JEE Main 2024 Jan 30 Shift 2 — Question Paper

If the domain of the function $f(x)=\log_e\left(\frac{2x+3}{4x^2+x-3}\right)+\cos^{-1}\left(\frac{2x-1}{x+2}\right)$ is $(\alpha, \beta]$, then the value of $5\beta-4\alpha$ is equal to

If $z$ is a complex number, then the number of common roots of the equations $z^{1985}+z^{100}+1=0$ and $z^3+2z^2+2z+1=0$, is equal to

Let $R=\begin{pmatrix} x & 0 & 0 \\ 0 & y & 0 \\ 0 & 0 & z \end{pmatrix}$ be a non-zero $3\times 3$ matrix, where $x\sin\theta = y\sin\left(\theta+\frac{2\pi}{3}\right) = z\sin\left(\theta+\frac{4\pi}{3}\right) \neq 0,\ \theta \in (0, 2\pi)$. For a square matrix $M$, let trace $(M)$ denote the sum of all the diagonal entries of $M$. Then, among the statements:

(I) Trace $(R)=0$

(II) If trace $(\mathrm{adj}(\mathrm{adj}(R)))=0$, then $R$ has exactly one non-zero entry.

Consider the system of linear equations $x+y+z=5$, $x+2y+\lambda^2 z=9$, $x+3y+\lambda z=\mu$, where $\lambda, \mu \in \mathbb{R}$. Then, which of the following statement is NOT correct?

Suppose $2-p,\ p,\ 2-\alpha,\ \alpha$ are the coefficients of four consecutive terms in the expansion of $(1+x)^n$. Then the value of $p^2-\alpha^2+6\alpha+2p$ equals

Let $a$ and $b$ be be two distinct positive real numbers. Let $11^{\text{th}}$ term of a GP, whose first term is $a$ and third term is $b$, is equal to $p^{\text{th}}$ term of another GP, whose first term is $a$ and fifth term is $b$. Then $p$ is equal to

Let $f: \mathbb{R}-\{0\} \to \mathbb{R}$ be a function satisfying $f\left(\frac{x}{y}\right)=\frac{f(x)}{f(y)}$ for all $x, y,\ f(y) \neq 0$. If $f'(1)=2024$, then

Let $a$ and $b$ be real constants such that the function $f$ defined by $f(x)=\begin{cases} x^2+3x+a & ,\ x \leq 1 \\ bx+2 & ,\ x > 1 \end{cases}$ be differentiable on $\mathbb{R}$. Then, the value of $\int_{-2}^{2} f(x)\,dx$ equals

Let $f(x)=(x+3)^2(x-2)^3,\ x \in [-4, 4]$. If $M$ and $m$ are the maximum and minimum values of $f$, respectively in $[-4, 4]$, then the value of $M-m$ is

Let $f: \mathbb{R} \to \mathbb{R}$ be defined as $f(x)=ae^{2x}+be^x+cx$. If $f(0)=-1$, $f'(\log_e 2)=21$ and $\int_0^{\log_e 4}(f(x)-cx)\,dx=\frac{39}{2}$, then the value of $|a+b+c|$ equals

For $\alpha, \beta \in (0, \pi/2)$, let $3\sin(\alpha+\beta)=2\sin(\alpha-\beta)$ and a real number $k$ be such that $\tan\alpha = k\tan\beta$. Then, the value of $k$ is equal to

Let $y=f(x)$ be a thrice differentiable function in $(-5, 5)$. Let the tangents to the curve $y=f(x)$ at $(1, f(1))$ and $(3, f(3))$ make angles $\pi/6$ and $\pi/4$, respectively with positive $x$-axis. If $27\int_1^3 \left(\left(f'(t)\right)^2+1\right)f''(t)\,dt = \alpha+\beta\sqrt{3}$ where $\alpha, \beta$ are integers, then the value of $\alpha+\beta$ equals

Let $f: \mathbb{R} \to \mathbb{R}$ be a function defined by $f(x)=\frac{x}{\left(1+x^4\right)^{1/4}}$, and $g(x)=f(f(f(f(x))))$. Then, $18\int_0^{\sqrt{2\sqrt{5}}} x^2 g(x)\,dx$ is equal to

Let $A(\alpha, 0)$ and $B(0, \beta)$ be the points on the line $5x+7y=50$. Let the point $P$ divide the line segment $AB$ internally in the ratio $7:3$. Let $3x-25=0$ be a directrix of the ellipse $E: \frac{x^2}{a^2}+\frac{y^2}{b^2}=1$ and the corresponding focus be $S$. If from $S$, the perpendicular on the $x$-axis passes through $P$, then the length of the latus rectum of $E$ is equal to,

Let $P$ be a point on the hyperbola $H: \frac{x^2}{9}-\frac{y^2}{4}=1$, in the first quadrant such that the area of triangle formed by $P$ and the two foci of $H$ is $2\sqrt{13}$. Then, the square of the distance of $P$ from the origin is

If $x^2-y^2+2hxy+2gx+2fy+c=0$ is the locus of a point, which moves such that it is always equidistant from the lines $x+2y+7=0$ and $2x-y+8=0$, then the value of $g+c+h-f$ equals

Let $L_1: \vec{r}=(\hat{i}-\hat{j}+2\hat{k})+\lambda(\hat{i}-\hat{j}+2\hat{k}),\ \lambda \in \mathbb{R}$,

$L_2: \vec{r}=(\hat{j}-\hat{k})+\mu(3\hat{i}+\hat{j}+p\hat{k}),\ \mu \in \mathbb{R}$, and $L_3: \vec{r}=\delta(\ell\hat{i}+m\hat{j}+n\hat{k}),\ \delta \in \mathbb{R}$

be three lines such that $L_1$ is perpendicular to $L_2$ and $L_3$ is perpendicular to both $L_1$ and $L_2$. Then, the point which lies on $L_3$ is

Let $\vec{a}$ and $\vec{b}$ be two vectors such that $|\vec{b}|=1$ and $|\vec{b}\times\vec{a}|=2$. Then $\left|(\vec{b}\times\vec{a})-\vec{b}\right|^2$ is equal to

Let $\vec{a}=\hat{i}+\alpha\hat{j}+\beta\hat{k},\ \alpha, \beta \in \mathbb{R}$. Let a vector $\vec{b}$ be such that the angle between $\vec{a}$ and $\vec{b}$ is $\frac{\pi}{4}$ and $|\vec{b}|^2=6$. If $\vec{a}\cdot\vec{b}=3\sqrt{2}$, then the value of $(\alpha^2+\beta^2)\left|\vec{a}\times\vec{b}\right|^2$ is equal to

Bag A contains 3 white, 7 red balls and Bag B contains 3 white, 2 red balls. One bag is selected at random and a ball is drawn from it. The probability of drawing the ball from the bag A, if the ball drawn is white, is

The number of symmetric relations defined on the set $\{1, 2, 3, 4\}$ which are not reflexive is ______.

The number of real solutions of the equation $x\left(x^2+3|x|+5|x-1|+6|x-2|\right)=0$ is ________.

In an examination of Mathematics paper, there are 20 questions of equal marks and the question paper is divided into three sections : $A$, $B$ and $C$. A student is required to attempt total 15 questions taking at least 4 questions from each section. If section $A$ has 8 questions, section $B$ has 6 questions and section $C$ has 6 questions, then the total number of ways a student can select 15 questions is ______.

Let $\alpha=\sum_{k=0}^{n}\left(\frac{\left({}^{n}C_{k}\right)^2}{k+1}\right)$ and $\beta=\sum_{k=0}^{n-1}\left(\frac{{}^{n}C_{k}\ {}^{n}C_{k+1}}{k+2}\right)$. If $5\alpha=6\beta$, then $n$ equals ______.

Let $S_n$ be the sum to $n$-terms of an arithmetic progression $3, 7, 11, \ldots$ If $40<\left(\frac{6}{n(n+1)}\sum_{k=1}^{n} S_k\right)<42$, then $n$ equals ______.

The area of the region enclosed by the parabola $(y-2)^2=x-1$, the line $x-2y+4=0$ and the positive coordinate axes is _______.

Let $Y=Y(X)$ be a curve lying in the first quadrant such that the area enclosed by the line $Y-y=Y'(x)(X-x)$ and the co-ordinate axes, where $(x, y)$ is any point on the curve, is always $\frac{-y^2}{2Y'(x)}+1,\ Y'(x)\neq 0$. If $Y(1)=1$, then $12Y(2)$ equals _______.

Consider two circles $C_1: x^2+y^2=25$ and $C_2: (x-\alpha)^2+y^2=16$, where $\alpha \in (5, 9)$. Let the angle between the two radii (one to each circle) drawn from one of the intersection points of $C_1$ and $C_2$ be $\sin^{-1}\left(\frac{\sqrt{63}}{8}\right)$. If the length of common chord of $C_1$ and $C_2$ is $\beta$, then the value of $(\alpha\beta)^2$ equals _______.

Let a line passing through the point $(-1, 2, 3)$ intersect the lines $L_1: \frac{x-1}{3}=\frac{y-2}{2}=\frac{z+1}{-2}$ at $M(\alpha, \beta, \gamma)$ and $L_2: \frac{x+2}{-3}=\frac{y-2}{-2}=\frac{z-1}{4}$ at $N(a, b, c)$. Then, the value of $\frac{(\alpha+\beta+\gamma)^2}{(a+b+c)^2}$ equals ________.

The variance $\sigma^2$ of the data

$x_i$	0	1	5	6	10	12	17
$f_i$	3	2	3	2	6	3	3

is ______.

If mass is written as $m = k\, c^{P}\, G^{-1/2}\, h^{1/2}$ then the value of $P$ will be : (Constants have their usual meaning with $k$ a dimensionless constant)

Projectiles A and B are thrown at angles of $45°$ and $60°$ with vertical respectively from top of a $400$ m high tower. If their ranges and times of flight are same, the ratio of their speeds of projection $v_{\mathrm{A}} : v_{\mathrm{B}}$ is :

[Take $g = 10\ ms^{-2}$]

Three blocks $A$, $B$ and $C$ are pulled on a horizontal smooth surface by a force of $80N$ as shown in figure

[Figure: blocks A (5 kg), B (3 kg) and C (2 kg) in a row connected by strings with tensions $T_1$ (between A and B) and $T_2$ (between B and C); force $F = 80N$ applied to C towards the right]

The tensions $T_1$ and $T_2$ in the string are respectively:

A block of mass $m$ is placed on a surface having vertical crossection given by $y = x^2/4$. If coefficient of friction is $0.5$, the maximum height above the ground at which block can be placed without slipping is:

A block of mass $1$ kg is pushed up a surface inclined to horizontal at an angle of $60°$ by a force of $10\ N$ parallel to the inclined surface as shown in figure. When the block is pushed up by $10\ m$ along inclined surface, the work done against frictional force is :

[$g = 10\ m/s^2$]

[Figure: incline at $60°$ to the horizontal with block $M$ on it, coefficient of static friction $\mu_s = 0.1$, force of $10\ N$ applied up along the incline]

Escape velocity of a body from earth is $11.2\ km/s$. If the radius of a planet be one-third the radius of earth and mass be one-sixth that of earth, the escape velocity from the planet is :

A block of ice at $-10°C$ is slowly heated and converted to steam at $100°C$. Which of the following curves represent the phenomenon qualitatively:

Choose the correct statement for processes $A$ & $B$ shown in figure.

[Figure: P–V diagram with two downward-sloping curves; curve A is lower and less steep, curve B is higher and steeper, both with arrows directed toward increasing V]

If three moles of monoatomic gas $\left(\gamma = \frac{5}{3}\right)$ is mixed with two moles of a diatomic gas $\left(\gamma = \frac{7}{5}\right)$, the value of adiabatic exponent $\gamma$ for the mixture is

A particle of charge '$-q$' and mass '$m$' moves in a circle of radius '$r$' around an infinitely long line charge of linear charge density '$+\lambda$'. Then time period will be given as :

(Consider $k$ as Coulomb's constant)

When a potential difference $V$ is applied across a wire of resistance $R$, it dissipates energy at a rate $W$. If the wire is cut into two halves and these halves are connected mutually parallel across the same supply, the energy dissipation rate will become:

Match List I with List II

	List I		List II
A.	Gauss's law of magnetostatics	I.	$\oint \vec{E} \cdot \vec{da} = \dfrac{1}{\varepsilon_o} \int \rho \, dV$
B.	Faraday's law of electro magnetic induction	II.	$\oint \vec{B} \cdot \vec{da} = 0$
C.	Ampere's law	III.	$\int \vec{E} \cdot \vec{dl} = \dfrac{-d}{dt} \int \vec{B} \cdot \vec{da}$
D.	Gauss's law of electrostatics	IV.	$\oint \vec{B} \cdot \vec{dl} = \mu_0 I$

Choose the correct answer from the options given below:

An alternating voltage $V(t) = 220 \sin 100\pi t$ volt is applied to a purely resistive load of $50\ \Omega$. The time taken for the current to rise from half of the peak value to the peak value is:

If the total energy transferred to a surface in time $t$ is $6.48 \times 10^5\ J$, then the magnitude of the total momentum delivered to this surface for complete absorption will be:

A beam of unpolarised light of intensity $I_o$ is passed through a polaroid $A$ and then through another polaroid $B$ which is oriented so that its principal plane makes an angle of $45°$ relative to that of $A$. The intensity of emergent light is:

For the photoelectric effect, the maximum kinetic energy $(E_k)$ of the photoelectrons is plotted against the frequency $(\nu)$ of the incident photons as shown in figure. The slope of the graph gives

[Figure: straight-line graph of $E_k$ (y-axis) versus $\nu$ (x-axis), intersecting the $\nu$-axis at a positive intercept and making angle $\theta$ with it]

In a nuclear fission reaction of an isotope of mass $M$, three similar daughter nuclei of same mass are formed. The speed of a daughter nuclei in terms of mass defect $\Delta M$ will be :

An electron revolving in $n^{\text{th}}$ Bohr orbit has magnetic moment $\mu_n$. If $\mu_n \propto n^x$, the value of $x$ is

[Figure: series circuit with a 15 V battery, diode $D_1$ (Ge) and diode $D_2$ (Si) both forward biased, a $1.5\ k\Omega$ resistor, and a load resistance $R_L = 2.5\ k\Omega$ across which the voltage is measured]

In the given circuit, the voltage across load resistance $(R_L)$ is:

If $50$ Vernier divisions are equal to $49$ main scale divisions of a traveling microscope and one smallest reading of main scale is $0.5\ mm$, the Vernier constant of traveling microscope is

A vector has magnitude same as that of $\vec{A} = 3\hat{i} + 4\hat{j}$ and is parallel to $\vec{B} = 4\hat{i} + 3\hat{j}$. The $x$ and $y$ components of this vector in first quadrant are $x$ and $3$ respectively where $x =$ ________.

Two discs of moment of inertia $I_1 = 4\ kg\ m^2$ and $I_2 = 2\ kg\ m^2$, about their central axes & normal to their planes, rotating with angular speeds $10\ rad/s$ & $4\ rad/s$ respectively are brought into contact face to face with their axes of rotation coincident. The loss in kinetic energy of the system in the process is ________ J.

A big drop is formed by coalescing $1000$ small identical drops of water. If $E_1$ be the total surface energy of $1000$ small drops of water and $E_2$ be the surface energy of single big drop of water, then $E_1 : E_2$ is $x : 1$ where $x =$ _______.

A point source is emitting sound waves of intensity $16 \times 10^{-8}\ Wm^{-2}$ at the origin. The difference in intensity (magnitude only) at two points located at a distances of $2m$ and $4m$ from the origin respectively will be _______ $\times\ 10^{-8}\ Wm^{-2}$.

Two identical charged spheres are suspended by strings of equal lengths. The strings make an angle of $37°$ with each other. When suspended in a liquid of density $0.7\ g/cm^3$, the angle remains same. If density of material of the sphere is $1.4\ g/cm^3$, the dielectric constant of the liquid is _______ $\left(\tan 37° = \frac{3}{4}\right)$

Two resistance of $100\ \Omega$ and $200\ \Omega$ are connected in series with a battery of $4\ V$ and negligible internal resistance . A voltmeter is used to measure voltage across $100\ \Omega$ resistance, which gives reading as $1\ V$. The resistance of voltmeter must be _______ $\Omega$.

The current of $5A$ flows in a square loop of sides $1\ m$ is placed in air. The magnetic field at the centre of the loop is $X\sqrt{2} \times 10^{-7}\ T$. The value of $X$ is ________.

A power transmission line feeds input power at $2.3\ kV$ to a step down transformer with its primary winding having $3000$ turns. The output power is delivered at $230\ V$ by the transformer. The current in the primary of the transformer is $5A$ and its efficiency is $90\%$. The winding of transformer is made of copper. The output current of transformer is _______ $A$.

In an experiment to measure the focal length $(f)$ of a convex lens, the magnitude of object distance $(x)$ and the image distance $(y)$ are measured with reference to the focal point of the lens. The $y$-$x$ plot is shown in figure.

The focal length of the lens is ________ $cm$.

[Figure: y–x graph with y (cm) and x (cm) axes marked 10 to 40; data points A, B, C, D, E lie on a downward-sloping curve/line, with point C near (20, 20)]

A simple pendulum is placed at a place where its distance from the earth's surface is equal to the radius of the earth. If the length of the string is $4m$, then the time period of small oscillations will be _______ $s$. [take $g = \pi^2\ ms^{-2}$]

If a substance 'A' dissolves in solution of a mixture of 'B' and 'C' with their respective number of moles as $\mathrm{n_A}$, $\mathrm{n_B}$ and $\mathrm{n_C}$, Mole fraction of C in the solution is :

Given below are two statements:

Statement - I: Since Fluorine is more electronegative than nitrogen, the net dipole moment of $\mathrm{NF_3}$ is greater than $\mathrm{NH_3}$.

Statement - II: In $\mathrm{NH_3}$, the orbital dipole due to lone pair and the dipole moment of NH bonds are in opposite direction, but in $\mathrm{NF_3}$ the orbital dipole due to lone pair and dipole moments of N-F bonds are in same direction.

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

The molecule / ion with square pyramidal shape is

The solution from the following with highest depression in freezing point/lowest freezing point is

Reduction potential of ions are given below:

$\mathrm{ClO_4^-}$	$\mathrm{IO_4^-}$	$\mathrm{BrO_4^-}$
$\mathrm{E^\circ = 1.19\ V}$	$\mathrm{E^\circ = 1.65\ V}$	$\mathrm{E^\circ = 1.74\ V}$

The correct order of their oxidising power is :

Given below are two statements:

Statement - I: Along the period, the chemical reactivity of the elements gradually increases from group 1 to group 18.

Statement - II: The nature of oxides formed by group 1 elements is basic while that of group 17 elements is acidic.

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

Choose the correct statements about the hydrides of group 15 elements.

A. The stability of the hydrides decreases in the order $\mathrm{NH_3} > \mathrm{PH_3} > \mathrm{AsH_3} > \mathrm{SbH_3} > \mathrm{BiH_3}$.

B. The reducing ability of the hydride increases in the order $\mathrm{NH_3} < \mathrm{PH_3} < \mathrm{AsH_3} < \mathrm{SbH_3} < \mathrm{BiH_3}$.

C. Among the hydrides, $\mathrm{NH_3}$ is strong reducing agent while $\mathrm{BiH_3}$ is mild reducing agent.

D. The basicity of the hydrides increases in the order $\mathrm{NH_3} < \mathrm{PH_3} < \mathrm{AsH_3} < \mathrm{SbH_3} < \mathrm{BiH_3}$.

Choose the most appropriate 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: $\mathrm{H_2Te}$ is more acidic than $\mathrm{H_2S}$.

Reason R: Bond dissociation enthalpy of $\mathrm{H_2Te}$ is lower than $\mathrm{H_2S}$.

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

Alkaline oxidative fusion of $\mathrm{MnO_2}$ gives "A" which on electrolytic oxidation in alkaline solution produces B. A and B respectively are

The orange colour of $\mathrm{K_2Cr_2O_7}$ and purple colour of $\mathrm{KMnO_4}$ is due to

The coordination geometry around the manganese in decacarbonyldimanganese (0) is

Which among the following purification methods is based on the principle of "Solubility" in two different solvents?

The correct stability order of carbocations is

IUPAC name of following compound is :

$$\mathrm{CH_3 - \underset{\underset{NH_2}{|}}{CH} - CH_2 - CN}$$

Given below are two statements:

Statement - I: High concentration of strong nucleophilic reagent with secondary alkyl halides which do not have bulky substituents will follow $\mathrm{S_N^2}$ mechanism.

Statement - II: A secondary alkyl halide when treated with a large excess of ethanol follows $\mathrm{S_N^1}$ mechanism.

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

Salicylaldehyde is synthesized from phenol, when reacted with