JEE Main 2023 · April session

JEE Main 6 April 2023, Shift 2

89 questions (1 more dropped by NTA): 30 numerical-answer and 59 multiple choice. 15 were rated hard and 29 easy. Its hard-question count was the 4th highest of the 12 2023 shifts Lumi analysed. The biggest chapter was Binomial Theorem, with 3 questions.

Questions: 89; 1 dropped by NTA
Numerical answer: 30; 34% of the shift
Hard: 15; 17% of the shift
Chapters asked: 67; 11 questions with a figure

Key findings · 6 Apr 2023, Shift 2

1
15 hard questions (17%): the 4th highest count of the 12 2023 shifts Lumi analysed. Across those shifts the hard share ran from 9% to 24%.
2
Maths was the hardest section, with 11 of 29 rated hard; Physics had the most easy questions (13 of 30).
3
19 questions tested a new idea, one no earlier year had asked: 25% of the 76 questions tagged to an idea.
4
39 came back from 2022: the same idea had been asked in the 2022 shifts Lumi analysed.
5
32 shared an idea with another 2023 shift (42% of tagged questions). Shifts in the same year can test the same idea in a new setting.
6
Biggest chapters: Current Electricity in Physics (3), The s-Block Elements in Chemistry (3) and Binomial Theorem in Maths (3).

The shift at a glance

Subjects, difficulty and chapters

Physics

30 Q · 10 numerical

Easy 13Medium 16Hard 1

Top chapters

Chemistry

30 Q · 10 numerical

Easy 13Medium 14Hard 3

Top chapters

Maths

29 Q · 10 numerical

Easy 3Medium 15Hard 11

Top chapters

Question format

Single correct43 Q
Numerical value30 Q
Diagram based6 Q
Two statements4 Q
Assertion–reason4 Q
Other1 Q
Match the list1 Q

What each question asks you to do

Calculation35 Q
Multi-step18 Q
Conceptual13 Q
Recall13 Q
Application8 Q
Data or graph1 Q
Diagram reading1 Q

Most common: Calculation.

Against the other 2023 shifts

This shift next to the 11 other 2023 shifts Lumi analysed

Shares are of each shift’s scored questions. The highlighted row is this shift.

Shift	Questions	Easy	Hard	Numerical	New ideas	Multi-step
6 Apr, Shift 1	90	31%	11%	33%	30%	27%
6 Apr, Shift 2	89	33%	17%	34%	25%	20%
8 Apr, Shift 1	89	38%	9%	34%	29%	21%
8 Apr, Shift 2	89	42%	16%	34%	23%	30%
10 Apr, Shift 1	88	39%	9%	33%	12%	25%
10 Apr, Shift 2	90	38%	12%	33%	20%	21%
11 Apr, Shift 1	90	34%	16%	33%	25%	33%
11 Apr, Shift 2	88	24%	18%	34%	28%	39%
12 Apr, Shift 1	89	30%	24%	34%	29%	35%
13 Apr, Shift 1	89	24%	19%	34%	21%	37%
13 Apr, Shift 2	90	33%	13%	33%	35%	32%
15 Apr, Shift 1	90	43%	14%	33%	18%	30%

New ideas: share of a shift’s questions testing an idea no earlier year had asked. Lumi analyses 108 of the 174 JEE Main shifts held from 2017 to 2026.

Drill down

What this shift asked, and its history

Every chapter this shift asked, with all its topics and ideas. Counts and squares cover every analysed shift from 2017 to 2026; rows this shift did not ask are faded, and each row lists this shift’s questions with the most recent from other shifts.

subjectAskedShareYears’17’20’21’22’23’24’25’26

Open30 this shift3092 · 33%33%8/830299720720358540200225
Open30 this shift3084 · 33%33%8/830298716720359538199224
Open29 this shift3065 · 33%33%8/829299715710354533200225

Squares are the 8 years with analysed shifts, 2017 to 2026. A number is the questions that year when there is more than one; the shade is scaled to one shift, so a year with more analysed shifts (1 to 24 a year) does not look darker. Share is of the level above.

Every question

All 89 questions of this shift

The opening words of each question, with its chapter, topic, idea and how hard it was. Open one for the full text, options and the official answer. “New idea” means no earlier year had tested it.

Subject

Difficulty

Idea

Find

89 of 89 questions

Physics30 questions

13 easy16 medium1 hard

Q31
A 2 meter long scale with least count of 0.2 cm is used to measure the locations of objects on an optical bench. While measuring the focal length of a convex lens, the object pin and the convex lens are placed at 80 cm mark and 1m mark, respectively. The…
Units and Measurements
MediumSingle correct
Q32
As shown in the figure, a particle is moving with constant speed $\pi$ m/s. Considering its motion from A to B, the magnitude of the average velocity is:
Motion in a Plane Kinematics in 2D Uniform Circular Motion
MediumDiagram based · figureIdea asked in 2 yearsAsked the year before
Q33
A particle starts with an initial velocity of 10.0 ms $^{-1}$ along $x$ -direction and accelerates uniformly at the rate of 2.0 ms $^{-2}$ . The time taken by the particle to reach the velocity of 60.0 ms $^{-1}$ is __________.
Motion in a Straight Line Kinematics in 1D Uniform Acceleration Equations
EasySingle correctIdea asked in 4 yearsAsked the year before
Q34
A child of mass 5 kg is going round a merry-go-round that makes 1 rotation in 3.14 s. The radius of the merry-go-round is 2 m. The centrifugal force on the child will be
Laws of Motion Circular Motion Centrifugal Pseudo-Force
EasySingle correctIdea asked in 3 years
Q35
Choose the incorrect statement from the following:
Gravitation Orbital Mechanics Kepler's Laws
EasySingle correctIdea asked in 4 yearsAsked the year before
Q36
The weight of a body on the surface of the earth is 100 N. The gravitational force on it when taken at a height , from the surface of earth , equal to one-fourth the radius of the earth is:
Gravitation
EasySingle correct
Q37
Given below are two statements: one is labelled as Assertion A and the other is labelled as Reason R Assertion A: When you squeeze one end of a tube to get toothpaste out from the other end, Pascal's principle is observed. Reason R: A change in the pressure…
Mechanical Properties of Fluids Fluid Statics Pascal's Law
EasyAssertion–reasonIdea asked in 3 years
Q38
A body cools in 7 minutes from $60^\circ$ C to $40^\circ$ C. The temperature of the surrounding is $10^\circ$ C. The temperature of the body after the next 7 minutes will be:
Thermal Properties of Matter Thermal Radiation Newton's Law of Cooling
MediumSingle correctIdea asked in 3 yearsAsked the year before
Q39
The temperature of an ideal gas is increased from 200 K to 800 K. If r.m.s. speed of gas at 200 K is $v_0$ . Then, r.m.s. speed of the gas at 800 K will be:
Kinetic Theory Kinetic Theory of Gases Molecular Speed Types
EasySingle correctIdea asked in 3 yearsAsked the year before
Q40
The ratio of speed of sound in hydrogen gas to the speed of sound in oxygen gas at the same temperature is:
Waves Sound Velocity General Formula for Speed of Longitudinal Waves
EasySingle correctNew idea
Q41
For an amplitude modulated wave the minimum amplitude is 3 V, while the modulation index is 60%. The maximum amplitude of the modulated wave is:
Communication Systems
EasySingle correct
Q42
Given below are two statements: one is labelled as Assertion A and the other is labelled as Reason R Assertion A: Diffusion current in a p-n junction is greater than the drift current in magnitude if the junction is forward biased. Reason R: Diffusion…
Semiconductor Electronics: Materials, Devices and Simple Circuits Semiconductor Diodes Forward and Reverse Biasing
MediumAssertion–reasonIdea asked in 4 yearsAsked the year before
Q43
A small particle of mass $m$ moves in such a way that its potential energy $U=\frac{1}{2}m\omega^2 r^2$ where $\omega$ is constant and $r$ is the distance of the particle from origin. Assuming Bohr's quantization of momentum and circular orbit, the radius of…
Atoms Bohr's Atomic Model Angular Momentum Quantization
MediumSingle correctIdea asked in 5 yearsAsked the year before
Q44
The work functions of Aluminium and Gold are 4.1 eV and and 5.1 eV respectively. The ratio of the slope of the stopping potential versus frequency plot for Gold to that of Aluminium is
Dual Nature of Radiation and Matter Photoelectric Effect Stopping Potential vs Frequency Graph
EasySingle correctIdea asked in 2 years
Q45
Given below are two statements: one is labelled as Assertion A and the other is labelled as Reason R Assertion A: The phase difference of two light waves change if they travel through different media having same thickness, but different indices of…
Wave Optics Refraction and Snell's Law Optical Path Length Calculation
EasyAssertion–reasonNew idea
Q46
The energy density associated with electric field $\vec{E}$ and magnetic field $\vec{B}$ of an electromagnetic wave in free space is given by ( $\epsilon_0-$ permittivity of free space, $\mu_0=$ permeability of free space)
Electromagnetic Waves
EasySingle correct
Q47
A capacitor of capacitance 150.0 $\mu$ F is connected to an alternating source of emf given by $E=36\sin(120\pi t)$ V. The maximum value of current in the circuit is approximately equal to :
Alternating Current Phasor Diagrams Phase Relations in RLC
EasySingle correctIdea asked in 5 yearsAsked the year before
Q48
A student is provided with a variable voltage source V, a test resistor $R_T=10\Omega$ , two identical galvanometers $G_1$ and $G_2$ and two additional resistors, $R_1=10M\Omega$ and $R_2=0.001\Omega$ . For conducting an experiment to verify ohm's law, the…
Current Electricity
MediumDiagram based · figure
Q49
Figure shows a part of an electric circuit. The potentials at points $a$ , $b$ and $c$ are 30 V, 12 V and 2 V respectively. The current through the 20 $\Omega$ resistor will be,
Current Electricity Kirchhoff's Laws Nodal Analysis Technique
MediumSingle correct · figureIdea asked in 2 years
Q50
A dipole comprises of two charged particles of identical magnitude q and opposite in nature. The mass 'm' of the positive charged particle is half of the mass of the negative charged particle. The two charges are separated by a distance 'l'. If the dipole is…
Electric Charges and Fields Electric Dipoles Angular SHM and Time Period of Dipole
HardSingle correctNew ideaIdea asked in 2 years
Q51
A body is dropped on ground from a height ' $h_1$ ' and after hitting the ground, it rebounds to a height ' $h_2$ '. If the ratio of velocities of the body just before and after hitting ground is 4, then percentage loss in kinetic energy of the body is…
Work, Energy and Power Collisions Coefficient of Restitution
MediumNumerical valueNew idea
Q52
A ring and a solid sphere rotating about an axis passing through their centers have same radii of gyration. The axis of rotation is perpendicular to plane of ring. The ratio of radius of ring to that of sphere is $\sqrt{\frac{2}{x}}$ . The value of $x$ is…
System of Particles and Rotational Motion Moment of Inertia Radius of Gyration
MediumNumerical valueIdea asked in 4 yearsAsked the year before
Q53
A metal block of mass m is suspended from a rigid support through a metal wire of diameter 14 mm. The tensile stress developed in the wire under equilibrium state is $7\times10^{5}$ Nm $^{-2}$ . The value of mass m is __________ kg. (Take, g = 9.8 ms $^{-2}$ …
Mechanical Properties of Solids Elasticity Stress and Strain Types
EasyNumerical valueIdea asked in 4 yearsAsked the year before
Q54
Two concentric circular coils with radii 1 cm and 1000 cm, and number of turns 10 and 200 respectively are placed coaxially with centers coinciding. The mutual inductance of this arrangement will be __________ $\times 10^{-8}$ H. (Take, $\pi^2=10$ )
Electromagnetic Induction Inductance Mutual Inductance of Coaxial Coils
MediumNumerical valueIdea asked in 2 years
Q55
A beam of light consisting of two wavelengths 7000 Å and 5500 Å is used to obtain interference pattern in Young's double slit experiment. The distance between the slits is 2.5 mm and the distance between the plane of slits and the screen is 150 cm. The least…
Wave Optics Young's Double Slit Coinciding Maxima of Two Wavelengths
MediumNumerical valueNew ideaIdea asked in 2 years
Q56
Experimentally it is found that 12.8 eV energy is required to separate a hydrogen atom into a proton and an electron. So the orbital radius of the electron in a hydrogen atom is $\frac{9}{x}\times10^{-10}$ m. The value of the $x$ is : __________. (1 eV =…
Atoms Bohr's Atomic Model Bohr Orbit Radius
MediumNumerical valueIdea asked in 4 yearsAsked the year before
Q57
A proton with a kinetic energy of 2.0 eV moves into a region of uniform magnetic field of magnitude $\frac{\pi}{2}\times10^{-3}$ T. The angle between the direction of magnetic field and velocity of proton is $60^\circ$ . The pitch of the helical path taken by…
Moving Charges and Magnetism Lorentz Force Helical Motion and Pitch
MediumNumerical valueIdea asked in 4 years
Q58
As shown in the figure, the voltmeter reads 2 V across 5 $\Omega$ resistor. The resistance of the voltmeter is __________ $\Omega$ .
Current Electricity Kirchhoff's Laws Kirchhoff's Voltage Law (KVL)
MediumNumerical value · figureIdea asked in 6 yearsAsked the year before
Q59
As shown in the figure, two parallel plate capacitors having equal plate area of 200 cm $^2$ are joined in such a way that $a \neq b$ . The equivalent capacitance of the combination is $x\,\epsilon_0$ F. The value of $x$ is __________.
Electrostatic Potential and Capacitance Parallel Plate Capacitors Series and Parallel Combinations
MediumNumerical value · figureIdea asked in 6 yearsAsked the year before
Q60
A simple pendulum with length 100 cm and bob of mass 250 g is executing S.H.M. of amplitude 10 cm. The maximum tension in the string is found to be $\frac{x}{40}$ N. The value of $x$ is __________.
Oscillations Pendulums String Tension at Mean Position
MediumNumerical valueNew idea

Chemistry30 questions

13 easy14 medium3 hard

Q61
The volume of 0.02 M aqueous HBr required to neutralize 10.0 mL of 0.01 M aqueous $\mathrm{Ba(OH)_2}$ is (Assume complete neutralization)
Some Basic Concepts of Chemistry Mole Concept Normality and n-factor Calculation
EasySingle correctIdea asked in 4 yearsAsked the year before
Q62
If the radius of the first orbit of hydrogen atom is $a_o$ , then de Broglie's wavelength of electron in $3^{rd}$ orbit is
Structure of Atom Bohr's Atomic Theory Circumference and De Broglie Wavelength
EasySingle correctIdea asked in 3 years
Q63
Consider the following reaction that goes from A to B in three steps as shown below: Choose the correct option Number of intermediates | Number of Activated complexes | Rate determining step
Chemical Kinetics Arrhenius Equation Reaction Energy Profiles
MediumDiagram based · figureIdea asked in 3 years
Q64
Which one of the following elements will remain as liquid inside pure boiling water?
Classification of Elements and Periodicity in Properties Group 13 & 14 Gallium Melting Point Anomaly
MediumSingle correctNew idea
Q65
The product, which is not obtained during the electrolysis of brine solution is
The s-Block Elements Hydrogen and Hydrides Dihydrogen from Brine Electrolysis
EasySingle correctIdea asked in 2 yearsAsked the year before
Q66
Structures of $\mathrm{BeCl_2}$ in solid state, vapour phase and at very high temperature respectively are:
The s-Block Elements s-Block Elements Phase-Dependent Structure of BeCl2
MediumSingle correctIdea asked in 2 years
Q67
Ion having highest hydration enthalpy among the given alkaline earth metal ions is:
The s-Block Elements s-Block Elements Hydration Enthalpy Trends
EasySingle correctIdea asked in 3 yearsAsked the year before
Q68
Group-13 elements react with $\mathrm{O_2}$ in amorphous form to form oxides of type $\mathrm{M_2O_3}$ (M = element). Which among the following is the most basic oxide?
The p-Block Elements Group 13 & 14 Acid-Base Nature of Group 13 Oxides
EasySingle correctIdea asked in 2 years
Q69
During the reaction of permanganate with thiosulphate, the change in oxidation of manganese occurs by value of 3. Identify which of the below medium will favour the reaction.
Redox Reactions Permanganate n-Factor Across Media
MediumSingle correctIdea asked in 3 yearsAsked the year before
Q70
Given below are two statements: one is labelled as "Assertion A" and the other is labelled as "Reason R" Assertion A: In the complex $\mathrm{Ni(CO)_4}$ and $\mathrm{Fe(CO)_5}$ , the metals have zero oxidation state. Reason R: Low oxidation states are found…
Coordination Compounds Redox Reactions Oxidation States in Metal Carbonyls
MediumAssertion–reasonNew idea
Q71
The group of chemicals used as pesticide is
Environmental Chemistry Pesticides and Herbicides Toxicity
EasySingle correctIdea asked in 2 yearsAsked the year before
Q72
The IUPAC name of $\mathrm{K_3[Co(C_2O_4)_3]}$ is:-
Coordination Compounds Coordination Nomenclature Anionic Complex Suffixes
EasySingle correctNew idea
Q73
Formation of which complex, among the following, is not a confirmatory test of $\mathrm{Pb^{2+}}$ ions
Organic Chemistry: Some Basic Principles and Techniques Cation Analysis Group I: Lead Chromate Test
EasySingle correctNew idea
Q74
Element not present in Nessler's reagent is
Amines Cation Analysis Group 0: Nessler's Reagent Complex Formula
EasySingle correctNew ideaIdea asked in 2 years
Q75
In the following reaction, 'B' is
Alcohols, Phenols and Ethers Alkenes and Alkynes Carbocation Rearrangements
HardDiagram based · figureIdea asked in 3 years
Q76
From the figure of column chromatography given below, identify incorrect statements. A. Compound 'c' is more polar than 'a' and 'b' B. Compound 'a' is least polar C. Compound 'b' comes out of the column before 'c' and after 'a' D. Compound 'a' spends more…
Organic Chemistry: Some Basic Principles and Techniques Adsorption Adsorption in Chromatography
MediumOther · figureNew ideaIdea asked in 2 years
Q77
Find out the major product from the following reaction.
Aldehydes, Ketones and Carboxylic Acids Nucleophilic Addition Grignard Addition to Alpha,Beta-Unsaturated Carbonyls
HardDiagram based · figureIdea asked in 2 years
Q78
The strongest acid from the following is
Alcohols, Phenols and Ethers Alcohols and Phenols Phenol Acidity Comparison
EasyDiagram based · figureIdea asked in 3 yearsAsked the year before
Q79
Match List I with List II List I (Natural Amino acid) | List II (One Letter Code) (A) Arginine | (I) D (B) Aspartic acid | (II) N (C) Asparagine | (III) A (D) Alanine | (IV) R Choose the correct answer from the options given below:
Biomolecules
EasyMatch the list
Q80
Given below are two statements: Statement I: Morphine is a narcotic analgesic. It helps in relieving pain without producing sleep. Statement II: Morphine and its derivatives are obtained from opium poppy. In the light of the above statements, choose the…
Chemistry in Everyday Life Narcotic Analgesics and Clinical Use
EasyTwo statementsNew idea
Q81
Number of crystal systems from the following where body centred unit cell can be found, is ____________ Cubic, tetragonal, orthorhombic, hexagonal, rhombohedral, monoclinic, triclinic
The Solid State Crystal Lattices 14 Bravais Lattices Distribution
MediumNumerical valueNew idea
Q82
The number of species having a square planar shape from the following is ______________. $\mathrm{XeF_4}$ , $\mathrm{SF_4}$ , $\mathrm{SiF_4}$ , $\mathrm{BF_4^-}$ , $\mathrm{BrF_4^-}$ , $\mathrm{[Cu(NH_3)_4]^{2+}}$ , $\mathrm{[FeCl_4]^{2-}}$ ,…
Chemical Bonding and Molecular Structure VSEPR Theory Octahedral and Square Planar
MediumNumerical valueIdea asked in 3 years
Q83
Consider the following data Heat of combustion of $\mathrm{H_2(g)}$ = $-241.8$ kJ mol $^{-1}$ Heat of combustion of $\mathrm{C(s)}$ = $-393.5$ kJ mol $^{-1}$ Heat of combustion of $\mathrm{C_2H_5OH(l)}$ = $-1234.7$ kJ mol $^{-1}$ The heat of formation of…
Thermodynamics (Chemistry)Enthalpy and Hess's Law Formation and Combustion
MediumNumerical valueIdea asked in 5 yearsAsked the year before
Q84
Consider the following pairs of solution which will be isotonic at the same temperature. The number of pairs of solutions is / are ___________. A. 1 M aq. NaCl and 2 M aq. urea B. 1 M aq. $\mathrm{CaCl_2}$ and 1.5 M aq. KCl C. 1.5 M aq. $\mathrm{AlCl_3}$ and…
Solutions Colligative Properties Isotonic Solutions
MediumNumerical valueIdea asked in 3 yearsAsked the year before
Q85
The equilibrium composition for the reaction $\mathrm{PCl_3 + Cl_2 \rightleftharpoons PCl_5}$ at 298 K is given below: $[\mathrm{PCl_3}]_{eq}$ = 0.2 mol L $^{-1}$ , $[\mathrm{Cl_2}]_{eq}$ = 0.1 mol L $^{-1}$ , $[\mathrm{PCl_5}]_{eq}$ = 0.40 mol L $^{-1}$ If 0.2…
Equilibrium Chemical Equilibrium Le Chatelier's Principle
MediumNumerical valueIdea asked in 2 years
Q86
The standard reduction potentials at 298 K for the following half cells are given below: $\mathrm{NO_3^- + 4H^+ + 3e^- \rightarrow NO(g) + 2H_2O}$ $E^\theta$ = 0.97 V $\mathrm{V^{2+}(aq) + 2e^- \rightarrow V}$ $E^\theta$ = $-1.19$ V…
Electrochemistry Nernst Equation Standard Cell Potential
MediumNumerical valueIdea asked in 7 yearsAsked the year before
Q87
The number of colloidal systems from the following, which will have 'liquid' as the dispersion medium, is ______________. Gem stones, paints, smoke, cheese, milk, hair cream, insecticide sprays, froth, soap lather
Surface Chemistry Colloids Colloid Classification by State
MediumNumerical valueIdea asked in 4 yearsAsked the year before
Q88
In an ice crystal, each water molecule is hydrogen bonded to ______________ neighbouring molecules.
Hydrogen Hydrogen and Hydrides Structure and Density of Ice
EasyNumerical valueNew idea
Q89
Among the following, the number of compounds which will give positive iodoform reaction is ______________. (a) 1-Phenylbutan-2-one (b) 2-Methylbutan-2-ol (c) 3-Methylbutan-2-ol (d) 1-Phenylethanol (e) 3,3-dimethylbutan-2-one (f) 1-Phenylpropan-2-ol
Aldehydes, Ketones and Carboxylic Acids Alcohols and Phenols Iodoform Test for Specific Alcohols
MediumNumerical valueIdea asked in 4 yearsAsked the year before
Q90
Number of isomeric aromatic amines with molecular formula $\mathrm{C_8H_{11}N}$ , which can be synthesized by Gabriel Phthalimide synthesis is ______________.
Amines Gabriel Phthalimide Synthesis
HardNumerical valueIdea asked in 6 yearsAsked the year before

Maths29 questions

3 easy15 medium11 hard

Q1
[English stem not rendered in the PDF; reconstructed from the Hindi copy] Let the function $f(x)=\frac{1}{\sqrt{\lceil x\rceil-x}}$ , where $\lceil x\rceil$ is the smallest integer $\geq x$ , have domain $A$ and range $B$ . Then the statements (S1) :…
Relations and Functions Functions and Domain Domain Constraints of Greatest Integer Function
MediumTwo statementsIdea asked in 2 years
Q2
In a group of 100 persons 75 speak English and 40 speak Hindi. Each person speaks at least one of the two languages. If the number of persons, who speak only English is $\alpha$ and the number of persons who speak only Hindi is $\beta$ , then the eccentricity…
Conic Sections Ellipse Eccentricity Eccentricity calculation
EasySingle correctIdea asked in 6 yearsAsked the year before
Q4
Let $P$ be a square matrix such that $P^2=I-P$ . For $\alpha,\beta,\gamma,\delta\in\mathbb{N}$ , if $P^\alpha+P^\beta=\gamma I-29P$ and $P^\alpha-P^\beta=\delta I-13P$ , then $\alpha+\beta+\gamma-\delta$ is equal to
Matrices Matrix Multiplication Powers of Square Matrices
HardSingle correctIdea asked in 6 yearsAsked the year before
Q5
For the system of equations $x+y+z=6$ $x+2y+\alpha z=10$ $x+3y+5z=\beta$ , which one of the following is NOT true?
Determinants Cramer's Rule Evaluating Parameters for Consistency
MediumSingle correctIdea asked in 6 yearsAsked the year before
Q6
Among the statements: (S1) : $2023^{2022}-1999^{2022}$ is divisible by 8 (S2) : $13(13)^n-11n-13$ is divisible by 144 for infinitely many $n\in\mathbb{N}$
Binomial Theorem
HardTwo statements
Q7
If the coefficients of $x^7$ in $\left(ax^2+\frac{1}{2bx}\right)^{11}$ and $x^{-7}$ in $\left(ax-\frac{1}{3bx^2}\right)^{11}$ are equal, then
Binomial Theorem
MediumSingle correct
Q8
All the letters of the word PUBLIC are written in all possible orders and these words are written as in a dictionary with serial numbers. Then the serial number of the word PUBLIC is
Permutations and Combinations Lexicographic Rank of a Word
MediumSingle correctIdea asked in 4 yearsAsked the year before
Q9
$\lim_{n\to\infty}\left\{\left(2^{\frac{1}{2}}-2^{\frac{1}{3}}\right)\left(2^{\frac{1}{2}}-2^{\frac{1}{5}}\right)\cdots\cdots\left(2^{\frac{1}{2}}-2^{\frac{1}{2n+1}}\right)\right\}$ is equal to
Limits and Derivatives
HardSingle correct
Q10
If gcd $(m, n)=1$ and $1^2-2^2+3^2-4^2+\cdots+(2021)^2-(2022)^2+(2023)^2=1012\,m^2n$ then $m^2-n^2$ is equal to
Sequences and Series
MediumSingle correct
Q11
Let $f(x)$ be a function satisfying $f(x)+f(\pi-x)=\pi^2,\ \forall x\in\mathbb{R}$ . Then $\int_0^{\pi}f(x)\sin x\,dx$ is equal to
Integrals Definite Integral Properties King's Property
MediumSingle correctIdea asked in 6 yearsAsked the year before
Q12
If the solution curve $f(x,y)=0$ of the differential equation $\left(1+\log_e x\right)\frac{dx}{dy}-x\log_e x=e^y,\ x>0$ , passes through the points $(1,0)$ and $(\alpha,2)$ , then $\alpha^\alpha$ is equal to
Differential Equations Linear Differential Equations Generalized Substitution for LDEs
HardSingle correctIdea asked in 4 years
Q13
If the tangents at the points P and Q on the circle $x^2+y^2-2x+y=5$ meet at the point $R\left(\frac{9}{4},2\right)$ , then the area of the triangle PQR is:
Conic Sections Circle Tangents Area of Tangent-Chord Triangle
MediumSingle correctNew idea
Q14
The area bounded by the curves $y=|x-1|+|x-2|$ and $y=3$ is equal to
Application of Integrals Area Under Curves Area with Modulus
EasySingle correctIdea asked in 4 yearsAsked the year before
Q15
A plane P contains the line of intersection of the plane $\vec{r}\cdot\left(\hat{i}+\hat{j}+\hat{k}\right)=6$ and $\vec{r}\cdot\left(2\hat{i}+3\hat{j}+4\hat{k}\right)=-5$ . If P passes through the point $(0, 2, -2)$ , then the square of distance of the point…
Three Dimensional Geometry Planes in 3D Family of Planes
MediumSingle correctIdea asked in 4 yearsAsked the year before
Q16
Let the line L pass through the point $(0, 1, 2)$ , intersect the line $\frac{x-1}{2}=\frac{y-2}{3}=\frac{z-3}{4}$ and be parallel to the plane $2x+y-3z=4$ . Then the distance of the point $P(1, -9, 2)$ from the line L is
Three Dimensional Geometry Lines in 3D Perpendicular Distance from Point to Line
HardSingle correctIdea asked in 6 yearsAsked the year before
Q17
The sum of all values of $\alpha$ , for which the points whose position vectors are $\hat{i}-2\hat{j}+3\hat{k}$ , $2\hat{i}-3\hat{j}+4\hat{k}$ , $(\alpha+1)\hat{i}+2\hat{k}$ and $9\hat{i}+(\alpha-8)\hat{j}+6\hat{k}$ are coplanar, is equal to
Vector Algebra Triple Products Coplanarity of Four Points
MediumSingle correctIdea asked in 3 yearsAsked the year before
Q18
Let the vectors $\vec{a},\vec{b},\vec{c}$ represent three coterminous edges of a parallelopiped of volume V. Then the volume of the parallelopiped, whose coterminous edges are represented by $\vec{a}$ , $\vec{b}+\vec{c}$ and $\vec{a}+2\vec{b}+3\vec{c}$ is…
Vector Algebra Triple Products STP Geometric Interpretation
MediumSingle correctIdea asked in 2 years
Q19
Three dice are rolled. If the probability of getting different numbers on the three dice is $\frac{p}{q}$ , where $p$ and $q$ are co-prime, then $q-p$ is equal to
Probability
EasySingle correct
Q20
Among the statements (S1): $(p\Rightarrow q)\vee\left((\sim p)\wedge q\right)$ is a tautology (S2): $(q\Rightarrow p)\Rightarrow\left((\sim p)\wedge q\right)$ is a contradiction
Mathematical Reasoning
MediumTwo statements
Q21
For $\alpha,\beta,z\in\mathbb{C}$ and $\lambda>1$ , if $\sqrt{\lambda-1}$ is the radius of the circle $|z-\alpha|^2+|z-\beta|^2=2\lambda$ , then $|\alpha-\beta|$ is equal to ________.
Complex Numbers and Quadratic Equations Complex Numbers - Polar Form Distance Formula in Polar Coordinates
MediumNumerical valueNew idea
Q22
The number of 4-letter words, with or without meaning, each consisting of 2 vowels and 2 consonants, which can be formed from the letters of the word UNIVERSE without repetition is ______________.
Permutations and Combinations Permutations of Distinct Objects
MediumNumerical valueIdea asked in 4 yearsAsked the year before
Q23
If $(20)^{19}+2(21)(20)^{18}+3(21)^2(20)^{17}+\ldots+20(21)^{19}=k(20)^{19}$ , then $k$ is equal to ___________.
Binomial Theorem Arithmetico-Geometric Series Sum of Finite AGP
HardNumerical valueIdea asked in 2 yearsAsked the year before
Q24
The number of points, where the curve $y=x^5-20x^3+50x+2$ crosses the x-axis, is ________.
Application of Derivatives Curve Sketching Graphical Root Counting
MediumNumerical valueNew ideaIdea asked in 2 years
Q25
Let a curve $y=f(x),\ x\in(0,\infty)$ pass through the points $P\left(1,\frac{3}{2}\right)$ and $Q\left(a,\frac{1}{2}\right)$ . If the tangent at any point $R(b,f(b))$ to the given curve cuts the y-axis at the point $S(0,c)$ such that $bc=3$ , then $(PQ)^2$ is…
Differential Equations Linear Differential Equations Integrating Factor Calculation
HardNumerical valueIdea asked in 7 yearsAsked the year before
Q26
Let $f(x)=\frac{x}{(1+x^n)^{\frac{1}{n}}},\ x\in\mathbb{R}-\{-1\},\ n\in\mathbb{N},\ n>2$ . If $f^n(x)=(fofof\ldots\text{ upto }n\text{ times})(x)$ , then $\lim_{n\to\infty}\int_0^1 x^{n-2}\left(f^n(x)\right)dx$ is equal to ___________.
Integrals
HardNumerical value
Q27
Let the eccentricity of an ellipse $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$ is reciprocal to that of the hyperbola $2x^2-2y^2=1$ . If the ellipse intersects the hyperbola at right angles, then square of length of the latus-rectum of the ellipse is ___________.
Conic Sections Ellipse Eccentricity Length of Latus Rectum
HardNumerical valueIdea asked in 6 yearsAsked the year before
Q28
If the lines $\frac{x-1}{2}=\frac{2-y}{-3}=\frac{z-3}{\alpha}$ and $\frac{x-4}{5}=\frac{y-1}{2}=\frac{z}{\beta}$ intersect, then the magnitude of the minimum value of $8\alpha\beta$ is ___________.
Three Dimensional Geometry Skew Lines Coplanarity Determinant Test
HardNumerical valueIdea asked in 3 yearsAsked the year before
Q29
If the mean and variance of the frequency distribution $x_i$ : 2, 4, 6, 8, 10, 12, 14, 16 $f_i$ : 4, 4, $\alpha$ , 15, 8, $\beta$ , 4, 5 are 9 and 15.08 respectively, then the value of $\alpha^2+\beta^2-\alpha\beta$ is ___________.
Statistics Variance and SD Finding Unknown Frequency via Variance
MediumNumerical value · figureIdea asked in 2 years
Q30
The value of $\tan 9^\circ-\tan 27^\circ-\tan 63^\circ+\tan 81^\circ$ is ___________.
Trigonometric Functions Trigonometric Identities Products of Tangents of Complementary Angles
HardNumerical valueNew idea

Dropped from the final key and left out of every count: Q3.

Questions from the official paper published by NTA, answers from the final key. Lumi analyses 108 of the 174 JEE Main shifts held from 2017 to 2026; see the other 2023 shifts.