JEE Main 2024 Apr 6 Shift 1 — Question Paper

Let the relations $R_1$ and $R_2$ on the set $X = \{1, 2, 3, ...., 20\}$ be given by $R_1 = \{(x, y): 2x - 3y = 2\}$ and $R_2 = \{(x, y): -5x + 4y = 0\}$. If $M$ and $N$ be the minimum number of elements required to be added in $R_1$ and $R_2$, respectively, in order to make the relations symmetric, then $M+N$ equals

The function $f(x) = \dfrac{x^2 + 2x - 15}{x^2 - 4x + 9}$, $x \in \mathbb{R}$ is

The number of triangles whose vertices are at the vertices of a regular octagon but none of whose sides is a side of the octagon is

Let $\alpha$, $\beta$ be the distinct roots of the equation $x^2 - (t^2 - 5t + 6)x + 1 = 0$, $t \in \mathbb{R}$ and $a_n = \alpha^n + \beta^n$. Then the minimum value of $\dfrac{a_{2023} + a_{2025}}{a_{2024}}$ is

For $\alpha, \beta \in \mathbb{R}$ and a natural number $n$, let $A_r = \begin{vmatrix} r & 1 & \frac{n^2}{2} + \alpha \\ 2r & 2 & n^2 - \beta \\ 3r - 2 & 3 & \frac{n(3n-1)}{2} \end{vmatrix}$. Then $2A_{10} - A_8$ is

Let $f : (-\infty, \infty) - \{0\} \to \mathbb{R}$ be a differentiable function such that $f'(1) = \lim\limits_{a \to \infty} a^2 f\left(\dfrac{1}{a}\right)$. Then $\lim\limits_{a \to \infty} \dfrac{a(a+1)}{2} \tan^{-1}\left(\dfrac{1}{a}\right) + a^2 - 2\log_e a$ is equal to

Let $A = \{n \in [100, 700] \cap \mathbb{N}: n$ is neither a multiple of 3 nor a multiple of 4$\}$. Then the number of elements in $A$ is

If $f(x) = \begin{cases} x^3 \sin\left(\dfrac{1}{x}\right), & x \neq 0 \\ 0, & x = 0 \end{cases}$, then

The interval in which the function $f(x) = x^x$, $x > 0$, is strictly increasing is

Let a variable line of slope $m > 0$ passing through the point $(4, -9)$ intersect the coordinate axes at the points $A$ and $B$. The minimum value of the sum of the distances of $A$ and $B$ from the origin is

$\displaystyle\int_0^{\pi/4} \dfrac{\cos^2 x \, \sin^2 x}{\left(\cos^3 x + \sin^3 x\right)^2} \, dx$ is equal to

Let the area of the region enclosed by the curves $y = 3x$, $2y = 27 - 3x$ and $y = 3x - x\sqrt{x}$ be $A$. Then $10A$ is equal to

Let $y = y(x)$ be the solution of the differential equation $\left(1 + x^2\right) \dfrac{dy}{dx} + y = e^{\tan^{-1} x}$, $y(1) = 0$. Then $y(0)$ is

Let $y = y(x)$ be the solution of the differential equation $(2x \log_e x) \dfrac{dy}{dx} + 2y = \dfrac{3}{x} \log_e x$, $x > 0$ and $y(e^{-1}) = 0$. Then, $y(e)$ is equal to

A circle is inscribed in an equilateral triangle of side of length 12. If the area and perimeter of any square inscribed in this circle are $m$ and $n$, respectively, then $m + n^2$ is equal to

Let $C$ be the circle of minimum area touching the parabola $y = 6 - x^2$ and the lines $y = \sqrt{3}\,|x|$. Then, which one of the following points lies on the circle $C$?

The shortest distance between the lines $\dfrac{x-3}{2} = \dfrac{y+15}{-7} = \dfrac{z-9}{5}$ and $\dfrac{x+1}{2} = \dfrac{y-1}{1} = \dfrac{z-9}{-3}$ is

The mean and standard deviation of 20 observations are found to be 10 and 2, respectively. On rechecking, it was found that an observation by mistake was taken 8 instead of 12. The correct standard deviation is

If $A(3, 1, -1)$, $B\left(\dfrac{5}{3}, \dfrac{7}{3}, \dfrac{1}{3}\right)$, $C(2, 2, 1)$ and $D\left(\dfrac{10}{3}, \dfrac{2}{3}, \dfrac{-1}{3}\right)$ are the vertices of a quadrilateral $ABCD$, then its area is

A company has two plants $A$ and $B$ to manufacture motorcycles. 60% motorcycles are manufactured at plant $A$ and the remaining are manufactured at plant $B$. 80% of the motorcycles manufactured at plant $A$ are rated of the standard quality, while 90% of the motorcycles manufactured at plant $B$ are rated of the standard quality. A motorcycle picked up randomly from the total production is found to be of the standard quality. If $p$ is the probability that it was manufactured at plant $B$, then $126p$ is

Let $x_1, x_2, x_3, x_4$ be the solution of the equation $4x^4 + 8x^3 - 17x^2 - 12x + 9 = 0$ and $\left(4 + x_1^2\right)\left(4 + x_2^2\right)\left(4 + x_3^2\right)\left(4 + x_4^2\right) = \dfrac{125}{16}m$. Then the value of $m$ is ________

Let $\alpha\beta\gamma = 45$; $\alpha, \beta, \gamma \in \mathbb{R}$. If $x(\alpha, 1, 2) + y(1, \beta, 2) + z(2, 3, \gamma) = (0, 0, 0)$ for some $x, y, z \in \mathbb{R}$, $xyz \neq 0$, then $6\alpha + 4\beta + \gamma$ is equal to ________

If the second, third and fourth terms in the expansion of $(x + y)^n$ are 135, 30 and $\dfrac{10}{3}$, respectively, then $6(n^3 + x^2 + y)$ is equal to ________

Let the first term of a series be $T_1 = 6$ and its $r^{\text{th}}$ term $T_r = 3T_{r-1} + 6^r$, $r = 2, 3, ......, n$. If the sum of the first $n$ terms of this series is $\dfrac{1}{5}\left(n^2 - 12n + 39\right)\left(4 \cdot 6^n - 5 \cdot 3^n + 1\right)$, then $n$ is equal to ________.

Let $r_k = \dfrac{\int_0^1 \left(1 - x^7\right)^k \, dx}{\int_0^1 \left(1 - x^7\right)^{k+1} \, dx}$, $k \in \mathbb{N}$. Then the value of $\displaystyle\sum_{k=1}^{10} \dfrac{1}{7\left(r_k - 1\right)}$ is equal to ________

Let $L_1$, $L_2$ be the lines passing through the point $P(0, 1)$ and touching the parabola $9x^2 + 12x + 18y - 14 = 0$. Let $Q$ and $R$ be the points on the lines $L_1$ and $L_2$ such that the $\triangle PQR$ is an isosceles triangle with base $QR$. If the slopes of the lines $QR$ are $m_1$ and $m_2$, then $16\left(m_1^2 + m_2^2\right)$ is equal to ________.

Let a conic $C$ pass through the point $(4, -2)$ and $P(x, y)$, $x \geq 3$, be any point on $C$. Let the slope of the line touching the conic $C$ only at a single point $P$ be half the slope of the line joining the points $P$ and $(3, -5)$. If the focal distance of the point $(7, 1)$ on $C$ is $d$, then $12d$ equals ________.

Let $P$ be the point $(10, -2, -1)$ and $Q$ be the foot of the perpendicular drawn from the point $R(1, 7, 6)$ on the line passing through the points $(2, -5, 11)$ and $(-6, 7, -5)$. Then the length of the line segment $PQ$ is equal to ________.

Let $\vec{a} = 2\hat{i} - 3\hat{j} + 4\hat{k}$, $\vec{b} = 3\hat{i} + 4\hat{j} - 5\hat{k}$ and a vector $\vec{c}$ be such that $\vec{a} \times \left(\vec{b} + \vec{c}\right) + \vec{b} \times \vec{c} = \hat{i} + 8\hat{j} + 13\hat{k}$. If $\vec{a} \cdot \vec{c} = 13$, then $\left(24 - \vec{b} \cdot \vec{c}\right)$ is equal to ________.

For $n \in \mathbb{N}$, if $\cot^{-1} 3 + \cot^{-1} 4 + \cot^{-1} 5 + \cot^{-1} n = \dfrac{\pi}{4}$, then $n$ is equal to ________.

Four particles $A, B, C, D$ of mass $\frac{m}{2}, m, 2m, 4m$, have same momentum, respectively. The particle with maximum kinetic energy is :

A light string passing over a smooth light pulley connects two blocks of masses $m_1$ and $m_2$ (where $m_2 > m_1$). If the acceleration of the system is $\frac{g}{\sqrt{2}}$, then the ratio of the masses $\frac{m_1}{m_2}$ is:

To project a body of mass m from earth's surface to infinity, the required kinetic energy is (assume, the radius of earth is $R_E$, $g$ = acceleration due to gravity on the surface of earth):

A train starting from rest first accelerates uniformly up to a speed of 80 $km/h$ for time $t$, then it moves with a constant speed for time $3t$. The average speed of the train for this duration of journey will be (in $km/h$) :

A small ball of mass m and density $\rho$ is dropped in a viscous liquid of density $\rho_0$. After sometime, the ball falls with constant velocity. The viscous force on the ball is :

The specific heat at constant pressure of a real gas obeying $PV^2 = RT$ equation is:

A sample contains mixture of helium and oxygen gas. The ratio of root mean square speed of helium and oxygen in the sample, is :

$\sigma$ is the uniform surface charge density of a thin spherical shell of radius $R$. The electric field at any point on the surface of the spherical shell is :

Electromagnetic waves travel in a medium with speed of $1.5\times10^{8}\ \mathrm{m\ s^{-1}}$. The relative permeability of the medium is 2.0. The relative permittivity will be:

The value of unknown resistance $(x)$ for which the potential difference between $B$ and $D$ will be zero in the arrangement shown, is :

[Figure: Wheatstone-bridge-like network between A and C. Upper-left arm A to B: $24\,\Omega$ in parallel with series $12\,\Omega$ and $12\,\Omega$; upper-right arm B to C: $1\,\Omega$ in parallel with $1\,\Omega$; lower-left arm A to D: $x\,\Omega$ in parallel with series $12\,\Omega$ and $12\,\Omega$; lower-right arm D to C: $0.5\,\Omega$; battery of $14.5\,V$ connected across A and C]

While measuring diameter of wire using screw gauge the following readings were noted. Main scale reading is 1 mm and circular scale reading is equal to 42 divisions. Pitch of screw gauge is 1 mm and it has 100 divisions on circular scale. The diameter of the wire is $\frac{x}{50}\ mm$. The value of $x$ is :

Match List I with List II

	LIST I		LIST II
A.	Torque	I.	$[M^1L^1T^{-2}A^{-2}]$
B.	Magnetic field	II.	$[L^2A^1]$
C.	Magnetic moment	III.	$[M^1T^{-2}A^{-1}]$
D.	Permeability of free space	IV.	$[M^1L^2T^{-2}]$

Choose the correct answer from the options given below:

An element $\Delta l = \Delta x\,\hat{i}$ is placed at the origin and carries a large current $I = 10\mathrm{A}$. The magnetic field on the $y$-axis at a distance of 0.5m from the elements $\Delta x$ of 1cm length is :

[Figure: current element $\Delta x$ along the x-axis at the origin; point $P$ on the y-axis at a distance $0.5\,m$ above the element]

Given below are two statements :

Statement I : In an LCR series circuit, current is maximum at resonance.

Statement II : Current in a purely resistive circuit can never be less than that in a series LCR circuit when connected to same voltage source.

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

To find the spring constant $(k)$ of a spring experimentally, a student commits 2% positive error in the measurement of time and 1% negative error in measurement of mass. The percentage error in determining value of $k$ is :

Which of the following phenomena does not explain by wave nature of light.

A. reflection B. diffraction C. photoelectric effect D. interference E. polarization

Choose the most appropriate answer from the options given below:

In photoelectric experiment energy of 2.48 eV irradiates a photo sensitive material. The stopping potential was measured to be 0.5 V. Work function of the photo sensitive material is :

The ratio of the shortest wavelength of Balmer series to the shortest wavelength of Lyman series for hydrogen atom is :

The correct truth table for the following logic circuit is :

[Figure: logic circuit — inputs A and B feed an AND gate; input A also passes through a NOT gate; the NOT-gate output and the AND-gate output feed an OR gate whose output is Y]

A bullet of mass $50g$ is fired with a speed 100 m/s on a plywood and emerges with 40 m/s. The percentage loss of kinetic energy is :

The refractive index of prism is $\mu = \sqrt{3}$ and the ratio of the angle of minimum deviation to the angle of prism is one. The value of angle of prism is _________ $^\circ$.

A particle is doing simple harmonic motion of amplitude 0.06 m and time period 3.14 s. The maximum velocity of the particle is _________ cm/s.

For three vectors $\vec{A} = (-x\hat{i} - 6\hat{j} - 2\hat{k})$, $\vec{B} = (-\hat{i} + 4\hat{j} + 3\hat{k})$ and $\vec{C} = (-8\hat{i} - \hat{j} + 3\hat{k})$, if $\vec{A}\cdot(\vec{B}\times\vec{C}) = 0$, then value of $x$ is _________.

A big drop is formed by coalescing 1000 small droplets of water. The ratio of surface energy of 1000 droplets to that of energy of big drop is $\frac{10}{x}$. The value of $x$ is _________.

When a $dc$ voltage of 100V is applied to an inductor, a $dc$ current of 5A flows through it. When an $ac$ voltage of 200V peak value is connected to inductor, its inductive reactance is found to be $20\sqrt{3}\,\Omega$. The power dissipated in the circuit is _________ $W$.

If the radius of earth is reduced to three-fourth of its present value without change in its mass then value of duration of the day of earth will be _________ hours 30 minutes.

A circular coil having 200 turns, $2.5\times10^{-4}\ \mathrm{m^2}$ area and carrying $100\ \mu A$ current is placed in a uniform magnetic field of 1T. Initially the magnetic dipole moment $(\vec{M})$ was directed along $\vec{B}$. Amount of work, required to rotate the coil through $90^\circ$ from its initial orientation such that $\vec{M}$ becomes perpendicular to $\vec{B}$, is _________ $\mu J$.

A wire of resistance $R$ and radius $r$ is stretched till its radius became $r/2$. If new resistance of the stretched wire is $x\,R$, then value of $x$ is _________.

Three infinitely long charged thin sheets are placed as shown in figure. The magnitude of electric field at the point $P$ is $\frac{x\sigma}{\epsilon_o}$. The value of $x$ is _________ (all quantities are measured in SI units).

[Figure: three vertical sheets parallel to the YZ plane — sheet with charge density $-\sigma$ at $X=-a$, sheet with $-2\sigma$ at $X=a$, sheet with $\sigma$ at $X=3a$; point $P$ lies between $X=a$ and $X=3a$]

Radius of a certain orbit of hydrogen atom is 8.48 Å. If energy of electron in this orbit is $E/x$, then $x =$ _________.

(Given $\mathrm{a}_o = 0.529$ Å, $E$ = energy of electron in ground state).

The density of '$x$' M solution ('$x$' molar) of NaOH is $1.12 \mathrm{~g~mL^{-1}}$, while in molality, the concentration of the solution is $3 \mathrm{~m}$ (3 molal). Then $x$ is

(Given : Molar mass of NaOH is $40 \mathrm{~g/mol}$)

Match List I with List II

LIST I (Molecule / Species)	LIST II (Property / Shape)
A. $\mathrm{SO_2Cl_2}$	I. Paramagnetic
B. $\mathrm{NO}$	II. Diamagnetic
C. $\mathrm{NO_2^-}$	III. Tetrahedral
D. $\mathrm{I_3^-}$	IV. Linear

Choose the correct answer from the options given below:

At $-20\ ^{\circ}\mathrm{C}$ and 1 atm pressure, a cylinder is filled with equal number of $\mathrm{H_2}$, $\mathrm{I_2}$ and $\mathrm{HI}$ molecules for the reaction

$\mathrm{H_2(g) + I_2(g) \rightleftharpoons 2HI(g)}$, the $\mathrm{K_p}$ for the process is $x \times 10^{-1}$.

$x =$ ________.

[Given : $\mathrm{R} = 0.082 \mathrm{~L~atm~K^{-1}~mol^{-1}}$]

Which of the following material is not a semiconductor.

A conductivity cell with two electrodes (dark side) are half filled with infinitely dilute aqueous solution of a weak electrolyte. If volume is doubled by adding more water at constant temperature, the molar conductivity of the cell will -

[Figure: a cubic conductivity cell of dimensions 1 cm × 1 cm × 1 cm with two shaded (dark) electrode plates on opposite faces, marked (+) and (−)]

The electron affinity value are negative for

A. $\mathrm{Be \rightarrow Be^-}$ B. $\mathrm{N \rightarrow N^-}$ C. $\mathrm{O \rightarrow O^{2-}}$ D. $\mathrm{Na \rightarrow Na^-}$ E. $\mathrm{Al \rightarrow Al^-}$

Choose the most appropriate answer from the options given below :

Given below are two statements :

Statement I : Gallium is used in the manufacturing of thermometers.

Statement II : A thermometer containing gallium is useful for measuring the freezing point (256 K) of brine solution.

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

Match List I with List II

LIST I (Compound/Species)	LIST II (Shape/Geometry)
A. $\mathrm{SF_4}$	I. Tetrahedral
B. $\mathrm{BrF_3}$	II. Pyramidal
C. $\mathrm{BrO_3^-}$	III. See saw
D. $\mathrm{NH_4^+}$	IV. Bent T-Shape

Choose the correct answer from the options given below:

The number of element from the following that do not belong to lanthanoids is

Eu, Cm, Er, Tb, Yb and Lu

Match List I with List II

LIST I (Hybridization)	LIST II (Orientation in Space)
A. $\mathrm{sp^3}$	I. Trigonal bipyramidal
B. $\mathrm{dsp^2}$	II. Octahedral
C. $\mathrm{sp^3d}$	III. Tetrahedral
D. $\mathrm{sp^3d^2}$	IV. Square planar

Choose the correct answer from the options given below:

Consider the following complexes

$\mathrm{[CoCl(NH_3)_5]^{2+}}$, $\mathrm{[Co(CN)_6]^{3-}}$, $\mathrm{[Co(NH_3)_5(H_2O)]^{3+}}$, $\mathrm{[Cu(H_2O)_4]^{2+}}$

(A)     (B)     (C)     (D)

The correct order of A, B, C and D in terms of wavenumber of light absorbed is :

Match List I with List II

LIST I (Precipitating reagent and conditions)	LIST II (Cation)
A. $\mathrm{NH_4Cl + NH_4OH}$	I. $\mathrm{Mn^{2+}}$
B. $\mathrm{NH_4OH + Na_2CO_3}$	II. $\mathrm{Pb^{2+}}$
C. $\mathrm{NH_4OH + NH_4Cl + H_2S}$ gas	III. $\mathrm{Al^{3+}}$
D. dilute HCl	IV. $\mathrm{Sr^{2+}}$

Choose the correct answer from the options given below:

Which of the following statements are correct?

A. Glycerol is purified by vacuum distillation because it decomposes at its normal boiling point. B. Aniline can be purified by steam distillation as aniline is miscible in water. C. Ethanol can be separated from ethanol water mixture by azeotropic distillation because it forms azeotrope. D. An organic compound is pure, if mixed M.P. is remained same.

Choose the most appropriate answer from the options given below :

Which of the following is metamer of the given compound (X)?

[Figure: compound (X) — a benzene ring attached through NH to a carbonyl bonded to a cyclohexane ring, i.e. $\mathrm{C_6H_5{-}NH{-}CO{-}C_6H_{11}}$ (N-phenyl cyclohexanecarboxamide)]

Functional group present in sulphonic acids is :

Match List I with List II

LIST I (Compound)	LIST II (Uses)
A. Iodoform	I. Fire extinguisher
B. Carbon tetrachloride	II. Insecticide
C. CFC	III. Antiseptic
D. DDT	IV. Refrigerants

Choose the correct answer from the options given below:

Given below are two statements :

Statement I : Piciric acid is 2,4,6 - trinitrotoluene.

Statement II : Phenol - 2,4 - disulphonic acid is treated with Conc. $\mathrm{HNO_3}$ to get picric acid.

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

In Reimer - Tiemann reaction, phenol is converted into salicylaldehyde through an intermediate. The structure of intermediate is ________.

Which among the following aldehydes is most reactive towards nucleophilic addition reactions?

DNA molecule contains 4 bases whose structure are shown below. One of the structures is not correct, identify the incorrect base structure.

Frequency of the de-Broglie wave of electron in Bohr's first orbit of hydrogen atom is __________ $\times 10^{13}$ Hz (nearest integer).

[Given : $\mathrm{R_H}$ (Rydberg constant) $= 2.18 \times 10^{-18}$ J, $h$ (Plank's constant) $= 6.6 \times 10^{-34}$ J.s.]

Number of molecules from the following which can exhibit hydrogen bonding is ________. (nearest integer)

$\mathrm{CH_3OH}$, $\mathrm{H_2O}$, $\mathrm{C_2H_6}$, $\mathrm{C_6H_6}$, [Figure: benzene ring with $\mathrm{-NO_2}$ and $\mathrm{-OH}$ on adjacent positions (o-nitrophenol)], $\mathrm{HF}$, $\mathrm{NH_3}$

An ideal gas, $\bar{C}_v = \frac{5}{2}R$, is expanded adiabatically against a constant pressure of 1 atm untill it doubles in volume. If the initial temperature and pressure is 298 K and 5 atm, respectively then the final temperature is _________ K (nearest integer).

[$\bar{C}_v$ is the molar heat capacity at constant volume]

Consider the dissociation of the weak acid HX as given below

$\mathrm{HX(aq) \rightleftharpoons H^+(aq) + X^-(aq)}$, $\mathrm{K_a} = 1.2 \times 10^{-5}$

[$\mathrm{K_a}$ : dissociation constant]

The osmotic pressure of 0.03 M aqueous solution of HX at 300 K is _________ $\times 10^{-2}$ bar (nearest integer).

[Given : $\mathrm{R} = 0.083 \mathrm{~L~bar~mol^{-1}~K^{-1}}$]

Time required for 99.9% completion of a first order reaction is _________ times the time required for completion of 90% reaction.(nearest integer)

Among $\mathrm{CrO}$, $\mathrm{Cr_2O_3}$ and $\mathrm{CrO_3}$, the sum of spin-only magnetic moment values of basic and amphoteric oxides is _______ $10^{-2}$ BM (nearest integer).

(Given atomic number of Cr is 24)

The difference in the 'spin-only' magnetic moment values of $\mathrm{KMnO_4}$ and the manganese product formed during titration of $\mathrm{KMnO_4}$ against oxalic acid in acidic medium is ________ BM. (nearest integer)

The major product of the following reaction is P.

$$\mathrm{CH_3C \equiv C{-}CH_3} \xrightarrow[\substack{\text{(ii) dil. } \mathrm{KMnO_4} \\ 273\ \mathrm{K}}]{\text{(i) Na/liq. } \mathrm{NH_3}} \text{'P'}$$

Number of oxygen atoms present in product 'P' is ________. (nearest integer)

The major products from the following reaction sequence are product A and product B.

[Figure: cyclohexene at the centre undergoing two separate reaction sequences — to the right: (i) $\mathrm{Br_2}$, (ii) $\mathrm{HC{\equiv}C{-}CH_2{-}O^-Na^+}$ (1.0 eq.) giving A; to the left: (i) $\mathrm{Br_2}$, (ii) alc. KOH (3 eq.) giving B]

The total sum of $\pi$ electrons in product A and product B are _________ (nearest integer)

9.3 g of pure aniline upon diazotisation followed by coupling with phenol gives an orange dye. The mass of orange dye produced (assume 100% yield/conversion) is _________ g. (nearest integer)