary coil of a transformer is connected to 220 V ac. Primary and secondary turns of the transforms are 100 and 10 respectively. Secondary coil of transformer is connected to two series resistances shown in figure. The output voltage $(\mathrm{V}_0)$ is :

[Figure: 220 V ac source on transformer primary; secondary connected to a $15~\mathrm{k}\Omega$ resistor in series with a $7~\mathrm{k}\Omega$ resistor; $\mathrm{V}_0$ taken across the $7~\mathrm{k}\Omega$ resistor]

The displacement and the increase in the velocity of a moving particle in the time interval of t to $(t+1)$ s are 125 m and 50 m/s, respectively. The distance travelled by the particle in $(t+2)^{\text{th}}$ s is ________ m.

Consider a Disc of mass 5 kg, radius 2 m, rotating with angular velocity of 10 rad/s about an axis perpendicular to the plane of rotation. An identical disc is kept gently over the rotating disc along the same axis. The energy dissipated so that both the discs continue to rotate together without slipping is ________ J.

[Figure: a disc of mass 5 kg and radius 2 m rotating at $\omega = 10~\mathrm{rad/s}$ about a vertical axis through its centre]

Each of three blocks P, Q and R shown in figure has a mass of 3 kg. Each of the wires A and B has cross-sectional area $0.005~\mathrm{cm^2}$ and Young's modulus $2 \times 10^{11}~\mathrm{N~m^{-2}}$. Neglecting friction, the longitudinal strain on wire B is ________ $\times 10^{-4}$. (Take $\mathrm{g} = 10~\mathrm{m/s^2}$)

[Figure: blocks P and Q on a horizontal table connected by wire A; wire B runs from Q over a pulley at the table edge down to hanging block R]

In a closed organ pipe, the frequency of fundamental note is 30 Hz. A certain amount of water is now poured in the organ pipe so that the fundamental frequency is increased to 110 Hz. If the organ pipe has a cross-sectional area of $2~\mathrm{cm^2}$, the amount of water poured in the organ tube is ________ g. (Take speed of sound in air is 330 m/s)

A capacitor of capacitance C and potential V has energy E. It is connected to another capacitor of capacitance 2 C and potential 2 V. Then the loss of energy is $\frac{x}{3}\,\mathrm{E}$, where $x$ is ________.

Two cells are connected in opposition as shown. Cell $\mathrm{E}_1$ is of 8 V emf and $2~\Omega$ internal resistance; the cell $\mathrm{E}_2$ is of 2 V emf and $4~\Omega$ internal resistance. The terminal potential difference of cell $\mathrm{E}_2$ is ________ V.

[Figure: a closed loop with cells $\mathrm{E}_1$ and $\mathrm{E}_2$ connected in opposition between points A, B and C]

The horizontal component of earth's magnetic field at a place is $3.5 \times 10^{-5}~\mathrm{T}$. A very long straight conductor carrying current of $\sqrt{2}~\mathrm{A}$ in the direction from South east to North West is placed. The force per unit length experienced by the conductor is ________ $\times 10^{-6}~\mathrm{N/m}$.

A ceiling fan having 3 blades of length 80 cm each is rotating with an angular velocity of 1200 rpm. The magnetic field of earth in that region is 0.5 G and angle of dip is $30°$. The emf induced across the blades is $\mathrm{N}\pi \times 10^{-5}~\mathrm{V}$. The value of N is ________.

The distance between object and its two times magnified real image as produced by a convex lens is 45 cm. The focal length of the lens used is ________ cm.

A electron of hydrogen atom on an excited state is having energy $\mathrm{E_n} = -0.85~\mathrm{eV}$. The maximum number of allowed transitions to lower energy level is ________.

Given below are two statements :

Statement (I) : The orbitals having same energy are called as degenerate orbitals.

Statement (II) : In hydrogen atom, 3p and 3d orbitals are not degenerate orbitals.

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

Match List - I with List - II.

List - I (Molecule)	List - II (Shape)
(A) $\mathrm{BrF_5}$	(I) T-shape
(B) $\mathrm{H_2O}$	(II) See saw
(C) $\mathrm{ClF_3}$	(III) Bent
(D) $\mathrm{SF_4}$	(IV) Square pyramidal

Choose the correct answer from the options given below :

What happens to freezing point of benzene when small quantity of napthalene is added to benzene ?

Aluminium chloride in acidified aqueous solution forms an ion having geometry

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

Assertion (A) : There is a considerable increase in covalent radius from N to P. However from As to Bi only a small increase in covalent radius is observed.

Reason (R) : Covalent and ionic radii in a particular oxidation state increases down the group.

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

Match List - I with List - II.

List - I (Species)	List - II (Electronic distribution)
(A) $\mathrm{Cr^{+2}}$	(I) $\mathrm{3d^8}$
(B) $\mathrm{Mn^{+}}$	(II) $\mathrm{3d^3 4s^1}$
(C) $\mathrm{Ni^{+2}}$	(III) $\mathrm{3d^4}$
(D) $\mathrm{V^{+}}$	(IV) $\mathrm{3d^5 4s^1}$

Choose the correct answer from the options given below :

Diamagnetic Lanthanoid ions are :

Choose the correct statements from the following :

(A) Ethane-1, 2-diamine is a chelating ligand.

(B) Metallic aluminium is produced by electrolysis of aluminium oxide in presence of cryolite.

(C) Cyanide ion is used as ligand for leaching of silver.

(D) Phosphine act as a ligand in Wilkinson catalyst.

(E) The stability constants of $\mathrm{Ca^{2+}}$ and $\mathrm{Mg^{2+}}$ are similar with EDTA complexes.

Choose the correct answer from the options given below :

The Lassiagne's extract is boiled with dil $\mathrm{HNO_3}$ before testing for halogens because,

Structure of 4-Methylpent-2-enal is :

Which of the following molecule/species is most stable ?

Compound A formed in the following reaction reacts with B gives the product C. Find out A and B.

$$\mathrm{CH_3{-}C{\equiv}CH + Na \longrightarrow A \xrightarrow{B} \underset{(C)}{CH_3{-}C{\equiv}C{-}CH_2{-}CH_2{-}CH_3} + NaBr}$$

Example of vinylic halide is :

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

Assertion (A) : $\mathrm{CH_2{=}CH{-}CH_2{-}Cl}$ is an example of allyl halide.

Reason (R) : Allyl halides are the compounds in which the halogen atom is attached to sp$^2$ hybridised carbon atom.

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

In the given reactions, identify the reagent A and reagent B.

[Figure: toluene ($\mathrm{C_6H_5CH_3}$) converted to benzaldehyde ($\mathrm{C_6H_5CHO}$) by two routes — Route 1: "A" + $\mathrm{(CH_3CO)_2O}$, 273–283 K $\rightarrow$ [Intermediate] $\xrightarrow{\mathrm{H_3O^+},\ \Delta}$ benzaldehyde; Route 2: "B" + $\mathrm{CS_2}$ $\rightarrow$ [Intermediate] $\xrightarrow{\mathrm{H_3O^+}}$ benzaldehyde]

$$\mathrm{C_6H_5COCl \xrightarrow[\text{Pd-BaSO}_4]{H_2} C_6H_5CHO}$$

This reduction reaction is known as :

The final product A, formed in the following multistep reaction sequence is :

$$\mathrm{C_6H_5Br} \xrightarrow{\substack{\text{(i) Mg, ether then } \mathrm{CO_2},\ \mathrm{H^+} \\ \text{(ii) } \mathrm{NH_3},\ \Delta \\ \text{(iii) } \mathrm{Br_2},\ \mathrm{NaOH}}} A$$

Sugar which does not give reddish brown precipitate with Fehling's reagent, is :

Given below are two statements :

Statement (I) : The gas liberated on warming a salt with dil $\mathrm{H_2SO_4}$, turns a piece of paper dipped in lead acetate into black, it is a confirmatory test for sulphide ion.

Statement (II) : In statement-I the colour of paper turns black because of formation of lead sulphite.

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

Following is a confirmatory test for aromatic primary amines. Identify reagent (A) and (B).

$$\mathrm{C_6H_5NH_2 \xrightarrow{(A)} C_6H_5\overset{\oplus}{N_2}\overset{\ominus}{Cl} \xrightarrow[\text{NaOH}]{(B)} \text{Scarlet red dye}}$$

The mass of sodium acetate ($\mathrm{CH_3COONa}$) required to prepare 250 mL of 0.35 M aqueous solution is ________ g. (Molar mass of $\mathrm{CH_3COONa}$ is 82.02 $\mathrm{g\ mol^{-1}}$)

The total number of molecular orbitals formed from 2s and 2p atomic orbitals of a diatomic molecule is ________.

[Figure: P–V diagram with V ($\mathrm{dm^3}$) on the y-axis and P (kPa) on the x-axis; point A at (10 kPa, 10 $\mathrm{dm^3}$), point B at (10 kPa, 30 $\mathrm{dm^3}$), point C at (30 kPa, 10 $\mathrm{dm^3}$); path A$\to$B vertical (constant P), B$\to$C straight diagonal, C$\to$A horizontal (constant V)]

An ideal gas undergoes a cyclic transformation starting from the point A and coming back to the same point by tracing the path A$\to$B$\to$C$\to$A as shown in the diagram above. The total work done in the process is ________ J.

The pH at which $\mathrm{Mg(OH)_2}$ $[\mathrm{K_{sp}} = 1 \times 10^{-11}]$ begins to precipitate from a solution containing 0.10 M $\mathrm{Mg^{2+}}$ ions is ________.