JEE Main 2024 Apr 4 Shift 1 — Question Paper

If the domain of the function $\sin^{-1}\left(\frac{3x-22}{2x-19}\right)+\log_{e}\left(\frac{3x^{2}-8x+5}{x^{2}-3x-10}\right)$ is $(\alpha, \beta]$, then $3\alpha+10\beta$ is equal to :

Let $\alpha$ and $\beta$ be the sum and the product of all the non-zero solutions of the equation $(\bar{z})^{2}+|z|=0$, $z \in \mathbf{C}$. Then $4(\alpha^{2}+\beta^{2})$ is equal to :

If 2 and 6 are the roots of the equation $ax^{2}+bx+1=0$, then the quadratic equation, whose roots are $\frac{1}{2a+b}$ and $\frac{1}{6a+b}$, is :

If the system of equations

$$x+(\sqrt{2}\sin\alpha)y+(\sqrt{2}\cos\alpha)z=0$$ $$x+(\cos\alpha)y+(\sin\alpha)z=0$$ $$x+(\sin\alpha)y-(\cos\alpha)z=0$$

has a non-trivial solution, then $\alpha \in \left(0, \frac{\pi}{2}\right)$ is equal to :

Let $\alpha \in (0, \infty)$ and $A=\begin{bmatrix} 1 & 2 & \alpha \\ 1 & 0 & 1 \\ 0 & 1 & 2 \end{bmatrix}$. If $\det(\mathrm{adj}(2A-A^{T})\cdot\mathrm{adj}(A-2A^{T}))=2^{8}$, then $(\det(A))^{2}$ is equal to :

Let $f: \mathbf{R} \to \mathbf{R}$ be a function given by

$$f(x)=\begin{cases} \frac{1-\cos 2x}{x^{2}}, & x<0 \\ \alpha, & x=0, \\ \frac{\beta\sqrt{1-\cos x}}{x}, & x>0 \end{cases}$$

where $\alpha, \beta \in \mathbf{R}$. If $f$ is continuous at $x=0$, then $\alpha^{2}+\beta^{2}$ is equal to :

There are 5 points $P_{1}, P_{2}, P_{3}, P_{4}, P_{5}$ on the side AB, excluding A and B, of a triangle ABC. Similarly there are 6 points $P_{6}, P_{7}, \ldots, P_{11}$ on the side BC and 7 points $P_{12}, P_{13}, \ldots, P_{18}$ on the side CA of the triangle. The number of triangles, that can be formed using the points $P_{1}, P_{2}, \ldots, P_{18}$ as vertices, is :

The sum of all rational terms in the expansion of $\left(2^{\frac{1}{5}}+5^{\frac{1}{3}}\right)^{15}$ is equal to :

Let the first three terms 2, $p$ and $q$, with $q \neq 2$, of a G.P. be respectively the $7^{\text{th}}$, $8^{\text{th}}$ and $13^{\text{th}}$ terms of an A.P. If the $5^{\text{th}}$ term of the G.P. is the $n^{\text{th}}$ term of the A.P., then $n$ is equal to :

Let $f(x)=\begin{cases} -2, & -2 \leq x \leq 0 \\ x-2, & 0<x \leq 2 \end{cases}$ and $h(x)=f(|x|)+|f(x)|$. Then $\int\limits_{-2}^{2} h(x)\, dx$ is equal to :

Let $f(x)=x^{5}+2e^{x/4}$ for all $x \in \mathbf{R}$. Consider a function $g(x)$ such that $(g \circ f)(x)=x$ for all $x \in \mathbf{R}$. Then the value of $8g'(2)$ is :

Let the sum of the maximum and the minimum values of the function $f(x)=\frac{2x^{2}-3x+8}{2x^{2}+3x+8}$ be $\frac{m}{n}$, where $\gcd(m, n)=1$. Then $m+n$ is equal to :

One of the points of intersection of the curves $y=1+3x-2x^{2}$ and $y=\frac{1}{x}$ is $\left(\frac{1}{2}, 2\right)$. Let the area of the region enclosed by these curves be $\frac{1}{24}(l\sqrt{5}+m)-n\log_{e}(1+\sqrt{5})$, where $l$, $m$, $n \in \mathbf{N}$. Then $l+m+n$ is equal to

If the solution $y=y(x)$ of the differential equation $(x^{4}+2x^{3}+3x^{2}+2x+2)dy-(2x^{2}+2x+3)dx=0$ satisfies $y(-1)=-\frac{\pi}{4}$, then $y(0)$ is equal to :

A square is inscribed in the circle $x^{2}+y^{2}-10x-6y+30=0$. One side of this square is parallel to $y=x+3$. If $(x_{i}, y_{i})$ are the vertices of the square, then $\Sigma(x_{i}^{2}+y_{i}^{2})$ is equal to :

The vertices of a triangle are $A(-1, 3)$, $B(-2, 2)$ and $C(3, -1)$. A new triangle is formed by shifting the sides of the triangle by one unit inwards. Then the equation of the side of the new triangle nearest to origin is :

Let the point, on the line passing through the points $P(1, -2, 3)$ and $Q(5, -4, 7)$, farther from the origin and at a distance of 9 units from the point P, be $(\alpha, \beta, \gamma)$. Then $\alpha^{2}+\beta^{2}+\gamma^{2}$ is equal to :

Let a unit vector which makes an angle of $60°$ with $2\hat{i}+2\hat{j}-\hat{k}$ and an angle of $45°$ with $\hat{i}-\hat{k}$ be $\vec{C}$. Then $\vec{C}+\left(-\frac{1}{2}\hat{i}+\frac{1}{3\sqrt{2}}\hat{j}-\frac{\sqrt{2}}{3}\hat{k}\right)$ is :

Three urns A, B and C contain 7 red, 5 black; 5 red, 7 black and 6 red, 6 black balls, respectively. One of the urn is selected at random and a ball is drawn from it. If the ball drawn is black, then the probability that it is drawn from urn A is :

Let $\alpha, \beta \in \mathbf{R}$. Let the mean and the variance of 6 observations $-3, 4, 7, -6, \alpha, \beta$ be 2 and 23, respectively. The mean deviation about the mean of these 6 observations is :

In a survey of 220 students of a higher secondary school, it was found that at least 125 and at most 130 students studied Mathematics; at least 85 and at most 95 studied Physics; at least 75 and at most 90 studied Chemistry; 30 studied both Physics and Chemistry; 50 studied both Chemistry and Mathematics; 40 studied both Mathematics and Physics and 10 studied none of these subjects. Let $m$ and $n$ respectively be the least and the most number of students who studied all the three subjects. Then $m+n$ is equal to ________.

Let A be a $3 \times 3$ matrix of non-negative real elements such that $A\begin{bmatrix} 1 \\ 1 \\ 1 \end{bmatrix}=3\begin{bmatrix} 1 \\ 1 \\ 1 \end{bmatrix}$. Then the maximum value of $\det(A)$ is ________.

Let $a=1+\frac{^{2}C_{2}}{3!}+\frac{^{3}C_{2}}{4!}+\frac{^{4}C_{2}}{5!}+\ldots$,

$b=1+\frac{^{1}C_{0}+{}^{1}C_{1}}{1!}+\frac{^{2}C_{0}+{}^{2}C_{1}+{}^{2}C_{2}}{2!}+\frac{^{3}C_{0}+{}^{3}C_{1}+{}^{3}C_{2}+{}^{3}C_{3}}{3!}+\ldots$

Then $\frac{2b}{a^{2}}$ is equal to ________.

If $\lim\limits_{x \to 1} \frac{(5x+1)^{1/3}-(x+5)^{1/3}}{(2x+3)^{1/2}-(x+4)^{1/2}}=\frac{m\sqrt{5}}{n(2n)^{2/3}}$, where $\gcd(m, n)=1$, then $8m+12n$ is equal to ________.

If $\int\limits_{0}^{\frac{\pi}{4}} \frac{\sin^{2}x}{1+\sin x \cos x}\, dx=\frac{1}{a}\log_{e}\left(\frac{a}{3}\right)+\frac{\pi}{b\sqrt{3}}$, where $a, b \in \mathbf{N}$, then $a+b$ is equal to ________.

Let the solution $y=y(x)$ of the differential equation $\frac{dy}{dx}-y=1+4\sin x$ satisfy $y(\pi)=1$. Then $y\left(\frac{\pi}{2}\right)+10$ is equal to ________.

Let A be a square matrix of order 2 such that $|A|=2$ and the sum of its diagonal elements is $-3$. If the points $(x, y)$ satisfying $A^{2}+xA+y\,I=O$ lie on a hyperbola, whose transverse axis is parallel to the $x$-axis, eccentricity is $e$ and the length of the latus rectum is $l$, then $e^{4}+l^{4}$ is equal to ________.

Let the length of the focal chord PQ of the parabola $y^{2}=12x$ be 15 units. If the distance of PQ from the origin is $p$, then $10p^{2}$ is equal to ________.

If the shortest distance between the lines $\frac{x+2}{2}=\frac{y+3}{3}=\frac{z-5}{4}$ and $\frac{x-3}{1}=\frac{y-2}{-3}=\frac{z+4}{2}$ is $\frac{38}{3\sqrt{5}}k$, and $\int\limits_{0}^{k}[x^{2}]\, dx=\alpha-\sqrt{\alpha}$, where $[x]$ denotes the greatest integer function, then $6\alpha^{3}$ is equal to ________.

Let ABC be a triangle of area $15\sqrt{2}$ and the vectors $\vec{AB}=\hat{i}+2\hat{j}-7\hat{k}$, $\vec{BC}=a\hat{i}+b\hat{j}+c\hat{k}$ and $\vec{AC}=6\hat{i}+d\hat{j}-2\hat{k}$, $d>0$. Then the square of the length of the largest side of the triangle ABC is ________.

The equation of stationary wave is :

$$y = 2a \sin\left(\frac{2\pi nt}{\lambda}\right) \cos\left(\frac{2\pi x}{\lambda}\right).$$

Which of the following is NOT correct :

The co-ordinates of a particle moving in $x$-$y$ plane are given by : $x = 2 + 4t$, $y = 3t + 8t^2$. The motion of the particle is :

A wooden block, initially at rest on the ground, is pushed by a force which increases linearly with time t. Which of the following curve best describes acceleration of the block with time :

A body travels 102.5 m in $n^{\text{th}}$ second and 115.0 m in $(n+2)^{\text{th}}$ second. The acceleration is :

If a rubber ball falls from a height h and rebounds upto the height of h/2. The percentage loss of total energy of the initial system as well as velocity ball before it strikes the ground, respectively, are :

A metal wire of uniform mass density having length L and mass M is bent to form a semicircular arc and a particle of mass m is placed at the centre of the arc. The gravitational force on the particle by the wire is :

Given below are two statements :

Statement I : When speed of liquid is zero everywhere, pressure difference at any two points depends on equation $\mathrm{P_1 - P_2} = \rho g(\mathrm{h_2 - h_1})$.

Statement II : In ventury tube shown $2gh = v_1^2 - v_2^2$

[Figure: a venturi tube with a wide section (pressure $P_1$, area $A_1$, speed $v_1$) and a narrow section (pressure $P_2$, area $A_2$, speed $v_2$); two vertical tubes rise from the two sections with liquid level difference h]

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

On celcius scale the temperature of body increases by 40°C. The increase in temperature on Fahrenheit scale is :

P-T diagram of an ideal gas having three different densities $\rho_1$, $\rho_2$, $\rho_3$ (in three different cases) is shown in the figure. Which of the following is correct :

[Figure: P vs T graph with three straight lines from the origin, with slopes decreasing in the order $\rho_1$, $\rho_2$, $\rho_3$ (line $\rho_1$ steepest)]

An infinitely long positively charged straight thread has a linear charge density $\lambda~\mathrm{Cm^{-1}}$. An electron revolves along a circular path having axis along the length of the wire. The graph that correctly represents the variation of the kinetic energy of electron as a function of radius of circular path from the wire is :

The resistances of the platinum wire of a platinum resistance thermometer at the ice point and steam point are $8~\Omega$ and $10~\Omega$ respectively. After inserting in a hot bath of temperature 400°C, the resistance of platinum wire is :

An electron is projected with uniform velocity along the axis inside a current carrying long solenoid. Then :

In an ac circuit, the instantaneous current is zero, when the instantaneous voltage is maximum. In this case, the source may be connected to :

A. pure inductor.

B. pure capacitor.

C. pure resistor.

D. combination of an inductor and capacitor.

Choose the correct answer from the options given below :

The electric field in an electromagnetic wave is given by $\vec{\mathrm{E}} = \hat{i}~40 \cos\omega\left(\mathrm{t} - \frac{z}{c}\right)~\mathrm{N\,C^{-1}}$. The magnetic field induction of this wave is (in SI unit) :

An effective power of a combination of 5 identical convex lenses which are kept in contact along the principal axis is 25 D. Focal length of each of the convex lens is :

Which figure shows the correct variation of applied potential difference (V) with photoelectric current (I) at two different intensities of light ($\mathrm{I_1} < \mathrm{I_2}$) of same wavelengths :

Which of the following nuclear fragments corresponding to nuclear fission between neutron $\left({}^{1}_{0}\mathrm{n}\right)$ and uranium isotope $\left({}^{235}_{92}\mathrm{U}\right)$ is correct :

The value of net resistance of the network as shown in the given figure is :

[Figure: three parallel branches between terminals at $-6$ V and $-8$ V — top branch: 15 Ω resistor; middle branch: 10 Ω resistor in series with a diode pointing toward the $-8$ V terminal; bottom branch: 5 Ω resistor in series with a diode pointing toward the $-6$ V terminal]

In an experiment to measure focal length ($f$) of convex lens, the least counts of the measuring scales for the position of object ($u$) and for the position of image ($v$) are $\Delta u$ and $\Delta v$, respectively. The error in the measurement of the focal length of the convex lens will be :

To measure the internal resistance of a battery, potentiometer is used. For R = 10 Ω, the balance point is observed at $l = 500$ cm and for R = 1 Ω the balance point is observed at $l = 400$ cm. The internal resistance of the battery is approximately :

Two forces $\vec{\mathrm{F}}_1$ and $\vec{\mathrm{F}}_2$ are acting on a body. One force has magnitude thrice that of the other force and the resultant of the two forces is equal to the force of larger magnitude. The angle between $\vec{\mathrm{F}}_1$ and $\vec{\mathrm{F}}_2$ is $\cos^{-1}\left(\frac{1}{n}\right)$. The value of $|n|$ is ________.

A solid sphere and a hollow cylinder roll up without slipping on same inclined plane with same initial speed $v$. The sphere and the cylinder reaches upto maximum heights $\mathrm{h_1}$ and $\mathrm{h_2}$, respectively, above the initial level. The ratio $\mathrm{h_1} : \mathrm{h_2}$ is $\frac{n}{10}$. The value of n is ________.

A soap bubble is blown to a diameter of 7 cm. 36960 erg of work is done in blowing it further. If surface tension of soap solution is 40 dyne/cm then the new radius is ________ cm. Take $\left(\pi = \frac{22}{7}\right)$.

An elastic spring under tension of 3 N has a length $a$. Its length is $b$ under tension 2 N. For its length $(3a - 2b)$, the value of tension will be ________ N.

An infinite plane sheet of charge having uniform surface charge density $+\sigma_s~\mathrm{C/m^2}$ is placed on $x$-$y$ plane. Another infinitely long line charge having uniform linear charge density $+\lambda_e~\mathrm{C/m}$ is placed at $z = 4$ m plane and parallel to $y$-axis. If the magnitude values $|\sigma_s| = 2|\lambda_e|$ then at point (0, 0, 2), the ratio of magnitudes of electric field values due to sheet charge to that of line charge is $\pi\sqrt{n} : 1$. The value of n is ________.

Twelve wires each having resistance 2 Ω are joined to form a cube. A battery of 6 V emf is joined across point a and c. The voltage difference between e and f is ________ V.

[Figure: a cube with top face vertices a, b, c, d and bottom face vertices e, h, g, f (e below a, f below d, g below c, h below h-corner); a 6 V battery connected across the face diagonal between vertices a and c of the top face]

The magnetic field existing in a region is given by $\vec{\mathrm{B}} = 0.2(1 + 2x)\hat{k}~\mathrm{T}$. A square loop of edge 50 cm carrying 0.5 A current is placed in $x$-$y$ plane with its edges parallel to the $x$-$y$ axes, as shown in figure. The magnitude of the net magnetic force experienced by the loop is ________ mN.

[Figure: x-y coordinate axes with a square loop located at x = 2 m, y = 2 m, its sides parallel to the axes and current circulating around it as marked by arrows]

A alternating current at any instant is given by $i = \left[6 + \sqrt{56}\,\sin\left(100\pi \mathrm{t} + \pi/3\right)\right]$ A. The rms value of the current is ________ A.

Two wavelengths $\lambda_1$ and $\lambda_2$ are used in Young's double slit experiment. $\lambda_1 = 450$ nm and $\lambda_2 = 650$ nm. The minimum order of fringe produced by $\lambda_2$ which overlaps with the fringe produced by $\lambda_1$ is n. The value of n is ________.

A hydrogen atom changes its state from $n = 3$ to $n = 2$. Due to recoil, the percentage change in the wave length of emitted light is approximately $1 \times 10^{-n}$. The value of n is ________.

[Given Rhc = 13.6 eV, hc = 1242 eV nm, h = $6.6 \times 10^{-34}$ J s mass of the hydrogenatom = $1.6 \times 10^{-27}$ kg]

The Molarity (M) of an aqueous solution containing 5.85 g of NaCl in 500 mL water is : (Given : Molar Mass Na : 23 and Cl : 35.5 gmol$^{-1}$)

Which one of the following molecules has maximum dipole moment ?

What will be the decreasing order of basic strength of the following conjugate bases ?

$\mathrm{^-OH,\ R\bar{O},\ CH_3CO\bar{O},\ \bar{Cl}}$

One of the commonly used electrode is calomel electrode. Under which of the following categories, calomel electrode comes ?

What pressure (bar) of $\mathrm{H_2}$ would be required to make emf of hydrogen electrode zero in pure water at 25°C ?

The correct order of first ionization enthalpy values of the following elements is : (A) O (B) N (C) Be (D) F (E) B Choose the correct answer from the options given below :

Number of elements from the following that CANNOT form compounds with valencies which match with their respective group valencies is _________. B, C, N, S, O, F, P, Al, Si

Number of molecules/ions from the following in which the central atom is involved in $\mathrm{sp^3}$ hybridization is ________.

$\mathrm{NO_3^-,\ BCl_3,\ ClO_2^-,\ ClO_3}$

The element which shows only one oxidation state other than its elemental form is :

The correct sequence of ligands in the order of decreasing field strength is :

Number of complexes from the following with even number of unpaired "d" electrons is ________.

$\mathrm{[V(H_2O)_6]^{3+}, [Cr(H_2O)_6]^{2+}, [Fe(H_2O)_6]^{3+}, [Ni(H_2O)_6]^{3+}, [Cu(H_2O)_6]^{2+}}$

[Given atomic numbers : V = 23, Cr = 24, Fe = 26, Ni = 28 Cu = 29]

In the precipitation of the iron group (III) in qualitative analysis, ammonium chloride is added before adding ammonium hydroxide to :

Which of the following nitrogen containing compound does not give Lassaigne's test ?

Match List I with List II :

[Figure: List-I shows four curved-arrow mechanism steps: (A) aniline resonance — the $\mathrm{NH_2}$ lone pair delocalises into the benzene ring giving the $\mathrm{{}^+NH_2}$= iminium resonance structure with a ring carbanion; (B) cyclohexene $+ \mathrm{H^+} \rightarrow$ cyclohexyl carbocation; (C) cyclohexene $+ \mathrm{\bar{C}N} \rightarrow$ cyclohexyl carbanion bearing a CN group; (D) nitrobenzene resonance — the $\mathrm{-NO_2}$ group withdraws ring electrons giving a ring carbocation resonance structure.]

List - I (Mechanism steps)	List - II (Effect)
(A) aniline lone-pair delocalisation into ring	(I) $-$ E effect
(B) cyclohexene $+ \mathrm{H^+} \rightarrow$ carbocation	(II) $-$ R effect
(C) cyclohexene $+ \mathrm{\bar{C}N} \rightarrow$ carbanion with CN	(III) $+$ E effect
(D) nitrobenzene ring-electron withdrawal by $\mathrm{-NO_2}$	(IV) $+$ R effect

Choose the correct answer from the options given below :

Which among the following is incorrect statement ?

[Figure: compound (A) is 1-bromo-4-(2-bromoethyl)benzene — a benzene ring with Br at the para position and a $\mathrm{-CH_2CH_2Br}$ side chain]

$$\mathrm{(A) \xrightarrow{NaOH_{(alc)}} \ (B) \ \xrightarrow[Ether]{HBr} \ (C)}$$

Identify (B) and (C) and how are (A) and (C) related ?

Identify the correct set of reagents or reaction conditions 'X' and 'Y' in the following set of transformation.

$$\mathrm{CH_3-CH_2-CH_2-Br \xrightarrow{'X'} Product \xrightarrow{'Y'} \underset{\underset{\mathrm{Br}}{|}}{CH_3-CH}-CH_3}$$

Given below are two statements :

Statements I : Acidity of α-hydrogens of aldehydes and ketones is responsible for Aldol reaction.

Statement II : Reaction between benzaldehyde and ethanal will NOT give Cross - Aldol product.

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

Identify the product in the following reaction :

[Figure: 3-acetylbenzaldehyde — a benzene ring bearing a $\mathrm{-CHO}$ group and a $\mathrm{-COCH_3}$ group at meta positions]

$$\xrightarrow[\mathrm{HCl}]{\mathrm{Zn-Hg}} \text{Product}$$

Which of the following is the correct structure of L-Glucose ?

The de-Broglie's wavelength of an electron in the 4$^{th}$ orbit is ________ $\pi a_0$. ($a_0$ = Bohr's radius)