JEE Main 2024 Apr 9 Shift 1 — Question Paper

If the domain of the function $f(x)=\sin^{-1}\left(\frac{x-1}{2x+3}\right)$ is $\mathbf{R}-(\alpha, \beta)$, then $12\alpha\beta$ is equal to :

Let $\alpha, \beta$ be the roots of the equation $x^2+2\sqrt{2}\,x-1=0$. The quadratic equation, whose roots are $\alpha^4+\beta^4$ and $\frac{1}{10}\left(\alpha^6+\beta^6\right)$, is :

Let $\lambda, \mu \in \mathbf{R}$. If the system of equations

$$3x+5y+\lambda z=3$$ $$7x+11y-9z=2$$ $$97x+155y-189z=\mu$$

has infinitely many solutions, then $\mu+2\lambda$ is equal to :

The coefficient of $x^{70}$ in $x^2(1+x)^{98}+x^3(1+x)^{97}+x^4(1+x)^{96}+\ldots+x^{54}(1+x)^{46}$ is $^{99}C_p-{}^{46}C_q$. Then a possible value of $p+q$ is :

If the sum of the series $\frac{1}{1\cdot(1+\mathrm{d})}+\frac{1}{(1+\mathrm{d})(1+2\mathrm{d})}+\ldots+\frac{1}{(1+9\mathrm{d})(1+10\mathrm{d})}$ is equal to 5, then $50\mathrm{d}$ is equal to :

Let $f(x)=ax^3+bx^2+cx+41$ be such that $f(1)=40$, $f'(1)=2$ and $f''(1)=4$. Then $a^2+b^2+c^2$ is equal to :

A variable line L passes through the point $(3, 5)$ and intersects the positive coordinate axes at the points A and B. The minimum area of the triangle OAB, where O is the origin, is :

Let $\int \frac{2-\tan x}{3+\tan x}\,\mathrm{d}x = \frac{1}{2}\left(\alpha x+\log_{\mathrm{e}}|\beta \sin x+\gamma \cos x|\right)+\mathrm{C}$, where C is the constant of integration. Then $\alpha+\frac{\gamma}{\beta}$ is equal to :

The parabola $y^2=4x$ divides the area of the circle $x^2+y^2=5$ in two parts. The area of the smaller part is equal to :

The solution of the differential equation $(x^2+y^2)\mathrm{d}x-5xy\,\mathrm{d}y=0$, $y(1)=0$, is :

The solution curve, of the differential equation $2y\,\frac{\mathrm{d}y}{\mathrm{d}x}+3=5\,\frac{\mathrm{d}y}{\mathrm{d}x}$, passing through the point $(0, 1)$ is a conic, whose vertex lies on the line :

Let $f(x)=x^2+9$, $g(x)=\frac{x}{x-9}$ and $\mathrm{a}=f{\circ}g(10)$, $\mathrm{b}=g{\circ}f(3)$. If e and $l$ denote the eccentricity and the length of the latus rectum of the ellipse $\frac{x^2}{\mathrm{a}}+\frac{y^2}{\mathrm{b}}=1$, then $8\mathrm{e}^2+l^2$ is equal to.

Let a circle passing through $(2, 0)$ have its centre at the point $(\mathrm{h}, \mathrm{k})$. Let $(x_{\mathrm{c}}, y_{\mathrm{c}})$ be the point of intersection of the lines $3x+5y=1$ and $(2+\mathrm{c})x+5\mathrm{c}^2y=1$. If $\mathrm{h}=\lim\limits_{\mathrm{c}\to 1} x_{\mathrm{c}}$ and $\mathrm{k}=\lim\limits_{\mathrm{c}\to 1} y_{\mathrm{c}}$, then the equation of the circle is :

A ray of light coming from the point $\mathrm{P}(1, 2)$ gets reflected from the point Q on the $x$-axis and then passes through the point $\mathrm{R}(4, 3)$. If the point $\mathrm{S}(\mathrm{h}, \mathrm{k})$ is such that PQRS is a parallelogram, then $\mathrm{hk}^2$ is equal to :

Let the line L intersect the lines $x-2=-y=z-1$, $2(x+1)=2(y-1)=z+1$ and be parallel to the line $\frac{x-2}{3}=\frac{y-1}{1}=\frac{z-2}{2}$. Then which of the following points lies on L?

The shortest distance between the lines $\frac{x-3}{4}=\frac{y+7}{-11}=\frac{z-1}{5}$ and $\frac{x-5}{3}=\frac{y-9}{-6}=\frac{z+2}{1}$ is :

Let $\vec{\mathrm{OA}}=2\vec{a}$, $\vec{\mathrm{OB}}=6\vec{a}+5\vec{b}$ and $\vec{\mathrm{OC}}=3\vec{b}$, where O is the origin. If the area of the parallelogram with adjacent sides $\vec{\mathrm{OA}}$ and $\vec{\mathrm{OC}}$ is 15 sq. units, then the area (in sq. units) of the quadrilateral OABC is equal to :

Let three vectors $\vec{a}=\alpha\hat{i}+4\hat{j}+2\hat{k}$, $\vec{b}=5\hat{i}+3\hat{j}+4\hat{k}$, $\vec{c}=x\hat{i}+y\hat{j}+z\hat{k}$ form a triangle such that $\vec{c}=\vec{a}-\vec{b}$ and the area of the triangle is $5\sqrt{6}$. If $\alpha$ is a positive real number, then $|\vec{c}|^2$ is equal to :

The frequency distribution of the age of students in a class of 40 students is given below.

Age	15	16	17	18	19	20
No of Students	5	8	5	12	$x$	$y$

If the mean deviation about the median is 1.25, then $4x+5y$ is equal to :

Let $|\cos\theta\,\cos(60-\theta)\,\cos(60+\theta)|\le\frac{1}{8}$, $\theta\in[0, 2\pi]$. Then, the sum of all $\theta\in[0, 2\pi]$, where $\cos 3\theta$ attains its maximum value, is :

Let $\mathrm{A}=\{2, 3, 6, 7\}$ and $\mathrm{B}=\{4, 5, 6, 8\}$. Let R be a relation defined on $\mathrm{A}\times\mathrm{B}$ by $(\mathrm{a}_1, \mathrm{b}_1)\,\mathrm{R}\,(\mathrm{a}_2, \mathrm{b}_2)$ if and only if $\mathrm{a}_1+\mathrm{a}_2=\mathrm{b}_1+\mathrm{b}_2$. Then the number of elements in R is ________.

The sum of the square of the modulus of the elements in the set $\{z=\mathrm{a}+\mathrm{ib} : \mathrm{a}, \mathrm{b}\in\mathbf{Z},\ z\in\mathbf{C},\ |z-1|\le 1,\ |z-5|\le|z-5\mathrm{i}|\}$ is ________.

Let A be a non-singular matrix of order 3. If $\det(3\,\mathrm{adj}(2\,\mathrm{adj}((\det \mathrm{A})\mathrm{A})))=3^{-13}\cdot 2^{-10}$ and $\det(3\,\mathrm{adj}(2\mathrm{A}))=2^{\mathrm{m}}\cdot 3^{\mathrm{n}}$, then $|3\mathrm{m}+2\mathrm{n}|$ is equal to ________.

Let $f:(0, \pi)\to\mathbf{R}$ be a function given by $f(x)=\begin{cases}\left(\frac{8}{7}\right)^{\frac{\tan 8x}{\tan 7x}}, & 0<x<\frac{\pi}{2}\\ \mathrm{a}-8, & x=\frac{\pi}{2}\\ \left(1+|\cot x|\right)^{\frac{\mathrm{b}}{\mathrm{a}}|\tan x|}, & \frac{\pi}{2}<x<\pi\end{cases}$

where $\mathrm{a}, \mathrm{b}\in\mathbf{Z}$. If $f$ is continuous at $x=\frac{\pi}{2}$, then $\mathrm{a}^2+\mathrm{b}^2$ is equal to ________.

The remainder when $428^{2024}$ is divided by 21 is ________.

If a function $f$ satisfies $f(\mathrm{m}+\mathrm{n})=f(\mathrm{m})+f(\mathrm{n})$ for all $\mathrm{m}, \mathrm{n}\in\mathbf{N}$ and $f(1)=1$, then the largest natural number $\lambda$ such that $\sum\limits_{\mathrm{k}=1}^{2022} f(\lambda+\mathrm{k})\le(2022)^2$ is equal to ________.

Let the set of all positive values of $\lambda$, for which the point of local minimum of the function $(1+x(\lambda^2-x^2))$ satisfies $\frac{x^2+x+2}{x^2+5x+6}<0$, be $(\alpha, \beta)$. Then $\alpha^2+\beta^2$ is equal to ________.

Let $\lim\limits_{\mathrm{n}\to\infty}\left(\frac{\mathrm{n}}{\sqrt{\mathrm{n}^4+1}}-\frac{2\mathrm{n}}{(\mathrm{n}^2+1)\sqrt{\mathrm{n}^4+1}}+\frac{\mathrm{n}}{\sqrt{\mathrm{n}^4+16}}-\frac{8\mathrm{n}}{(\mathrm{n}^2+4)\sqrt{\mathrm{n}^4+16}}+\ldots+\frac{\mathrm{n}}{\sqrt{\mathrm{n}^4+\mathrm{n}^4}}-\frac{2\mathrm{n}\cdot\mathrm{n}^2}{(\mathrm{n}^2+\mathrm{n}^2)\sqrt{\mathrm{n}^4+\mathrm{n}^4}}\right)$ be $\frac{\pi}{\mathrm{k}}$, using only the principal values of the inverse trigonometric functions. Then $\mathrm{k}^2$ is equal to ________.

Let the centre of a circle, passing through the points $(0, 0)$, $(1, 0)$ and touching the circle $x^2+y^2=9$, be $(\mathrm{h}, \mathrm{k})$. Then for all possible values of the coordinates of the centre $(\mathrm{h}, \mathrm{k})$, $4(\mathrm{h}^2+\mathrm{k}^2)$ is equal to ________.

Let a, b and c denote the outcome of three independent rolls of a fair tetrahedral die, whose four faces are marked 1, 2, 3, 4. If the probability that $ax^2+bx+c=0$ has all real roots is $\frac{\mathrm{m}}{\mathrm{n}}$, $\gcd(\mathrm{m}, \mathrm{n})=1$, then $\mathrm{m}+\mathrm{n}$ is equal to ________.

The dimensional formula of latent heat is :

A particle moving in a straight line covers half the distance with speed 6 m/s. The other half is covered in two equal time intervals with speeds 9 m/s and 15 m/s respectively. The average speed of the particle during the motion is :

A heavy iron bar, of weight W is having its one end on the ground and the other on the shoulder of a person. The bar makes an angle $\theta$ with the horizontal. The weight experienced by the person is :

A light unstretchable string passing over a smooth light pulley connects two blocks of masses $m_1$ and $m_2$. If the acceleration of the system is $\frac{g}{8}$, then the ratio of the masses $\frac{m_2}{m_1}$ is :

A particle of mass m moves on a straight line with its velocity increasing with distance according to the equation $v = \alpha\sqrt{x}$, where $\alpha$ is a constant. The total work done by all the forces applied on the particle during its displacement from $x = 0$ to $x = d$, will be :

An astronaut takes a ball of mass m from earth to space. He throws the ball into a circular orbit about earth at an altitude of 318.5 km. From earth's surface to the orbit, the change in total mechanical energy of the ball is $x\dfrac{GM_e m}{21R_e}$. The value of $x$ is (take $R_e = 6370$ km) :

A sphere of relative density $\sigma$ and diameter D has concentric cavity of diameter d. The ratio of $\dfrac{D}{d}$, if it just floats on water in a tank is :

A sample of 1 mole gas at temperature T is adiabatically expanded to double its volume. If adiabatic constant for the gas is $\gamma = \dfrac{3}{2}$, then the work done by the gas in the process is :

The volume of an ideal gas ($\gamma = 1.5$) is changed adiabatically from 5 litres to 4 litres. The ratio of initial pressure to final pressure is :

A capacitor is made of a flat plate of area A and a second plate having a stair-like structure as shown in figure. If the area of each stair is $\dfrac{A}{3}$ and the height is d, the capacitance of the arrangement is :

[Figure: a flat bottom plate of area A; above it a stair-like top plate with three steps each of area A/3, at heights d, 2d and 3d from the bottom plate]

The equivalent resistance between A and B is :

[Figure: resistor network between terminals A and B: top row has 6 Ω and 10 Ω in series; left side has 8 Ω and 4 Ω; right side has 5 Ω and 7 Ω; bottom row has 11 Ω and 8 Ω in series; a middle vertical branch crosses over the horizontal middle wire]

Given below are two statements :

Statement (I) : When currents vary with time, Newton's third law is valid only if momentum carried by the electromagnetic field is taken into account.

Statement (II) : Ampere's circuital law does not depend on Biot-Savart's law.

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

A bulb and a capacitor are connected in series across an ac supply. A dielectric is then placed between the plates of the capacitor. The glow of the bulb :

A plane EM wave is propagating along $x$ direction. It has a wavelength of 4 mm. If electric field is in $y$ direction with the maximum magnitude of $60\ \mathrm{Vm^{-1}}$, the equation for magnetic field is :

Given below are two statements :

Statement (I) : When an object is placed at the centre of curvature of a concave lens, image is formed at the centre of curvature of the lens on the other side.

Statement (II) : Concave lens always forms a virtual and erect image.

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

A proton, an electron and an alpha particle have the same energies. Their de-Broglie wavelengths will be compared as :

The energy equivalent of 1 g of substance is :

A light emitting diode (LED) is fabricated using GaAs semiconducting material whose band gap is 1.42 eV. The wavelength of light emitted from the LED is :

One main scale division of a vernier caliper is equal to m units. If $n^{\text{th}}$ division of main scale coincides with $(n+1)^{\text{th}}$ division of vernier scale, the least count of the vernier caliper is :

A galvanmeter has a coil of resistance $200\ \Omega$ with a full scale deflection at $20\ \mu\mathrm{A}$. The value of resistance to be added to use it as an ammeter of range (0-20) mA is :

If $\vec{a}$ and $\vec{b}$ makes an angle $\cos^{-1}\left(\dfrac{5}{9}\right)$ with each other, then $\left|\vec{a} + \vec{b}\right| = \sqrt{2}\left|\vec{a} - \vec{b}\right|$ for $\left|\vec{a}\right| = n\left|\vec{b}\right|$. The integer value of n is ________.

A string is wrapped around the rim of a wheel of moment of inertia $0.40\ \mathrm{kgm^2}$ and radius 10 cm. The wheel is free to rotate about its axis. Initially the wheel is at rest. The string is now pulled by a force of 40 N. The angular velocity of the wheel after 10 s is $x$ rad/s, where $x$ is ________.

Two persons pull a wire towards themselves. Each person exerts a force of 200 N on the wire. Young's modulus of the material of wire is $1 \times 10^{11}\ \mathrm{N\,m^{-2}}$. Original length of the wire is 2 m and the area of cross section is $2\ \mathrm{cm^2}$. The wire will extend in length by ________ $\mu\mathrm{m}$.

The position, velocity and acceleration of a particle executing simple harmonic motion are found to have magnitudes of 4 m, $2\ \mathrm{ms^{-1}}$ and $16\ \mathrm{ms^{-2}}$ at a certain instant. The amplitude of the motion is $\sqrt{x}$ m where $x$ is ________.

At the centre of a half ring of radius $R = 10$ cm and linear charge density $4\ \mathrm{nC\ m^{-1}}$, the potential is $x\pi$ V. The value of $x$ is ________.

The current flowing through the $1\ \Omega$ resistor is $\dfrac{n}{10}$ A. The value of n is ________.

[Figure: diamond-shaped circuit A-B-C-D-E: a 10 V battery with 4 Ω and 1 Ω resistors on branch A to C (current I₁ marked through the 1 Ω), a 2 Ω resistor from B to C and another 2 Ω from C to D, 4 Ω on branch A to D, and a 5 V battery at E on the right connecting B and D]

A square loop of edge length 2 m carrying current of 2 A is placed with its edges parallel to the $x$-$y$ axis. A magnetic field is passing through the $x$-$y$ plane and expressed as $\vec{B} = B_0(1 + 4x)\hat{k}$, where $B_0 = 5$ T. The net magnetic force experienced by the loop is ________ N.

When a coil is connected across a 20 V dc supply, it draws a current of 5 A. When it is connected across 20 V, 50 Hz ac supply, it draws a current of 4 A. The self inductance of the coil is ________ mH. (Take $\pi = 3$)

In a Young's double slit experiment, the intensity at a point is $\left(\dfrac{1}{4}\right)^{\text{th}}$ of the maximum intensity, the minimum distance of the point from the central maximum is ________ $\mu\mathrm{m}$.

(Given : $\lambda = 600$ nm, $d = 1.0$ mm, $D = 1.0$ m)

A star has 100% helium composition. It starts to convert three $^{4}\mathrm{He}$ into one $^{12}\mathrm{C}$ via triple alpha process as $^{4}\mathrm{He} + {}^{4}\mathrm{He} + {}^{4}\mathrm{He} \rightarrow {}^{12}\mathrm{C} + Q$. The mass of the star is $2.0 \times 10^{32}$ kg and it generates energy at the rate of $5.808 \times 10^{30}$ W. The rate of converting these $^{4}\mathrm{He}$ to $^{12}\mathrm{C}$ is $n \times 10^{42}\ \mathrm{s^{-1}}$, where n is ________.

[ Take, mass of $^{4}\mathrm{He} = 4.0026$ u, mass of $^{12}\mathrm{C} = 12$ u]

Compare the energies of following sets of quantum numbers for multielectron system. (A) $n = 4, l = 1$ (B) $n = 4, l = 2$ (C) $n = 3, l = 1$ (D) $n = 3, l = 2$ (E) $n = 4, l = 0$ Choose the correct answer from the options given below :

In which one of the following pairs the central atoms exhibit $\mathrm{sp^2}$ hybridization ?

0.05M $\mathrm{CuSO_4}$ when treated with 0.01M $\mathrm{K_2Cr_2O_7}$ gives green colour solution of $\mathrm{Cu_2Cr_2O_7}$. The two solutions are separated as shown below : [SPM : Semi Permeable Membrane]

[Figure: two compartments separated by a semi permeable membrane; Side X contains $\mathrm{K_2Cr_2O_7}$ solution, Side Y contains $\mathrm{CuSO_4}$ solution]

Due to osmosis :

The molar conductivity for electrolytes A and B are plotted against $\mathrm{C^{1/2}}$ as shown below. Electrolytes A and B respectively are :

[Figure: plot of $\Lambda_m\ (\mathrm{S\ cm^2\ mol^{-1}})$ vs $\mathrm{C^{1/2}\ (mol\ L^{-1})^{1/2}}$; curve A falls steeply from about 400 at very low concentration, curve B decreases slowly and nearly linearly from about 200]

Given below are two statements : Statement (I) : The oxidation state of an element in a particular compound is the charge acquired by its atom on the basis of electron gain enthalpy consideration from other atoms in the molecule. Statement (II) : $\mathrm{p\pi - p\pi}$ bond formation is more prevalent in second period elements over other periods. In the light of the above statements, choose the most appropriate answer from the options given below :

Given below are two statements : one is labelled as Assertion (A) and the other is labelled as Reason (R). Assertion (A) : Both rhombic and monoclinic sulphur exist as $\mathrm{S_8}$ while oxygen exists as $\mathrm{O_2}$. Reason (R) : Oxygen forms $\mathrm{p\pi - p\pi}$ multiple bonds with itself and other elements having small size and high electronegativity like C, N, which is not possible for sulphur. In the light of the above statements, choose the most appropriate answer from the options given below :

The $\mathrm{F^-}$ ions make the enamel on teeth much harder by converting hydroxyapatite (the enamel on the surface of teeth) into much harder fluoroapatite having the formula.

The electronic configuration of Cu(II) is $\mathrm{3d^9}$ whereas that of Cu(I) is $\mathrm{3d^{10}}$. Which of the following is correct ?

Given below are two statements : one is labelled as Assertion (A) and the other is labelled as Reason (R). Assertion (A) : The total number of geometrical isomers shown by $\mathrm{[Co(en)_2Cl_2]^+}$ complex ion is three. Reason (R) : $\mathrm{[Co(en)_2Cl_2]^+}$ complex ion has an octahedral geometry. In the light of the above statements, choose the most appropriate answer from the options given below :

Methods used for purification of organic compounds are based on :

Relative stability of the contributing structures is :

$$\mathrm{CH_2{=}CH{-}CH{=}O\ \,(I)\ \longleftrightarrow\ \overset{+}{C}H_2{-}CH{=}CH{-}\overset{-}{O}\ \,(II)\ \longleftrightarrow\ \overset{-}{C}H_2{-}CH{=}CH{-}\overset{+}{O}\ \,(III)}$$

(Resonance structures of prop-2-enal: (I) neutral structure with $\mathrm{C{=}O}$; (II) positive charge on terminal $\mathrm{CH_2}$ carbon and negative charge on oxygen; (III) negative charge on terminal $\mathrm{CH_2}$ carbon and positive charge on oxygen)

[Figure: benzene + succinic anhydride $\xrightarrow{\mathrm{AlCl_3}}$ A $\xrightarrow[\mathrm{HCl}]{\mathrm{Zn{-}Hg}}$ B $\xrightarrow{\text{Conc. } \mathrm{H_2SO_4}}$ C]

What is the structure of C ?

Given below are two statements : one is labelled as Assertion (A) and the other is labelled as Reason (R). Assertion (A) : $\mathrm{S_N2}$ reaction of $\mathrm{C_6H_5CH_2Br}$ occurs more readily than the $\mathrm{S_N2}$ reaction of $\mathrm{CH_3CH_2Br}$. Reason (R) : The partially bonded unhybridized p-orbital that develops in the trigonal bipyramidal transition state is stabilized by conjugation with the phenyl ring. In the light of the above statements, choose the most appropriate answer from the options given below :

In the following sequence of reaction, the major products B and C respectively are :

[Figure: 1-bromo-3-chlorocyclobutane (Cl and Br on opposite corners of a cyclobutane ring) $\xrightarrow{\mathrm{Na/Et_2O}}$ A ; A $\xrightarrow[\mathrm{(ii)\ D_2O}]{\mathrm{(i)\ Mg/Et_2O}}$ B ; A $\xrightarrow{\mathrm{CoF_2}}$ C]

Identify major product "X" formed in the following reaction :

$$\mathrm{C_6H_6 \xrightarrow[\text{Anhydrous } AlCl_3/CuCl]{CO,\ HCl} X\ (Major\ Product)}$$

Identify the product A and product B in the following set of reactions.

$$\mathrm{CH_3{-}CH{=}CH_2 \xrightarrow{H_2O,\ H^+} major\ Product\ A}$$ $$\mathrm{CH_3{-}CH{=}CH_2 \xrightarrow[H_2O,\ H_2O_2,\ \bar{O}H]{(BH_3)_2} major\ Product\ B}$$