JEE Main 2024 — 5 April, Shift 1

Let d be the distance of the point of intersection of the lines $\quad \frac{\mathrm{x}+6}{3}=\frac{\mathrm{y}}{2}=\frac{\mathrm{z}+1}{1} \quad$ and $\frac{x-7}{4}=\frac{y-9}{3}=\frac{z-4}{2}$ from the point $(7,8,9)$. Then $d^{2}+6$ is equal to :

Let a rectangle ABCD of sides 2 and 4 be inscribed in another rectangle $P Q R S$ such that the vertices of the rectangle $A B C D$ lie on the sides of the rectangle PQRS. Let a and b be the sides of the rectangle PQRS when its area is maximum. Then $(\mathrm{a}+\mathrm{b})^{2}$ is equal to :

Let two straight lines drawn from the origin $O$ intersect the line $3 x+4 y=12$ at the points $P$ and Q such that $\triangle \mathrm{OPQ}$ is an isosceles triangle and $\angle \mathrm{POQ}=90^{\circ}$. If $l=\mathrm{OP}^{2}+\mathrm{PQ}^{2}+\mathrm{QO}^{2}$, then the greatest integer less than or equal

to $l$ is :

If $y=y(x)$ is the solution of the differential equation $\frac{d y}{d x}+2 y=\sin (2 x), y(0)=\frac{3}{4}$, then $y\left(\frac{\pi}{8}\right)$ is equal to :

For the function $\mathrm{f}(\mathrm{x})=\sin \mathrm{x}+3 \mathrm{x}-\frac{2}{\pi}\left(\mathrm{x}^{2}+\mathrm{x}\right)$, where $\mathrm{x} \in\left[0, \frac{\pi}{2}\right]$, consider the following two statements :

Statement I : f is increasing in $\left(0, \frac{\pi}{2}\right)$.

Statement II : $\mathrm{f}^{\prime}$ is decreasing in $\left(0, \frac{\pi}{2}\right)$.

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

If the system of equations

$11 x+y+\lambda z=-5$

$2 x+3 y+5 z=3$

$8 x-19 y-39 z=\mu$

has infinitely many solutions, then $\lambda^{4}-\mu$ is equal to :

Let $\mathrm{A}=\{1,3,7,9,11\}$ and $\mathrm{B}=\{2,4,5,7,8,10,12\}$. Then the total number of one-one maps $f: A \rightarrow B$,

such that $f(1)+f(3)=14$, is :

If the function $f(x)=\frac{\sin 3 x+\alpha \sin x-\beta \cos 3 x}{x^{3}}$, $x \in R$, is continuous at $x=0$, then $f(0)$ is equal to :

The integral $\int_{0}^{\frac{\pi}{4}} \frac{136 \sin x}{3 \sin x+5 \cos x} d x$ is equal to :

The coefficients $a, b, c$ in the quadratic equation $a x^{2}+b x+c=0$ are chosen from the set $\{1,2,3,4,5,6,7,8\}$. The probability of this equation having repeated roots is :

Let A and B be two square matrices of order 3 such that $|A|=3$ and $|B|=2$. Then

$\left|\mathrm{A}^{\mathrm{T}} \mathrm{A}(\operatorname{adj}(2 \mathrm{~A}))^{-1}(\operatorname{adj}(4 \mathrm{~B}))(\operatorname{adj}(\mathrm{AB}))^{-1} \mathrm{AA}^{\mathrm{T}}\right|$ is equal to :

Let a circle C of radius 1 and closer to the origin be such that the lines passing through the point $(3,2)$ and parallel to the coordinate axes touch it. Then the shortest distance of the circle C from the point $(5,5)$ is :

Let the line $2 \mathrm{x}+3 \mathrm{y}-\mathrm{k}=0, \mathrm{k}>0$, intersect the x -axis and y -axis at the points A and B , respectively. If the equation of the circle having the line segment $A B$ as a diameter is $x^{2}+y^{2}-3 x-2 y=0$ and the length of the latus rectum of the ellipse $\mathrm{x}^{2}+9 \mathrm{y}^{2}=\mathrm{k}^{2}$ is $\frac{\mathrm{m}}{\mathrm{n}}$, where m and n are coprime, then $2 \mathrm{~m}+\mathrm{n}$ is equal to

Consider the following two statements :

Statement I : For any two non-zero complex numbers $\mathrm{z}_{1}, \mathrm{z}_{2}$ $\left(\left|z_{1}\right|+\left|z_{2}\right|\right)\left|\frac{z_{1}}{\left|z_{1}\right|}+\frac{z_{2}}{\left|z_{2}\right|}\right| \leq 2\left(\left|z_{1}\right|+\left|z_{2}\right|\right)$ and

Statement II : If $\mathrm{x}, \mathrm{y}, \mathrm{z}$ are three distinct complex numbers and $\mathrm{a}, \mathrm{b}, \mathrm{c}$ are three positive real numbers such that $\frac{a}{|y-z|}=\frac{b}{|z-x|}=\frac{c}{|x-y|}$, then $\frac{a^{2}}{y-z}+\frac{b^{2}}{z-x}+\frac{c^{2}}{x-y}=1$ Between the above two statements,

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

Suppose $\theta \in\left[0, \frac{\pi}{4}\right]$ is a solution of $4 \cos \theta-3 \sin \theta=1$. Then $\cos \theta$ is equal to :

If $\frac{1}{\sqrt{1}+\sqrt{2}}+\frac{1}{\sqrt{2}+\sqrt{3}}+\ldots+\frac{1}{\sqrt{99}+\sqrt{100}}=m$ and $\frac{1}{1 \cdot 2}+\frac{1}{2 \cdot 3}+\ldots+\frac{1}{99 \cdot 100}=n$,

then the point $(m, n)$ lies on the line

Let $f(x)=x^{5}+2 x^{3}+3 x+1, x \in R$, and $g(x)$ be a function such that $g(f(x))=x$ for all $x \in R$. Then $\frac{g(7)}{g^{\prime}(7)}$ is equal to :

If $\mathrm{A}(1,-1,2), \mathrm{B}(5,7,-6), \mathrm{C}(3,4,-10)$ and $\mathrm{D}(-1,-4,-2)$ are the vertices of a quadrilateral ABCD ,

then its area is :

The value of $\int_{-\pi}^{\pi} \frac{2 y(1+\sin y)}{1+\cos ^{2} y} d y$ is :

If the line $\frac{2-x}{3}=\frac{3 y-2}{4 \lambda+1}=4-z$ makes a right angle with the line $\frac{x+3}{3 \mu}=\frac{1-2 y}{6}=\frac{5-z}{7}$, then $4 \lambda+9 \mu$ is equal to :

From a lot of 10 items, which include 3 defective items, a sample of 5 items is drawn at random. Let the random variable X denote the number of defective items in the sample. If the variance of $X$ is $\sigma^{2}$, then $96 \sigma^{2}$ is equal to $\qquad$

If the constant term in the expansion of $\left(1+2 x-3 x^{3}\right)\left(\frac{3}{2} x^{2}-\frac{1}{3 x}\right)^{9}$ is $p$, then $108 p$ is equal to

The area of the region enclosed by the parabolas $y=x^{2}-5 x$ and $y=7 x-x^{2}$ is $\_\_\_\_\_\_\_\_$

The number of ways of getting a sum 16 on throwing a dice four times is $\qquad$

If $S=\{\mathrm{a} \in \mathrm{R}:|2 \mathrm{a}-1|=3[\mathrm{a}]+2\{\mathrm{a}\}\}$, where [t] denotes the greatest integer less than or equal to $t$ and $\{t\}$

represents the fractional part of $t$, then $72 \sum_{\mathrm{a} \in \mathrm{S}} \mathrm{a}$ is equal to $\qquad$

Let f be a differentiable function in the interval $(0, \infty)$ such that $f(1)=1$ and $\lim _{t \rightarrow x} \frac{t^{2} f(x)-x^{2} f(t)}{t-x}=1$

for each $x>0$. Then $2 f(2)+3 f(3)$ is equal to

Let $a_{1}, a_{2}, a_{3}, \ldots$ be in an arithmetic progression of positive terms.

Let $A_{k}=a_{1}{ }^{2}-a_{2}{ }^{2}+a_{3}{ }^{2}-a_{4}{ }^{2}+\ldots+a_{2 k-1}{ }^{2}-a_{2 k}{ }^{2}$. If $\mathrm{A}_{3}=-153, \mathrm{~A}_{5}=-435$ and

$\mathrm{a}_{1}{ }^{2}+\mathrm{a}_{2}{ }^{2}+\mathrm{a}_{3}{ }^{2}=66$, then $\mathrm{a}_{17}-\mathrm{A}_{7}$ is equal to $\qquad$

Let $\vec{a}=\hat{i}-3 \hat{j}+7 \hat{k}, \quad \vec{b}=2 \hat{i}-\hat{j}+\hat{k}$ and $\vec{c}$ be $a$ vector such that $(\vec{a}+2 \vec{b}) \times \vec{c}=3(\vec{c} \times \vec{a})$. If $\vec{a} \cdot \vec{c}=130$,

then $\vec{b} \cdot \vec{c}$ is equal to $\qquad$

The number of distinct real roots of the equation $|x||x+2|-5|x+1|-1=0$ is $\qquad$

Suppose $A B$ is a focal chord of the parabola $\mathrm{y}^{2}=12 \mathrm{x}$ of length $l$ and slope $\mathrm{m}<\sqrt{3}$. If the distance of the chord $A B$ from the origin is $d$, then $l \mathrm{~d}^{2}$ is equal to $\qquad$

Light emerges out of a convex lens when a source of light kept at its focus. The shape of wavefront of the light is:

If G be the gravitational constant and u be the energy density then which of the following quantity have the dimension as that the $\sqrt{\mathrm{uG}}$ :

Given below are two statements : one is labelled as Statement I and the other is labelled as Statement II :

Statement-I: When a capillary tube is dipped into a liquid, the liquid neither rises nor falls in the capillary. The contact angle may be $0^{\circ}$.

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

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

Given below are two statements : one is labelled as Statement I and the other is labelled as Statement II :

Statement-I: Figure shows the variation of stopping potential with frequency ( $v$ ) for the two photosensitive materials $\mathrm{M}_{1}$ and $\mathrm{M}_{2}$. The slope gives value of $\frac{\mathrm{h}}{\mathrm{e}}$, where h is Planck's constant, e is the charge of electron.

Statement-II: $\mathrm{M}_{2}$ will emit photoelectrons of greater kinetic energy for the incident radiation having same frequency.

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

The angle between vector $\overrightarrow{\mathrm{Q}}$ and the resultant of $(2 \overrightarrow{\mathrm{Q}}+2 \overrightarrow{\mathrm{P}})$ and $(2 \overrightarrow{\mathrm{Q}}-2 \overrightarrow{\mathrm{P}})$ is:

In hydrogen like system the ratio of coulombian force and gravitational force between an electron and a proton is in the order of:

In a co-axial straight cable, the central conductor and the outer conductor carry equal currents in opposite directions. The magnetic field is zero

An electron rotates in a circle around a nucleus having positive charge Ze. Correct relation between total energy (E) of electron to its potential energy (U) is:

If the collision frequency of hydrogen molecules in a closed chamber at $27^{\circ} \mathrm{C}$ is Z , then the collision frequency of the same system at $127^{\circ} \mathrm{C}$ is

Ratio of radius of gyration of a hollow sphere to that of a solid cylinder of equal mass, for moment of Inertia about their

diameter axis AB as shown in figure is $\sqrt{\frac{8}{x}}$. The value of $x$ is:

Two conducting circular loops A and B are placed in the same plane with their centres coinciding as shown in figure. The mutual inductance between them is:

Match list-I with list-II:

(Where a= radius of planet orbite, r=radius of planet, M=mass of sun, m=mass of planet)

Choose the correct answer from the options given below:

A wooden block of mass 5 kg rests on soft horizontal floor. When an iron cylinder of mass 25 kg is placed on the top of the block, the floor yields and the block and the cylinder together go down with an acceleration of $0.1 \mathrm{~ms}^{-2}$. The action force of the system on the floor is equal to:

A simple pendulum doing small oscillations at a place $R$ height above earth surface has time period of $T_{1}=4 \mathrm{~s} . \mathrm{T}_{2}$ would be it's time period if it is brought to a point which is at a height 2 R from earth surface. Choose the correct relation $[R=$ radius of Earth]:

A body of mass 50 kg is lifted to a height of 20 m from the ground in the two different ways as shown in the figures. The ratio of work done against the gravity in both the respective cases, will be:

The heat absorbed by a system in going through the given cyclic process is :

In the given figure $\mathrm{R}_{1}=10 \Omega, \mathrm{R}_{2}=8 \Omega, \mathrm{R}_{3}=4 \Omega$ and $\mathrm{R}_{4}=8 \Omega$. Battery is ideal with emf 12 V . Equivalent

resistant of the circuit and current supplied by battery are respectively.

An alternating voltage of amplitude 40 V and frequency 4 kHz is applied directly across the capacitor of $12 \mu \mathrm{~F}$. The maximum displacement current between the plates of the capacitor is nearly:

In Young's double slit experiment, carried out with light of wavelength $5000 A$, the distance between the slits is 0.3 mm and the screen is at 200 cm from the slits. The central maximum is at $x=0 \mathrm{~cm}$. The value of $x$ for third maxima is $\qquad$ mm

A 2 A current carrying straight metal wire of resistance $1 \Omega$, resistivity $2 \times 10^{-6} \Omega \mathrm{~m}$, area of cross-section $10 \mathrm{~mm}^{2}$ and mass 500 g is suspended horizontally in mid air by applying a uniform magnetic field $\overrightarrow{\mathrm{B}}$. The magnitude of B is $\qquad$ $\times 10^{-1} \mathrm{~T}$ (given, $\mathrm{g}=10 \mathrm{~m} / \mathrm{s}^{2}$ )

The electric field between the two parallel plates of a capacitor of $1.5 \mu \mathrm{~F}$ capacitance drops to one third of its initial value in

$6.6 \mu \mathrm{~s}$ when the plates are connected by a thin wire. The resistance of this wire is $\qquad$ $\Omega$. (Given, $\log 3=1.1)$

Three blocks $M_{1}, M_{2}, M_{3}$ having masses $4 \mathrm{~kg}, 6 \mathrm{~kg}$ and 10 kg respectively are hanging from a smooth pully using rope 1,2 and 3 as shown in figure. The tension in the rope $1, \mathrm{~T}_{1}$ when they are moving upward with acceleration of $2 \mathrm{~ms}^{-2}$ is $\qquad$ N (if $g=10 \mathrm{~m} / \mathrm{s}^{2}$ )

The density and breaking stress of a wire are $6 \times$ $10^{4} \mathrm{~kg} / \mathrm{m}^{3}$ and $1.2 \times 10^{8} \mathrm{~N} / \mathrm{m}^{2}$ respectively. The wire is suspended from a rigid support on a planet where acceleration due to gravity is $\frac{1^{\mathrm{rd}}}{3}$ of the value on the surface of earth. The maximum length of the wire with breaking is $\qquad$ m (take, $\mathrm{g}=$ $\left.10 \mathrm{~m} / \mathrm{s}^{2}\right)$

A body moves on a frictionless plane starting from rest. If $\mathrm{S}_{\mathrm{n}}$ is distance moved between $\mathrm{t}=\mathrm{n}-1$ and t $=\mathrm{n}$ and $\mathrm{S}_{\mathrm{n}-1}$ is distance moved between $\mathrm{t}=\mathrm{n}-2$ and $\mathrm{t}=\mathrm{n}-1$, then the ratio $\frac{\mathrm{S}_{\mathrm{n}-1}}{\mathrm{~S}_{\mathrm{n}}}$ is $\left(1-\frac{2}{\mathrm{x}}\right)$ for n $=10$. The value of $x$ is $\qquad$

If three helium nuclei combine to form a carbon nucleus then the energy released in this reaction is $\qquad$ $\times 10^{-2} \mathrm{MeV}$. (Given $1 \mathrm{u}=931 \mathrm{MeV} / \mathrm{c}^{2}$, atomic mass of helium $=4.002603 \mathrm{u}$ )

An ac source is connected in given series LCR circuit. The rms potential difference across the capacitor of $20 \mu \mathrm{~F}$ is $\qquad$ V. $V=50 \sqrt{2} \sin 100 t \text { volt }$

In the experiment to determine the galvanometer resistance by half-deflection method, the plot of $\frac{1}{\theta}$ vs the resistance $(\mathrm{R})$ of the resistance box is shown in the figure. The figure of merit of the galvanometer is $\qquad$ $\times 10^{-1} \mathrm{~A} /$ division. [The source has emf 2V]

Three capacitors of capacitances $25 \mu \mathrm{~F}, 30 \mu \mathrm{~F}$ and $45 \mu \mathrm{~F}$ are connected in parallel to a supply of 100 V. Energy stored in the above combination is E. When these capacitors are connected in series to the same supply, the stored energy is $\frac{9}{x} E$. The value of $x$ is

The incorrect postulates of Dalton's atomic theory are:

(A) Atoms of different elements differ in mass. (B) Matter consists of divisible atoms. (C) Compounds are formed when atoms of different elements combine in a fixed ratio. (D) All the atoms of a given element have different properties including mass. (E) Chemical reactions involve reorganisation of atoms.

Choose the correct answer from the options given below:

The following reaction occurs in the Blast furnance where iron ore is reduced to iron metal

$\mathrm{Fe}_{2} \mathrm{O}_{3(\mathrm{~s})}+3 \mathrm{CO}_{(\mathrm{g})} \rightleftharpoons \mathrm{Fe}_{(\mathrm{l})}+3 \mathrm{CO}_{2(\mathrm{~g})}$ Using the Le-chatelier's principle, predict which one of the

following will not disturb the equilibrium.

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

Assertion (A): Enthalpy of neutralisation of strong monobasic acid with strong monoacidic base is always $-57 \mathrm{~kJ} \mathrm{~mol}^{-1}$ .

Reason (R): Enthalpy of neutralisation is the amount of heat liberated when one mole of $\mathrm{H}^{+}$ ions furnished by acid combine with one mole of $\mathrm{OH}^{-}$ ions furnished by base to form one mole of water.

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

The statement(s) that are correct about the species $\mathrm{O}^{2-}, \mathrm{F}^{-}, \mathrm{Na}^{+}$and $\mathrm{Mg}^{2+}$.

(A) All are isoelectronic

(B) All have the same nuclear charge

(C) $\mathrm{O}^{2-}$ has the largest ionic radii

(D) $\mathrm{Mg}^{2+}$ has the smallest ionic radii

Choose the most appropriate answer from the options given below :

For the compounds:

(A) $\mathrm{H}_{3} \mathrm{C}-\mathrm{CH}_{2}-\mathrm{O}-\mathrm{CH}_{2}-\mathrm{CH}_{2}-\mathrm{CH}_{3}$

(B) $\mathrm{H}_{3} \mathrm{C}-\mathrm{CH}_{2}-\mathrm{CH}_{2}-\mathrm{CH}_{2}-\mathrm{CH}_{3}$

The increasing order of boiling point is : Choose the correct answer from the options given below :

Given below are two statements, one is labelled as Statement I and the other is labelled as Statement II.

Statement I : In group 13, the stability of +1 oxidation state increases down the group.

Statement II : The atomic size of gallium is greater than that of aluminium.

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

Identify 'A' in the following reaction :

The reaction at cathode in the cells commonly used in clocks involves.

Which one of the following complexes will exhibit the least paramagnetic behaviour?

[Atomic number, $\mathrm{Cr}=24, \mathrm{Mn}=25, \mathrm{Fe}=26, \mathrm{Co}=27$ ]

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

Assertion (A): Cis form of alkene is found to be more polar than the trans form.

Reason (R): Dipole moment of trans isomer of 2-butene is zero.

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

Given below are two statements, one is labelled as Statement I and the other is labelled as Statement II.

Statement I : Nitration of benzene involves the following step as shown below

Statement II : Use of Lewis base promotes the electrophilic substitution of benzene. In the light of the above statements, choose the most appropriate answer from the options given below :

The correct order of ligands arranged in increasing field strength.

Which of the following gives a positive test with ninhydrin?

The metal that shows highest and maximum number of oxidation state is:

All organic compound has $42.1 \%$ carbon, $6.4 \%$ hydrogen and remainder is oxygen. If its molecular weight is 342 g/mol , then its molecular formula is

Given below are two statements, one is labelled as Statement I and the other is labelled as Statement II.

Statement I : Bromination of phenol in solvent with low polarity such as $\mathrm{CHCl}_{3}$ or $\mathrm{CS}_{2}$ requires Lewis acid catalyst.

Statement II : The lewis acid catalyst polarises the bromine to generate $\mathrm{Br}^{+}$.

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

Molar ionic conductivities of divalent cation and anion are $57 \mathrm{~S} \mathrm{~cm}^{2} \mathrm{~mol}^{-1}$ and $73 \mathrm{~S} \mathrm{~cm}^{2} \mathrm{~mol}^{-1}$ respectively. The molar conductivity of solution of an electrolyte with the above cation and anion will be :

The number of neutrons present in the more abundant isotope of boron is $x$. Amorphous boron upon heating with air forms a product in which the oxidation state of boron is $y$. The value of $x+y$ is:

The value of Rydberg constant $R_H = 2.18 \times 10^{-18}\,\mathrm{J}$. The velocity of an electron of mass $9.1 \times 10^{-31}\,\mathrm{kg}$ in Bohr's first orbit of hydrogen atom is $\_\_\_\_\times 10^{5}\,\mathrm{m\,s^{-1}}$ (nearest integer).

In a borax bead test under hot condition, a metal salt (one from the given) is heated at point $B$ of the flame, resulted in green colour salt bead. The spin-only magnetic moment value of the salt is $\qquad$ BM (Nearest integer) [Given atomic number of $\mathrm{Cu}=29, \mathrm{Ni}=28$, $\mathrm{Mn}=25, \mathrm{Fe}=26]$

The heat of combustion of solid benzoic acid at constant volume is -321.30 $\mathrm{kJ}$ at $27^{\circ} \mathrm{C}$. The heat of combustion at constant pressure is $(-321.30-x R)$ $\mathrm{kJ}$ , the value of $x$ is $\qquad$

The spin only magnetic moment value of the ion among $\mathrm{Ti}^{2+}, \mathrm{V}^{2+}, \mathrm{Co}^{3+}$ and $\mathrm{Cr}^{2+}$, that acts as strong oxidising agent in aqueous solution is $\qquad$ BM (Near integer). (Given atomic numbers : Ti : 22, V : 23, Cr : 24, Co : 27)

During kinetic study of reaction $2 \mathrm{~A}+\mathrm{B} \rightarrow \mathrm{C}+\mathrm{D}$, the following results were obtained :

Based on above data, the overall order of the reaction is $\qquad$

An artificial cell is made by encapsulating 0.2 M glucose solution within a semipermeable membrane. The osmotic pressure

developed when the artificial cell is placed within a 0.05 M solution of NaCl at 300 K is $\qquad$ $\times 10^{-1}$ bar. (Nearest Integer)

[Given : $\mathrm{R}=0.083 \mathrm{~L} \mathrm{bar} \mathrm{mol}^{-1} \mathrm{~K}^{-1}$ ]

Assume complete dissociation of NaCl

In the lewis dot structure for $\mathrm{NO}_{2}^{-}$, total number of valence electrons around nitrogen is $\qquad$

$9.3 \mathrm{~g}$ of pure aniline is treated with bromine water at room temperature to give a white precipitate of the product ' P '. The mass of product ' P ' obtained is 26.4 g . The percentage yield is $\qquad$ $\%$.