JEE Main 2023 · April session

JEE Main 8 April 2023, Shift 1

89 questions (1 more dropped by NTA): 30 numerical-answer and 59 multiple choice. 8 were rated hard and 34 easy. Its hard-question count was the joint 11th highest of the 12 2023 shifts Lumi analysed. The biggest chapter was Permutations and Combinations, with 4 questions.

Questions: 89; 1 dropped by NTA
Numerical answer: 30; 34% of the shift
Hard: 8; 9% of the shift
Chapters asked: 66; 12 questions with a figure

Key findings · 8 Apr 2023, Shift 1

1
8 hard questions (9%): the joint 11th 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 6 of 30 rated hard; Physics had the most easy questions (20 of 30).
3
21 questions tested a new idea, one no earlier year had asked: 29% of the 72 questions tagged to an idea.
4
39 came back from 2022: the same idea had been asked in the 2022 shifts Lumi analysed.
5
26 shared an idea with another 2023 shift (36% of tagged questions). Shifts in the same year can test the same idea in a new setting.
6
Biggest chapters: Moving Charges and Magnetism in Physics (3), The p-Block Elements in Chemistry (3) and Permutations and Combinations in Maths (4).

The shift at a glance

Subjects, difficulty and chapters

Physics

30 Q · 10 numerical

Easy 20Medium 9Hard 1

Top chapters

Chemistry

29 Q · 10 numerical

Easy 11Medium 17Hard 1

Top chapters

The p-Block Elements3 Q
Amines2 Q
Coordination Compounds2 Q
Alcohols, Phenols and Ethers1 Q
Aldehydes, Ketones and Carboxylic Acids1 Q
Biomolecules1 Q

Maths

30 Q · 10 numerical

Easy 3Medium 21Hard 6

Top chapters

Question format

Single correct45 Q
Numerical value30 Q
Diagram based6 Q
Two statements3 Q
Match the list3 Q
Other1 Q
Assertion–reason1 Q

What each question asks you to do

Calculation38 Q
Multi-step19 Q
Conceptual15 Q
Recall12 Q
Data or graph4 Q
Application1 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 shift3084 · 33%33%8/830298716720359538199224
Open30 this shift3065 · 33%33%8/829299715710354533200225
Open29 this shift3092 · 33%33%8/830299720720358540200225

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

20 easy9 medium1 hard

Q31
The engine of a train moving with speed $10\ \mathrm{ms^{-1}}$ towards a platform sounds a whistle at frequency $400\ \mathrm{Hz}$ . The frequency heard by a passenger inside the train is: (neglect air speed. Speed of sound in air = $330\ \mathrm{ms^{-1}}$ )
Waves Doppler Effect Moving Source Cases
EasySingle correctIdea asked in 2 yearsAsked the year before
Q32
An air bubble of volume $1\ \mathrm{cm^3}$ rises from the bottom of a lake $40\ \mathrm{m}$ deep to the surface at a temperature of $12^\circ\mathrm{C}$ . The atmospheric pressure is $1\times10^5\ \mathrm{Pa}$ , the density of water is $1000\ \mathrm{kg/m^3}$ …
Mechanical Properties of Fluids Fluid Statics Hydrostatic Pressure
MediumSingle correctNew ideaIdea asked in 2 years
Q33
Given below are two statements: Statement I: If heat is added to a system, its temperature must increase. Statement II: If positive work is done by a system in a thermodynamic process, its volume must increase. In the light of the above statements, choose…
Thermodynamics First Law of Thermodynamics Thermodynamic Sign Conventions
MediumTwo statementsNew idea
Q34
An aluminium rod with Young's modulus $Y = 7.0\times10^{10}\ \mathrm{N/m^2}$ undergoes elastic strain of $0.04\%$ . The energy per unit volume stored in the rod in SI unit is:
Mechanical Properties of Solids Elasticity Elastic Energy Density
EasySingle correctIdea asked in 4 yearsAsked the year before
Q35
Given below are two statements: Statement I: If E be the total energy of a satellite moving around the earth, then its potential energy will be $\frac{E}{2}$ . Statement II: The kinetic energy of a satellite revolving in an orbit is equal to the half the…
Gravitation Orbital Mechanics Kinetic to Potential Energy Ratios
EasyTwo statementsNew idea
Q36
At any instant the velocity of a particle of mass $500\ \mathrm{g}$ is $(2t\hat{i}+3t^2\hat{j})\ \mathrm{ms^{-1}}$ . If the force acting on the particle at $t=1\mathrm{s}$ is $(\hat{i}+x\hat{j})\mathrm{N}$ . Then the value of $x$ will be:
Laws of Motion Newton's Laws of Motion F=ma for Single Body
EasySingle correctIdea asked in 6 yearsAsked the year before
Q37
Two forces having magnitude $A$ and $\frac{A}{2}$ are perpendicular to each other. The magnitude of their resultant is:
Motion in a Plane
EasySingle correct
Q38
The weight of a body on the earth is $400\ \mathrm{N}$ . Then weight of the body when taken to a depth half of the radius of the earth will be:
Gravitation
EasySingle correct
Q39
Two projectiles A and B are thrown with initial velocities of $40\ \mathrm{m/s}$ and $60\ \mathrm{m/s}$ at angles $30^\circ$ and $60^\circ$ with the horizontal respectively. The ratio of their ranges respectively is ( $g = 10\ \mathrm{m/s^2}$ )
Motion in a Plane Projectile Motion Ground-to-Ground Parameters
MediumSingle correctIdea asked in 6 yearsAsked the year before
Q40
A cylindrical wire of mass $(0.4\pm0.01)\ \mathrm{g}$ has length $(8\pm0.04)\ \mathrm{cm}$ and radius $(6\pm0.03)\ \mathrm{mm}$ . The maximum error in its density will be:
Units and Measurements
MediumSingle correct
Q41
A TV transmitting antenna is $98\ \mathrm{m}$ high and the receiving antenna is at the ground level. If the radius of the earth is $6400\ \mathrm{km}$ , the surface area covered by the transmitting antenna is approximately:
Communication Systems
MediumSingle correct
Q42
For the logic circuit shown, the output waveform at Y is:
Semiconductor Electronics: Materials, Devices and Simple Circuits Logic Gates Timing Diagrams and Waveform Analysis
MediumDiagram based · figureIdea asked in 4 yearsAsked the year before
Q43
For a nucleus ${}^{A}_{Z}\mathrm{X}$ having mass number A and atomic number Z A. The surface energy per nucleon $(\mathrm{b_s}) = -a_1 A^{2/3}$ . B. The Coulomb contribution to the binding energy $\mathrm{b_c} = -a_2\frac{Z(Z-1)}{A^{4/3}}$ C. The volume…
Nuclei Nuclear Binding Energy Liquid Drop Model Surface Energy Term
MediumOtherNew idea
Q44
Proton (P) and electron (e) will have same de-Broglie wavelength when the ratio of their momentum is (assume, $\mathrm{m_p} = 1849\ \mathrm{m_e}$ ):
Dual Nature of Radiation and Matter Matter Waves Basic Wavelength-Momentum Relation
EasySingle correctIdea asked in 5 yearsAsked the year before
Q45
In a reflecting telescope, a secondary mirror is used to:
Ray Optics and Optical Instruments Optical Instruments Reflecting Telescopes
EasySingle correctNew idea
Q46
Dimension of $\frac{1}{\mu_0 \epsilon_0}$ should be equal to
Electromagnetic Waves Coulomb's Law Dimensional Analysis of Permittivity
EasySingle correctNew ideaIdea asked in 3 years
Q47
Certain galvanometers have a fixed core made of non magnetic metallic material. The function of this metallic material is
Moving Charges and Magnetism
EasySingle correct
Q48
A charge particle moving in magnetic field B, has the components of velocity along B as well as perpendicular to B. The path of the charge particle will be
Moving Charges and Magnetism Lorentz Force Helical Motion and Pitch
EasySingle correctIdea asked in 4 years
Q49
In this figure the resistance of the coil of galvanometer G is $2\ \Omega$ . The emf of the cell is $4\ \mathrm{V}$ . The ratio of potential difference across $\mathrm{C_1}$ and $\mathrm{C_2}$ is:
Current Electricity RC Circuits Initial and Steady State Behavior
MediumDiagram based · figureIdea asked in 5 years
Q50
Graphical variation of electric field due to a uniformly charged insulating solid sphere of radius R, with distance r from the centre O is represented by:
Electric Charges and Fields Gauss's Law Nonconducting Sphere
EasyDiagram based · figureNew idea
Q51
An organ pipe $40\ \mathrm{cm}$ long is open at both ends. The speed of sound in air is $360\ \mathrm{ms^{-1}}$ . The frequency of the second harmonic is ________ Hz.
Waves Resonance in Air Columns Open Organ Pipe Harmonics
EasyNumerical valueNew ideaIdea asked in 2 years
Q52
An air bubble of diameter $6\ \mathrm{mm}$ rises steadily through a solution of density $1750\ \mathrm{kg/m^3}$ at the rate of $0.35\ \mathrm{cm/s}$ . The co-efficient of viscosity of the solution (neglect density of air) is ________ Pas (given,…
Mechanical Properties of Fluids Viscosity Terminal Velocity of Rising Bubbles
MediumNumerical valueIdea asked in 3 yearsAsked the year before
Q53
The moment of inertia of a semicircular ring about an axis, passing through the center and perpendicular to the plane of ring, is $\frac{1}{x}\mathrm{MR^2}$ , where R is the radius and $M$ is the mass of the semicircular ring. The value of $x$ will be ________.
System of Particles and Rotational Motion Moment of Inertia MOI of Sectors and Arcs
EasyNumerical valueNew idea
Q54
The momentum of a body is increased by $50\%$ . The percentage increase in the kinetic energy of the body is ________ %.
Work, Energy and Power Work and Kinetic Energy Momentum and Kinetic Energy Relation
EasyNumerical valueIdea asked in 4 yearsAsked the year before
Q55
A nucleus with mass number $242$ and binding energy per nucleon as $7.6\ \mathrm{MeV}$ breaks into two fragment each with mass number $121$ . If each fragment nucleus has binding energy per nucleon as $8.1\ \mathrm{MeV}$ , the total gain in binding energy is…
Nuclei Nuclear Binding Energy Binding Energy per Nucleon
EasyNumerical valueIdea asked in 6 yearsAsked the year before
Q56
Two vertical parallel mirrors A and B are separated by $10\ \mathrm{cm}$ . A point object O is placed at a distance of $2\ \mathrm{cm}$ from mirror A. The distance of the second nearest image behind mirror A from the mirror A is ________ cm.
Ray Optics and Optical Instruments
HardNumerical value · figure
Q57
An oscillating LC circuit consists of a $75\ \mathrm{mH}$ inductor and a $1.2\ \mu\mathrm{F}$ capacitor. If the maximum charge to the capacitor is $2.7\ \mu\mathrm{C}$ . The maximum current in the circuit will be ________ mA.
Alternating Current LCR Resonance LC Energy Oscillation at Resonance
EasyNumerical valueIdea asked in 3 yearsAsked the year before
Q58
The magnetic intensity at the center of a long current carrying solenoid is found to be $1.6\times10^3\ \mathrm{Am^{-1}}$ . If the number of turns is 8 per cm, then the current flowing through the solenoid is ________ A.
Moving Charges and Magnetism Ampere's Circuital Law Ideal Infinite Solenoid
EasyNumerical valueIdea asked in 4 yearsAsked the year before
Q59
A current of $2\ \mathrm{A}$ flows through a wire of cross-sectional area $25.0\ \mathrm{mm^2}$ . The number of free electrons in a cubic meter are $2.0\times10^{28}$ . The drift velocity of the electrons is ________ $\times10^{-6}\ \mathrm{ms^{-1}}$ (given,…
Current Electricity Ohm's Law and Resistance Drift Velocity and Current
EasyNumerical valueIdea asked in 3 yearsAsked the year before
Q60
An electric dipole of dipole moment is $6.0\times10^{-6}\mathrm{C\,m}$ placed in a uniform electric field of $1.5\times10^{3}\mathrm{NC^{-1}}$ in such a way that dipole moment is along electric field. The work done in rotating dipole by $180^\circ$ in this…
Electrostatic Potential and Capacitance Electric Dipoles Work Done in Rotating a Dipole
EasyNumerical valueNew idea

Chemistry29 questions

11 easy17 medium1 hard

Q61
$2\mathrm{IO_3^-} + x\mathrm{I^-} + 12\mathrm{H^+} \rightarrow 6\mathrm{I_2} + 6\mathrm{H_2O}$ What is the value of $x$ ?
Redox Reactions Ion-Electron Acidic
EasySingle correctNew ideaIdea asked in 2 years
Q62
The reaction $\frac{1}{2}\mathrm{H_2(g)} + \mathrm{AgCl(s)} \rightleftharpoons \mathrm{H^+(aq)} + \mathrm{Cl^-(aq)} + \mathrm{Ag(s)}$ occurs in which of the given galvanic cell.
Electrochemistry
MediumSingle correct
Q63
Which of the following represent the Freundlich adsorption isotherms? A. [graph: $\frac{x}{m}$ vs $p$ ] B. [graph: $\log\frac{x}{m}$ vs $\log p$ ] C. [graph: $\frac{x}{m}$ vs $c$ ] D. [graph: $\frac{x}{m}$ vs $p^{1/n}$ ] Choose the correct answer from the…
Surface Chemistry Adsorption Freundlich Logarithmic Plot
MediumDiagram based · figureIdea asked in 4 yearsAsked the year before
Q64
The correct order of electronegativity for given elements is:
Classification of Elements and Periodicity in Properties Periodic Trends Electronegativity Scales
MediumSingle correctNew ideaIdea asked in 4 years
Q65
Which of the following metals can be extracted through alkali leaching technique?
General Principles and Processes of Isolation of Elements
MediumSingle correct
Q66
The water gas on reacting with cobalt as a catalyst forms
Hydrogen Hydrogen and Hydrides Syngas Production via Reforming
EasySingle correctIdea asked in 2 years
Q67
What is the purpose of adding gypsum to cement?
The p-Block Elements s-Block Elements Role of Gypsum in Portland Cement
EasySingle correctNew idea
Q68
Given below are two statements: Statement I: Lithium and Magnesium do not form superoxide Statement II: The ionic radius of $\mathrm{Li^+}$ is larger than ionic radius of $\mathrm{Mg^{2+}}$ In the light of the above statements, choose the most appropriate…
The s-Block Elements s-Block Elements Formation of Oxides, Peroxides, and Superoxides
MediumTwo statementsIdea asked in 4 yearsAsked the year before
Q69
Which halogen is known to cause the reaction given below: $2\mathrm{Cu^{2+}} + 4\mathrm{X^-} \rightarrow \mathrm{Cu_2X_2(s)} + \mathrm{X_2}$
The p-Block Elements
MediumSingle correct
Q70
Which of the following complex is octahedral, diamagnetic and the most stable?
Coordination Compounds Crystal Field Theory High and Low Spin States
MediumSingle correctIdea asked in 7 yearsAsked the year before
Q72
The correct order of spin only magnetic moments for the following complex ions is
Coordination Compounds Crystal Field Theory Spin-only Magnetic Moment
MediumSingle correctIdea asked in 7 yearsAsked the year before
Q73
Choose the halogen which is most reactive towards $\mathrm{S_N1}$ reaction in the given compounds (A, B, C & D) A. [structure] B. [structure] C. [structure] D. [structure]
Haloalkanes and Haloarenes SN1 and SN2 Mechanisms SN1 Carbocation Intermediate
HardDiagram based · figureIdea asked in 2 years
Q74
The major product formed in the following reaction is: [reaction: ortho-disubstituted benzene bearing $\mathrm{CH_2CH_2CO_2H}$ and $\mathrm{CH_2CO_2Et}$ chains] $\xrightarrow[\text{EtOH}]{\text{(i) LiBH}_4}$ $\xrightarrow{\text{(ii) H}_3\mathrm{O}^+}$ major…
Aldehydes, Ketones and Carboxylic Acids Carboxylic Acids and Derivatives Reduction of Acids
MediumDiagram based · figureIdea asked in 5 yearsAsked the year before
Q75
Given below are two statements: One is labelled as Assertion A and the other is labelled as Reason R. Assertion A: Butan $-1-$ ol has higher boiling point than ethoxyethane. Reason R: Extensive hydrogen bonding leads to stronger association of molecules. In…
Alcohols, Phenols and Ethers Alcohols and Phenols Hydrogen Bonding Trends in Alcohols
EasyAssertion–reasonIdea asked in 3 years
Q76
Match List I with List II: [benzenediazonium chloride, $\mathrm{C_6H_5N_2Cl}$ ] is reacted with reagents in List I to form products in List II. List I (Reagent) | List II (Product) A. [aniline, benzene ring with $\mathrm{NH_2}$ ] | I. [fluorobenzene, benzene…
Amines Diazonium Salts Balz-Schiemann Reaction
MediumMatch the list · figureIdea asked in 2 yearsAsked the year before
Q77
Sulphur (S) containing amino acids from the following are: (a) isoleucine (b) cysteine (c) lysine (d) methionine (e) glutamic acid
Biomolecules
EasySingle correct
Q78
Match List I with List II: List I | List II A. Saccharin | I. High potency sweetener B. Aspartame | II. First artificial sweetening agent C. Alitame | III. Stable at cooking temperature D. Sucralose | IV. Unstable at cooking temperature Choose the correct…
Chemistry in Everyday Life Artificial Sweetener Structural Properties and Stability
EasyMatch the listIdea asked in 3 yearsAsked the year before
Q79
Match List I with List II: List I (Reagents used) | List II (Compound with Functional group detected) A. Alkaline solution of copper sulphate and sodium citrate | I. [1-phenylethanol: benzene ring bearing $\mathrm{CH(OH)CH_3}$ ] B. Neutral $\mathrm{FeCl_3}$ …
Amines
MediumMatch the list · figure
Q80
In chromyl chloride, the number of d-electrons present on chromium is same as in (Given at no. of Ti : 22, V : 23, Cr : 24, Mn : 25, Fe : 26)
The d- and f-Block Elements d-Block Elements Electronic Configuration
EasySingle correctIdea asked in 4 yearsAsked the year before
Q81
[figure: three bulbs joined by pipes with stopcocks] Three bulbs are filled with $\mathrm{CH_4}$ , $\mathrm{CO_2}$ and Ne as shown in the picture. The bulbs are connected through pipes of zero volume. When the stopcocks are opened and the temperature is kept…
States of Matter Ideal Gas Laws Connected Gas Bulbs Equilibrium
MediumNumerical value · figureIdea asked in 4 years
Q82
The number of following statement/s which is/are incorrect is ________ (A) Line emission spectra are used to study the electronic structure (B) The emission spectra of atoms in the gas phase show a continuous spread of wavelength from red to violet (C) An…
Structure of Atom
EasyNumerical value
Q83
The number of following factors which affect the percent covalent character of the ionic bond is ________ (A) Polarising power of cation (B) Extent of distortion of anion (C) Polarisability of the anion (D) Polarising power of anion
Chemical Bonding and Molecular Structure
MediumNumerical value
Q84
When a 60 W electric heater is immersed in a gas for 100s in a constant volume container with adiabatic walls, the temperature of the gas rises by $5^\circ\mathrm{C}$ . The heat capacity of the given gas is ________ $\mathrm{J\,K^{-1}}$ (Nearest integer)
Thermodynamics (Chemistry)Enthalpy and Hess's Law Enthalpy and Heat Capacity
EasyNumerical valueNew idea
Q85
The vapour pressure vs. temperature curve for a solution solvent system is shown below. [graph] The boiling point of the solvent is ______ $^\circ\mathrm{C}$ .
Solutions Colligative Properties Phase Diagram Interpretation: Boiling Point
EasyNumerical value · figureNew idea
Q86
The titration curve of weak acid vs. strong base with phenolphthalein as indictor) is shown below. The $\mathrm{K_{phenolphthalein}} = 4\times10^{-10}$ . Given: $\log 2 = 0.3$ [graph] The number of following statement/s which is/are correct about…
Equilibrium pH and Buffers pH of Acid-Base Indicators
MediumNumerical value · figureNew idea
Q87
The number of given statement/s which is/are correct is ________ (A) The stronger the temperature dependence of the rate constant, the higher is the activation energy. (B) If a reaction has zero activation energy, its rate is independent of temperature. (C)…
Chemical Kinetics Arrhenius Equation Activation Energy Calculation
MediumNumerical valueIdea asked in 3 yearsAsked the year before
Q88
$\mathrm{XeF_4}$ reacts with $\mathrm{SbF_5}$ to form $\mathrm{[XeF_m]^{n+}[SbF_y]^{z-}}$ . $\mathrm{m + n + y + z}$ = ________
The p-Block Elements Group 17 & 18 Xenon Fluorides as Fluoride Donors
MediumNumerical valueIdea asked in 2 years
Q89
Molar mass of the hydrocarbon (X) which on ozonolysis consumes one mole of $\mathrm{O_3}$ per mole of (X) and gives one mole each of ethanal and propanone is ________ $\mathrm{g\,mol^{-1}}$ (Molar mass of C : 12 $\mathrm{g\,mol^{-1}}$ , H : 1…
Hydrocarbons Alkenes and Alkynes Reductive Ozonolysis Products
MediumNumerical valueIdea asked in 7 yearsAsked the year before
Q90
0.5 g of an organic compound (X) with 60% carbon will produce ______ $\times10^{-1}$ g of $\mathrm{CO_2}$ on complete combustion.
Some Basic Concepts of Chemistry Stoichiometry Empirical Formula from Combustion Data
EasyNumerical valueIdea asked in 4 yearsAsked the year before

Maths30 questions

3 easy21 medium6 hard

Q1
Let the number of elements in sets $A$ and $B$ be five and two respectively. Then the number of subsets of $A \times B$ each having at least 3 and at most 6 elements is :
Permutations and Combinations Selection of One or More Objects
EasySingle correctIdea asked in 3 years
Q2
If for $z=\alpha+i\beta$ , $|z+2|=z+4(1+i)$ , then $\alpha+\beta$ and $\alpha\beta$ are the roots of the equation
Complex Numbers and Quadratic Equations Complex Numbers - Polar Form Modulus and Argument
MediumSingle correctIdea asked in 5 yearsAsked the year before
Q3
Let $\alpha,\beta,\gamma$ be the three roots of the equation $x^3+bx+c=0$ . If $\beta\gamma=1=-\alpha$ , then $b^3+2c^3-3\alpha^3-6\beta^3-8\gamma^3$ is equal to
Complex Numbers and Quadratic Equations Quadratic Roots Root-Coefficient Relation
MediumSingle correctIdea asked in 6 yearsAsked the year before
Q4
Let $A=\begin{bmatrix}2&1&0\\1&2&-1\\0&-1&2\end{bmatrix}$ . If $|adj(adj(adj\,2A))|=(16)^n$ , then $n$ is equal to
Determinants Matrix Inverse Inverse Relation to Double Adjoint
MediumSingle correctIdea asked in 6 yearsAsked the year before
Q5
Let $P=\begin{bmatrix}\frac{\sqrt{3}}{2}&\frac{1}{2}\\-\frac{1}{2}&\frac{\sqrt{3}}{2}\end{bmatrix}$ , $A=\begin{bmatrix}1&1\\0&1\end{bmatrix}$ and $Q=PAP^T$ . If $P^TQ^{2007}P=\begin{bmatrix}a&b\\c&d\end{bmatrix}$ , then $2a+b-3c-4d$ equal to
Matrices Matrix Multiplication Powers of Square Matrices
HardSingle correctIdea asked in 6 yearsAsked the year before
Q6
The number of arrangements of the letters of the word "INDEPENDENCE" in which all the vowels always occur together is
Permutations and Combinations The Block/Tie Method
MediumSingle correctIdea asked in 4 years
Q7
The number of ways, in which 5 girls and 7 boys can be seated at a round table so that no two girls sit together, is
Permutations and Combinations Circular Permutations
MediumSingle correctNew idea
Q8
If the coefficients of three consecutive terms in the expansion of $(1+x)^n$ are in the ratio $1:5:20$ , then the coefficient of the fourth term is
Binomial Theorem Binomial Coefficients Ratio of Consecutive Coefficients
MediumSingle correctIdea asked in 5 yearsAsked the year before
Q9
Let $S_K=\frac{1+2+\ldots+K}{K}$ and $\sum_{j=1}^{n}S_j^2=\frac{n}{A}(Bn^2+Cn+D)$ , where $A,B,C,D\in\mathbb{N}$ and $A$ has least value. Then
Sequences and Series
MediumSingle correct
Q10
$\lim_{x\to0}\left(\left(\frac{1-\cos^2(3x)}{\cos^3(4x)}\right)\left(\frac{\sin^3(4x)}{\left(\log_e(2x+1)\right)^5}\right)\right)$ is equal to ________
Limits and Derivatives L'Hopital's Rule Simplifying with Standard Limits Before L'Hopital
MediumSingle correctIdea asked in 6 yearsAsked the year before
Q11
Let $I(x)=\int\frac{(x+1)}{x(1+xe^x)^2}dx$ , $x>0$ . If $\lim_{x\to\infty}I(x)=0$ , then $I(1)$ is equal to
Integrals Integration by Substitution Implicit Derivative Recognition
HardSingle correctIdea asked in 2 years
Q12
The area of the region $\{(x,y):x^2\le y\le 8-x^2,\ y\le 7\}$ is
Application of Integrals Area Under Curves Regions with Multiple Bounding Curves
MediumSingle correctIdea asked in 3 years
Q13
Let $C(\alpha,\beta)$ be the circumcenter of the triangle formed by the lines $4x+3y=69$ , $4y-3x=17$ , and $x+7y=61$ . Then $(\alpha-\beta)^2+\alpha+\beta$ is equal to
Straight Lines
MediumSingle correct
Q14
Let $R$ be the focus of the parabola $y^2=20x$ and the line $y=mx+c$ intersect the parabola at two points $P$ and $Q$ . Let the point $G(10,10)$ be the centroid of the triangle $PQR$ . If $c-m=6$ , then $(PQ)^2$ is
Conic Sections Parabola Focal Chords Equation of focal chord
HardSingle correctIdea asked in 2 yearsAsked the year before
Q15
If the equation of the plane containing the line $x+2y+3z-4=0=2x+y-z+5$ and perpendicular to the plane $\vec{r}=(\hat{i}-\hat{j})+\lambda(\hat{i}+\hat{j}+\hat{k})+\mu(\hat{i}-2\hat{j}+3\hat{k})$ is $ax+by+cz=4$ , then $(a-b+c)$ is equal to
Three Dimensional Geometry Planes in 3D Family of Planes Perpendicular to a Third Plane
MediumSingle correctIdea asked in 3 yearsAsked the year before
Q16
The shortest distance between the lines $\frac{x-4}{4}=\frac{y+2}{5}=\frac{z+3}{3}$ and $\frac{x-1}{3}=\frac{y-3}{4}=\frac{z-4}{2}$ is
Three Dimensional Geometry Lines in 3D Shortest Distance Between Skew Lines
MediumSingle correctNew idea
Q17
If the points with position vectors $\alpha\hat{i}+10\hat{j}+13\hat{k}$ , $6\hat{i}+11\hat{j}+11\hat{k}$ , $\frac{9}{2}\hat{i}+\beta\hat{j}-8\hat{k}$ are collinear, then $(19\alpha-6\beta)^2$ is equal to
Vector Algebra Linear Independence Collinearity of Three Position Vectors
MediumSingle correctIdea asked in 2 yearsAsked the year before
Q18
In a bolt factory, machines $A$ , $B$ and $C$ manufacture respectively 20%, 30% and 50% of the total bolts. Of their output 3, 4 and 2 percent are respectively defective bolts. A bolt is drawn at random from the product. If the bolt drawn is found the…
Probability Bayes' Theorem Bayes' Theorem Application
EasySingle correctIdea asked in 6 yearsAsked the year before
Q19
Let $f(x)=\frac{\sin x+\cos x-\sqrt{2}}{\sin x-\cos x}$ , $x\in[0,\pi]-\left\{\frac{\pi}{4}\right\}$ . Then $f\left(\frac{7\pi}{12}\right)f''\left(\frac{7\pi}{12}\right)$ is equal to
Continuity and Differentiability Chain Rule Second Derivative of Composite Functions
HardSingle correctNew ideaIdea asked in 2 years
Q20
Negation of $(p\Rightarrow q)\Rightarrow(q\Rightarrow p)$ is
Mathematical Reasoning
MediumSingle correct
Q21
Let $A=\{0,3,4,6,7,8,9,10\}$ and $R$ be the relation defined on $A$ such that $R=\{(x,y)\in A\times A: x-y$ is odd positive integer or $x-y=2\}$ . The minimum number of elements that must be added to the relation $R$ , so that it is a symmetric relation, is…
Relations and Functions Sets and Relations Counting Relations
MediumNumerical valueIdea asked in 6 yearsAsked the year before
Q22
Let $[t]$ denote the greatest integer $\le t$ . If the constant term in the expansion of $\left(3x^2-\frac{1}{2x^5}\right)^7$ is $\alpha$ , then $[\alpha]$ is equal to ____________.
Binomial Theorem
MediumNumerical value
Q23
If $a_\alpha$ is the greatest term in the sequence $a_n=\frac{n^3}{n^4+147}$ , $n=1,2,3,\ldots,$ then $\alpha$ is equal to ____________.
Application of Derivatives Maxima and Minima Extrema of Discrete Sequences via Calculus
MediumNumerical valueNew idea
Q24
Let $[t]$ denote the greatest integer $\le t$ . Then $\frac{2}{\pi}\int_{\pi/6}^{5\pi/6}(8[\operatorname{cosec}x]-5[\cot x])\,dx$ is equal to ____________.
Integrals Definite Integral Properties Greatest Integer Function Integrals
HardNumerical valueIdea asked in 7 yearsAsked the year before
Q25
If the solution curve of the differential equation $(y-2\log_e x)dx+(x\log_e x^2)dy=0$ , $x>1$ passes through the points $\left(e,\frac{4}{3}\right)$ and $(e^4,\alpha)$ , then $\alpha$ is equal to __________.
Differential Equations Linear Differential Equations Integrating Factor Calculation
MediumNumerical valueIdea asked in 7 yearsAsked the year before
Q26
The largest natural number $n$ such that $3^n$ divides $66!$ is ______________.
Permutations and Combinations Exponent of Prime in n! (Legendre's Formula)
EasyNumerical valueNew ideaIdea asked in 2 years
Q27
Consider a circle $C_1:x^2+y^2-4x-2y=\alpha-5$ . Let its mirror image in the line $y=2x+1$ be another circle $C_2:5x^2+5y^2-10fx-10gy+36=0$ . Let $r$ be the radius of $C_2$ . Then $\alpha+r$ is equal to __________.
Conic Sections General Conic Section Condition for a Circle
MediumNumerical valueIdea asked in 3 yearsAsked the year before
Q28
Let $\lambda_1,\lambda_2$ be the values of $\lambda$ for which the points $\left(\frac{5}{2},1,\lambda\right)$ and $(-2,0,1)$ are at equal distance from the plane $2x+3y-6z+7=0$ . If $\lambda_1>\lambda_2$ , then the distance of the point…
Three Dimensional Geometry Point-Plane Distance Distance from Point
MediumNumerical valueIdea asked in 2 years
Q29
Let $\vec{a}=6\hat{i}+9\hat{j}+12\hat{k}$ , $\vec{b}=\alpha\hat{i}+11\hat{j}-2\hat{k}$ and $\vec{c}$ be vectors such that $\vec{a}\times\vec{c}=\vec{a}\times\vec{b}$ . If $\vec{a}\cdot\vec{c}=-12$ , $\vec{c}\cdot(\hat{i}-2\hat{j}+\hat{k})=5$ , then…
Vector Algebra Cross Product Solving Vector Equations with Cross Products
HardNumerical valueIdea asked in 6 yearsAsked the year before
Q30
Let the mean and variance of 8 numbers $x,y,10,12,6,12,4,8$ be 9 and 9.25 respectively. If $x>y$ , then $3x-2y$ is equal to __________.
Statistics Variance and SD Finding Unknown Observations from Mean and Variance
MediumNumerical valueIdea asked in 7 yearsAsked the year before

Dropped from the final key and left out of every count: Q71.

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.