JEE Main 2023 · April session

JEE Main 15 April 2023, Shift 1

90 questions: 30 numerical-answer and 60 multiple choice. 13 were rated hard and 39 easy. Its hard-question count was the 7th highest of the 12 2023 shifts Lumi analysed. The biggest chapter was Coordination Compounds, with 3 questions.

Questions: 90; none dropped
Numerical answer: 30; 33% of the shift
Hard: 13; 14% of the shift
Chapters asked: 67; 12 questions with a figure

Key findings · 15 Apr 2023, Shift 1

1
13 hard questions (14%): the 7th 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 30 rated hard; Physics had the most easy questions (19 of 30).
3
13 questions tested a new idea, one no earlier year had asked: 18% of the 74 questions tagged to an idea.
4
44 came back from 2022: the same idea had been asked in the 2022 shifts Lumi analysed.
5
34 shared an idea with another 2023 shift (46% of tagged questions). Shifts in the same year can test the same idea in a new setting.
6
Biggest chapters: Current Electricity in Physics (2), Coordination Compounds in Chemistry (3) and Permutations and Combinations in Maths (3).

The shift at a glance

Subjects, difficulty and chapters

Physics

30 Q · 10 numerical

Easy 19Medium 11Hard 0

Top chapters

Current Electricity2 Q
Electromagnetic Induction2 Q
Electrostatic Potential and Capacitance2 Q
Gravitation2 Q
Moving Charges and Magnetism2 Q
Atoms1 Q

Chemistry

30 Q · 10 numerical

Easy 17Medium 11Hard 2

Top chapters

Maths

30 Q · 10 numerical

Easy 3Medium 16Hard 11

Top chapters

Permutations and Combinations3 Q
Three Dimensional Geometry3 Q
Complex Numbers and Quadratic Equations2 Q
Conic Sections2 Q
Determinants2 Q
Integrals2 Q

Question format

Single correct43 Q
Numerical value30 Q
Other6 Q
Diagram based6 Q
Match the list2 Q
Two statements2 Q
Assertion–reason1 Q

What each question asks you to do

Calculation28 Q
Multi-step27 Q
Application13 Q
Conceptual13 Q
Recall8 Q
Data or graph1 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

19 easy11 medium0 hard

Q31
In a linear Simple Harmonic Motion (SHM) (A) Restoring force is directly proportional to the displacement. (B) The acceleration and displacement are opposite in direction. (C) The velocity is maximum at mean position. (D) The acceleration is minimum at…
Oscillations SHM Equations Extrema of Kinematic and Energy Variables
EasyOtherNew ideaIdea asked in 2 years
Q32
A flask contains Hydrogen and Argon in the ratio 2:1 by mass. The temperature of the mixture is 30°C. The ratio of average kinetic energy per molecule of the two gases (K argon/K hydrogen) is : (Given: Atomic Weight of Ar = 39.9)
Kinetic Theory Kinetic Theory of Gases Kinetic Energy and Temp
EasySingle correctIdea asked in 7 yearsAsked the year before
Q33
A thermodynamic system is taken through cyclic process. The total work done in the process is :
Thermodynamics Thermodynamic Processes Cyclic Process Work
EasyDiagram based · figureIdea asked in 4 years
Q34
A wire of length 'L' and radius 'r' is clamped rigidly at one end. When the other end of the wire is pulled by a force $f$ , its length increases by 'l'. Another wire of same material of length '2L' and radius '2r' is pulled by a force '2f'. Then the increase…
Mechanical Properties of Solids Elasticity Hooke's Law
MediumSingle correctIdea asked in 3 years
Q35
A body is released from a height equal to the radius (R) of the earth. The velocity of the body when it strikes the surface of the earth will be : (Given $g$ = acceleration due to gravity on the earth.)
Gravitation Gravitational Potential Exact Potential Energy Change at Large Heights
MediumSingle correctIdea asked in 3 yearsAsked the year before
Q36
Two identical particles each of mass 'm' go round a circle of radius $a$ under the action of their mutual gravitational attraction. The angular speed of each particle will be :
Gravitation Orbital Mechanics Binary Star Systems
MediumSingle correctIdea asked in 2 years
Q37
A vector in $x-y$ plane makes an angle of 30° with y-axis. The magnitude of y-component of vector is $2\sqrt{3}$ . The magnitude of $x$ -component of the vector will be :
Motion in a Plane Kinematics in 2D Vector Component Resolution
EasySingle correctIdea asked in 4 years
Q38
The position of a particle related to time is given by $x = (5t^2 - 4t + 5)\,m$ . The magnitude of velocity of the particle at $t = 2s$ will be :
Motion in a Straight Line Kinematics in 1D Calculus in Kinematics
EasySingle correctIdea asked in 5 yearsAsked the year before
Q39
The speed of a wave produced in water is given by $v = \lambda^a g^b \rho^c$ . Where $\lambda$ , $g$ and $\rho$ are wavelength of wave, acceleration due to gravity and density of water respectively. The values of $a$ , $b$ and $c$ respectively, are :
Units and Measurements
EasySingle correct
Q40
The position vector of a particle related to time $t$ is given by $\vec{r} = (10t\hat{i} + 15t^2\hat{j} + 7\hat{k})\,m$ The direction of net force experienced by the particle is :
Laws of Motion Newton's Laws of Motion Momentum Formulation (F = dp/dt)
EasySingle correctIdea asked in 4 yearsAsked the year before
Q41
The electric field due to a short electric dipole at a large distance ( $r$ ) from center of dipole on the equatorial plane varies with distance as :
Electrostatic Potential and Capacitance Electric Dipoles Electric Field on Equatorial Line
EasySingle correctIdea asked in 3 years
Q42
The height of transmitting antenna is 180 m and the height of the receiving antenna is 245 m. The maximum distance between them for satisfactory communication in line of sight will be : (given R = 6400 km)
Communication Systems
EasySingle correct
Q43
In the given circuit, the current (I) through the battery will be
Semiconductor Electronics: Materials, Devices and Simple Circuits Semiconductor Diodes Ideal vs. Practical Diode Circuit Analysis
MediumDiagram based · figureIdea asked in 7 yearsAsked the year before
Q44
The half-life of a radioactive nucleus is 5 years. The fraction of the original sample that would decay in 15 years is:
Nuclei Radioactive Decay Half-Life Calculations
EasySingle correctIdea asked in 4 yearsAsked the year before
Q45
The de Broglie wavelength of an electron having kinetic energy E is $\lambda$ . If the kinetic energy of electron becomes $\frac{E}{4}$ , then its de-Broglie wavelength will be :
Dual Nature of Radiation and Matter Matter Waves Wavelength from Kinetic Energy
EasySingle correctIdea asked in 4 years
Q46
A single slit of width $a$ is illuminated by a monochromatic light of wavelength 600 nm. The value of ' $a$ ' for which first minimum appears at $\theta = 30^{\circ}$ on the screen will be :
Wave Optics Single Slit Diffraction Single Slit Minima Condition
EasySingle correctNew ideaIdea asked in 4 years
Q47
Match List I with List II of Electromagnetic waves with corresponding wavelength range : List I | List II (A) Microwave | (I) 400 nm to 1 nm (B) Ultraviolet | (II) 1 nm to $10^{-3}$ nm (C) X-Ray | (III) 1 mm to 700 nm (D) Infra-red | (IV) 0.1 m to 1 mm…
Electromagnetic Waves
EasyMatch the list
Q48
A 12 V battery connected to a coil of resistance 6 $\Omega$ through a switch, drives a constant current in the circuit. The switch is opened in 1 ms. The emf induced across the coil is 20 V. The inductance of the coil is :
Electromagnetic Induction Inductance Self-Inductance and Back EMF
EasySingle correctIdea asked in 5 yearsAsked the year before
Q49
For designing a voltmeter of range 50 V and an ammeter of range 10 mA using a galvanometer which has a coil of resistance 54 $\Omega$ showing a full scale deflection for 1 mA as in figure. (A) for voltmeter $R \approx 50\,\mathrm{k}\Omega$ (B) for ammeter r…
Moving Charges and Magnetism
MediumOther · figure
Q50
Given below are two statements: Statement I : The equivalent resistance of resistors in a series combination is smaller than least resistance used in the combination. Statement II : The resistivity of the material is independent of temperature. In the light…
Current Electricity Ohm's Law and Resistance Temperature Coefficient of Resistance
EasyTwo statementsIdea asked in 3 yearsAsked the year before
Q51
The fundamental frequency of vibration of a string stretched between two rigid support is 50 Hz. The mass of the string is 18 g and its linear mass density is 20 g/m. The speed of the transverse waves so produced in the string is ______________ ms $^{-1}$
Waves Waves on Strings Wave Speed on String
EasyNumerical valueIdea asked in 5 yearsAsked the year before
Q52
There is an air bubble of radius 1.0 mm in a liquid of surface tension 0.075 Nm $^{-1}$ and density 1000 kg m $^{-3}$ at a depth of 10 cm below the free surface. The amount by which the pressure inside the bubble is greater than the atmospheric pressure is…
Mechanical Properties of Fluids Surface Tension Excess Pressure in Submerged Air Bubbles
MediumNumerical valueNew idea
Q53
A solid sphere and a solid cylinder of same mass and radius are rolling on a horizontal surface without slipping. The ratio of their radius of gyrations respectively ( $k_{sph}$ : $k_{cyl}$ ) is $2 : \sqrt{x}$ . The value of $x$ is ______________
System of Particles and Rotational Motion Moment of Inertia Radius of Gyration
EasyNumerical valueIdea asked in 4 yearsAsked the year before
Q54
A block of mass 10 kg is moving along x-axis under the action of force $F = 5x\ N$ . The work done by the force in moving the block from $x = 2m$ to $4m$ will be ____________ J.
Work, Energy and Power Work and Kinetic Energy Work by Variable Force
EasyNumerical valueIdea asked in 6 yearsAsked the year before
Q55
In the given figure the total charge stored in the combination of capacitors is 100 $\mu$ C. The value of ' $x$ ' is ______________.
Electrostatic Potential and Capacitance Parallel Plate Capacitors Series and Parallel Combinations
EasyNumerical value · figureIdea asked in 6 yearsAsked the year before
Q56
As per given figure A, B and C are the first, second and third excited energy levels of hydrogen atom respectively. If the ratio of the two wavelengths (i.e. $\frac{\lambda_1}{\lambda_2}$ ) is $\frac{7}{4n}$ , then the value of $n$ will be ______________.
Atoms Bohr's Atomic Model Rydberg Formula and Spectra
MediumNumerical value · figureIdea asked in 6 yearsAsked the year before
Q57
The refractive index of a transparent liquid filled in an equilateral hollow prism is $\sqrt{2}$ . The angle of minimum deviation for the liquid will be ______________ $^{\circ}$ .
Ray Optics and Optical Instruments Prism Deviation Liquid-Filled Equilateral Prism
MediumNumerical valueNew idea
Q58
A 20 cm long metallic rod is rotated with 210 rpm about an axis normal to the rod passing through its one end. The other end of the rod is in contact with a circular metallic ring. A constant and uniform magnetic field 0.2T parallel to the axis exists…
Electromagnetic Induction Faraday's Law Rotational Motional EMF
MediumNumerical valueIdea asked in 4 yearsAsked the year before
Q59
An election in a hydrogen atom revolves around its nucleus with a speed of $6.76 \times 10^6$ ms $^{-1}$ in an orbit of radius 0.52 A $^{o}$ . The magnetic field produced at the nucleus of the hydrogen atom is ____________ T.
Moving Charges and Magnetism Biot-Savart Law Field on axis of circular loop
MediumNumerical valueIdea asked in 2 years
Q60
A network of four resistances is connected to 9 V battery, as shown in figure. The magnitude of voltage difference between the points A and B is ____________ V.
Current Electricity Wheatstone Bridge Unbalanced Bridge Analysis
MediumNumerical value · figureIdea asked in 3 years

Chemistry30 questions

17 easy11 medium2 hard

Q61
Which of the following expressions is correct in case of a CsCl unit cell (edge length 'a')?
The Solid State Crystal Lattices Edge-Radius Relationships
EasySingle correctIdea asked in 2 yearsAsked the year before
Q62
Given below are two statements: Statement I : According to Bohr's model of hydrogen atom, the angular momentum of an electron in a given stationary state is quantised. Statement II : The concept of electron in Bohr's orbit, violates the Heisenberg…
Structure of Atom Bohr's Atomic Theory Angular Momentum Quantization
EasyTwo statementsIdea asked in 3 years
Q63
Consider the following statements: (A) $\mathrm{NF_3}$ molecule has a trigonal planar structure. (B) Bond length of $\mathrm{N_2}$ is shorter than $\mathrm{O_2}$ . (C) Isoelectronic molecules or ions have identical bond order. (D) Dipole moment of…
Chemical Bonding and Molecular Structure VSEPR Theory Steric Number 4 Shapes
MediumOtherIdea asked in 3 yearsAsked the year before
Q64
Which of the following statement(s) is/are correct? (A) The pH of $1 \times 10^{-8}$ M HCl solution is 8. (B) The conjugate base of $\mathrm{H_2PO_4^-}$ is $\mathrm{HPO_4^{2-}}$ . (C) $K_w$ increases with increase in temperature. (D) When a solution of a weak…
Equilibrium pH and Buffers pH of Extremely Dilute Strong Acids/Bases
MediumOtherNew idea
Q65
Which one of the following is not an example of calcination?
General Principles and Processes of Isolation of Elements Metallurgical Extraction Calcination vs. Roasting
EasySingle correctIdea asked in 3 years
Q66
During water-gas shift reaction
Hydrogen Hydrogen and Hydrides Water-Gas Shift Reaction
EasySingle correctIdea asked in 2 years
Q67
Given below are two statements: One is labelled as Assertion A and the other is labelled as Reason R : Assertion (A) : $\mathrm{BeCl_2}$ and $\mathrm{MgCl_2}$ produce characteristic flame Reason (R) : The excitation energy is high in $\mathrm{BeCl_2}$ and…
The s-Block Elements s-Block Elements Flame Coloration
EasyAssertion–reasonIdea asked in 4 yearsAsked the year before
Q68
For a good quality cement, the ratio of silica to alumina is found to be
Environmental Chemistry
EasySingle correct
Q69
The number of P - O - P bonds in $\mathrm{H_4P_2O_7}$ , $\mathrm{(HPO_3)_3}$ and $\mathrm{P_4O_{10}}$ are respectively
The p-Block Elements Group 15 & 16 Phosphorus Oxoacids
MediumSingle correctIdea asked in 4 yearsAsked the year before
Q70
The complex with highest magnitude of crystal field splitting energy ( $\Delta_0$ ) is
Coordination Compounds Crystal Field Theory Effect of Metal Oxidation State on Splitting
MediumSingle correctNew idea
Q71
The possibility of photochemical smog formation will be minimum at
Environmental Chemistry Primary Reactions of Photochemical Smog
EasySingle correctIdea asked in 2 years
Q72
[figure: ninhydrin (benzene ring fused to a five-membered ring bearing three C=O groups) $\xrightarrow{\mathrm{H_2O}}$ 'A' major product] 'A' formed in the above reaction is
Aldehydes, Ketones and Carboxylic Acids Nucleophilic Addition Stability of Chloral Hydrate
MediumDiagram based · figureNew idea
Q73
Decreasing order of reactivity towards electrophilic substitution for the following compounds is : (a) toluene (benzene bearing $\mathrm{CH_3}$ ) (b) benzene (c) benzene bearing $\mathrm{CF_3}$ (d) benzene bearing $\mathrm{OCH_3}$ (e) benzene bearing…
Haloalkanes and Haloarenes Aromatic Hydrocarbons Ranking Activating Groups
MediumOther · figureIdea asked in 3 yearsAsked the year before
Q74
The major product formed in the Friedel-Craft acylation of chlorobenzene is
Haloalkanes and Haloarenes Aromatic Hydrocarbons Halogen Directing Exception
EasyDiagram based · figureNew idea
Q75
[figure: 4-nitrotoluene (benzene ring with $\mathrm{CH_3}$ at top and $\mathrm{NO_2}$ at bottom) $\longrightarrow\longrightarrow$ benzoic acid ring bearing $\mathrm{COOH}$ at top, $\mathrm{Br}$ on the adjacent carbon at upper right and $\mathrm{Cl}$ at…
Amines Diazonium Salts Sandmeyer Reaction
HardDiagram based · figureIdea asked in 4 years
Q76
The product formed in the following multistep reaction is : $\mathrm{CH_3-CH=CH_2}$ $\xrightarrow[\text{iii) PCC}\;\text{iv) }\mathrm{CH_3MgBr}]{\text{i) }\mathrm{B_2H_6}\;\text{ii) }\mathrm{H_2O_2,\,NaOH}}$
Alcohols, Phenols and Ethers Alkenes and Alkynes Hydroboration-Oxidation
MediumOther · figureIdea asked in 4 yearsAsked the year before
Q77
Consider the following sequence of reactions: [figure: aniline ( $\mathrm{C_6H_5NH_2}$ ) $\xrightarrow{\mathrm{NaNO_2},\ 0\text{-}5^\circ\mathrm{C}}$ 'A' $\xrightarrow{\text{N, N-Dimethylaniline}}$ 'B'] The product 'B' is
Amines Diazonium Salts Azo Coupling Reactions
EasyDiagram based · figureIdea asked in 5 yearsAsked the year before
Q78
Match List I with List II: List I - (Monomer) (A) Tetrafluoroethene (B) Acrylonitrile (C) Caprolactam (D) Isoprene List II - (Polymer) (I) Orlon (II) Natural rubber (III) Teflon (IV) Nylon-6 Choose the correct answer from the options given below :
Polymers Common Condensation Polymers
EasyMatch the listIdea asked in 2 yearsAsked the year before
Q79
Which is not true for arginine?
Biomolecules
EasySingle correct
Q80
Which of the following statement is correct for paper chromatography?
Organic Chemistry: Some Basic Principles and Techniques
EasySingle correct
Q81
The volume (in mL) of 0.1 M $\mathrm{AgNO_3}$ required for complete precipitation of chloride ions present in 20 mL of 0.01 M solution of $[\mathrm{Cr(H_2O)_5Cl}]\mathrm{Cl_2}$ as silver chloride is __________
Coordination Compounds Coordination Isomerism Precipitation Tests for Ionization Isomers
MediumNumerical valueIdea asked in 7 yearsAsked the year before
Q82
30.4 kJ of heat is required to melt one mole of sodium chloride and the entropy change at the melting point is 28.4 J $\mathrm{K^{-1}\,mol^{-1}}$ at 1 atm. The melting point of sodium chloride is __________ K (Nearest Integer)
Thermodynamics (Chemistry)Entropy Entropy Change at Equilibrium Phase Transition
EasyNumerical valueNew idea
Q83
The vapour pressure of 30% (w/v) aqueous solution of glucose is __________ mm Hg at $25^\circ\mathrm{C}$ . [Given : The density of 30% (w/v), aqueous solution of glucose is 1.2 g $\mathrm{cm^{-3}}$ and vapour pressure of pure water is 24 mm Hg.] (Molar mass…
Solutions Colligative Properties Relative Lowering of VP
MediumNumerical valueIdea asked in 7 yearsAsked the year before
Q84
The number of correct statements from the following is __________ (A) Conductivity always decreases with decrease in concentration for both strong and weak electrolytes. (B) The number of ions per unit volume that carry current in a solution increases on…
Electrochemistry
MediumNumerical value
Q85
For a reversible reaction $\mathrm{A \rightleftharpoons B}$ , the $\Delta H_{\text{forward reaction}} = 20$ kJ $\mathrm{mol^{-1}}$ . The activation energy of the uncatalysed forward reaction is 300 kJ $\mathrm{mol^{-1}}$ . When the reaction is catalysed keeping…
Chemical Kinetics Arrhenius Equation Catalyst Effect on Arrhenius Parameters
HardNumerical valueIdea asked in 3 yearsAsked the year before
Q86
20 mL of 0.5 M NaCl is required to coagulate 200 mL of $\mathrm{As_2S_3}$ solution in 2 hours. The coagulating value of NaCl is __________.
Surface Chemistry Colloids Hardy-Schulze Rule
EasyNumerical valueIdea asked in 3 years
Q87
The total number of isoelectronic species from the given set is __________. $\mathrm{O^{2-}, F^-, Al, Mg^{2+}, Na^+, O^+, Mg, Al^{3+}, F}$
Classification of Elements and Periodicity in Properties Periodic Trends Isoelectronic Species Radii Comparison
EasyNumerical valueIdea asked in 5 yearsAsked the year before
Q88
The total change in the oxidation state of manganese involved in the reaction of $\mathrm{KMnO_4}$ and potassium iodide in the acidic medium is __________.
Redox Reactions Permanganate n-Factor Across Media
EasyNumerical valueIdea asked in 3 yearsAsked the year before
Q89
The homoleptic and octahedral complex of $\mathrm{Co^{2+}}$ and $\mathrm{H_2O}$ has __________ unpaired electron(s) in the $\mathrm{t_{2g}}$ set of orbitals.
Coordination Compounds Crystal Field Theory Octahedral Field Splitting
MediumNumerical valueIdea asked in 3 yearsAsked the year before
Q90
In Chromyl chloride, the oxidation state of chromium is (+) __________.
Redox Reactions Oxidation Number Assignment
EasyNumerical valueIdea asked in 4 yearsAsked the year before

Maths30 questions

3 easy16 medium11 hard

Q1
If the domain of the function $f(x)=\log_e\left(4x^2+11x+6\right)+\sin^{-1}(4x+3)+\cos^{-1}\left(\frac{10x+6}{3}\right)$ is $(\alpha,\beta]$ , then $36|\alpha+\beta|$ is equal to
Relations and Functions Functions and Domain Inverse Trig Function Domain
MediumSingle correctIdea asked in 4 yearsAsked the year before
Q2
The number of real roots of the equation $x|x|-5|x+2|+6=0$ , is
Complex Numbers and Quadratic Equations Quadratic Roots Roots of Modulus Quadratics
MediumSingle correctIdea asked in 6 yearsAsked the year before
Q3
If the set $\left\{\mathrm{Re}\left(\frac{z-\bar{z}+z\bar{z}}{2-3z+5\bar{z}}\right): z\in\mathbb{C}, \mathrm{Re}(z)=3\right\}$ is equal to the interval $(\alpha,\beta]$ , then $24(\beta-\alpha)$ is equal to
Complex Numbers and Quadratic Equations
HardSingle correct
Q4
Let the determinant of a square matrix A of order $m$ be $m-n$ , where $m$ and $n$ satisfy $4m+n=22$ and $17m+4n=93$ . If $det\left(n\,adj\left(adj(mA)\right)\right)=3^a5^b6^c$ , then $a+b+c$ is equal to
Determinants Matrix Inverse Inverse Relation to Double Adjoint
MediumSingle correctIdea asked in 6 yearsAsked the year before
Q5
Let the system of linear equations $-x+2y-9z=7$ $-x+3y+7z=9$ $-2x+y+5z=8$ $-3x+y+13z=\lambda$ has a unique solution $x=\alpha, y=\beta, z=\gamma$ . Then the distance of the point $(\alpha,\beta,\gamma)$ from the plane $2x-2y+z=\lambda$ is
Determinants Cramer's Rule Consistency of Overdetermined Systems (3 eq, 2 var)
MediumSingle correctNew idea
Q6
The total number of three-digit numbers, divisible by 3, which can be formed using the digits 1, 3, 5, 8, if repetition of digits is allowed, is
Permutations and Combinations Permutations with Repetition
MediumSingle correctIdea asked in 4 yearsAsked the year before
Q7
Let $\left(a+bx+cx^2\right)^{10}=\sum_{i=0}^{20}p_ix^i$ , $a,b,c\in\mathbb{N}$ . If $p_1=20$ and $p_2=210$ , then $2(a+b+c)$ is equal to
Binomial Theorem
MediumSingle correct
Q8
Let $A_1$ and $A_2$ be two arithmetic means and $G_1, G_2, G_3$ be three geometric means of two distinct positive numbers. Then $G_1^4+G_2^4+G_3^4+G_1^2G_3^2$ is equal to
Sequences and Series AP and GP Arithmetic and Geometric Means
MediumSingle correctIdea asked in 4 yearsAsked the year before
Q9
Let $[x]$ denote the greatest integer function and $f(x)=\max\{1+x+[x], 2+x, x+2[x]\}$ , $0\le x\le 2$ . Let $m$ be the number of points in $[0,2]$ , where $f$ is not continuous and $n$ be the number of points in $(0,2)$ , where $f$ is not differentiable. Then…
Continuity and Differentiability Differentiability of Max/Min Functions
HardSingle correctIdea asked in 5 yearsAsked the year before
Q10
If $\int_0^1\frac{1}{\left(5+2x-2x^2\right)\left(1+e^{(2-4x)}\right)}dx=\frac{1}{\alpha}\log_e\left(\frac{\alpha+1}{\beta}\right)$ , $\alpha,\beta>0$ , then $\alpha^4-\beta^4$ is equal to
Integrals Definite Integral Properties King's Property
HardSingle correctIdea asked in 6 yearsAsked the year before
Q11
Let $x=x(y)$ be the solution of the differential equation $2(y+2)\log_e(y+2)dx+\left(x+4-2\log_e(y+2)\right)dy=0$ , $y>-1$ with $x\left(e^4-2\right)=1$ . Then $x\left(e^9-2\right)$ is equal to
Differential Equations Linear Differential Equations Linear in x Form
HardSingle correctIdea asked in 4 yearsAsked the year before
Q12
If $(\alpha,\beta)$ is the orthocenter of the triangle ABC with vertices $A(3,-7)$ , $B(-1,2)$ and $C(4,5)$ , then $9\alpha-6\beta+60$ is equal to
Straight Lines
MediumSingle correct
Q13
The number of common tangents, to the circles $x^2+y^2-18x-15y+131=0$ and $x^2+y^2-6x-6y-7=0$ , is
Conic Sections Circle Tangents Number of Common Tangents
EasySingle correctIdea asked in 4 years
Q14
Let the foot of perpendicular of the point $P(3,-2,-9)$ on the plane passing through the points $(-1,-2,-3)$ , $(9,3,4)$ , $(9,-2,1)$ be $Q(\alpha,\beta,\gamma)$ . Then the distance of $Q$ from the origin is
Three Dimensional Geometry Planes in 3D Foot of Perpendicular to a Plane
MediumSingle correctNew idea
Q15
Let S be the set of all values of $\lambda$ , for which the shortest distance between the lines $\frac{x-\lambda}{0}=\frac{y-3}{4}=\frac{z+6}{1}$ and $\frac{x+\lambda}{3}=\frac{y}{-4}=\frac{z-6}{0}$ is 13. Then $8\left|\sum_{\lambda\in S}\lambda\right|$ is…
Three Dimensional Geometry Skew Lines Cartesian Shortest Distance
MediumSingle correctIdea asked in 6 yearsAsked the year before
Q16
Let $S$ be the set of all $(\lambda,\mu)$ for which the vectors $\lambda\hat{i}-\hat{j}+\hat{k}$ , $\hat{i}+2\hat{j}+\mu\hat{k}$ and $3\hat{i}-4\hat{j}+5\hat{k}$ , where $\lambda-\mu=5$ , are coplanar, then…
Vector Algebra Triple Products Solving for Unknown in Coplanarity
MediumSingle correctIdea asked in 4 yearsAsked the year before
Q17
Let ABCD be a quadrilateral. If E and F are the mid points of the diagonals AC and BD respectively and $\left(\overrightarrow{AB}-\overrightarrow{BC}\right)+\left(\overrightarrow{AD}-\overrightarrow{DC}\right)=k\,\overrightarrow{FE}$ , then $k$ is equal to
Vector Algebra
EasySingle correct
Q18
A bag contains 6 white and 4 black balls. A die is rolled once and the number of balls equal to the number obtained on the die are drawn from the bag at random. The probability that all the balls drawn are white is
Probability Bayes' Theorem Theorem of Total Probability
MediumSingle correctIdea asked in 3 years
Q19
The mean and standard deviation of 10 observations are 20 and 8 respectively. Later on, it was observed that one observation was recorded as 50 instead of 40. Then the correct variance is
Statistics Variance and SD Correcting Incorrect Variance
EasySingle correctIdea asked in 6 yearsAsked the year before
Q20
Negation of $p\wedge\left(q\wedge\sim(p\wedge q)\right)$ is
Mathematical Reasoning
MediumSingle correct
Q21
Let $A=\{1,2,3,4\}$ and R be a relation on the set $A\times A$ defined by $R=\left\{((a,b),(c,d)): 2a+3b=4c+5d\right\}$ . Then the number of elements in R is ______.
Relations and Functions Sets and Relations Relation Types
MediumNumerical valueIdea asked in 5 years
Q22
A person forgets his 4-digit ATM pin code. But he remembers that in the code all the digits are different, the greatest digit is 7 and the sum of the first two digits is equal to the sum of the last two digits. Then the maximum number of trials necessary to…
Permutations and Combinations Permutations of Distinct Objects
HardNumerical valueIdea asked in 4 yearsAsked the year before
Q23
The number of elements in the set $\{n\in\mathbb{N}: 10\le n\le 100$ and $3^n-3$ is a multiple of $7\}$ is ______.
Permutations and Combinations
MediumNumerical value
Q24
If the sum of the series…
Sequences and Series AP and GP Infinite GP Sum
HardNumerical valueIdea asked in 6 yearsAsked the year before
Q25
Consider the triangles with vertices $A(2,1)$ , $B(0,0)$ and $C(t,4)$ , $t\in[0,4]$ . If the maximum and the minimum perimeters of such triangles are obtained at $t=\alpha$ and $t=\beta$ respectively, then $6\alpha+21\beta$ is equal to ______.
Straight Lines
HardNumerical value
Q26
Let $f(x)=\int\frac{dx}{\left(3+4x^2\right)\sqrt{4-3x^2}}$ , $|x|<\frac{2}{\sqrt{3}}$ . If $f(0)=0$ and $f(1)=\frac{1}{\alpha\beta}\tan^{-1}\left(\frac{\alpha}{\beta}\right)$ , $\alpha,\beta>0$ , then $\alpha^2+\beta^2$ is equal to ______.
Integrals Integration by Substitution Algebraic Manipulation
HardNumerical valueNew idea
Q27
If the area bounded by the curve $2y^2=3x$ , lines $x+y=3$ , $y=0$ and outside the circle $(x-3)^2+y^2=2$ is A, then $4(\pi+4A)$ is equal to ______.
Application of Integrals Area Under Curves Area Between Curves
HardNumerical valueIdea asked in 7 yearsAsked the year before
Q28
Let an ellipse with centre $(1,0)$ and latus rectum of length $\frac{1}{2}$ have its major axis along x-axis. If its minor axis subtends an angle $60^\circ$ at the foci, then the square of the sum of the lengths of its minor and major axes is equal to ______.
Conic Sections Ellipse Eccentricity Eccentricity from Minor Axis Subtended Angle
MediumNumerical valueNew idea
Q29
Let the plane P contain the line $2x+y-z-3=0=5x-3y+4z+9$ and be parallel to the line $\frac{x+2}{2}=\frac{3-y}{-4}=\frac{z-7}{5}$ . Then the distance of the point A $(8,-1,-19)$ from the plane P measured parallel to the line…
Three Dimensional Geometry Point-Plane Distance Distance Along a Line
HardNumerical valueIdea asked in 5 yearsAsked the year before
Q30
If the line $x=y=z$ intersects the line $x\sin A+y\sin B+z\sin C-18=0=x\sin 2A+y\sin 2B+z\sin 2C-9$ , where $A,B,C$ are the angles of a triangle $ABC$ , then $80\left(\sin\frac{A}{2}\sin\frac{B}{2}\sin\frac{C}{2}\right)$ is equal to ______.
Trigonometric Functions
HardNumerical value

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.