JEE Main 2023 (13 Apr, Shift 2) — JEE Main 2023 Question Paper with Solutions

Q1 MATHS Unit 14: Trigonometry

The range of $f(x)=4\sin^{-1}\left(\dfrac{x^2}{x^2+1}\right)$ is

A. $[0,\pi]$
B. $[0,\pi]$
C. $[0,2\pi]$
D. $[0,2\pi]$

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Q2 MATHS Unit 2: Complex Numbers and Quadratic Equations

Let $S=\{z\in\mathbb{C}: \bar{z}=i(z^2+\operatorname{Re}(\bar{z}))\}$. Then $\sum_{z\in S}|z|^2$ is equal to

A. $\dfrac{5}{2}$
B. $3$
C. $\dfrac{7}{2}$
D. $4$

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Q3 MATHS Unit 2: Complex Numbers and Quadratic Equations

Let $\alpha,\beta$ be the roots of the equation $x^2-\sqrt{2}x+2=0$. Then $\alpha^{14}+\beta^{14}$ is equal to

A. $-64$
B. $-64\sqrt{2}$
C. $-128$
D. $-128\sqrt{2}$

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Q4 MATHS Unit 3: Matrices and Determinants

If the system of equations $$2x+y-z=5$$ $$2x-5y+\lambda z=\mu$$ $$x+2y-5z=7$$ has infinitely many solutions, then $(\lambda+\mu)^2+(\lambda-\mu)^2$ is equal to

A. $904$
B. $912$
C. $916$
D. $920$

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Q5 MATHS Unit 3: Matrices and Determinants

Let for $A=\begin{bmatrix}1&2&3\\ \alpha&3&1\\ 1&1&2\end{bmatrix}$, $|A|=2$. If $|2\,\operatorname{adj}(2\,\operatorname{adj}(2A))|=32^n$, then $3n+\alpha$ is equal to

A. $9$
B. $10$
C. $11$
D. $12$

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Q6 MATHS Unit 6: Sequence and Series

All words, with or without meaning, are made using all the letters of the word MONDAY. These words are written in a dictionary with serial numbers. The serial number of the word MONDAY is

A. $324$
B. $326$
C. $327$
D. $328$

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Q7 MATHS Unit 5: Binomial Theorem and its Simple Applications

The coefficient of $x^5$ in the expansion of $\left(2x^3-\dfrac{1}{3x^2}\right)^5$ is

A. $8$
B. $9$
C. $\dfrac{80}{9}$
D. $\dfrac{26}{3}$

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Q8 MATHS Unit 6: Sequence and Series

Let $a_1,a_2,a_3,\ldots$ be a G.P. of increasing positive numbers. Let the sum of its $6^{\text{th}}$ and $8^{\text{th}}$ terms be 2 and the product of its $3^{\text{rd}}$ and $5^{\text{th}}$ terms be $\dfrac{1}{9}$. Then $6(a_2+a_4)(a_4+a_6)$ is equal to

A. $2$
B. $2\sqrt{2}$
C. $3\sqrt{3}$
D. $3$

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Q9 MATHS Unit 7: Limit Continuity and Differentiability

If $\lim_{x\to 0} \frac{e^{ax} - \cos(bx) - \frac{c x e^{-cx}}{2}}{1 - \cos(2x)} = 17$, then $5a^2 + b^2$ is equal to

A. 68
B. 64
C. 72
D. 76

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Q10 MATHS Unit 8: Integral Calculus

The value of $\frac{e^{\pi/4}+\int_0^{\pi/4} e^{-x}\tan^{50}x\,dx}{\int_0^{\pi/4} e^{-x}\left(\tan^{49}x+\tan^{51}x\right)\,dx}$ is

A. 25
B. 49
C. 50
D. 51

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Q11 MATHS Unit 10: Co-ordinate Geometry

The area of the region $\{(x,y): x^2 \le y \le |x^2-4|,\ y \ge 1\}$ is

↻ Repeated question — also appeared in 2023 (1 set)

A. $\frac{4}{3}(4\sqrt{2}-1)$
B. $\frac{4}{3}(4\sqrt{2}+1)$
C. $\frac{3}{4}(4\sqrt{2}-1)$
D. $\frac{3}{4}(4\sqrt{2}+1)$

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Q12 MATHS Unit 11: Three Dimensional Geometry

Let $(\alpha,\beta)$ be the centroid of the triangle formed by the lines $15x-y=82$, $6x-5y=-4$ and $9x+4y=17$. Then $\alpha+2\beta$ and $2\alpha-\beta$ are the roots of the equation

A. $x^2-14x+48=0$
B. $x^2-10x+25=0$
C. $x^2-7x+12=0$
D. $x^2-13x+42=0$

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Q13 MATHS Unit 11: Three Dimensional Geometry

Let the centre of a circle $C$ be $(\alpha,\beta)$ and its radius $r<8$. Let $3x+4y=24$ and $3x-4y=32$ be two tangents and $4x+3y=1$ be a normal to $C$. Then $(\alpha-\beta+r)$ is equal to

A. 5
B. 6
C. 7
D. 9

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Q14 MATHS Unit 11: Three Dimensional Geometry

The plane, passing through the points $(0,-1,2)$ and $(-1,2,1)$ and parallel to the line passing through $(5,1,-7)$ and $(1,-1,-1)$, also passes through the point

A. $(1,-2,1)$
B. $(-2,5,0)$
C. $(2,0,1)$
D. $(0,5,-2)$

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Q15 MATHS Unit 11: Three Dimensional Geometry

The line, that is coplanar to the line $\frac{x+3}{-3}=\frac{y-1}{1}=\frac{z-5}{5}$, is

A. $\frac{x-1}{-1}=\frac{y-2}{2}=\frac{z-5}{5}$
B. $\frac{x+1}{-1}=\frac{y-2}{2}=\frac{z-5}{5}$
C. $\frac{x+1}{-1}=\frac{y-2}{2}=\frac{z-5}{4}$
D. $\frac{x+1}{1}=\frac{y-2}{2}=\frac{z-5}{5}$

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Q16 MATHS Unit 11: Three Dimensional Geometry

Let $N$ be the foot of perpendicular from the point $P(1,-2,3)$ on the line passing through the points $(4,5,8)$ and $(1,-7,5)$. Then the distance of $N$ from the plane $2x-2y+z+5=0$ is

A. 6
B. 7
C. 8
D. 9

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Q17 MATHS Unit 12: Vector Algebra

Let $|\vec{a}| = 2$, $|\vec{b}| = 3$ and the angle between the vectors $\vec{a}$ and $\vec{b}$ be $\pi/4$. Then $\left|(\vec{a} + 2\vec{b}) \times (2\vec{a} - 3\vec{b})\right|^2$ is equal to

A. 441
B. 882
C. 482
D. 841

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Q18 MATHS Unit 12: Vector Algebra

Let for a triangle $ABC$, $\overrightarrow{AB} = -2\hat{i} + \hat{j} + 3\hat{k}$ $\overrightarrow{CB} = \alpha\hat{i} + \beta\hat{j} + \gamma\hat{k}$ $\overrightarrow{CA} = 4\hat{i} + 3\hat{j} + \delta\hat{k}$ If $\delta > 0$ and the area of the triangle $ABC$ is $5\sqrt{6}$, then $\overrightarrow{CB}\cdot\overrightarrow{CA}$ is equal to

A. 60
B. 54
C. 120
D. 108

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Q19 MATHS Unit 2: Complex Numbers and Quadratic Equations

The random variable $X$ follows binomial distribution $B(n,p)$, for which the difference of the mean and the variance is $1$. If $2P(X=2)=3P(X=1)$, then $n^2P(X>1)$ is equal to

A. 15
B. 16
C. 12
D. 11

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Q20 CHEMISTRY Unit 14: Some Basic Principles of Organic Chemistry

The statement $(p \wedge (\sim q)) \vee ((\sim p) \wedge q) \vee ((\sim p) \wedge (\sim q))$ is equivalent to ____

A. $(\sim p) \vee (\sim q)$
B. $p \vee (\sim q)$
C. $(\sim p) \vee q$
D. $p \vee q$

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Q31 PHYSICS Unit 12: Current Electricity

In the equation $\left[X + \dfrac{a}{Y^2}\right][Y-b] = RT$, $X$ is pressure, $Y$ is volume, $R$ is universal gas constant and $T$ is temperature. The physical quantity equivalent to the ratio $a/b$ is:

A. Energy
B. Pressure gradient
C. Impulse
D. Coefficient of viscosity

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Q32 PHYSICS Unit 2: Kinematics

The distance travelled by an object in time $t$ is given by $s = (2.5)t^2$. The instantaneous speed of the object at $t = 5\ \text{s}$ will be:

A. $5\ \text{m s}^{-1}$
B. $12.5\ \text{m s}^{-1}$
C. $25\ \text{m s}^{-1}$
D. $62.5\ \text{m s}^{-1}$

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Q33 PHYSICS Unit 2: Kinematics

A passenger sitting in a train A moving at $90\ \text{km/h}$ observes another train B moving in the opposite direction for $8\ \text{s}$. If the velocity of the train B is $54\ \text{km/h}$, then length of train B is:

A. $80\ \text{m}$
B. $120\ \text{m}$
C. $200\ \text{m}$
D. $320\ \text{m}$

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Q34 PHYSICS Unit 3: Laws of Motion

A vehicle of mass $200\ \text{kg}$ is moving along a levelled curved road of radius $70\ \text{m}$ with angular velocity $0.2\ \text{rad/s}$. The centripetal force acting on the vehicle is:

A. $14\ \text{N}$
B. $2240\ \text{N}$
C. $560\ \text{N}$
D. $2800\ \text{N}$

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Q35 PHYSICS Unit 6: Gravitation

Given below are two statements: Statement I : For a planet, if the ratio of mass of the planet to its radius increases, the escape velocity from the planet also increases. Statement II : Escape velocity is independent of the radius of the planet. In the light of above statements, choose the most appropriate answer form the options given below

A. Both Statement I and Statement II are correct
B. Both Statement I and Statement II are incorrect
C. Statement I is correct but statement II is incorrect
D. Statement I is incorrect but statement II is correct

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Q36 PHYSICS Unit 20: Experimental Skills

Given below are two statements: one is labelled as Assertion A and the other is labelled as Reason R Assertion A : A spherical body of radius $(5 \pm 0.1)$ mm having a particular density is falling through a liquid of constant density. The percentage error in the calculation of its terminal velocity is 4%. Reason R : The terminal velocity of the spherical body falling through the liquid is inversely proportional to its radius. In the light of the above statements, choose the correct answer from the options given below

A. Both A and R are true and R is the correct explanation of A
B. Both A and R are true but R is NOT the correct explanation of A
C. A is true but R is false
D. A is false but R is true

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Q37 PHYSICS Unit 6: Gravitation

Two planets A and B of radii $R$ and $1.5\,R$ have densities $\rho$ and $\rho/2$ respectively. The ratio of acceleration due to gravity at the surface of B to A is:

A. 2:1
B. 2:3
C. 3:4
D. 4:3

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Q38 PHYSICS Unit 8: Thermodynamics

The initial pressure and volume of an ideal gas are $P_0$ and $V_0$. The final pressure of the gas when the gas is suddenly compressed to volume $\frac{V_0}{4}$ will be : (Given $\gamma =$ ratio of specific heats at constant pressure and constant volume)

A. $P_0$
B. $4P_0$
C. $P_0(4)^{\gamma}$
D. $P_0(4)^{\frac{1}{\gamma}}$

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Q39 PHYSICS Unit 9: Kinetic Theory of Gases

The mean free path of molecules of a certain gas at STP is $1500d$, where $d$ is the diameter of the gas molecules. While maintaining the standard pressure, the mean free path of the molecules at $373\,\text{K}$ is approximately:

A. $1500d$
B. $750d$
C. $2049d$
D. $1098d$

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Q40 PHYSICS Unit 4: Work Energy and Power

A particle executes SHM of amplitude $A$. The distance from the mean position when it's kinetic energy becomes equal to its potential energy is:

A. $2A$
B. $\frac{A}{2}$
C. $\frac{A}{\sqrt{2}}$
D. $\sqrt{2}A$

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Q41 PHYSICS Unit 11: Electrostatics

A $10\,\mu\text{C}$ charge is divided into two parts and placed at $1\,\text{cm}$ distance so that the repulsive force between them is maximum. The charges of the two parts are:

A. $8\,\mu\text{C},\,2\,\mu\text{C}$
B. $7\,\mu\text{C},\,3\,\mu\text{C}$
C. $5\,\mu\text{C},\,5\,\mu\text{C}$
D. $9\,\mu\text{C},\,1\,\mu\text{C}$

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Q42 PHYSICS Unit 12: Current Electricity

In the network shown below, the charge accumulated in the capacitor in steady state will be:

A. $10.3\,\mu\text{C}$
B. $4.8\,\mu\text{C}$
C. $7.2\,\mu\text{C}$
D. $12\,\mu\text{C}$

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Q43 PHYSICS Unit 13: Magnetic Effects of Current and Magnetism

An electron is moving along the positive x-axis. If the uniform magnetic field is applied parallel to the negative z-axis, then A. The electron will experience magnetic force along positive y-axis B. The electron will experience magnetic force along negative y-axis C. The electron will not experience any force in magnetic field D. The electron will continue to move along the positive x-axis E. The electron will move along circular path in magnetic field Choose the correct answer from the options given below:

A. A and E only
B. C and D only
C. B and D only
D. B and E only

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Q44 PHYSICS Unit 10: Oscillations and Waves

Given below are two statements: Statement I : An AC circuit undergoes electrical resonance if it contains either a capacitor or an inductor. Statement II : An AC circuit containing a pure capacitor or a pure inductor consumes high power due to its non-zero power factor. In the light of above statements, choose the correct answer form the options given below:

A. Both Statement I and Statement II are true
B. Both Statement I and Statement II are false
C. Statement I is true but statement II is false
D. Statement I is false but statement II is true

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Q45 PHYSICS Unit 15: Electromagnetic Waves

In an electromagnetic wave, at an instant and at a particular position, the electric field is along the negative z-axis and magnetic field is along the positive x-axis. Then the direction of propagation of electromagnetic wave is:

A. positive y-axis
B. negative y-axis
C. positive z-axis
D. at 45° angle from positive y-axis

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Q46 PHYSICS Unit 10: Oscillations and Waves

In a Young's double slits experiment, the ratio of amplitude of light coming from slits is 2:1. The ratio of the maximum to minimum intensity in the interference pattern is:

A. 2:1
B. 9:4
C. 9:1
D. 25:9

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Q47 PHYSICS Unit 17: Dual Nature of Matter and Radiation

Given below are two statements: Statement I : Out of microwaves, infrared rays and ultraviolet rays, ultraviolet rays are the most effective for the emission of electrons from a metallic surface. Statement II : Above the threshold frequency, the maximum kinetic energy of photoelectrons is inversely proportional to the frequency of the incident light. In the light of above statements, choose the correct answer form the options given below

A. Both Statement I and Statement II are true
B. Both Statement I and Statement II are false
C. Statement I is true but Statement II is false
D. Statement I is false but Statement II is true

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Q48 PHYSICS Unit 18: Atoms and Nuclei

Given below are two statements: one is labelled as Assertion A and the other is labelled as Reason R Assertion A : The binding energy per nucleon is practically independent of the atomic number for nuclei of mass number in the range 30 to 170. Reason R : Nuclear force is short ranged. In the light of the above statements, choose the correct answer from the options given below

A. Both A and R are true and R is the correct explanation of A
B. Both A and R are true but R is NOT the correct explanation of A
C. A is true but R is false
D. A is false but R is true

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Q49 PHYSICS Unit 19: Electronic Devices

The output from a NAND gate having inputs A and B given below will be,

A. Waveform shown in option A
B. Waveform shown in option B
C. Waveform shown in option C
D. Waveform shown in option D

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Q50 PHYSICS Unit 15: Electromagnetic Waves

To radiate EM signal of wavelength $\lambda$ with high efficiency, the antennas should have a minimum size equal to :

A. $2\lambda$
B. $\lambda$
C. $\lambda/2$
D. $\lambda/4$

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Q61 CHEMISTRY Unit 12: Coordination Compounds

Given below are two statements : Statement I : $\mathrm{SO_2}$ and $\mathrm{H_2O}$ both possess V-shaped structure. Statement II : The bond angle of $\mathrm{SO_2}$ is less than that of $\mathrm{H_2O}$. In the light of the above statements, choose the most appropriate answer from the options given below:

A. Both Statement I and Statement II are correct
B. Both Statement I and Statement II are incorrect
C. Statement I is correct but Statement II is incorrect
D. Statement I is incorrect but Statement II is correct

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Q62 CHEMISTRY Unit 4: Chemical Thermodynamics

What happens when methane undergoes combustion in systems A and B respectively? System A: Adiabatic system System B: Diathermic container

A. System A: Temperature rises; System B: Temperature remains same
B. System A: Temperature remains same; System B: Temperature rises
C. System A: Temperature falls; System B: Temperature rises
D. System A: Temperature falls; System B: Temperature remains same

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Q63 CHEMISTRY Unit 5: Solutions

Given below are two statements, one is labelled as Assertion A and the other is labelled as Reason R. Assertion A : The diameter of colloidal particles in solution should not be much smaller than wavelength of light to show Tyndall effect. Reason R : The light scatters in all directions when the size of particles is large enough. In the light of the above statements, choose the correct answer from the options given below:

A. Both A and R are correct and R is the correct explanation of A
B. Both A and R are correct but R is NOT the correct explanation of A
C. A is true but R is false
D. A is false but R is true

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Q64 CHEMISTRY Unit 12: Coordination Compounds

The covalency and oxidation state respectively of boron in $[\mathrm{BF_4}]^{-}$, are

A. 4 and 4
B. 4 and 3
C. 3 and 4
D. 3 and 5

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Q65 CHEMISTRY Unit 4: Chemical Thermodynamics

Given below are two statements related to Ellingham diagram: Statement I : Ellingham diagrams can be constructed for formation of oxides, sulfides and halides of metals. Statement II : It consists of plots of $\Delta_f H^\circ$ vs $T$ for formation of oxides of elements. In the light of the above statements, choose the most appropriate answer from the options given below:

A. Both Statement I and Statement II are correct
B. Both Statement I and Statement II are incorrect
C. Statement I is correct but Statement II is incorrect
D. Statement I is incorrect but Statement II is correct

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Q66 CHEMISTRY Unit 2: Atomic Structure

Given below are two statements, one is labelled as Assertion A and the other is labelled as Reason R. Assertion A : Isotopes of hydrogen have almost same chemical properties, but difference in their rates of reaction. Reason R : Isotopes of hydrogen have different enthalpy of bond dissociation. In the light of the above statements, choose the most appropriate answer from the options given below:

A. Both A and R are correct and R is the correct explanation of A
B. Both A and R are correct but R is NOT the correct explanation of A
C. A is correct but R is not correct
D. A is not correct but R is correct

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Q67 CHEMISTRY Unit 10: p-Block Elements

Better method for preparation of $\mathrm{BeF_2}$, among the following is

A. $\mathrm{Be + F_2 \xrightarrow{\Delta} BeF_2}$
B. $\mathrm{(NH_4)_2BeF_4 \xrightarrow{\Delta} BeF_2}$
C. $\mathrm{BeO + C + F_2 \xrightarrow{\Delta} BeF_2}$
D. $\mathrm{BeH_2 + F_2 \xrightarrow{\Delta} BeF_2}$

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Q68 CHEMISTRY Unit 4: Chemical Thermodynamics

Identify the correct order of standard enthalpy of formation of sodium halides.

A. $\mathrm{NaF < NaCl < NaBr < NaI}$
B. $\mathrm{NaI < NaBr < NaCl < NaF}$
C. $\mathrm{NaI < NaBr < NaF < NaCl}$
D. $\mathrm{NaCl < NaF < NaBr < NaI}$

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Q69 CHEMISTRY Unit 10: p-Block Elements

The correct group of halide ions which can be oxidised by oxygen in acidic medium is

A. $\mathrm{Br^-}$ and $\mathrm{I^-}$ only
B. $\mathrm{Cl^-}$, $\mathrm{Br^-}$ and $\mathrm{I^-}$ only
C. $\mathrm{I^-}$ only
D. $\mathrm{Br^-}$ only

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Q70 CHEMISTRY Unit 12: Coordination Compounds

Which of the following complexes will exhibit maximum attraction to an applied magnetic field?

A. $[\mathrm{Co(en)_3}]^{3+}$
B. $[\mathrm{Co(H_2O)_6}]^{2+}$
C. $[\mathrm{Zn(H_2O)_6}]^{2+}$
D. $[\mathrm{Ni(H_2O)_6}]^{2+}$

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Q71 CHEMISTRY Unit 10: p-Block Elements

Which of the following are the Green house gases? A. Water vapour B. Ozone C. $\mathrm{I_2}$ D. Molecular hydrogen Choose the most appropriate answer from the options given below:

A. A and B only
B. A and D only
C. C and D only
D. B and C only

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Q72 CHEMISTRY Unit 12: Coordination Compounds

The total number of stereoisomers for the complex $[\mathrm{Cr(ox)_2ClBr}]^{3-}$ (where ox = oxalate) is:

A. 2
B. 3
C. 4
D. 1

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Q73 CHEMISTRY Unit 13: Purification and Characterisation of Organic Compounds

In the wet tests for detection of various cations by precipitation, $\mathrm{Ba^{2+}}$ cations are detected by obtaining precipitate of

A. $\mathrm{BaSO_4}$
B. $\mathrm{Ba(OAc)_2}$
C. $\mathrm{BaCO_3}$
D. $\mathrm{Ba(ox)}$ : Barium oxalate

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Q74 CHEMISTRY Unit 14: Some Basic Principles of Organic Chemistry

Given below are two statements : Statement I : Tropolone is an aromatic compound and has $8\,\pi$ electrons. Statement II : $\pi$ electrons of $>\!\mathrm{C=O}$ group in tropolone is involved in aromaticity. In the light of the above statements, choose the correct answer from the options given below:

A. Both Statement I and Statement II are true
B. Both Statement I and Statement II are false
C. Statement I is true but Statement II is false
D. Statement I is false but Statement II is true

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Q75 CHEMISTRY Unit 16: Organic Compounds Containing Halogens

Match List I with List II 1 - Bromopropane is reacted with reagents in List I to give product in List II LIST I - Reagent LIST II - Product A. KOH (alc) I. Nitrile B. KCN (alc) II. Ester C. AgNO$_2$ III. Alkene D. H$_3$CCOOAg IV. Nitroalkane Choose the correct answer from the options given below:

A. A-IV, B-III, C-II, D-I
B. A-III, B-I, C-IV, D-II
C. A-I, B-III, C-IV, D-II
D. A-I, B-II, C-III, D-IV

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Q76 CHEMISTRY Unit 17: Organic Compounds Containing Oxygen

Given below are two statements, one is labelled as Assertion A and the other is labelled as Reason R. Assertion A : Order of acidic nature of the following compounds is A > B > C. Reason R : Fluoro is a stronger electron withdrawing group than Chloro group. In the light of the above statements, choose the correct answer from the options given below: