JEE Main 2022 · July session

JEE Main 25 July 2022, Shift 1

88 questions (2 more dropped by NTA): 29 numerical-answer and 59 multiple choice. 13 were rated hard and 25 easy. Its hard-question count was the joint 9th highest of the 24 2022 shifts Lumi analysed. The biggest chapter was Conic Sections, with 3 questions.

Questions: 88; 2 dropped by NTA
Numerical answer: 29; 33% of the shift
Hard: 13; 15% of the shift
Chapters asked: 71; 13 questions with a figure

Key findings · 25 Jul 2022, Shift 1

1
13 hard questions (15%): the joint 9th highest count of the 24 2022 shifts Lumi analysed. Across those shifts the hard share ran from 8% to 22%.
2
Maths was the hardest section, with 10 of 28 rated hard; Physics had the most easy questions (12 of 30).
3
28 questions tested a new idea, one no earlier year had asked: 38% of the 73 questions tagged to an idea.
4
39 came back from 2021: the same idea had been asked in the 2021 shifts Lumi analysed.
5
36 shared an idea with another 2022 shift (49% of tagged questions). Shifts in the same year can test the same idea in a new setting.
6
Biggest chapters: Alternating Current in Physics (2), Aldehydes, Ketones and Carboxylic Acids in Chemistry (2) and Conic Sections in Maths (3).

The shift at a glance

Subjects, difficulty and chapters

Physics

30 Q · 10 numerical

Easy 12Medium 17Hard 1

Top chapters

Chemistry

30 Q · 10 numerical

Easy 11Medium 17Hard 2

Top chapters

Maths

28 Q · 9 numerical

Easy 2Medium 16Hard 10

Top chapters

Question format

Single correct45 Q
Numerical value29 Q
Diagram based9 Q
Two statements2 Q
Other2 Q
Match the list1 Q

What each question asks you to do

Multi-step33 Q
Calculation28 Q
Application11 Q
Recall11 Q
Conceptual3 Q
Data or graph1 Q
Diagram reading1 Q

Most common: Multi-step.

Against the other 2022 shifts

This shift next to the 23 other 2022 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
24 Jun, Shift 1	89	29%	17%	34%	31%	37%
24 Jun, Shift 2	90	40%	14%	33%	39%	36%
25 Jun, Shift 1	90	44%	13%	33%	33%	30%
25 Jun, Shift 2	90	30%	13%	33%	34%	22%
26 Jun, Shift 1	89	34%	16%	34%	37%	32%
26 Jun, Shift 2	90	30%	13%	33%	36%	27%
27 Jun, Shift 1	90	34%	8%	33%	38%	31%
27 Jun, Shift 2	90	32%	17%	33%	34%	46%
28 Jun, Shift 1	90	32%	12%	33%	38%	27%
28 Jun, Shift 2	90	30%	17%	33%	33%	31%
29 Jun, Shift 1	90	41%	11%	33%	36%	30%
29 Jun, Shift 2	88	35%	9%	33%	41%	30%
30 Jun, Shift 1	90	36%	13%	33%	27%	27%
25 Jul, Shift 1	88	28%	15%	33%	38%	38%
25 Jul, Shift 2	90	37%	12%	33%	33%	33%
26 Jul, Shift 1	90	27%	13%	33%	34%	31%
26 Jul, Shift 2	89	37%	16%	34%	41%	38%
27 Jul, Shift 1	89	29%	15%	34%	30%	32%
27 Jul, Shift 2	88	19%	15%	33%	40%	35%
28 Jul, Shift 1	90	36%	17%	33%	36%	30%
28 Jul, Shift 2	90	28%	14%	33%	40%	28%
29 Jul, Shift 1	90	31%	22%	33%	41%	37%
29 Jul, Shift 2	90	39%	12%	33%	48%	32%
30 Jul, Shift 2	90	30%	17%	33%	36%	33%

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
Open28 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 88 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

88 of 88 questions

Physics30 questions

12 easy17 medium1 hard

Q31
If momentum [P], area [A] and time [T] are taken as fundamental quantities, then the dimensional formula for coefficient of viscosity is :
Units and Measurements Viscosity Units and Dimensions of Viscosity
MediumSingle correctIdea asked in 2 yearsAsked the year before
Q32
Which of the following physical quantities have the same dimensions ?
Units and Measurements
MediumSingle correct
Q33
A person moved from A to B on a circular path as shown in figure. If the distance travelled by him is $60$ m, then the magnitude of displacement would be : (Given $\cos135^\circ=-0.7$ )
Motion in a Plane
MediumDiagram based · figure
Q34
A body of mass $0.5$ kg travels on straight line path with velocity $v=(3x^2+4)$ m/s. The net workdone by the force during its displacement from $x=0$ to $x=2$ m is :
Work, Energy and Power Work and Kinetic Energy Work-Energy Theorem
EasySingle correctIdea asked in 6 years
Q35
A solid cylinder and a solid sphere, having same mass $M$ and radius $R$ , roll down the same inclined plane from top without slipping. They start from rest. The ratio of velocity of the solid cylinder to that of the solid sphere, with which they reach the…
System of Particles and Rotational Motion
MediumSingle correct
Q36
Three identical particles A, B and C of mass $100$ kg each are placed in a straight line with $AB=BC=13$ m. The gravitational force on a fourth particle P of the same mass is $F$ , when placed at a distance $13$ m from the particle B on the perpendicular…
Gravitation
MediumSingle correct
Q37
A certain amount of gas of volume $V$ at $27^\circ$ C temperature and pressure $2\times10^{7}\ \mathrm{Nm^{-2}}$ expands isothermally until its volume gets doubled. Later it expands adiabatically until its volume gets redoubled. The final pressure of the gas…
Thermodynamics Thermodynamic Processes Adiabatic Process Analysis
MediumSingle correctIdea asked in 7 yearsAsked the year before
Q38
Following statements are given : (A) The average kinetic energy of a gas molecule decreases when the temperature is reduced. (B) The average kinetic energy of a gas molecule increases with increase in pressure at constant temperature. (C) The average kinetic…
Kinetic Theory Kinetic Theory of Gases Kinetic Energy and Temp
EasyOtherIdea asked in 7 yearsAsked the year before
Q39
In figure (A), mass '2 m' is fixed on mass 'm' which is attached to two springs of spring constant k. In figure (B), mass 'm' is attached to two springs of spring constant 'k' and '2k'. If mass 'm' in (A) and in (B) are displaced by distance 'x' horizontally…
Oscillations SHM Equations Springs in Parallel
MediumDiagram based · figureIdea asked in 3 yearsAsked the year before
Q40
A condenser of $2\ \mu$ F capacitance is charged steadily from $0$ to $5$ C. Which of the following graph represents correctly the variation of potential difference (V) across it's plates with respect to the charge (Q) on the condenser ?
Electrostatic Potential and Capacitance Parallel Plate Capacitors Parallel Plate Capacitance Formula
EasyDiagram based · figureNew idea
Q41
Two charged particles, having same kinetic energy, are allowed to pass through a uniform magnetic field perpendicular to the direction of motion. If the ratio of radii of their circular paths is $6:5$ and their respective masses ratio is $9:4$ . Then, the…
Moving Charges and Magnetism Lorentz Force Radius-Kinetic Energy Proportionality
MediumSingle correctNew ideaIdea asked in 2 years
Q42
To increase the resonant frequency in series LCR circuit,
Alternating Current LCR Resonance Resonant Frequency Calculation
EasySingle correctIdea asked in 4 yearsAsked the year before
Q43
A small square loop of wire of side $l$ is placed inside a large square loop of wire $L$ ( $L\gg l$ ). Both loops are coplanar and their centres coincide at point O as shown in figure. The mutual inductance of the system is :
Electromagnetic Induction Inductance Mutual Inductance of Concentric Square Loops
HardDiagram based · figureNew ideaIdea asked in 2 years
Q44
The rms value of conduction current in a parallel plate capacitor is $6.9\ \mu$ A. The capacity of this capacitor, if it is connected to $230$ V ac supply with an angular frequency of $600$ rad/s, will be :
Alternating Current
MediumSingle correct
Q45
Which of the following statement is correct ?
Ray Optics and Optical Instruments
MediumSingle correct
Q46
Time taken by light to travel in two different materials A and B of refractive indices $\mu_A$ and $\mu_B$ of same thickness is $t_1$ and $t_2$ respectively. If $t_2-t_1=5\times10^{-10}$ s and the ratio of $\mu_A$ to $\mu_B$ is $1:2$ . Then, the thickness of…
Ray Optics and Optical Instruments Refraction and Snell's Law Time of Flight in a Medium
MediumSingle correctNew idea
Q47
A metal exposed to light of wavelength $800$ nm and and emits photoelectrons with a certain kinetic energy. The maximum kinetic energy of photo-electron doubles when light of wavelength $500$ nm is used. The workfunction of the metal is : (Take $hc=1230$ …
Dual Nature of Radiation and Matter Photoelectric Effect Solving via Two Wavelength Data Points
MediumSingle correctNew ideaIdea asked in 2 years
Q48
The momentum of an electron revolving in $n^{\mathrm{th}}$ orbit is given by : (Symbols have their usual meanings)
Atoms Bohr's Atomic Model Angular Momentum Quantization
EasySingle correctIdea asked in 5 yearsAsked the year before
Q49
The magnetic moment of an electron (e) revolving in an orbit around nucleus with an orbital angular momentum is given by :
Magnetism and Matter Magnetic Dipole Moment Gyromagnetic ratio and Bohr magneton
EasySingle correctNew idea
Q50
In the circuit, the logical value of $A=1$ or $B=1$ when potential at A or B is $5$ V and the logical value of $A=0$ or $B=0$ when potential at A or B is $0$ V. The truth table of the given circuit will be :
Semiconductor Electronics: Materials, Devices and Simple Circuits Logic Gates Diode Realization of OR and AND Gates
MediumDiagram based · figureIdea asked in 2 yearsAsked the year before
Q51
A car is moving with speed of $150$ km/h and after applying the break it will move $27$ m before it stops. If the same car is moving with a speed of one third the reported speed then it will stop after travelling ______ m distance.
Motion in a Straight Line Kinematics in 1D Stopping Distance Proportionality
EasyNumerical valueNew ideaIdea asked in 2 years
Q52
Four forces are acting at a point P in equilibrium as shown in figure. The ratio of force $F_1$ to $F_2$ is $1:x$ where $x=$ ______.
Laws of Motion Newton's Laws of Motion Static Equilibrium
EasyNumerical value · figureIdea asked in 4 yearsAsked the year before
Q53
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 $5$ cm. Another wire of the same material of length $4L$ and radius $4r$ is pulled by a force $4F$ under same…
Mechanical Properties of Solids Elasticity Elastic Moduli
MediumNumerical valueIdea asked in 7 yearsAsked the year before
Q54
A unit scale is to be prepared whose length does not change with temperature and remains $20$ cm, using a bimetallic strip made of brass and iron each of different length. The length of both components would change in such a way that difference between their…
Thermal Properties of Matter Thermal Expansion Constant Length Difference
EasyNumerical valueNew idea
Q55
An observer is riding on a bicycle and moving towards a hill at $18\ \mathrm{km\,h^{-1}}$ . He hears a sound from a source at some distance behind him directly as well as after its reflection from the hill. If the original frequency of the sound as emitted by…
Waves Doppler Effect Doppler Beat Frequency Calculation
MediumNumerical valueIdea asked in 2 yearsAsked the year before
Q56
The volume charge density of a sphere of radius $6$ m is $2\ \mu\mathrm{C\,cm^{-3}}$ . The number of lines of force per unit surface area coming out from the surface of the sphere is ______ $\times10^{10}\ \mathrm{NC^{-1}}$ . [Given : Permittivity of vacuum…
Electric Charges and Fields Electric Field Field of a Solid Non-Conducting Sphere
MediumNumerical valueNew idea
Q57
In the given figure, the value of $V_o$ will be ______ V.
Current Electricity Kirchhoff's Laws Millman's Theorem for Node Voltage
EasyNumerical value · figureNew ideaIdea asked in 2 years
Q58
Eight copper wire of length $l$ and diameter $d$ are joined in parallel to form a single composite conductor of resistance $R$ . If a single copper wire of length $2l$ have the same resistance ( $R$ ) then its diameter will be ______ $d$ .
Current Electricity Ohm's Law and Resistance Resistivity and Geometry
EasyNumerical valueIdea asked in 4 yearsAsked the year before
Q59
The energy band gap of semiconducting material to produce violet (wavelength $=4000$ Å) LED is ______ eV. (Round off to the nearest integer).
Semiconductor Electronics: Materials, Devices and Simple Circuits Semiconductor Diodes LED Bandgap and Emission Wavelength
EasyNumerical valueIdea asked in 3 years
Q60
The required height of a TV tower which can cover the population of $6.03$ lakh is $h$ . If the average population density is $100$ per square km and the radius of earth is $6400$ km, then the value of $h$ will be ______ m.
Communication Systems
MediumNumerical value

Chemistry30 questions

11 easy17 medium2 hard

Q61
$\mathrm{SO_2Cl_2}$ on reaction with excess of water results into acidic mixture $\mathrm{SO_2Cl_2+2H_2O\rightarrow H_2SO_4+2HCl}$ $16$ moles of NaOH is required for the complete neutralisation of the resultant acidic mixture. The number of moles of…
Some Basic Concepts of Chemistry Stoichiometry Mole-to-Mole Relationships
EasySingle correctNew ideaIdea asked in 3 years
Q62
Which of the following sets of quantum numbers is not allowed ?
Structure of Atom Quantum Numbers Validity of Quantum Number Sets
EasySingle correctNew idea
Q63
The depression in freezing point observed for a formic acid solution of concentration $0.5\ \mathrm{mL\,L^{-1}}$ is $0.0405^\circ$ C. Density of formic acid is $1.05\ \mathrm{g\,mL^{-1}}$ . The Van't Hoff factor of the formic acid solution is nearly : (Given…
Solutions Colligative Properties Van't Hoff Factor (i)
MediumSingle correctIdea asked in 4 yearsAsked the year before
Q64
$20$ mL of $0.1$ M $\mathrm{NH_4OH}$ is mixed with $40$ mL of $0.05$ M HCl. The pH of the mixture is nearest to : (Given : $K_b(\mathrm{NH_4OH})=1\times10^{-5}$ , $\log2=0.30$ , $\log3=0.48$ , $\log5=0.69$ , $\log7=0.84$ , $\log11=1.04$ )
Equilibrium pH and Buffers pH of Salt Solutions
MediumSingle correctNew idea
Q65
Match List - I with List - II List - I (A) $\mathrm{N_2(g)+3H_2(g)\rightarrow2NH_3(g)}$ (B) $\mathrm{CO(g)+3H_2(g)\rightarrow CH_4(g)+H_2O(g)}$ (C) $\mathrm{CO(g)+H_2(g)\rightarrow HCHO(g)}$ (D) $\mathrm{CO(g)+2H_2(g)\rightarrow CH_3OH(g)}$ List - II (I)…
Surface Chemistry
MediumMatch the list
Q66
The IUPAC nomenclature of an element with electronic configuration $[\mathrm{Rn}]\,5f^{14}6d^{1}7s^{2}$ is :
Classification of Elements and Periodicity in Properties Periodic Trends IUPAC Nomenclature for Z > 100
MediumSingle correctIdea asked in 3 years
Q67
The compound(s) that is(are) removed as slag during the extraction of copper is : (A) $\mathrm{CaO}$ (B) $\mathrm{FeO}$ (C) $\mathrm{Al_2O_3}$ (D) $\mathrm{ZnO}$ (E) $\mathrm{NiO}$ Choose the correct answer from the options given below :
General Principles and Processes of Isolation of Elements Metallurgical Extraction Bessemerization of Copper
MediumOtherNew idea
Q68
The reaction of $\mathrm{H_2O_2}$ with potassium permanganate in acidic medium leads to the formation of mainly :
The d- and f-Block Elements d-Block Elements Specific KMnO4 Oxidations in Acidic Medium
EasySingle correctNew idea
Q69
Choose the correct order of density of the alkali metals :
The s-Block Elements s-Block Elements Density Trends and Anomalies
EasySingle correctNew idea
Q70
The geometry around boron in the product 'B' formed from the following reaction is $\mathrm{BF_3+NaH\xrightarrow{450\ K}A+NaF}$ $\mathrm{A+NMe_3\rightarrow B}$
The p-Block Elements Group 13 & 14 Symmetrical vs. Asymmetrical Cleavage of Diborane
MediumSingle correctNew idea
Q71
The interhalogen compound formed from the reaction of bromine with excess of fluorine is a :
The p-Block Elements Group 17 & 18 Controlled Preparation of Interhalogens
MediumSingle correctNew idea
Q72
The photochemical smog does not generally contain :
Environmental Chemistry Classical vs Photochemical Smog
EasySingle correctIdea asked in 3 yearsAsked the year before
Q73
A compound 'A' on reaction with 'X' and 'Y' produces the same major product but different by product 'a' and 'b'. Oxidation of 'a' gives a substance produced by ants. Compound 'A' : $\mathrm{H_2C{=}C(CH_3){-}CH_2{-}C(CH_3)_2{-}CH_3}$ With X :…
Hydrocarbons Alkenes and Alkynes Reductive Ozonolysis Products
MediumSingle correct · figureIdea asked in 7 yearsAsked the year before
Q74
Most stable product of the following reaction is : (i) p-toluenesulfonyl chloride ( $\mathrm{H_3C{-}C_6H_4{-}SO_2Cl}$ ), Pyridine (ii) NaCN, DMF
Haloalkanes and Haloarenes SN1 and SN2 Mechanisms SN2 Transition State
MediumDiagram based · figureNew ideaIdea asked in 2 years
Q75
Which one of the following reactions does not represent correct combination of substrate and product under the given conditions ?
Aldehydes, Ketones and Carboxylic Acids Carboxylic Acids and Derivatives DIBAL-H Selective Reduction
MediumDiagram based · figureIdea asked in 3 yearsAsked the year before
Q76
An organic compound 'A' on reaction with $\mathrm{NH_3}$ followed by heating gives compound B. Which on further strong heating gives compound C ( $\mathrm{C_8H_5NO_2}$ ). Compound C on sequential reaction with ethanolic KOH, alkyl chloride and hydrolysis with…
Amines Gabriel Phthalimide Synthesis
MediumDiagram based · figureIdea asked in 6 yearsAsked the year before
Q77
Melamine polymer is formed by the condensation of :
Polymers Melamine-Formaldehyde Polymer
EasyDiagram based · figureNew idea
Q78
During the denaturation of proteins, which of these structures will remain intact ?
Biomolecules
EasySingle correct
Q79
Drugs used to bind to receptors, inhibiting its natural function and blocking a message are called :
Chemistry in Everyday Life Receptor Agonists and Antagonists
EasySingle correctNew ideaIdea asked in 2 years
Q80
Given below are two statements : Statement I : On heating with $\mathrm{KHSO_4}$ , glycerol is dehydrated and acrolein is formed. Statement II : Acrolein has fruity odour and can be used to test glycerol's presence. Choose the correct option.
Alcohols, Phenols and Ethers
MediumTwo statements
Q81
Among the following species $\mathrm{N_2,\ N_2^{+},\ N_2^{-},\ N_2^{2-},\ O_2,\ O_2^{+},\ O_2^{-},\ O_2^{2-}}$ the number of species showing diamagnesim is ______ .
Chemical Bonding and Molecular Structure Molecular Orbital Theory Magnetic Property Prediction
MediumNumerical valueIdea asked in 6 yearsAsked the year before
Q82
The enthalpy of combustion of propane, graphite and dihydrogen at 298 K are $-2220.0\ \mathrm{kJ\,mol^{-1}}$ , $-393.5\ \mathrm{kJ\,mol^{-1}}$ and $-285.8\ \mathrm{kJ\,mol^{-1}}$ respectively. The magnitude of enthalpy of formation of propane…
Thermodynamics (Chemistry)Enthalpy and Hess's Law Hess's Law
EasyNumerical valueIdea asked in 5 yearsAsked the year before
Q83
The pressure of a moist gas at $27^\circ$ C is $4$ atm. The volume of the container is doubled at the same temperature. The new pressure of the moist gas is ______ $\times10^{-1}$ atm. (Nearest integer) (Given : The vapour pressure of water at $27^\circ$ C is…
States of Matter Ideal Gas Laws Dalton's Law
MediumNumerical valueIdea asked in 4 yearsAsked the year before
Q84
The cell potential for $\mathrm{Zn|Zn^{2+}(aq)\|Sn^{x+}|\,Sn}$ is $0.801$ V at 298 K. The reaction quotient for the above reaction is $10^{-2}$ . The number of electrons involved in the given electrochemical cell reaction is ______. (Given :…
Electrochemistry Nernst Equation Full Cell Nernst Equation
MediumNumerical valueIdea asked in 6 yearsAsked the year before
Q85
The half life for the decomposition of gaseous compound A is $240$ s when the gaseous pressure was $500$ Torr initially. When the pressure was $250$ Torr, the half life was found to be $4.0$ min. The order of the reaction is ______. (Nearest integer)
Chemical Kinetics Integrated Rate Equations Nth-Order Half-Life Proportionality
EasyNumerical valueNew idea
Q86
Consider the following metal complexes : $[\mathrm{Co(NH_3)_6}]^{3+}$ $[\mathrm{CoCl(NH_3)_5}]^{2+}$ $[\mathrm{Co(CN)_6}]^{3-}$ $[\mathrm{Co(NH_3)_5(H_2O)}]^{3+}$ The spin-only magnetic moment value of the complex that absorbs light with shortest wavelength…
Coordination Compounds Crystal Field Theory Color and d-d Transitions
MediumNumerical valueIdea asked in 5 yearsAsked the year before
Q87
Among $\mathrm{Co^{3+},\ Ti^{2+},\ V^{2+}}$ and $\mathrm{Cr^{2+}}$ ions, one if used as a reagent cannot liberate $\mathrm{H_2}$ from dilute mineral acid solution, its spin-only magnetic moment in gaseous state is ______ B.M. (Nearest integer)
The d- and f-Block Elements d-Block Elements Standard Electrode Potential Trends
HardNumerical valueIdea asked in 3 yearsAsked the year before
Q88
While estimating the nitrogen present in an organic compound by Kjeldahl's method, the ammonia evolved from $0.25$ g of the compound neutralized $2.5$ mL of $2$ M $\mathrm{H_2SO_4}$ . The percentage of nitrogen present in organic compound is ______.
Organic Chemistry: Some Basic Principles and Techniques Stoichiometry Stoichiometry of Kjeldahl Method
EasyNumerical valueIdea asked in 4 yearsAsked the year before
Q89
The number of $\mathrm{sp^3}$ hybridised carbons in an acyclic neutral compound with molecular formula $\mathrm{C_4H_5N}$ is ______.
Organic Chemistry: Some Basic Principles and Techniques
MediumNumerical value
Q90
In the given reaction [cyclopentane ring bearing, on the same carbon, a $\mathrm{CH_3}$ group and a $\mathrm{C(=O)OEt}$ group, and on the adjacent ring carbon a $\mathrm{CH_2{-}C(=O){-}CH_3}$ side chain] $\xrightarrow{\ \bar{\mathrm{O}}\mathrm{Et}\ }$ A…
Aldehydes, Ketones and Carboxylic Acids Carboxylic Acids and Derivatives Dieckmann Condensation
HardNumerical value · figureNew idea

Maths28 questions

2 easy16 medium10 hard

Q1
The total number of functions, $f:\{1,2,3,4\}\rightarrow\{1,2,3,4,5,6\}$ such that $f(1)+f(2)=f(3)$ , is equal to :
Relations and Functions Permutations and Combinations Total Number of Functions
MediumSingle correctIdea asked in 4 yearsAsked the year before
Q2
If $\alpha,\ \beta,\ \gamma,\ \delta$ are the roots of the equation $x^4+x^3+x^2+x+1=0$ , then $\alpha^{2021}+\beta^{2021}+\gamma^{2021}+\delta^{2021}$ is equal to :
Complex Numbers and Quadratic Equations Roots of Unity Sum of p-th Powers of Roots
MediumSingle correctIdea asked in 2 yearsAsked the year before
Q4
The number of $\theta\in(0,4\pi)$ for which the system of linear equations $3(\sin 3\theta)x-y+z=2$ $3(\cos 2\theta)x+4y+3z=3$ $6x+7y+7z=9$ has no solution, is :
Determinants Cramer's Rule Trigonometric Coefficients and Solution Counting
MediumSingle correctNew idea
Q5
If $\lim_{n\to\infty}\left(\sqrt{n^2-n-1}+n\alpha+\beta\right)=0$ , then $8(\alpha+\beta)$ is equal to :
Limits and Derivatives L'Hopital's Rule Determining Unknown Constants from Finite Limits
MediumSingle correctIdea asked in 4 yearsAsked the year before
Q6
If the absolute maximum value of the function $f(x)=(x^2-2x+7)\,\mathrm{e}^{(4x^3-12x^2-180x+31)}$ in the interval $[-3,0]$ is $f(\alpha)$ , then :
Application of Derivatives Maxima and Minima Closed Interval Extrema
MediumSingle correctIdea asked in 6 yearsAsked the year before
Q7
The curve $y(x)=ax^3+bx^2+cx+5$ touches the $x$ -axis at the point $\mathrm{P}(-2,0)$ and cuts the $y$ -axis at the point $\mathrm{Q}$ , where $y'$ is equal to $3$ . Then the local maximum value of $y(x)$ is :
Application of Derivatives Maxima and Minima First Derivative Test
MediumSingle correctIdea asked in 3 years
Q8
The area of the region given by $A=\{(x,y):x^2\le y\le\min\{x+2,\,4-3x\}\}$ is :
Application of Integrals Area Under Curves Area Involving Max/Min Functions
MediumSingle correctIdea asked in 6 yearsAsked the year before
Q9
For any real number $x$ , let $[x]$ denote the largest integer less than equal to $x$ . Let $f$ be a real valued function defined on the interval $[-10,10]$ by…
Integrals Definite Integral Properties Periodic Function Property
HardSingle correctIdea asked in 3 yearsAsked the year before
Q10
The slope of the tangent to a curve $C:y=y(x)$ at any point $(x,y)$ on it is $\frac{2\mathrm{e}^{2x}-6\mathrm{e}^{-x}+9}{2+9\mathrm{e}^{-2x}}$ . If $C$ passes through the points $\left(0,\frac{1}{2}+\frac{\pi}{2\sqrt{2}}\right)$ and…
Differential Equations Integration by Substitution Exponential Factor Manipulation
HardSingle correctNew ideaIdea asked in 4 years
Q11
The general solution of the differential equation $(x-y^2)\,dx+y(5x+y^2)\,dy=0$ is :
Differential Equations Homogeneous Equations Identifying Equations Reducible to Homogeneous Form
HardSingle correctNew idea
Q12
A line, with the slope greater than one, passes through the point $\mathrm{A}(4,3)$ and intersects the line $x-y-2=0$ at the point $\mathrm{B}$ . If the length of the line segment $\mathrm{AB}$ is $\frac{\sqrt{29}}{3}$ , then $\mathrm{B}$ also lies on the line :
Straight Lines Straight Line Distance Distance Measured Parallel to a Line
MediumSingle correctNew ideaIdea asked in 2 years
Q13
Let the locus of the centre $(\alpha,\beta)$ , $\beta>0$ , of the circle which touches the circle $x^2+(y-1)^2=1$ externally and also touches the $x$ -axis be $L$ . Then the area bounded by $L$ and the line $y=4$ is :
Conic Sections Area Under Curves Area of Locus-Defined Regions
MediumSingle correctNew idea
Q14
Let $P$ be the plane containing the straight line $\frac{x-3}{9}=\frac{y+4}{-1}=\frac{z-7}{-5}$ and perpendicular to the plane containing the straight lines $\frac{x}{2}=\frac{y}{3}=\frac{z}{5}$ and $\frac{x}{3}=\frac{y}{7}=\frac{z}{8}$ . If $d$ is the…
Three Dimensional Geometry Planes in 3D Plane Containing a Line and a Point
HardSingle correctIdea asked in 3 yearsAsked the year before
Q15
Let $ABC$ be a triangle such that $\overrightarrow{BC}=\vec a$ , $\overrightarrow{CA}=\vec b$ , $\overrightarrow{AB}=\vec c$ , $|\vec a|=6\sqrt2$ , $|\vec b|=2\sqrt3$ and $\vec b\cdot\vec c=12$ . Consider the statements : (S1) :…
Vector Algebra Cross Product Magnitude and Sine Formula
HardTwo statementsIdea asked in 7 yearsAsked the year before
Q16
If the sum and the product of mean and variance of a binomial distribution are $24$ and $128$ respectively, then the probability of one or two successes is :
Probability
MediumSingle correct
Q17
If the numbers appeared on the two throws of a fair six faced die are $\alpha$ and $\beta$ , then the probability that $x^2+\alpha x+\beta>0$ , for all $x\in\mathbf{R}$ , is :
Probability
MediumSingle correct
Q18
The number of solutions of $|\cos x|=\sin x$ , such that $-4\pi\le x\le4\pi$ is :
Trigonometric Functions General Solutions Modulus Trigonometric Equations
EasySingle correctIdea asked in 2 yearsAsked the year before
Q19
A tower $PQ$ stands on a horizontal ground with base $Q$ on the ground. The point $R$ divides the tower in two parts such that $QR=15$ m. If from a point $A$ on the ground the angle of elevation of $R$ is $60^\circ$ and the part $PR$ of the tower subtends an…
Trigonometric Functions
MediumSingle correct
Q20
Which of the following statements is a tautology ?
Mathematical Reasoning
EasySingle correct
Q21
Let $A=\begin{pmatrix}2&-1&-1\\1&0&-1\\1&-1&0\end{pmatrix}$ and $B=A-I$ . If $\omega=\frac{\sqrt3\,i-1}{2}$ , then the number of elements in the set $\{n\in\{1,2,\dots,100\}:A^n+(\omega B)^n=A+B\}$ is equal to ______.
Matrices Matrix Multiplication Powers of Square Matrices
HardNumerical valueIdea asked in 6 yearsAsked the year before
Q22
The letters of the word 'MANKIND' are written in all possible orders and arranged in serial order as in an English dictionary. Then the serial number of the word 'MANKIND' is ______.
Permutations and Combinations Lexicographic Rank of a Word
MediumNumerical valueIdea asked in 4 years
Q23
If the maximum value of the term independent of $t$ in the expansion of $\left(t^2x^{\frac{1}{5}}+\frac{(1-x)^{\frac{1}{10}}}{t}\right)^{15}$ , $x\ge0$ , is $K$ , then $8K$ is equal to ______.
Binomial Theorem Binomial Coefficients Maximum Binomial Coefficient
HardNumerical valueIdea asked in 3 years
Q24
Let $a,\ b$ be two non-zero real numbers. If $p$ and $r$ are the roots of the equation $x^2-8ax+2a=0$ and $q$ and $s$ are the roots of the equation $x^2+12bx+6b=0$ , such that $\frac{1}{p},\frac{1}{q},\frac{1}{r},\frac{1}{s}$ are in A.P., then $a^{-1}-b^{-1}$ …
Sequences and Series AP and GP AP General Term
HardNumerical valueIdea asked in 4 yearsAsked the year before
Q25
Let $a_1=b_1=1$ , $a_n=a_{n-1}+2$ and $b_n=a_n+b_{n-1}$ for every natural number $n\ge2$ . Then $\sum_{n=1}^{15}a_n\cdot b_n$ is equal to ______.
Sequences and Series AP and GP AP Sum of n terms
MediumNumerical valueIdea asked in 7 yearsAsked the year before
Q26
Let $f(x)=\begin{cases}\left|\,4x^2-8x+5\,\right|, & \text{if } 8x^2-6x+1\ge0\\ [\,4x^2-8x+5\,], & \text{if } 8x^2-6x+1<0,\end{cases}$ where $[\alpha]$ denotes the greatest integer less than or equal to $\alpha$ . Then the number of points in $\mathbf{R}$ …
Continuity and Differentiability Piecewise Differentiability
HardNumerical valueIdea asked in 4 yearsAsked the year before
Q27
If $\lim_{n\to\infty}\frac{(n+1)^{k-1}}{n^{k+1}}\left[(nk+1)+(nk+2)+\dots+(nk+n)\right]=33\cdot\lim_{n\to\infty}\frac{1}{n^{k+1}}\cdot\left[1^k+2^k+3^k+\dots+n^k\right]$ , then the integral value of $k$ is equal to ______.
Integrals Integral as Limit of Sum Summation to Riemann Form
MediumNumerical valueIdea asked in 4 yearsAsked the year before
Q28
Let the equation of two diameters of a circle $x^2+y^2-2x+2fy+1=0$ be $2px-y=1$ and $2x+py=4p$ . Then the slope $m\in(0,\infty)$ of the tangent to the hyperbola $3x^2-y^2=3$ passing through the centre of the circle is equal to ______.
Conic Sections General Conic Section Condition for Line to be Tangent
MediumNumerical valueIdea asked in 3 yearsAsked the year before
Q29
The sum of diameters of the circles that touch (i) the parabola $75x^2=64(5y-3)$ at the point $\left(\frac{8}{5},\frac{6}{5}\right)$ and (ii) the $y$ -axis, is equal to ______.
Conic Sections Circle Tangents Family of Circles Touching a Line
HardNumerical valueIdea asked in 5 yearsAsked the year before

Dropped from the final key and left out of every count: Q3, Q30.

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 2022 shifts.