JEE Main 2024 Jan 31 Shift 1 — Question Paper

If $f(x)=\frac{4x+3}{6x-4}, x \neq \frac{2}{3}$ and $(fof)(x)=g(x)$, where $g: \mathbb{R}-\left\{\frac{2}{3}\right\} \rightarrow \mathbb{R}-\left\{\frac{2}{3}\right\}$, then $(gogog)(4)$ is equal to

Let S be the set of postive integral values of $a$ for which $\frac{ax^2+2(a+1)x+9a+4}{x^2-8x+32}<0, \forall x \in \mathbb{R}$. Then, the number of elements in S is :

If $f(x)=\begin{vmatrix} x^3 & 2x^2+1 & 1+3x \\ 3x^2+2 & 2x & x^3+6 \\ x^3-x & 4 & x^2-2 \end{vmatrix}$ for all $x \in \mathbb{R}$, then $2f(0)+f'(0)$ is equal to

If the system of linear equations

$x-2y+z=-4$ $2x+\alpha y+3z=5$ $3x-y+\beta z=3$

has infinitely many solutions, then $12\alpha+13\beta$ is equal to

The sum of the series $\frac{1}{1-3 \cdot 1^2+1^4}+\frac{2}{1-3 \cdot 2^2+2^4}+\frac{3}{1-3 \cdot 3^2+3^4}+\ldots$ up to 10-terms is

$\lim_{x \to 0} \frac{e^{2|\sin x|}-2|\sin x|-1}{x^2}$

Let $a$ be the sum of all coefficients in the expansion of $(1-2x+2x^2)^{2023}(3-4x^2+2x^3)^{2024}$ and $b=\lim_{x \to 0}\left(\frac{\int_0^x \frac{\log(1+t)}{t^{2024}+1}dt}{x^2}\right)$. If the equations $cx^2+dx+e=0$ and $2bx^2+ax+4=0$ have a common root, where $c, d, e \in \mathbb{R}$, then $\mathrm{d}:\mathrm{c}:\mathrm{e}$ equals

For $0<c<b<a$, let $(a+b-2c)x^2+(b+c-2a)x+(c+a-2b)=0$ and $\alpha \neq 1$ be one of its root. Then, among the two statements

(I) If $\alpha \in (-1, 0)$, then $b$ cannot be the geometric mean of $a$ and $c$

(II) If $\alpha \in (0, 1)$, then $b$ may be the geometric mean of $a$ and $c$

Let $g(x)$ be a linear function and $f(x)=\begin{cases} g(x) & , x \leq 0 \\ \left(\frac{1+x}{2+x}\right)^{\frac{1}{x}} & , x>0 \end{cases}$, is continuous at $x=0$. If $f'(1)=f(-1)$, then the value $g(3)$ is

The area of the region $\left\{(x, y): y^2 \leq 4x, x<4, \frac{xy(x-1)(x-2)}{(x-3)(x-4)}>0, x \neq 3\right\}$ is

Let $y=y(x)$ be the solution of the differential equation $\frac{dy}{dx}=\frac{(\tan x)+y}{\sin x(\sec x-\sin x \tan x)}, x \in \left(0, \frac{\pi}{2}\right)$ satisfying the condition $y\left(\frac{\pi}{4}\right)=2$. Then, $y\left(\frac{\pi}{3}\right)$ is

The solution curve of the differential equation $y\frac{dx}{dy}=x(\log_e x-\log_e y+1), x>0, y>0$ passing through the point $(e, 1)$ is

If one of the diameters of the circle $x^2+y^2-10x+4y+13=0$ is a chord of another circle C, whose center is the point of intersection of the lines $2x+3y=12$ and $3x-2y=5$, then the radius of the circle C is :

Let $\alpha, \beta, \gamma, \delta \in \mathbb{Z}$ and let $A(\alpha, \beta)$, $B(1,0)$, $C(\gamma, \delta)$ and $D(1,2)$ be the vertices of a parallelogram ABCD. If $AB=\sqrt{10}$ and the points A and C lie on the line $3y=2x+1$, then $2(\alpha+\beta+\gamma+\delta)$ is equal to

The distance of the point Q(0,2,−2) form the line passing through the point P(5, −4, 3) and perpendicular to the lines $\vec{r}=(-3\hat{i}+2\hat{k})+\lambda(2\hat{i}+3\hat{j}+5\hat{k})$, $\lambda \in \mathbb{R}$ and $\vec{r}=(\hat{i}-2\hat{j}+\hat{k})+\mu(-\hat{i}+3\hat{j}+2\hat{k})$, $\mu \in \mathbb{R}$ is :

If the foci of a hyperbola are same as that of the ellipse $\frac{x^2}{9}+\frac{y^2}{25}=1$ and the eccentricity of the hyperbola is $\frac{15}{8}$ times the eccentricity of the ellipse, then the smaller focal distance of the point $\left(\sqrt{2}, \frac{14}{3}\sqrt{\frac{2}{5}}\right)$ on the hyperbola, is equal to

Three rotten apples are accidently mixed with fifteen good apples. Assuming the random variable $x$ to be the number of rotten apples in a draw of two apples, the variance of $x$ is

Let $\vec{a}=3\hat{i}+\hat{j}-2\hat{k}$, $\vec{b}=4\hat{i}+\hat{j}+7\hat{k}$ and $\vec{c}=\hat{i}-3\hat{j}+4\hat{k}$ be three vectors. If a vectors $\vec{p}$ satisfies $\vec{p} \times \vec{b}=\vec{c} \times \vec{b}$ and $\vec{p} \cdot \vec{a}=0$, then $\vec{p} \cdot (\hat{i}-\hat{j}-\hat{k})$ is equal to

Two marbles are drawn in succession from a box containing 10 red, 30 white, 20 blue and 15 orange marbles, with replacement being made after each drawing. Then the probability, that first drawn marble is red and second drawn marble is white, is

For $\alpha, \beta, \gamma \neq 0$, if $\sin^{-1}\alpha+\sin^{-1}\beta+\sin^{-1}\gamma=\pi$ and $(\alpha+\beta+\gamma)(\alpha-\gamma+\beta)=3\alpha\beta$, then $\gamma$ equals

Let $A=\{1, 2, 3, 4\}$ and $R=\{(1, 2), (2,3), (1,4)\}$ be a relation on A. Let S be the equivalence relation on A such that $R \subset S$ and the number of elements in S is n. Then, the minimum value of n is ______

If $\alpha$ denotes the number of solutions of $|1-i|^x=2^x$ and $\beta=\left(\frac{|z|}{\arg(z)}\right)$, where $z=\frac{\pi}{4}(1+i)^4\left[\frac{1-\sqrt{\pi}\,i}{\sqrt{\pi}+i}+\frac{\sqrt{\pi}-i}{1+\sqrt{\pi}\,i}\right]$, $i=\sqrt{-1}$, then the distance of the point $(\alpha, \beta)$ from the line $4x-3y=7$ is _____

The total number of words (with or without meaning) that can be formed out of the letters of the word 'DISTRIBUTION' taken four at a time, is equal to _____

In the expansion of $(1+x)(1-x^2)\left(1+\frac{3}{x}+\frac{3}{x^2}+\frac{1}{x^3}\right)^5, x \neq 0$, the sum of the coefficients of $x^3$ and $x^{-13}$ is equal to _____

Let $S=(-1, \infty)$ and $f: S \rightarrow \mathbb{R}$ be defined as

$$f(x)=\int_{-1}^{x}\left(e^t-1\right)^{11}(2t-1)^5(t-2)^7(t-3)^{12}(2t-10)^{61}\,dt,$$

Let p = Sum of squares of the values of $x$, where $f(x)$ attains local maxima on S, and q = Sum of the values of $x$, where $f(x)$ attains local minima on S. Then, the value of $\mathrm{p}^2+2\mathrm{q}$ is ________

Let $f: \mathbb{R} \rightarrow \mathbb{R}$ be a function defined by $f(x)=\frac{4^x}{4^x+2}$ and $M=\int_{f(a)}^{f(1-a)} x\sin^4(x(1-x))\,dx$, $N=\int_{f(a)}^{f(1-a)} \sin^4(x(1-x))\,dx$; $a \neq \frac{1}{2}$. If $\alpha M=\beta N, \alpha, \beta \in \mathbb{N}$, then the least value of $\alpha^2+\beta^2$ is equal to ______

If the integral $525\int_{0}^{\frac{\pi}{2}} \sin 2x \cos^{\frac{11}{2}}x\left(1+Cos^{\frac{5}{2}}x\right)^{\frac{1}{2}} dx$ is equal to $(n\sqrt{2}-64)$, then $n$ is equal to _______

Let the foci and length of the latus rectum of an ellipse $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1, a>b$ be $(\pm 5, 0)$ and $\sqrt{50}$, respectively. Then, the square of the eccentricity of the hyperbola $\frac{x^2}{b^2}-\frac{y^2}{a^2b^2}=1$ equals

Let Q and R be the feet of perpendiculars from the point P($a$, $a$, $a$) on the lines $x=y, z=1$ and $x=-y, z=-1$ respectively. If $\angle$QPR is a right angle, then $12a^2$ is equal to _____

Let $\vec{a}$ and $\vec{b}$ be two vectors such that $|\vec{a}|=1, |\vec{b}|=4$, and $\vec{a} \cdot \vec{b}=2$. If $\vec{c}=(2\vec{a} \times \vec{b})-3\vec{b}$ and the angle between $\vec{b}$ and $\vec{c}$ is $\alpha$, then $192\sin^2\alpha$ is equal to ______

A force is represented by $F = ax^2 + bt^{\frac{1}{2}}$

where $x$ = distance and $t$ = time. The dimensions of $b^2/a$ are:

The relation between time $'t'$ and distance $'x'$ is $t = \alpha x^2 + \beta x$, where $\alpha$ and $\beta$ are constants. The relation between acceleration $(a)$ and velocity $(v)$ is :

A coin is placed on a disc. The coefficient of friction between the coin and the disc is $\mu$. If the distance of the coin from the center of the disc is $r$, the maximum angular velocity which can be given to the disc, so that the coin does not slip away, is :

In the given arrangement of a doubly inclined plane two blocks of masses $M$ and $m$ are placed. The blocks are connected by a light string passing over an ideal pulley as shown. The coefficient of friction between the surface of the plane and the blocks is 0.25. The value of $m$, for which $M = 10$ kg will move down with an acceleration of $2\ \mathrm{m/s^2}$, is: (take $g = 10\ \mathrm{m/s^2}$ and $\tan 37° = 3/4$)

[Figure: double inclined plane with block M on the 53° incline and block m on the 37° incline, connected by a string over a pulley at the apex]

An artillery piece of mass $M_1$ fires a shell of mass $M_2$ horizontally. Instantaneously after the firing, the ratio of kinetic energy of the artillery and that of the shell is:

Four identical particles of mass $m$ are kept at the four corners of a square. If the gravitational force exerted on one of the masses by the other masses is $\left(\frac{2\sqrt{2}+1}{32}\right)\frac{Gm^2}{L^2}$, the length of the sides of the square is

A small steel ball is dropped into a long cylinder containing glycerine. Which one of the following is the correct representation of the velocity time graph for the transit of the ball?

The given figure represents two isobaric processes for the same mass of an ideal gas, then

[Figure: V–T graph with two straight lines from origin O, line $P_2$ steeper (above) line $P_1$]

The parameter that remains the same for molecules of all gases at a given temperature is :

Two charges $q$ and $3q$ are separated by a distance $'r'$ in air. At a distance $x$ from charge $q$, the resultant electric field is zero. The value of $x$ is :

Two conductors have the same resistances at $0°\mathrm{C}$ but their temperature coefficients of resistance are $\alpha_1$ and $\alpha_2$. The respective temperature coefficients for their series and parallel combinations are :

A rigid wire consists of a semicircular portion of radius $R$ and two straight sections. The wire is partially immerged in a perpendicular magnetic field $B = B_0\,\hat{j}$ as shown in figure. The magnetic force on the wire if it has a current $i$ is:

[Figure: U-shaped wire with semicircular top of radius R inside a region of magnetic field pointing out of the page (dots), with x–y axes shown; the two straight sections extend downward out of the field region]

A coil is places perpendicular to a magnetic field of 5000 T. When the field is changed to 3000 T in 2 s, an induced emf of 22 V is produced in the coil. If the diameter of the coil is 0.02 m, then the number of turns in the coil is:

In a plane EM wave, the electric field oscillates sinusoidally at a frequency of $5 \times 10^{10}$ Hz and an amplitude of $50\ \mathrm{Vm^{-1}}$. The total average energy density of the electromagnetic field of the wave is : [Use $\varepsilon_0 = 8.85 \times 10^{-12}\ \mathrm{C^2/Nm^2}$]

The refractive index of a prism with apex angle $A$ is $\cot A/2$. The angle of minimum deviation is :

When a metal surface is illuminated by light of wavelength $\lambda$, the stopping potential is 8V. When the same surface is illuminated by light of wavelength $3\lambda$, stopping potential is 2V. The threshold wavelength for this surface is:

If the wavelength of the first member of Lyman series of hydrogen is $\lambda$. The wavelength of the second member will be

Identify the logic operation performed by the given circuit.

[Figure: logic circuit — inputs A and B each pass through a NOT gate (inverter), whose outputs feed a NAND gate giving output Y]

The fundamental frequency of a closed organ pipe is equal to the first overtone frequency of an open organ pipe. If length of the open pipe is 60 cm, the length of the closed pipe will be:

If the percentage errors in measuring the length and the diameter of a wire are 0.1% each. The percentage error in measuring its resistance will be:

A body starts falling freely from height $H$ hits an inclined plane in its path at height $h$. As a result of this perfectly elastic impact, the direction of the velocity of the body becomes horizontal. The value of $\frac{H}{h}$ for which the body will take the maximum time to reach the ground is ________.

A solid circular disc of mass 50 kg rolls along a horizontal floor so that its center of mass has a speed of 0.4 m/s. The absolute value of work done on the disc to stop it is __________ J.

The depth below the surface of sea to which a rubber ball be taken so as to decrease its volume by 0.02% is _______ $m$.

(Take density of sea water $= 10^3\ \mathrm{kgm^{-3}}$, Bulk modulus of rubber $= 9 \times 10^8\ \mathrm{Nm^{-2}}$, and $g = 10\ \mathrm{ms^{-2}}$)

A particle performs simple harmonic motion with amplitude $A$. Its speed is increased to three times at an instant when its displacement is $\frac{2A}{3}$. The new amplitude of motion is $\frac{nA}{3}$. The value of $n$ is ________.

A parallel plate capacitor with plate separation 5 mm is charged up by a battery. It is found that on introducing a dielectric sheet of thickness 2 mm , while keeping the battery connections intact, the capacitor draws 25% more charge from the battery than before. The dielectric constant of the sheet is ________.

Equivalent resistance of the following network is ________ $\Omega$.

[Figure: resistor network between terminals A and B — three parallel branches ($R_1$ 6 $\Omega$ shown with $R_2$, $R_3$ 2 $\Omega$, and $R_5$ 3 $\Omega$ with $R_6$ 3 $\Omega$) linked by series resistors of 2 $\Omega$ ($R_2$ top-left) and 2 $\Omega$ ($R_4$ top-right)]

An electron moves through a uniform magnetic field $\vec{B} = B_0\,\hat{i} + 2B_0\,\hat{j}\ T$. At a particular instant of time, the velocity of electron is $\vec{u} = 3\hat{i} + 5\hat{j}\ \mathrm{m/s}$. If the magnetic force acting on electron is $\vec{F} = 5e\,\hat{k}\ N$, where $e$ is the charge of electron, then the value of $B_0$ is ________ $T$.

A small square loop of wire of side $l$ is placed inside a large square loop of wire of side $L$ $(L = l^2)$. The loops are coplanar and their centers coincide. The value of the mutual inductance of the system is $\sqrt{x} \times 10^{-7}\ H$, where $x =$ ______.

Two waves of intensity ratio 1:9 cross each other at a point. The resultant intensities at that point, when (a) Waves are incoherent is $I_1$ (b) Waves are coherent is $I_2$ and differ in phase by $60°$. If $\frac{I_1}{I_2} = \frac{10}{x}$ then $x =$ ______.

The mass defect in a particular reaction is 0.4g. The amount of energy liberated is $n \times 10^7$ kWh, where $n =$ ______.

(speed of light $= 3 \times 10^8$ m/s)

The linear combination of atomic orbitals to form molecular orbitals takes place only when the combining atomic orbitals

A. have the same energy B. have the minimum overlap C. have same symmetry about the molecular axis D. have different symmetry about the molecular axis

Choose the most appropriate from the options given below:

Identify the mixture that shows positive deviations from Raoult's Law

For the given reaction, choose the correct expression of $\mathrm{K_C}$ from the following :-

$$\mathrm{Fe^{3+}_{(aq)} + SCN^-_{(aq)} \rightleftharpoons (FeSCN)^{2+}_{(aq)}}$$

Identify the factor from the following that does not affect electrolytic conductance of a solution.

Integrated rate law equation for a first order gas phase reaction is given by

(where $\mathrm{P_i}$ is initial pressure and $\mathrm{P_t}$ is total pressure at time $t$)

The correct sequence of electron gain enthalpy of the elements listed below is

A. Ar B. Br C. F D. S

Choose the most appropriate from the options given below:

Consider the oxides of group 14 elements

$\mathrm{SiO_2}$, $\mathrm{GeO_2}$, $\mathrm{SnO_2}$, $\mathrm{PbO_2}$, $\mathrm{CO}$ and $\mathrm{GeO}$. The amphoteric oxides are

Identify correct statements from below:

A. The chromate ion is square planar. B. Dichromates are generally prepared from chromates. C. The green manganate ion is diamagnetic. D. Dark green coloured $\mathrm{K_2MnO_4}$ disproportionates in a neutral or acidic medium to give permanganate. E. With increasing oxidation number of transition metal, ionic character of the oxides decreases.

Choose the correct answer from the options given below:

The metals that are employed in the battery industries are

A. Fe B. Mn C. Ni D. Cr E. Cd

Choose the correct answer from the options given below:

The correct statements from following are:

A. The strength of anionic ligands can be explained by crystal field theory. B. Valence bond theory does not give a quantitative interpretation of kinetic stability of coordination compounds. C. The hybridization involved in formation of $\mathrm{[Ni(CN)_4]^{2-}}$ complex is $\mathrm{dsp^2}$. D. The number of possible isomer(s) of cis-$\mathrm{[PtCl_2(en)_2]^{2+}}$ is one

Choose the correct answer from the options given below:

'Adsorption' principle is used for which of the following purification method?

Match List I with List II

	LIST I (Technique)		LIST II (Application)
A.	Distillation	I.	Separation of glycerol from spent-lye
B.	Fractional distillation	II.	Aniline - Water mixture
C.	Steam distillation	III.	Separation of crude oil fractions
D.	Distillation under reduced pressure	IV.	Chloroform - Aniline

Choose the correct answer from the options given below:

Give below are two statements: Statement - I: Noble gases have very high boiling points. Statement - II: Noble gases are monoatomic gases. They are held together by strong dispersion forces. Because of this they are liquefied at very low temperature. Hence, they have very high boiling points. In the light of the above statements, choose the correct answer from the options given below:

A species having carbon with sextet of electrons and can act as electrophile is called

Given below are two statements: Statement I: IUPAC name of $\mathrm{HO-CH_2-(CH_2)_3-CH_2-COCH_3}$ is 7-hydroxyheptan-2-one. Statement II: 2-oxoheptan-7-ol is the correct IUPAC name for above compound. In the light of the above statements, choose the most appropriate answer from the options given below:

The product (C) in the below mentioned reaction is :

$$\mathrm{CH_3-CH_2-CH_2-Br \xrightarrow[\Delta]{KOH_{(alc)}} A \xrightarrow{HBr} B \xrightarrow[KOH_{(aq)}]{\Delta} C}$$

Given below are two statements: One is labelled as Assertion A and the other is labelled as Reason R: Assertion A: $\mathrm{pK_a}$ value of phenol is 10.0 while that of ethanol is 15.9. Reason R: Ethanol is stronger acid than phenol. 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 Assertion A and the other is labelled as Reason R: Assertion A: Alcohols react both as nucleophiles and electrophiles. Reason R: Alcohols react with active metals such as sodium, potassium and aluminum to yield corresponding alkoxides and liberate hydrogen. In the light of the above statements, choose the correct answer from the options given below:

Match List I with List II

	LIST I		LIST II
A.	Glucose/$\mathrm{NaHCO_3}$/$\Delta$	I.	Gluconic acid
B.	Glucose/$\mathrm{HNO_3}$	II.	No reaction
C.	Glucose/HI/$\Delta$	III.	n-hexane
D.	Glucose/Bromine water	IV.	Saccharic acid

Choose the correct answer from the options given below:

The compound that is white in color is

Number of moles of methane required to produce $22\,\mathrm{g}$ $\mathrm{CO_{2(g)}}$ after combustion is $x \times 10^{-2}$ moles. The value of $x$ is ________.

The ionization energy of sodium in $\mathrm{kJ\,mol^{-1}}$, if electromagnetic radiation of wavelength $242\,\mathrm{nm}$ is just sufficient to ionize sodium atom is _______.

The number of species from the following in which the central atom uses $\mathrm{sp^3}$ hybrid orbitals in its bonding is _______.

$\mathrm{NH_3}$, $\mathrm{SO_2}$, $\mathrm{SiO_2}$, $\mathrm{BeCl_2}$, $\mathrm{CO_2}$, $\mathrm{H_2O}$, $\mathrm{CH_4}$, $\mathrm{BF_3}$

Consider the following reaction at 298 K. $\frac{3}{2}\mathrm{O_{2(g)}} \rightleftharpoons \mathrm{O_{3(g)}}$. $\mathrm{K_P} = 2.47 \times 10^{-29}$. $\Delta_r G^{\ominus}$ for the reaction is ________ kJ. (Given $R = 8.314\,\mathrm{JK^{-1}\,mol^{-1}}$)