JEE Main 2024 Jan 31 Shift 2 — Question Paper

If the function $f:(-\infty,-1] \rightarrow(a, b]$ defined by $f(x)=e^{x^{3}-3 x+1}$ is one - one and onto, then the distance of the point $P(2 b+4, a+2)$ from the line $x+e^{-3} y=4$ is :

The number of solutions, of the equation $e^{\sin x}-2 e^{-\sin x}=2$, is :

Let $z_{1}$ and $z_{2}$ be two complex numbers such that $z_{1}+z_{2}=5$ and $z_{1}^{3}+z_{2}^{3}=20+15 i$. Then, $\left|z_{1}^{4}+z_{2}^{4}\right|$ equals -

Let $A$ be a $3 \times 3$ real matrix such that

$$A\begin{pmatrix}1 \\ 0 \\ 1\end{pmatrix}=2\begin{pmatrix}1 \\ 0 \\ 1\end{pmatrix},\ A\begin{pmatrix}-1 \\ 0 \\ 1\end{pmatrix}=4\begin{pmatrix}-1 \\ 0 \\ 1\end{pmatrix},\ A\begin{pmatrix}0 \\ 1 \\ 0\end{pmatrix}=2\begin{pmatrix}0 \\ 1 \\ 0\end{pmatrix}.$$

Then, the system $(A-3 I)\begin{pmatrix}x \\ y \\ z\end{pmatrix}=\begin{pmatrix}1 \\ 2 \\ 3\end{pmatrix}$ has

The number of ways in which 21 identical apples can be distributed among three children such that each child gets at least 2 apples, is

If for some $m, n$; ${ }^{6} C_{m}+2\left({ }^{6} C_{m+1}\right)+{ }^{6} C_{m+2}>{ }^{8} C_{3}$ and ${ }^{n-1} P_{3}:{ }^{n} P_{4}=1: 8$, then ${ }^{n} P_{m+1}+{ }^{\mathrm{n}+1} C_{m}$ is equal to

Let $2^{\text {nd }}, 8^{\text {th }}$ and $44^{\text {th }}$ terms of a non-constant A. P. be respectively the $1^{\text {st }}, 2^{\text {nd }}$ and $3^{\text {rd }}$ terms of a G. P. If the first term of the A. P. is 1, then the sum of its first 20 terms is equal to -

Consider the function $f:(0, \infty) \rightarrow \mathbb{R}$ defined by $f(x)=e^{-\left|\log _{e} x\right|}$. If $m$ and $n$ be respectively the number of points at which $f$ is not continuous and $f$ is not differentiable, then $m+n$ is

Let $f: \mathbb{R} \rightarrow(0, \infty)$ be strictly increasing function such that $\lim _{x \rightarrow \infty} \frac{f(7 x)}{f(x)}=1$. Then, the value of $\lim _{x \rightarrow \infty}\left[\frac{f(5 x)}{f(x)}-1\right]$ is equal to

Let a variable line passing through the centre of the circle $x^{2}+y^{2}-16 x-4 y=0$, meet the positive co-ordinate axes at the points $A$ and $B$. Then the minimum value of $OA+OB$, where $O$ is the origin, is equal to

The area of the region enclosed by the parabolas $y=4 x-x^{2}$ and $3 y=(x-4)^{2}$ is equal to

Let $f, g:(0, \infty) \rightarrow \mathbb{R}$ be two functions defined by $f(x)=\int_{-x}^{x}\left(|t|-t^{2}\right) e^{-t^{2}} d t$ and $g(x)=\int_{0}^{x^{2}} t^{1 / 2} e^{-t} d t$. Then, the value of $9\left(f\left(\sqrt{\log _{e} 9}\right)+g\left(\sqrt{\log _{e} 9}\right)\right)$ is equal to

The temperature $T(t)$ of a body at time $t=0$ is $160^{\circ} \mathrm{F}$ and it decreases continuously as per the differential equation $\frac{d T}{d t}=-K(T-80)$, where $K$ is a positive constant. If $T(15)=120^{\circ} \mathrm{F}$, then $T(45)$ is equal to

Let $P$ be a parabola with vertex $(2,3)$ and directrix $2 x+y=6$. Let an ellipse $E: \frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1, a>b$, of eccentricity $\frac{1}{\sqrt{2}}$ pass through the focus of the parabola $P$. Then, the square of the length of the latus rectum of $E$, is

Let $A(a, b), B(3,4)$ and $C(-6,-8)$ respectively denote the centroid, circumcentre and orthocentre of a triangle. Then, the distance of the point $P(2 a+3,7 b+5)$ from the line $2 x+3 y-4=0$ measured parallel to the line $x-2 y-1=0$ is

Let $(\alpha, \beta, \gamma)$ be the mirror image of the point $(2,3,5)$ in the line $\frac{x-1}{2}=\frac{y-2}{3}=\frac{z-3}{4}$. Then, $2 \alpha+3 \beta+4 \gamma$ is equal to

The shortest distance, between lines $L_{1}$ and $L_{2}$, where $L_{1}: \frac{x-1}{2}=\frac{y+1}{-3}=\frac{z+4}{2}$ and $L_{2}$ is the line, passing through the points $\mathrm{A}(-4,4,3), \mathrm{B}(-1,6,3)$ and perpendicular to the line $\frac{x-3}{-2}=\frac{y}{3}=\frac{z-1}{1}$, is

Let the mean and the variance of 6 observations $a, b, 68,44,48,60$ be 55 and 194, respectively. If $a>b$, then $a+3 b$ is

A coin is biased so that a head is twice as likely to occur as a tail. If the coin is tossed 3 times, then the probability of getting two tails and one head is

If $a=\sin ^{-1}(\sin (5))$ and $b=\cos ^{-1}(\cos (5))$, then $a^{2}+b^{2}$ is equal to

Let $A=\{1,2,3, \ldots \ldots \ldots, 100\}$. Let $R$ be a relation on A defined by $(x, y) \in R$ if and only if $2 x=3 y$. Let $R_{1}$ be a symmetric relation on $A$ such that $R \subset R_{1}$ and the number of elements in $R_{1}$ is n. Then, the minimum value of n is _________.

Let A be a $3 \times 3$ matrix and $\det(A)=2$. If $n=\det (\underbrace{a d j\left(a d j\left(\ldots \ldots(a d j\ A)\right)\right)}_{2024-\text{times}})$, then the remainder when $n$ is divided by 9 is equal to ________.

Let the coefficient of $x^{r}$ in the expansion of $(x+3)^{n-1}+(x+3)^{n-2}(x+2)+(x+3)^{n-3}(x+2)^{2}+\ldots \ldots \ldots+(x+2)^{n-1}$ be $\alpha_{r}$. If $\sum_{r=0}^{n} \alpha_{r}=\beta^{n}-\gamma^{n}$, $\beta, \gamma \in \mathbb{N}$, then the value of $\beta^{2}+\gamma^{2}$ equals ________.

Let $a, b, c$ be the lengths of three sides of a triangle satisfying the condition $\left(a^{2}+b^{2}\right) x^{2}-2 b(a+c) x+\left(b^{2}+c^{2}\right)=0$. If the set of all possible values of $x$ is the interval $(\alpha, \beta)$, then $12\left(\alpha^{2}+\beta^{2}\right)$ is equal to ______.

If $\lim _{x \rightarrow 0} \frac{a x^{2} e^{x}-b \log _{e}(1+x)+c x e^{-x}}{x^{2} \sin x}=1$, then $16\left(a^{2}+b^{2}+c^{2}\right)$ is equal to ______.

$\left|\frac{120}{\pi^{3}} \int_{0}^{\pi} \frac{x^{2} \sin x \cos x}{\sin ^{4} x+\cos ^{4} x}\ d x\right|$ is equal to ________.

Let $y=y(x)$ be the solution of the differential equation $\sec ^{2} x\ d x+\left(e^{2 y} \tan ^{2} x+\tan x\right) d y=0$, $0<x<\frac{\pi}{2}$, $y\left(\pi / 4\right)=0$. If $y\left(\pi / 6\right)=\alpha$, then $e^{8 \alpha}$ is equal to _______.

Let $A(-2,-1), B(1,0), C(\alpha, \beta)$ and $D(\gamma, \delta)$ be the vertices of a parallelogram $ABCD$. If the point $C$ lies on $2 x-y=5$ and the point $D$ lies on $3 x-2 y=6$, then the value of $|\alpha+\beta+\gamma+\delta|$ is equal to ________.

A line passes through $A(4,-6,-2)$ and $B(16,-2,4)$. The point $P(a, b, c)$, where $a, b, c$ are non-negative integers, on the line $AB$ lies at a distance of 21 units, from the point $A$. The distance between the points $P(a, b, c)$ and $Q(4,-12,3)$ is equal to __________.

Let $\vec{a}=3 \hat{i}+2 \hat{j}+\hat{k}$, $\vec{b}=2 \hat{i}-\hat{j}+3 \hat{k}$ and $\vec{c}$ be a vector such that $(\vec{a}+\vec{b}) \times \vec{c}=2(\vec{a} \times \vec{b})+24 \hat{j}-6 \hat{k}$ and $(\vec{a}-\vec{b}+\hat{i}) \cdot \vec{c}=-3$. Then $|\vec{c}|^{2}$ is equal to ______.

Consider two physical quantities $A$ and $B$ related to each other as $E = \frac{B - x^2}{At}$ where $E$, $x$ and $t$ have dimensions of energy, length and time respectively. The dimension of $AB$ is

If two vectors $\vec{A}$ and $\vec{B}$ having equal magnitude $R$ are inclined at an angle $\theta$, then

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

[Figure: a fixed pulley attached to the ceiling with masses $m_1$ and $m_2$ hanging on either side of the string]

[Figure: a 5 kg block on an inclined plane of angle $30°$ with coefficient of friction $\mu = 0.1$]

A block of mass 5 kg is placed on a rough inclined surface as shown in the figure. If $\vec{F_1}$ is the force required to just move the block up the inclined plane and $\vec{F_2}$ is the force required to just prevent the block from sliding down, then the value of $\left|\vec{F_1}\right| - \left|\vec{F_2}\right|$ is: [Use g = 10 m/s$^2$]

A body of mass 2 kg begins to move under the action of a time dependent force given by $\vec{F} = \left(6t\,\hat{i} + 6t^2\,\hat{j}\right) N$. The power developed by the force at the time $t$ is given by:

The mass of the moon is $\frac{1}{144}$ times the mass of a planet and its diameter is $\frac{1}{16}$ times the diameter of a planet. If the escape velocity on the planet is v, the escape velocity on the moon will be :

A small spherical ball of radius $r$, falling through a viscous medium of negligible density has terminal velocity 'v'. Another ball of the same mass but of radius $2r$, falling through the same viscous medium will have terminal velocity:

The speed of sound in oxygen at S.T.P. will be approximately:

(given, $R = 8.3\ \mathrm{JK^{-1}}$, $\gamma = 1.4$)

A gas mixture consists of 8 moles of argon and 6 moles of oxygen at temperature T. Neglecting all vibrational modes, the total internal energy of the system is:

Force between two point charges $q_1$ and $q_2$ placed in vacuum at '$r$' cm apart is $F$. Force between them when placed in a medium having dielectric constant $K=5$ at '$r/5$' cm apart will be:

By what percentage will the illumination of the lamp decrease if the current drops by 20%?

A uniform magnetic field of $2 \times 10^{-3}\ T$ acts along positive $Y$-direction. A rectangular loop of sides 20 cm and 10 cm with current of $5\ A$ is in $Y$–$Z$ plane. The current is in anticlockwise sense with reference to negative $X$ axis. Magnitude and direction of the torque is:

An AC voltage $V = 20 \sin 200\pi t$ is applied to a series LCR circuit which drives a current $I = 10 \sin\left(200\,\pi t + \frac{\pi}{3}\right)$. The average power dissipated is:

Given below are two statements:

Statement I: Electromagnetic waves carry energy as they travel through space and this energy is equally shared by the electric and magnetic fields.

Statement II: When electromagnetic waves strike a surface, a pressure is exerted on the surface.

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

When unpolarized light is incident at an angle of $60°$ on a transparent medium from air, the reflected ray is completely polarized. The angle of refraction in the medium is:

In a photoelectric effect experiment a light of frequency 1.5 times the threshold frequency is made to fall on the surface of photosensitive material. Now if the frequency is halved and intensity is doubled, the number of photo electrons emitted will be:

The mass number of nucleus having radius equal to half of the radius of nucleus with mass number 192 is:

[Figure: logic circuit — inputs A and B; A and NOT(B) feed an OR gate; B and NOT(A) feed a second OR gate; the two OR outputs feed a NOR gate whose output is Y]

The output of the given circuit diagram is -

The resistance per centimeter of a meter bridge wire is $r$, with $X\ \Omega$ resistance in left gap. Balancing length from left end is at 40 cm with $25\ \Omega$ resistance in right gap. Now the wire is replaced by another wire of $2r$ resistance per centimeter. The new balancing length for same settings will be at

The measured value of the length of a simple pendulum is 20 cm with 2 mm accuracy. The time for 50 oscillations was measured to be 40 seconds with 1 second resolution. From these measurements, the accuracy in the measurement of acceleration due to gravity is N%. The value of N is:

A body of mass '$m$' is projected with a speed '$u$' making an angle of $45°$ with the ground. The angular momentum of the body about the point of projection, at the highest point is expressed as $\frac{\sqrt{2}\,mu^3}{Xg}$. The value of '$X$' is ________.

Two identical spheres each of mass 2 kg and radius 50 cm are fixed at the ends of a light rod so that the separation between the centers is 150 cm. Then, moment of inertia of the system about an axis perpendicular to the rod and passing through its middle point is $\frac{x}{20}\ kg\ m^2$, where the value of $x$ is ________.

Two blocks of mass 2 kg and 4 kg are connected by a metal wire going over a smooth pulley as shown in figure. The radius of wire is $4.0 \times 10^{-5}$ m and Young's modulus of the metal is $2.0 \times 10^{11}$ N/m$^2$. The longitudinal strain developed in the wire is $\frac{1}{\alpha\pi}$. The value of $\alpha$ is ________. [Use $g = 10$ m/s$^2$]

[Figure: a smooth pulley fixed to the ceiling with a 2 kg block and a 4 kg block hanging from the two ends of the wire]

The time period of simple harmonic motion of mass $M$ in the given figure is $\pi\sqrt{\frac{\alpha M}{5k}}$, where the value of $\alpha$ is ________.

[Figure: mass M suspended from the ceiling by a combination of springs — two springs of constant k in parallel, in series with another spring of constant k, all in parallel with a spring of constant k]

The distance between charges $+q$ and $-q$ is $2l$ and between $+2q$ and $-2q$ is $4l$. The electrostatic potential at point $P$ at a distance $r$ from center $O$ is $-\alpha\left[\frac{ql}{r^2}\right] \times 10^9\ V$, where the value of $\alpha$ is ________. (Use $\frac{1}{4\pi\varepsilon_0} = 9 \times 10^9\ Nm^2C^{-2}$)

[Figure: charges $+2q$, $-q$, $+q$, $-2q$ placed on a straight line with center $O$; point $P$ at distance $r$ from $O$ along a line making $60°$ with the line of charges]

In the following circuit, the battery has an emf of 2V and an internal resistance of $\frac{2}{3}\ \Omega$. The power consumption in the entire circuit is ________ W.

[Figure: a square network of $2\ \Omega$ resistors — four resistors along the sides/diagonals meeting at the center and one $2\ \Omega$ resistor at the bottom, connected to a 2 V battery with internal resistance $\frac{2}{3}\ \Omega$]

Two circular coils $P$ and $Q$ of 100 turns each have same radius of $\pi$ cm. The currents in $P$ and $R$ are 1$A$ and 2$A$ respectively. $P$ and $Q$ are placed with their planes mutually perpendicular and their centers coincide. The resultant magnetic field induction at the center of the coils is $\sqrt{x}\ mT$, where $x =$ __________.

[Use $\mu_0 = 4\pi \times 10^{-7}\ TmA^{-1}$]

The magnetic flux $\phi$ (in weber) linked with a closed circuit of resistance $8\ \Omega$ varies with time (in seconds) as $\phi = 5t^2 - 36t + 1$. The induced current in the circuit at $t = 2$ s is ________ $A$.

Light from a point source in air falls on a convex curved surface of radius 20 cm and refractive index 1.5. If the source is located at 100 cm from the convex surface, the image will be formed at ________ cm from the object.

A nucleus has mass number $A_1$ and volume $V_1$. Another nucleus has mass number $A_2$ and Volume $V_2$. If relation between mass number is $A_2 = 4A_1$, then $\frac{V_2}{V_1} =$ ________.

A sample of $\mathrm{CaCO_3}$ and $\mathrm{MgCO_3}$ weighed $2.21\ \mathrm{g}$ is ignited to constant weight of $1.152\ \mathrm{g}$. The composition of mixture is :

(Given molar mass in $\mathrm{g\ mol^{-1}}$ $\mathrm{CaCO_3}$ : 100, $\mathrm{MgCO_3}$ : 84)

The four quantum numbers for the electron in the outer most orbital of potassium (atomic no. 19) are

Which of the following is least ionic?

$\mathrm{A_{(g)}} \rightleftharpoons \mathrm{B_{(g)}} + \frac{\mathrm{C}}{2}_{\mathrm{(g)}}$ The correct relationship between $\mathrm{K_P}$, $\alpha$ and equilibrium pressure $\mathrm{P}$ is

Given below are two statements :

Statement I : $\mathrm{S_8}$ solid undergoes disproportionation reaction under alkaline conditions to form $\mathrm{S^{2-}}$ and $\mathrm{S_2O_3^{2-}}$.

Statement II : $\mathrm{ClO_4^-}$ can undergo disproportionation reaction under acidic condition.

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

Consider the following elements.

$$\text{Group} \downarrow \quad \begin{matrix} \mathrm{A'} & \mathrm{B'} \\ \mathrm{C'} & \mathrm{D'} \end{matrix} \quad \rightarrow \text{Period}$$

(A′ and B′ lie in the same period with B′ to the right of A′; C′ and D′ lie in the next period directly below A′ and B′ respectively.)

Which of the following is/are true about A′, B′, C′ and D′?

A. Order of atomic radii: B′ < A′ < D′ < C′ B. Order of metallic character: B′ < A′ < D′ < C′ C. Size of the element: D′ < C′ < B′ < A′ D. Order of ionic radii: B′⁺ < A′⁺ < D′⁺ < C′⁺

Choose the correct answer from the options given below :

Given below are two statements :

Statement I : Group 13 trivalent halides get easily hydrolyzed by water due to their covalent nature.

Statement II : $\mathrm{AlCl_3}$ upon hydrolysis in acidified aqueous solution forms octahedral $\mathrm{[Al(H_2O)_6]^{3+}}$ ion.

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

Choose the correct statements from the following

A. All group 16 elements form oxides of general formula $\mathrm{EO_2}$ and $\mathrm{EO_3}$, where E = S, Se, Te and Po. Both the types of oxides are acidic in nature. B. $\mathrm{TeO_2}$ is an oxidising agent while $\mathrm{SO_2}$ is reducing in nature. C. The reducing property decreases from $\mathrm{H_2S}$ to $\mathrm{H_2Te}$ down the group. D. The ozone molecule contains five lone pairs of electrons.

Choose the correct answer from the options given below:

Choose the correct statements from the following

A. $\mathrm{Mn_2O_7}$ is an oil at room temperature B. $\mathrm{V_2O_4}$ reacts with acid to give $\mathrm{VO_2^{2+}}$ C. $\mathrm{CrO}$ is a basic oxide D. $\mathrm{V_2O_5}$ does not react with acid

Choose the correct answer from the options given below :

Select the option with correct property -

Match List I with List II

	LIST I (Complex ion)		LIST II (Electronic Configuration)
A.	$\mathrm{[Cr(H_2O)_6]^{3+}}$	I.	$t_{2g}^{2}\,e_g^{0}$
B.	$\mathrm{[Fe(H_2O)_6]^{3+}}$	II.	$t_{2g}^{3}\,e_g^{0}$
C.	$\mathrm{[Ni(H_2O)_6]^{2+}}$	III.	$t_{2g}^{3}\,e_g^{2}$
D.	$\mathrm{[V(H_2O)_6]^{3+}}$	IV.	$t_{2g}^{6}\,e_g^{2}$

Choose the correct answer from the options given below:

The fragrance of flowers is due to the presence of some steam volatile organic compounds called essential oils. These are generally insoluble in water at room temperature but are miscible with water vapour in vapour phase. A suitable method for the extraction of these oils from the flowers is -

The correct order of reactivity in electrophilic substitution reaction of the following compounds is :

[Figure: four benzene-ring structures — A: benzene, B: toluene ($\mathrm{C_6H_5CH_3}$), C: chlorobenzene ($\mathrm{C_6H_5Cl}$), D: nitrobenzene ($\mathrm{C_6H_5NO_2}$)]

Identify structure of 2,3–dibromo–1–phenylpentane.

Major product of the following reaction is -

[Figure: 1-methylcyclopent-1-ene treated with $\mathrm{D-Cl}$]

$$\text{1-methylcyclopentene} \xrightarrow{\mathrm{D-Cl}} ?$$

Identify A and B in the following reaction sequence.

$$\text{Bromobenzene} \xrightarrow{\text{Conc } \mathrm{HNO_3}} \mathrm{A} \xrightarrow[\text{(ii) } \mathrm{HCl}]{\text{(i) } \mathrm{NaOH}} \mathrm{B}$$

[Figure: reaction scheme starting from bromobenzene]

Identify major product 'P' formed in the following reaction.

$$\mathrm{C_6H_6} + \mathrm{C_6H_5COCl} \xrightarrow{\text{Anhydrous } \mathrm{AlCl_3}} \text{'P' (Major Product)}$$

[Figure: benzene plus benzoyl chloride reaction scheme]