JEE Main 2024 Jan 29 Shift 1 — Question Paper

If $f(x)=\begin{cases} 2+2x, & -1 \le x < 0 \\ 1-\dfrac{x}{3}, & 0 \le x \le 3 \end{cases}$ ; $g(x)=\begin{cases} -x, & -3 \le x \le 0 \\ x, & 0 < x \le 1 \end{cases}$, then range of $(fog)(x)$ is

Let R be a relation on $\mathbb{Z}\times\mathbb{Z}$ defined by (a, b) R (c, d) if and only if $ad - bc$ is divisible by 5. Then R is

If $z=\dfrac{1}{2}-2i$ is such that $|z+1| = \alpha z + \beta(1+i)$, $i=\sqrt{-1}$ and $\alpha, \beta \in \mathbb{R}$, then $\alpha + \beta$ is equal to

Let A be a square matrix such that $AA^T = I$. Then $\dfrac{1}{2} A\left[\left(A+A^T\right)^2 + \left(A-A^T\right)^2\right]$ is equal to

Let $A = \begin{bmatrix} 1 & 0 & 0 \\ 0 & \alpha & \beta \\ 0 & \beta & \alpha \end{bmatrix}$ and $|2A|^3 = 2^{21}$ where $\alpha, \beta \in \mathbb{Z}$. Then a value of $\alpha$ is

In an A.P., the sixth term $a_6 = 2$. If the product $a_1 a_4 a_5$ is the greatest, then the common difference of the A.P. is equal to

If in a G.P. of 64 terms, the sum of all the terms is 7 times the sum of the odd terms of the G.P, then the common ratio of the G.P. is equal to

$\lim\limits_{x \to \frac{\pi}{2}}\left(\dfrac{1}{\left(x-\frac{\pi}{2}\right)^2}\int\limits_{x^3}^{\left(\frac{\pi}{2}\right)^3} \cos\left(t^{\frac{1}{3}}\right)dt\right)$ is equal to

Suppose $f(x)=\dfrac{\left(2^x + 2^{-x}\right)\tan x\sqrt{\tan^{-1}\left(x^2 - x + 1\right)}}{\left(7x^2 + 3x + 1\right)^3}$. Then the value of $f'(0)$ is equal to

Consider the function $f:\left[\dfrac{1}{2},1\right] \to \mathbb{R}$ defined by $f(x)=4\sqrt{2}x^3 - 3\sqrt{2}x - 1$.

Consider the statements

(I) The curve $y=f(x)$ intersects the $x$-axis exactly at one point.

(II) The curve $y=f(x)$ intersects the $x$-axis at $x=\cos\dfrac{\pi}{12}$.

Then

If the value of the integral $\int\limits_{-\frac{\pi}{2}}^{\frac{\pi}{2}}\left(\dfrac{x^2\cos x}{1+\pi^x} + \dfrac{1+\sin^2 x}{1+e^{\sin x^{2023}}}\right)dx = \dfrac{\pi}{4}(\pi + a) - 2$, then the value of $a$ is

For $x \in \left(-\dfrac{\pi}{2},\dfrac{\pi}{2}\right)$, if $y(x)=\int \dfrac{\operatorname{cosec} x + \sin x}{\operatorname{cosec} x \sec x + \tan x \sin^2 x}\,dx$, and $\lim\limits_{x \to \left(\frac{\pi}{2}\right)^-} y(x) = 0$ then $y\left(\dfrac{\pi}{4}\right)$ is equal to

A function $y=f(x)$ satisfies $f(x)\sin 2x + \sin x - \left(1+\cos^2 x\right)f'(x) = 0$ with condition $f(0)=0$. Then, $f\left(\dfrac{\pi}{2}\right)$ is equal to

Let $\left(5,\dfrac{a}{4}\right)$ be the circumcenter of a triangle with vertices $A(a,-2)$, $B(a,6)$ and $C\left(\dfrac{a}{4},-2\right)$. Let $\alpha$ denote the circumradius, $\beta$ denote the area and $\gamma$ denote the perimeter of the triangle. Then $\alpha + \beta + \gamma$ is

In a $\triangle ABC$, suppose $y=x$ is the equation of the bisector of the angle $B$ and the equation of the side $AC$ is $2x - y = 2$. If $2AB = BC$ and the points $A$ and $B$ are respectively $(4, 6)$ and $(\alpha, \beta)$, then $\alpha + 2\beta$ is equal to

Let PQR be a triangle with R $(-1, 4, 2)$. Suppose M $(2, 1, 2)$ is the mid point of PQ. The distance of the centroid of $\triangle PQR$ from the point of intersection of the lines $\dfrac{x-2}{0}=\dfrac{y}{2}=\dfrac{z+3}{-1}$ and $\dfrac{x-1}{1}=\dfrac{y+3}{-3}=\dfrac{z+1}{1}$ is

Let $\vec{a}, \vec{b}$ and $\vec{c}$ be three non-zero vectors such that $\vec{b}$ and $\vec{c}$ are non-collinear. If $\vec{a}+5\vec{b}$ is collinear with $\vec{c}$, $\vec{b}+6\vec{c}$ is collinear with $\vec{a}$ and $\vec{a}+\alpha\vec{b}+\beta\vec{c}=\vec{0}$, then $\alpha + \beta$ is equal to

Let O be the origin and the position vectors of A and B be $2\hat{i}+2\hat{j}+\hat{k}$ and $2\hat{i}+4\hat{j}+4\hat{k}$ respectively. If the internal bisector of $\angle AOB$ meets the line AB at C, then the length of OC is

A fair die is thrown until 2 appears. Then the probability, that 2 appears in even number of throws, is

If $\alpha, -\dfrac{\pi}{2} < \alpha < \dfrac{\pi}{2}$ is the solution of $4\cos\theta + 5\sin\theta = 1$, then the value of $\tan\alpha$ is

Let $\alpha, \beta$ be the roots of the equation $x^2 - x + 2 = 0$ with $Im(\alpha) > Im(\beta)$.

Then $\alpha^6 + \alpha^4 + \beta^4 - 5\alpha^2$ is equal to __________.

All the letters of the word "GTWENTY" are written in all possible ways with or without meaning and these words are written as in a dictionary. The serial number of the word "GTWENTY" is __________.

If $\dfrac{{}^{11}C_1}{2} + \dfrac{{}^{11}C_2}{3} + \ldots + \dfrac{{}^{11}C_9}{10} = \dfrac{n}{m}$ with $\gcd(n,m)=1$, then $n + m$ is equal to __________.

Let $f(x) = 2^x - x^2$, $x \in \mathbb{R}$. If $m$ and $n$ are respectively the number of points at which the curves $y=f(x)$ and $y=f'(x)$ intersect the $x$-axis, then the value of $m + n$ is __________.

The area (in sq. units) of the part of the circle $x^2 + y^2 = 169$ which is below the line $5x - y = 13$ is $\dfrac{\pi\alpha}{2\beta} - \dfrac{65}{2} + \dfrac{\alpha}{\beta}\sin^{-1}\left(\dfrac{12}{13}\right)$, where $\alpha, \beta$ are coprime numbers. Then $\alpha + \beta$ is equal to _______

If the solution curve $y=y(x)$ of the differential equation $(1+y^2)(1+\log_e x)\,dx + x\,dy = 0$, $x > 0$ passes through the point $(1,1)$ and $y(e) = \dfrac{\alpha - \tan\left(\dfrac{3}{2}\right)}{\beta + \tan\left(\dfrac{3}{2}\right)}$, then $\alpha + 2\beta$ is __________.

Equations of two diameters of a circle are $2x - 3y = 5$ and $3x - 4y = 7$. The line joining the points $\left(-\dfrac{22}{7}, -4\right)$ and $\left(-\dfrac{1}{7}, 3\right)$ intersects the circle at only one point $P(\alpha, \beta)$. Then, $17\beta - \alpha$ is equal to _______.

If the points of intersection of two distinct conics $x^2 + y^2 = 4b$ and $\dfrac{x^2}{16} + \dfrac{y^2}{b^2} = 1$ lie on the curve $y^2 = 3x^2$, then $3\sqrt{3}$ times the area of the rectangle formed by the intersection points is __________.

A line with direction ratios 2, 1, 2 meets the lines $x = y + 2 = z$ and $x + 2 = 2y = 2z$ respectively at the points P and Q. If the length of the perpendicular from the point $(1, 2, 12)$ to the line PQ is $l$, then $l^2$ is __________.

If the mean and variance of the data 65, 68, 58, 44, 48, 45, 60, $\alpha$, $\beta$, 60 where $\alpha > \beta$, are 56 and 66.2 respectively, then $\alpha^2 + \beta^2$ is equal to ________.

The resistance $R = \frac{V}{I}$ where $V = (200 \pm 5)$ V and $I = (20 \pm 0.2)$ A, the percentage error in the measurement of R is :

A body starts moving from rest with constant acceleration covers displacement $S_1$ in first $(p - 1)$ seconds and $S_2$ in first $p$ seconds. The displacement $S_1 + S_2$ will be made in time :

If the radius of curvature of the path of two particles of same mass are in the ratio 3:4, then in order to have constant centripetal force, their velocities will be in the ratio of :

A block of mass 100 kg slides over a distance of 10 m on a horizontal surface. If the co-efficient of friction between the surfaces is 0.4, then the work done against friction (in $J$) is :

The potential energy function (in $J$) of a particle in a region of space is given as $U = (2x^2 + 3y^3 + 2z)$. Here $x$, $y$ and $z$ are in meter. The magnitude of $x$-component of force (in $N$) acting on the particle at point $P$ (1, 2, 3) m is :

At what distance above and below the surface of the earth a body will have same weight. (take radius of earth as $R$.)

Given below are two statements:

Statement I : If a capillary tube is immersed first in cold water and then in hot water, the height of capillary rise will be smaller in hot water.

Statement II : If a capillary tube is immersed first in cold water and then in hot water, the height of capillary rise will be smaller in cold water.

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

A thermodynamic system is taken from an original state A to an intermediate state B by a linear process as shown in the figure. It's volume is then reduced to the original value from B to C by an isobaric process. The total work done by the gas from A to B and B to C would be :

[Figure: P–V diagram; pressure axis in Dyne/cm² marked 8000 (state A) and 4000 (states B, C); volume axis in m³ marked 3 and 7; straight line from A (3, 8000) to B (7, 4000), then isobaric line from B back to C (3, 4000)]

Two vessels A and B are of the same size and are at same temperature. A contains 1g of hydrogen and B contains 1g of oxygen. $\mathrm{P_A}$ and $\mathrm{P_B}$ are the pressures of the gases in A and B respectively, then $\frac{P_A}{P_B}$ is :

Two charges of $5Q$ and $-2Q$ are situated at the points $(3a, 0)$ and $(-5a, 0)$ respectively. The electric flux through a sphere of radius '$4a$' having center at origin is :

The electric current through a wire varies with time as $I = I_0 + \beta t$, where $I_0 = 20$ A and $\beta = 3$ A/s. The amount of electric charge crossed through a section of the wire in 20 s is :

A galvanometer having coil resistance $10\,\Omega$ shows a full scale deflection for a current of 3 mA. For it to measure a current of 8 A, the value of the shunt should be:

A capacitor of capacitance 100 $\mu$F is charged to a potential of 12 V and connected to a 6.4 mH inductor to produce oscillations. The maximum current in the circuit would be :

Match List I with List II

	LIST I		LIST II
A.	$\oint \vec{B} \cdot \vec{dl} = \mu_o i_c + \mu_o \varepsilon_o \dfrac{d\phi_E}{dt}$	I.	Gauss' law for electricity
B.	$\oint \vec{E} \cdot \vec{dl} = \dfrac{d\phi_B}{dt}$	II.	Gauss' law for magnetism
C.	$\oint \vec{E} \cdot \vec{dA} = \dfrac{Q}{\varepsilon_o}$	III.	Faraday law
D.	$\oint \vec{B} \cdot \vec{dA} = 0$	IV.	Ampere - Maxwell law

Choose the correct answer from the options given below:

A biconvex lens of refractive index 1.5 has a focal length of 20 cm in air. Its focal length when immersed in a liquid of refractive index 1.6 will be:

The de-Broglie wavelength of an electron is the same as that of a photon. If velocity of electron is 25% of the velocity of light, then the ratio of K.E. of electron and K.E. of photon will be:

The explosive in a Hydrogen bomb is a mixture of $_1\mathrm{H}^2$, $_1\mathrm{H}^3$ and $_3\mathrm{Li}^6$ in some condensed form. The chain reaction is given by

$$_3\mathrm{Li}^6 + {_0\mathrm{n}^1} \rightarrow {_2\mathrm{He}^4} + {_1\mathrm{H}^3}$$

$$_1\mathrm{H}^2 + {_1\mathrm{H}^3} \rightarrow {_2\mathrm{He}^4} + {_0\mathrm{n}^1}$$

During the explosion the energy released is approximately

[Given ; M (Li) = 6.01690 amu, M ($_1\mathrm{H}^2$) = 2.01471 amu, M ($_2\mathrm{He}^4$) = 4.00388 amu, and 1 amu = 931.5 MeV]

In the given circuit, the breakdown voltage of the Zener diode is 3.0 V. What is the value of $\mathrm{I_z}$?

[Figure: circuit with a 10 V battery in series with a 1 kΩ resistor; a Zener diode (carrying current $I_z$) connected between nodes A and B in parallel with a 2 kΩ resistor connected between nodes C and D across the output terminals]

A convex mirror of radius of curvature 30 cm forms an image that is half the size of the object. The object distance is :

The deflection in moving coil galvanometer falls from 25 divisions to 5 division when a shunt of $24\,\Omega$ is applied. The resistance of galvanometer coil will be :

A ball rolls off the top of a stairway with horizontal velocity $u$. The steps are 0.1 m high and 0.1 m wide. The minimum velocity $u$ with which that ball just hits the step 5 of the stairway will be $\sqrt{x}\ \mathrm{ms}^{-1}$ where $x =$ __________ [use $\mathrm{g} = 10\ \mathrm{m/s^2}$].

A cylinder is rolling down on an inclined plane of inclination $60^\circ$. It's acceleration during rolling down will be $\frac{x}{\sqrt{3}}\ \mathrm{m/s^2}$, where $x =$ ________ (use $\mathrm{g} = 10\ \mathrm{m/s^2}$).

In a test experiment on a model aeroplane in wind tunnel, the flow speeds on the upper and lower surfaces of the wings are $70\ \mathrm{ms}^{-1}$ and $65\ \mathrm{ms}^{-1}$ respectively. If the wing area is $2\ \mathrm{m^2}$, the lift of the wing is _____ $N$.

(Given density of air $= 1.2\ \mathrm{kg\ m^{-3}}$)

When the displacement of a simple harmonic oscillator is one third of its amplitude, the ratio of total energy to the kinetic energy is $\frac{x}{8}$, where $x =$ ________.

An electron is moving under the influence of the electric field of a uniformly charged infinite plane sheet S having surface charge density $+\sigma$. The electron at $t=0$ is at a distance of 1 m from S and has a speed of 1 m/s. The maximum value of $\sigma$ if the electron strikes S at $t=1$ s is $\alpha\left[\frac{m \epsilon_0}{e}\right] \frac{\mathrm{C}}{\mathrm{m}^2}$, the value of $\alpha$ is ______.

A $16\,\Omega$ wire is bend to form a square loop. A 9V battery with internal resistance $1\,\Omega$ is connected across one of its sides. If a $4\,\mu F$ capacitor is connected across one of its diagonals, the energy stored by the capacitor will be $\frac{x}{2}\ \mu J$, where $x =$ __________.

The magnetic potential due to a magnetic dipole at a point on its axis situated at a distance of 20 cm from its center is $1.5 \times 10^{-5}\ T\,m$. The magnetic moment of the dipole is ________ $A\,m^2$. (Given : $\frac{\mu_o}{4\pi} = 10^{-7}\ T\,m\,A^{-1}$)

A square loop of side 10 cm and resistance $0.7\,\Omega$ is placed vertically in east-west plane. A uniform magnetic field of 0.20 $T$ is set up across the plane in north east direction. The magnetic field is decreased to zero in 1 s at a steady rate. Then, magnitude of induced emf is $\sqrt{x} \times 10^{-3}\,V$. The value of $x$ is __________.

In a double slit experiment shown in figure, when light of wavelength 400 nm is used, dark fringe is observed at $P$. If $D=0.2$ m, the minimum distance between the slits $S_1$ and $S_2$ is _______ mm.

[Figure: source on the left, slits $S_1$ and $S_2$ separated by distance $d$ on a vertical line; source is at distance $D$ from the slit plane and point $P$ is at distance $D$ on the other side, both on the line through $S_2$; dashed lines from source to $S_1$ and from $S_1$ to $P$ form a triangle]

When a hydrogen atom going from $n=2$ to $n=1$ emits a photon, its recoil speed is $\frac{x}{5}\ m/s$. Where $x =$ ________. (Use, mass of hydrogen atom $= 1.6 \times 10^{-27}$ kg)

The correct set of four quantum numbers for the valence electron of rubidium atom $(Z = 37)$ is :

Which of the following is not correct?

Chlorine undergoes disproportionation in alkaline medium as shown below :

$$\mathrm{a\, Cl_{2(g)} + b\, OH^-_{(aq)} \rightarrow c\, ClO^-_{(aq)} + d\, Cl^-_{(aq)} + e\, H_2O_{(l)}}$$

The values of a, b, c and d in a balanced redox reaction are respectively :

Given below are two statements : one is labelled as Assertion A and the other is labelled as Reason R :

Assertion A : The first ionisation enthalpy decreases across a period.

Reason R : The increasing nuclear charge outweighs the shielding across the period.

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

Identify the incorrect pair from the following :

Given below are two statements :

Statement I : The electronegativity of group 14 elements from Si to Pb, gradually decreases.

Statement II : Group 14 contains non-metallic, metallic, as well as metalloid elements.

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

$\mathrm{KMnO_4}$ decomposes on heating at 513K to form $\mathrm{O_2}$ along with

In alkaline medium, $\mathrm{MnO_4^-}$ oxidises $\mathrm{I^-}$ to

In which one of the following metal carbonyls, CO forms a bridge between metal atoms?

Match List I with List II

	LIST I (Substances)		LIST II (Element Present)
A.	Ziegler catalyst	I.	Rhodium
B.	Blood Pigment	II.	Cobalt
C.	Wilkinson catalyst	III.	Iron
D.	Vitamin $\mathrm{B_{12}}$	IV.	Titanium

Choose the correct answer from the options given below:

Appearance of blood red colour, on treatment of the sodium fusion extract of an organic compound with $\mathrm{FeSO_4}$ in presence of concentrated $\mathrm{H_2SO_4}$ indicates the presence of element/s

The interaction between $\pi$ bond and lone pair of electrons present on an adjacent atom is responsible for

The difference in energy between the actual structure and the lowest energy resonance structure for the given compound is

Identify product A and product B :

[Figure: a cyclohexadiene ring + $\mathrm{Cl_2}$; upper path with $h\nu$ gives Product A, lower path in $\mathrm{CCl_4}$ gives Product B]

Given below are two statements : one is labelled as Assertion A and the other is labelled as Reason R :

Assertion A : Aryl halides cannot be prepared by replacement of hydroxyl group of phenol by halogen atom.

Reason R : Phenols react with halogen acids violently.

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

The final product A formed in the following multistep reaction sequence is

[Figure: benzene ring bearing a $\mathrm{CH{=}CH_2}$ group (styrene) $\xrightarrow{\text{(i) } \mathrm{H_2O},\, \mathrm{H}^{\oplus} \text{; (ii) } \mathrm{CrO_3} \text{; (iii) } \mathrm{H_2N{-}NH_2},\, \mathrm{KOH},\, \text{Heating}}$ A]

The major product(P) in the following reaction is

[Figure: benzene ring with $\mathrm{OCH_2CH_3}$ at one position and $\mathrm{CH{=}CH_2}$ at the meta position $\xrightarrow[\text{Heat}]{\text{con} \cdot \mathrm{HBr}\,(excess)}$ (P) major product]

Type of amino acids obtained by hydrolysis of proteins is :

In chromyl chloride test for confirmation of $\mathrm{Cl^-}$ ion, a yellow solution is obtained. Acidification of the solution and addition of amyl alcohol and 10% $\mathrm{H_2O_2}$ turns organic layer blue indicating formation of chromium pentoxide. The oxidation state of chromium in that is

The arenium ion which is not involved in the bromination of Aniline is ________.

A solution of $\mathrm{H_2SO_4}$ is 31.4% $\mathrm{H_2SO_4}$ by mass and has a density of $1.25\,\mathrm{g/mL}$.

The molarity of the $\mathrm{H_2SO_4}$ solution is ________ M (nearest integer)

[Given molar mass of $\mathrm{H_2SO_4} = 98\,\mathrm{g\ mol^{-1}}$]

The number of species from the following which are paramagnetic and with bond order equal to one is ________.

$\mathrm{H_2},\ \mathrm{He_2^+},\ \mathrm{O_2^+},\ \mathrm{N_2^{2-}},\ \mathrm{O_2^{2-}},\ \mathrm{F_2},\ \mathrm{Ne_2^+},\ \mathrm{B_2}$

Number of compounds with one lone pair of electrons on central atom amongst following is __________

$\mathrm{O_3},\ \mathrm{H_2O},\ \mathrm{SF_4},\ \mathrm{ClF_3},\ \mathrm{NH_3},\ \mathrm{BrF_5},\ \mathrm{XeF_4}$

The osmotic pressure of a dilute solution is $7 \times 10^5\,\mathrm{Pa}$ at 273K. Osmotic pressure of the same solution at 283K is ________ $\times 10^4\,\mathrm{Nm^{-2}}$.

For the reaction $\mathrm{N_2O_{4(g)} \rightleftharpoons 2NO_{2(g)}}$, $K_p = 0.492$ atm at 300K. $K_c$ for the reaction at same temperature is ________ $\times 10^{-2}$.

(Given : $R = 0.082\,\mathrm{L\ atm\ mol^{-1}\ K^{-1}}$)

The mass of zinc produced by the electrolysis of zinc sulphate solution with a steady current of 0.015 A for 15 minutes is ________ $\times 10^{-4}$ g.

(Atomic mass of zinc = 65.4 amu)

For a reaction taking place in three steps at same temperature, overall rate constant $K = \frac{K_1 K_2}{K_3}$. If $E_{a_1}$, $E_{a_2}$ and $E_{a_3}$ are 40, 50 and 60 kJ/mol respectively, the overall $E_a$ is ______ kJ/mol.

Number of compounds among the following which contain sulphur as heteroatom is ________.

Furan, Thiophene, Pyridine, Pyrrole, Cysteine, Tyrosine

$$\mathrm{\underset{H}{\overset{CH_3}{\phantom{|}}}\!C{=}C\!\overset{H}{\underset{CH_3}{\phantom{|}}} \xrightarrow[\text{(ii) } Zn + H_2O]{\text{(i) } O_3} (P)\ \text{Product}}$$

[Figure: but-2-ene, $\mathrm{CH_3(H)C{=}C(H)CH_3}$, undergoing (i) $\mathrm{O_3}$ (ii) $\mathrm{Zn + H_2O}$ to give product (P)]

Consider the given reaction. The total number of oxygen atom/s present per molecule of the product (P) is ______.

From the compounds given below, number of compounds which give positive Fehling's test is ________.

Benzaldehyde, Acetaldehyde, Acetone, Acetophenone, Methanal, 4-nitrobenzaldehyde, cyclohexane carbaldehyde.