JEE Main 2023 · April session

JEE Main 6 April 2023, Shift 1

90 questions: 30 numerical-answer and 60 multiple choice. 10 were rated hard and 28 easy. Its hard-question count was the 10th highest of the 12 2023 shifts Lumi analysed. The biggest chapter was Amines, with 3 questions.

Questions: 90; none dropped
Numerical answer: 30; 33% of the shift
Hard: 10; 11% of the shift
Chapters asked: 66; 11 questions with a figure

Key findings · 6 Apr 2023, Shift 1

1
10 hard questions (11%): the 10th 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 5 of 30 rated hard; Physics had the most easy questions (18 of 30).
3
22 questions tested a new idea, one no earlier year had asked: 30% of the 73 questions tagged to an idea.
4
42 came back from 2022: the same idea had been asked in the 2022 shifts Lumi analysed.
5
25 shared an idea with another 2023 shift (34% of tagged questions). Shifts in the same year can test the same idea in a new setting.
6
Biggest chapters: Atoms in Physics (2), Amines in Chemistry (3) and Binomial Theorem in Maths (3).

The shift at a glance

Subjects, difficulty and chapters

Physics

30 Q · 10 numerical

Easy 18Medium 11Hard 1

Top chapters

Chemistry

30 Q · 10 numerical

Easy 8Medium 18Hard 4

Top chapters

Amines3 Q
Biomolecules2 Q
Coordination Compounds2 Q
Redox Reactions2 Q
Thermodynamics (Chemistry)2 Q
Aldehydes, Ketones and Carboxylic Acids1 Q

Maths

30 Q · 10 numerical

Easy 2Medium 23Hard 5

Top chapters

Question format

Single correct41 Q
Numerical value30 Q
Diagram based8 Q
Match the list5 Q
Assertion–reason4 Q
Other2 Q

What each question asks you to do

Calculation29 Q
Multi-step24 Q
Conceptual13 Q
Application11 Q
Recall8 Q
Data or graph3 Q
Diagram reading2 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
Open30 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 90 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

90 of 90 questions

Physics30 questions

18 easy11 medium1 hard

Q31
By what percentage will the transmission range of a TV tower be affected when the height of the tower is increased by 21% ?
Communication Systems
EasySingle correct
Q32
Name the logic gate equivalent to the diagram attached
Semiconductor Electronics: Materials, Devices and Simple Circuits Logic Gates Electrical Switch Analogies for Logic Gates
EasyDiagram based · figureNew idea
Q33
A small ball of mass M and density $\rho$ is dropped in a viscous liquid of density $\rho_0$ . After some time, the ball falls with a constant velocity. What is the viscous force on the ball ?
Mechanical Properties of Fluids Viscosity Terminal Velocity Calculation
EasySingle correctIdea asked in 3 yearsAsked the year before
Q34
The energy levels of an hydrogen atom are shown below. The transition corresponding to emission of shortest wavelength is
Atoms Bohr's Atomic Model Energy Level Calculations
EasyDiagram based · figureIdea asked in 6 yearsAsked the year before
Q35
The kinetic energy of an electron, $\alpha$ -particle and a proton are given as 4 K, 2 K and K respectively. The de-Broglie wavelength associated with electron ( $\lambda_e$ ), $\alpha$ -particle ( $\lambda_\alpha$ ) and the proton ( $\lambda_p$ ) are as follows:
Dual Nature of Radiation and Matter Matter Waves Wavelength from Kinetic Energy
MediumSingle correctIdea asked in 4 years
Q36
A monochromatic light wave with wavelength $\lambda_1$ and frequency $\nu_1$ in air enters another medium. If the angle of incidence and angle of refraction at the interface are $45^\circ$ and $30^\circ$ respectively, then the wavelength $\lambda_2$ and…
Ray Optics and Optical Instruments Refraction and Snell's Law Relative Refractive Index and Wave Parameters
EasySingle correctNew idea
Q37
For the plane electromagnetic wave given by $E=E_0\sin(\omega t-kx)$ and $B=B_0\sin(\omega t-kx)$ , the ratio of average electric energy density to average magnetic energy density is
Electromagnetic Waves
EasySingle correct
Q38
The induced emf can be produced in a coil by A. moving the coil with uniform speed inside uniform magnetic field B. moving the coil with non uniform speed inside uniform magnetic field C. rotating the coil inside the uniform magnetic field D. changing the…
Electromagnetic Induction Faraday's Law Magnetic Flux Calculation
EasyOtherIdea asked in 2 yearsAsked the year before
Q39
A particle is moving with constant speed in a circular path. When the particle turns by an angle $90^\circ$ , the ratio of instantaneous velocity to its average velocity is $\pi : x\sqrt2$ . The value of $x$ will be -
Motion in a Plane Kinematics in 2D Uniform Circular Motion
MediumSingle correctIdea asked in 2 yearsAsked the year before
Q40
A long straight wire of circular cross-section (radius a) is carrying steady current I. The current I is uniformly distributed across this cross-section. The magnetic field is
Moving Charges and Magnetism Ampere's Circuital Law B vs r Graph for Solid Cylinder
EasySingle correctNew idea
Q41
The resistivity ( $\rho$ ) of semiconductor varies with temperature. Which of the following curve represents the correct behaviour
Semiconductor Electronics: Materials, Devices and Simple Circuits Ohm's Law and Resistance Temperature Dependence in Semiconductors
EasyDiagram based · figureNew ideaIdea asked in 2 years
Q42
For a uniformly charged thin spherical shell, the electric potential (V) radially away from the centre (O) of shell can be graphically represented as -
Electrostatic Potential and Capacitance Electric Potential V-r Graphs for Spherical Geometries
EasyDiagram based · figureNew idea
Q43
A mass m is attached to two strings as shown in figure. The spring constants of two springs are $K_1$ and $K_2$ . For the frictionless surface, the time period of oscillation of mass m is
Oscillations SHM Equations Springs in Parallel
EasyDiagram based · figureIdea asked in 3 yearsAsked the year before
Q44
A source supplies heat to a system at the rate of 1000 W. If the system performs work at a rate of 200 W. The rate at which internal energy of the system increases is
Thermodynamics First Law of Thermodynamics First Law Energy Balance
EasySingle correctIdea asked in 3 years
Q45
The number of air molecules per $\mathrm{cm^3}$ increased from $3\times10^{19}$ to $12\times10^{19}$ . The ratio of collision frequency of air molecules before and after the increase in number respectively is :
Kinetic Theory Kinetic Theory of Gases Collision Frequency
EasySingle correctNew ideaIdea asked in 2 years
Q46
Two resistances are given as $R_1=(10\pm0.5)\,\Omega$ and $R_2=(15\pm0.5)\,\Omega$ . The percentage error in the measurement of equivalent resistance when they are connected in parallel is -
Units and Measurements Ohm's Law and Resistance Error Analysis in Ohm's Law
MediumSingle correctIdea asked in 5 yearsAsked the year before
Q47
Given below are two statements : one is labelled as Assertion A and the other is labelled as Reason R. Assertion A : When a body is projected at an angle $45^\circ$ , it's range is maximum. Reason R : For maximum range, the value of $\sin 2\theta$ should be…
Motion in a Plane Projectile Motion Ground-to-Ground Parameters
EasyAssertion–reasonIdea asked in 6 yearsAsked the year before
Q48
A small block of mass 100 g is tied to a spring of spring constant 7.5 N/m and length 20 cm. The other end of spring is fixed at a particular point A. If the block moves in a circular path on a smooth horizontal surface with constant angular velocity 5 rad/s…
Laws of Motion Circular Motion Spring-Mass in Circular Motion
MediumSingle correctIdea asked in 3 yearsAsked the year before
Q49
Given below are two statements : one is labelled as Assertion A and the other is labelled as Reason R. Assertion A : Earth has atmosphere whereas moon doesn't have any atmosphere. Reason R : The escape velocity on moon is very small as compared to that on…
Gravitation Kinetic Theory of Gases Planetary Atmosphere Retention
EasyAssertion–reasonNew idea
Q50
A planet has double the mass of the earth. Its average density is equal to that of the earth. An object weighing W on earth will weigh on that planet:
Gravitation
MediumSingle correct
Q51
A person driving car at a constant speed of 15 m/s is approaching a vertical wall. The person notices a change of 40 Hz in the frequency of his car's horn upon reflection from the wall. The frequency of horn is ____________ Hz. (Given: Speed of sound : 330…
Waves Doppler Effect Moving Reflector Echo
MediumNumerical valueIdea asked in 3 yearsAsked the year before
Q52
A pole is vertically submerged in swimming pool, such that it gives a length of shadow 2.15 m within water when sunlight is incident at an angle of $30^\circ$ with the surface of water. If swimming pool is filled to a height of 1.5 m, then the height of the…
Ray Optics and Optical Instruments Refraction and Snell's Law Snell's Law Application
HardNumerical valueIdea asked in 4 yearsAsked the year before
Q53
The radius of fifth orbit of the $\mathrm{Li^{++}}$ is ____________ $\times10^{-12}$ m. Take: radius of hydrogen atom = 0.51 Å
Atoms Bohr's Atomic Model Bohr Orbit Radius
EasyNumerical valueIdea asked in 4 yearsAsked the year before
Q54
An ideal transformer with purely resistive load operates at 12 kV on the primary side. It supplies electrical energy to a number of nearby houses at 120 V. The average rate of energy consumption in the houses served by the transformer is 60 kW. The value of…
Alternating Current
MediumNumerical value
Q55
The length of a metallic wire is increased by 20% and its area of cross section is reduced by 4%. The percentage change in resistance of the metallic wire is ____________
Current Electricity Ohm's Law and Resistance Resistivity and Geometry
EasyNumerical valueIdea asked in 4 yearsAsked the year before
Q56
A parallel plate capacitor with plate area A and plate separation d is filled with a dielectric material of dielectric constant $K=4$ . The thickness of the dielectric material is $x$ , where $x<d$ . Let $C_1$ and $C_2$ be the capacitance of the system for…
Electrostatic Potential and Capacitance Dielectrics Partial and Multiple Slabs
MediumNumerical value · figureIdea asked in 6 yearsAsked the year before
Q57
Two identical circular wires of radius 20 cm and carrying current $\sqrt2$ A are placed in perpendicular planes as shown in figure. The net magnetic field at the centre of the circular wires is ____________ $\times10^{-8}$ T. (Take $\pi=3.14$ )
Moving Charges and Magnetism Biot-Savart Law Superposition of magnetic fields
MediumNumerical value · figureIdea asked in 4 yearsAsked the year before
Q58
A steel rod has a radius of 20 mm and a length of 2.0 m. A force of 62.8 kN stretches it along its length. Young's modulus of steel is $2.0\times10^{11}\ \mathrm{N/m^2}$ . The longitudinal strain produced in the wire is ____________ $\times10^{-5}$
Mechanical Properties of Solids Elasticity Hooke's Law
EasyNumerical valueIdea asked in 3 years
Q59
A particle of mass 10 g moves in a straight line with retardation $2x$ , where $x$ is the displacement in SI units. Its loss of kinetic energy for above displacement is $\left(\frac{10}{x}\right)^{-n}$ J. The value of n will be ____________
Work, Energy and Power Work and Kinetic Energy Work-Energy Theorem
MediumNumerical valueIdea asked in 6 yearsAsked the year before
Q60
Two identical solid spheres each of mass 2 kg and radii 10 cm are fixed at the ends of a light rod. The separation between the centres of the spheres is 40 cm. The moment of inertia of the system about an axis perpendicular to the rod passing through its…
System of Particles and Rotational Motion Moment of Inertia MOI of Composite Bodies
MediumNumerical valueIdea asked in 5 yearsAsked the year before

Chemistry30 questions

8 easy18 medium4 hard

Q61
A compound is formed by two elements X and Y. The element Y forms cubic close packed arrangement and those of element X occupy one third of the tetrahedral voids. What is the formula of the compound ?
The Solid State Crystal Lattices Deriving Formula from Void Occupancy
MediumSingle correctNew idea
Q62
For a concentrated solution of a weak electrolyte ( $K_{eq}$ = equilibrium constant) $\mathrm{A_2B_3}$ of concentration 'c', the degree of dissociation ' $\alpha$ ' is
Equilibrium Chemical Equilibrium Degree of Dissociation
MediumSingle correctIdea asked in 4 yearsAsked the year before
Q63
Match List I with List II LIST I - Enzymatic reaction | LIST II - Enzyme A. Sucrose $\rightarrow$ Glucose and Fructose | I. Zymase B. Glucose $\rightarrow$ ethyl alcohol and $\mathrm{CO_2}$ | II. Pepsin C. Starch $\rightarrow$ Maltose | III. Invertase D.…
Biomolecules
EasyMatch the list
Q64
The difference between electron gain enthalpies will be maximum between:
Classification of Elements and Periodicity in Properties Periodic Trends Positive Electron Gain Enthalpy
MediumSingle correctIdea asked in 2 years
Q65
Which of the following options are correct for the reaction $2\,[\mathrm{Au(CN)_2}]^{-}(aq)+\mathrm{Zn}(s)\rightarrow 2\,\mathrm{Au}(s)+[\mathrm{Zn(CN)_4}]^{2-}(aq)$ A. Redox reaction B. Displacement reaction C. Decomposition reaction D. Combination reaction…
Redox Reactions Identifying Redox Agents
MediumOtherIdea asked in 3 years
Q66
Given below are two statements, one is labelled as Assertion A and the other is labelled as Reason R. Assertion A : Loss of electron from hydrogen atom results in nucleus of $\sim1.5\times10^{-3}$ pm size. Reason R : Proton ( $\mathrm{H^+}$ ) always exists in…
Hydrogen
MediumAssertion–reason
Q67
The standard electrode potential of $\mathrm{M^+/M}$ in aqueous solution does not depend on
Electrochemistry s-Block Elements Aqueous Reducing Power and E° Anomaly
MediumSingle correctNew idea
Q68
The setting time of Cement is increased by adding
The s-Block Elements s-Block Elements Role of Gypsum in Portland Cement
EasySingle correctNew idea
Q69
Match List I with List II LIST I (Oxide) | LIST II (Type of bond) A. $\mathrm{N_2O_4}$ | I. 1 N = O bond B. $\mathrm{NO_2}$ | II. 1 N – O – N bond C. $\mathrm{N_2O_5}$ | III. 1 N – N bond D. $\mathrm{N_2O}$ | IV. 1 N=N / N≡N bond Choose the correct answer…
The p-Block Elements Group 15 & 16 Nitrogen Oxides Structure
MediumMatch the listIdea asked in 3 yearsAsked the year before
Q70
Strong reducing and oxidizing agents among the following, respectively, are
The d- and f-Block Elements f-Block Elements Europium(II) and Ytterbium(II) as Reducing Agents
MediumSingle correctNew idea
Q71
The possibility of photochemical smog formation is more at
Environmental Chemistry Classical vs Photochemical Smog
EasySingle correctIdea asked in 3 yearsAsked the year before
Q72
Given below are two statements, one is labelled as Assertion A and the other is labelled as Reason R. Assertion A: The spin only magnetic moment value for $[\mathrm{Fe(CN)_6}]^{3-}$ is 1.74 BM, whereas for $[\mathrm{Fe(H_2O)_6}]^{3+}$ is 5.92 BM. Reason R :…
Coordination Compounds Crystal Field Theory Spin-only Magnetic Moment
MediumAssertion–reasonIdea asked in 7 yearsAsked the year before
Q73
Compound P (M.F. $\mathrm{C_{14}H_{13}ON}$ ; M.F. = MOLECULAR FORMULA) on treatment with HCl, $\Delta$ gives a Filter step: Residue Q and Filtrate; the Filtrate with NaOH gives Oily Liquid R. Compound P is neutral, Q gives effervescence with…
Amines Hinsberg Test
HardDiagram based · figureIdea asked in 5 yearsAsked the year before
Q74
Match List I with List II LIST I (Element detected) | LIST II (Reagent used / Product formed) A. Nitrogen | I. $\mathrm{Na_2[Fe(CN)_5NO]}$ B. Sulphur | II. $\mathrm{AgNO_3}$ C. Phosphorous | III. $\mathrm{Fe_4[Fe(CN)_6]_3}$ D. Halogen | IV.…
Organic Chemistry: Some Basic Principles and Techniques
MediumMatch the list
Q75
For the reaction $\mathrm{RCH_2Br + I^-}\xrightarrow{\text{Acetone}}\underset{\text{major}}{\mathrm{RCH_2I}}+\mathrm{Br^-}$ The correct statement is
Haloalkanes and Haloarenes SN1 and SN2 Mechanisms Finkelstein Reaction
HardSingle correctIdea asked in 4 yearsAsked the year before
Q76
The major product formed in the following reaction is [Reactant: benzene ring bearing ortho $\mathrm{CH_2CONH_2}$ and $\mathrm{COOCH_3}$ groups] $\xrightarrow[\Delta]{\mathrm{Br_2/NaOH}}$
Amines Hofmann Bromamide Degradation
HardDiagram based · figureIdea asked in 5 yearsAsked the year before
Q77
Match List I with List II LIST I (Name of reaction) | LIST II (Reagent used) A. Hell-Volhard-Zelinsky reaction | I. $\mathrm{NaOH + I_2}$ B. Iodoform reaction | II. (i) $\mathrm{CrO_2Cl_2}$ , $\mathrm{CS_2}$ (ii) $\mathrm{H_2O}$ C. Etard reaction | III. (i)…
Aldehydes, Ketones and Carboxylic Acids Carboxylic Acids and Derivatives Hell-Volhard-Zelinsky Reaction
MediumMatch the listIdea asked in 4 yearsAsked the year before
Q78
The major products A and B from the following reactions are: [Reactant: acetanilide, $\mathrm{C_6H_5NHCOCH_3}$ ; with $\mathrm{Br_2/AcOH}$ giving A and $\mathrm{LiAlH_4}$ giving B]
Amines Protection of Aniline for Mono-Substitution
MediumDiagram based · figureIdea asked in 3 years
Q79
Polymer used in orlon is:
Polymers Polyacrylonitrile (PAN / Orlon)
EasySingle correctNew idea
Q80
Match List I with List II LIST I (Vitamin) | LIST II (Deficiency disease) A. Vitamin A | I. Beri-Beri B. Thiamine | II. Cheilosis C. Ascorbic acid | III. Xeropthalmia D. Riboflavin | IV. Scurvy Choose the correct answer from the options given below:
Biomolecules
EasyMatch the list
Q81
If 5 moles of $\mathrm{BaCl_2}$ is mixed with 2 moles of $\mathrm{Na_3PO_4}$ , the maximum number of moles of $\mathrm{Ba_3(PO_4)_2}$ formed is ____________ (Nearest integer)
Some Basic Concepts of Chemistry Mole Concept Limiting Reagent and Reaction Yield
EasyNumerical valueIdea asked in 4 yearsAsked the year before
Q82
The wavelength of an electron of kinetic energy $4.50\times10^{-29}$ J is ________ $\times10^{-5}$ m. (Nearest integer) Given : mass of electron is $9\times10^{-31}$ kg, $h = 6.6\times10^{-34}$ J s
Structure of Atom
EasyNumerical value
Q83
The number of species from the following which have square pyramidal structure is ________ $\mathrm{PF_5, BrF_4^-, IF_5, BrF_5, XeOF_4, ICl_4^-}$
Chemical Bonding and Molecular Structure VSEPR Theory Square Pyramidal Geometry (SN=6, 1 LP)
MediumNumerical valueNew ideaIdea asked in 3 years
Q84
The value of log K for the reaction $\mathrm{A \rightleftharpoons B}$ at 298 K is ________. (Nearest integer) Given: $\Delta H^\circ = -54.07$ kJ mol $^{-1}$ $\Delta S^\circ = 10$ J K $^{-1}$ mol $^{-1}$ (Take $2.303\times8.314\times298 = 5705$ )
Thermodynamics (Chemistry)Gibbs Free Energy Gibbs Energy and Equil
MediumNumerical valueIdea asked in 5 yearsAsked the year before
Q85
Mass of Urea ( $\mathrm{NH_2CONH_2}$ ) required to be dissolved in 1000 g of water in order to reduce the vapour pressure of water by 25% is ________ g. (Nearest integer) Given: Molar mass of N, C, O and H are 14, 12, 16 and 1 g mol $^{-1}$ respectively.
Solutions Colligative Properties Relative Lowering of VP
MediumNumerical valueIdea asked in 7 yearsAsked the year before
Q86
Consider the graph of Gibbs free energy G vs Extent of reaction. The number of statement/s from the following which are true with respect to points (a), (b) and (c) is ________ A. Reaction is spontaneous at (a) and (b) B. Reaction is at equilibrium at point…
Thermodynamics (Chemistry)Gibbs Free Energy Gibbs Energy vs Extent of Reaction Curve
MediumNumerical value · figureNew idea
Q87
For the adsorption of hydrogen on platinum, the activation energy is 30 kJ mol $^{-1}$ and for the adsorption of hydrogen on nickel, the activation energy is 41.4 kJ mol $^{-1}$ . The logarithm of the ratio of the rates of chemisorption on equal areas of the…
Surface Chemistry Arrhenius Equation Activation Energy Calculation
MediumNumerical valueIdea asked in 3 yearsAsked the year before
Q88
Number of ambidentate ligands in a representative metal complex [ M(en)(SCN) $_4$ ] is ________. [en = ethylenediamine]
Coordination Compounds Coordination Isomerism Ambidentate Ligands in Linkage Isomerism
EasyNumerical valueIdea asked in 3 yearsAsked the year before
Q89
Number of bromo derivatives obtained on treating ethane with excess of $\mathrm{Br_2}$ in diffused sunlight is ________
Hydrocarbons Alkanes and Free Radicals Halogenation Chain Mechanism
HardNumerical valueIdea asked in 3 yearsAsked the year before
Q90
In ammonium – phosphomolybdate, the oxidation state of Mo is + ________
Redox Reactions Oxidation Number Assignment
MediumNumerical valueIdea asked in 4 yearsAsked the year before

Maths30 questions

2 easy23 medium5 hard

Q1
Let $A=\{x\in\mathbb{R}:[x+3]+[x+4]\le 3\}$ , $B=\left\{x\in\mathbb{R}: 3^x\left(\sum_{r=1}^{\infty}\frac{3}{10^r}\right)^{x-3}<3^{-3x}\right\}$ , where $[t]$ denotes greatest integer function. Then,
Relations and Functions Sets and Relations Roster and Set-Builder Forms
MediumSingle correctNew idea
Q2
The sum of all the roots of the equation $|x^2-8x+15|-2x+7=0$ is:
Complex Numbers and Quadratic Equations Quadratic Roots Quadratics with Absolute Values
MediumSingle correctNew idea
Q3
Let $A=[a_{ij}]_{2\times2}$ , where $a_{ij}\neq0$ for all i, j and $A^2=I$ . Let $a$ be the sum of all diagonal elements of $A$ and $b=|A|$ . Then $3a^2+4b^2$ is equal to
Matrices Matrix Multiplication Involutory Matrix Properties
MediumSingle correctNew idea
Q4
If the system of equations $x+y+az=b$ $2x+5y+2z=6$ $x+2y+3z=3$ has infinitely many solutions, then $2a+3b$ is equal to
Determinants Cramer's Rule Condition for Infinite Solutions
MediumSingle correctIdea asked in 7 yearsAsked the year before
Q5
If ${}^{2n}C_3 : {}^{n}C_3 = 10:1$ , then the ratio $(n^2+3n):(n^2-3n+4)$ is
Binomial Theorem Binomial Coefficients Basic nCr Properties
EasySingle correctNew ideaIdea asked in 2 years
Q6
If the ratio of the fifth term from the beginning to the fifth term from the end in the expansion of $\left(\sqrt[4]{2}+\frac{1}{\sqrt[4]{3}}\right)^n$ is $\sqrt6:1$ , then the third term from the beginning is:
Binomial Theorem
MediumSingle correct
Q7
The sum of the first 20 terms of the series $5+11+19+29+41+\ldots$ is
Sequences and Series AP and GP Method of Differences for Telescoping Series
MediumSingle correctIdea asked in 6 yearsAsked the year before
Q8
Let $a_1,a_2,a_3,\ldots,a_n$ be $n$ positive consecutive terms of an arithmetic progression. If $d>0$ is its common difference, then…
Sequences and Series AP and GP Method of Differences (Vn Method)
MediumSingle correctIdea asked in 5 yearsAsked the year before
Q9
If $2x^y+3y^x=20$ , then $\frac{dy}{dx}$ at $(2,2)$ is equal to:
Continuity and Differentiability Chain Rule Logarithmic Differentiation of Composite Products
MediumSingle correctNew idea
Q10
Let $5f(x)+4f\left(\frac{1}{x}\right)=\frac{1}{x}+3,\ x>0$ . Then $18\int_1^2 f(x)\,dx$ is equal to:
Integrals
MediumSingle correct
Q11
Let $I(x)=\int\frac{x^2(x\sec^2x+\tan x)}{(x\tan x+1)^2}\,dx$ . If $I(0)=0$ , then $I\left(\frac{\pi}{4}\right)$ is equal to
Integrals Integration by Parts Generalized Reverse Product Rule
HardSingle correctNew idea
Q12
The straight lines $l_1$ and $l_2$ pass through the origin and trisect the line segment of the line $L: 9x+5y=45$ between the axes. If $m_1$ and $m_2$ are the slopes of the lines $l_1$ and $l_2$ , then the point of intersection of the line $y=(m_1+m_2)x$ with…
Straight Lines Straight Line Distance Section Ratio by a Line
MediumSingle correctIdea asked in 3 yearsAsked the year before
Q13
If the equation of the plane passing through the line of intersection of the planes $2x-y+z=3$ , $4x-3y+5z+9=0$ and parallel to the line $\frac{x+1}{-2}=\frac{y+3}{4}=\frac{z-2}{5}$ is $ax+by+cz+6=0$ , then $a+b+c$ is equal to
Three Dimensional Geometry Planes in 3D Family of Planes
MediumSingle correctIdea asked in 4 yearsAsked the year before
Q14
One vertex of a rectangular parallelopiped is at the origin O and the lengths of its edges along $x$ , $y$ and $z$ axes are 3, 4 and 5 units respectively. Let P be the vertex $(3,4,5)$ . Then the shortest distance between the diagonal OP and an edge parallel…
Three Dimensional Geometry Skew Lines Shortest Distance Between Cube Diagonals
MediumSingle correctNew idea
Q15
Let the position vectors of the points A, B, C and D be $5\hat i+5\hat j+2\lambda\hat k$ , $\hat i+2\hat j+3\hat k$ , $-2\hat i+\lambda\hat j+4\hat k$ and $-\hat i+5\hat j+6\hat k$ . Let the set $S=\{\lambda\in\mathbb{R}:$ the points A, B, C and D are…
Vector Algebra Triple Products Coplanarity of Four Points
MediumSingle correctIdea asked in 3 yearsAsked the year before
Q16
Let $\vec a=2\hat i+3\hat j+4\hat k$ , $\vec b=\hat i-2\hat j-2\hat k$ and $\vec c=-\hat i+4\hat j+3\hat k$ . If $\vec d$ is a vector perpendicular to both $\vec b$ and $\vec c$ , and $\vec a\cdot\vec d=18$ , then $|\vec a\times\vec d|^2$ is equal to
Vector Algebra Cross Product Lagrange's Identity
MediumSingle correctIdea asked in 5 yearsAsked the year before
Q17
The mean and variance of a set of 15 numbers are 12 and 14 respectively. The mean and variance of another set of 15 numbers are 14 and $\sigma^2$ respectively. If the variance of all the 30 numbers in the two sets is 13, then $\sigma^2$ is equal to
Statistics Variance and SD Combined Variance of Two Sets
MediumSingle correctIdea asked in 3 years
Q18
A pair of dice is thrown 5 times. For each throw, a total of 5 is considered a success. If the probability of at least 4 successes is $\frac{k}{3^{11}}$ , then $k$ is equal to
Probability
MediumSingle correct
Q19
From the top A of a vertical wall AB of height 30 m, the angles of depression of the top P and bottom Q of a vertical tower PQ are $15^\circ$ and $60^\circ$ respectively, B and Q are on the same horizontal level. If C is a point on AB such that CB = PQ, then…
Trigonometric Functions
MediumSingle correct
Q20
Statement $(P\Rightarrow Q)\wedge(R\Rightarrow Q)$ is logically equivalent to
Mathematical Reasoning
EasySingle correct
Q21
Let $A=\{1,2,3,4,\ldots,10\}$ and $B=\{0,1,2,3,4\}$ . The number of elements in the relation $R=\{(a,b)\in A\times A: 2(a-b)^2+3(a-b)\in B\}$ is ____________
Relations and Functions Sets and Relations Counting Relations
MediumNumerical valueIdea asked in 6 yearsAsked the year before
Q22
Let the point $(p,p+1)$ lie inside the region $E=\left\{(x,y):3-x\le y\le\sqrt{9-x^2},\,0\le x\le3\right\}$ . If the set of all values of $p$ is the interval $(a,b)$ , then $b^2+b-a^2$ is equal to ____________
Application of Integrals
MediumNumerical value
Q23
The number of ways of giving 20 distinct oranges to 3 children such that each child gets at least one orange is ____________
Permutations and Combinations Number of Surjective (Onto) Functions
MediumNumerical valueIdea asked in 4 yearsAsked the year before
Q24
The coefficient of $x^{18}$ in the expansion of $\left(x^4-\frac{1}{x^3}\right)^{15}$ is ____________
Binomial Theorem
MediumNumerical value
Q25
Let $a\in\mathbb{Z}$ and $[t]$ be the greatest integer $\le t$ . Then the number of points, where the function $f(x)=[a+13\sin x]$ , $x\in(0,\pi)$ is not differentiable, is ____________
Continuity and Differentiability Continuity involving GIF and Fractional Part
HardNumerical valueIdea asked in 6 yearsAsked the year before
Q26
If the area of the region $S=\{(x,y):2y-y^2\le x^2\le 2y,\ x\ge y\}$ is equal to $\frac{n+2}{n+1}-\frac{\pi}{n-1}$ , then the natural number $n$ is equal to ____________
Application of Integrals Area Under Curves Area of Inequalities
HardNumerical valueIdea asked in 8 yearsAsked the year before
Q27
A circle passing through the point $P(\alpha,\beta)$ in the first quadrant touches the two coordinate axes at the points A and B. The point P is above the line AB. The point Q on the line segment AB is the foot of perpendicular from P on AB. If PQ is equal…
Conic Sections
HardNumerical value
Q28
Let the tangent to the curve $x^2+2x-4y+9=0$ at the point $P(1,3)$ on it meet the $y$ -axis at A. Let the line passing through P and parallel to the line $x-3y=6$ meet the parabola $y^2=4x$ at B. If B lies on the line $2x-3y=8$ , then $(AB)^2$ is equal to…
Conic Sections Tangents and Normals Equation of Tangent
MediumNumerical valueIdea asked in 2 years
Q29
Let $y=y(x)$ be a solution of the differential equation $(x\cos x)\,dy+(xy\sin x+y\cos x-1)\,dx=0,\ 0<x<\frac{\pi}{2}$ . If $\frac{\pi}{3}y\left(\frac{\pi}{3}\right)=\sqrt3$ , then…
Differential Equations Linear Differential Equations LDEs Disguised in Exact Differential Form
HardNumerical valueNew ideaIdea asked in 2 years
Q30
Let the image of the point $(1,2,3)$ in the plane $2x-y+z=9$ be Q. If the coordinates of the point R are $(6,10,7)$ , then the square of the area of the triangle PQR is ____________
Three Dimensional Geometry Planes in 3D Image of a Point in a Plane
MediumNumerical valueIdea asked in 2 years

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.