JEE Main 2022 · July session

JEE Main 28 July 2022, Shift 1

90 questions: 30 numerical-answer and 60 multiple choice. 15 were rated hard and 32 easy. Its hard-question count was the joint 2nd highest of the 24 2022 shifts Lumi analysed. The biggest chapter was Application of Derivatives, with 3 questions.

Questions: 90; none dropped
Numerical answer: 30; 33% of the shift
Hard: 15; 17% of the shift
Chapters asked: 67; 11 questions with a figure

Key findings · 28 Jul 2022, Shift 1

1
15 hard questions (17%): the joint 2nd 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 12 of 30 rated hard; Physics had the most easy questions (17 of 30).
3
27 questions tested a new idea, one no earlier year had asked: 36% of the 75 questions tagged to an idea.
4
41 came back from 2021: the same idea had been asked in the 2021 shifts Lumi analysed.
5
38 shared an idea with another 2022 shift (51% 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), Organic Chemistry: Some Basic Principles and Techniques in Chemistry (3) and Application of Derivatives in Maths (3).

The shift at a glance

Subjects, difficulty and chapters

Physics

30 Q · 10 numerical

Easy 17Medium 12Hard 1

Top chapters

Chemistry

30 Q · 10 numerical

Easy 12Medium 16Hard 2

Top chapters

Maths

30 Q · 10 numerical

Easy 3Medium 15Hard 12

Top chapters

Question format

Single correct46 Q
Numerical value30 Q
Other4 Q
Diagram based4 Q
Match the list4 Q
Two statements2 Q

What each question asks you to do

Multi-step27 Q
Calculation24 Q
Conceptual13 Q
Recall13 Q
Application11 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
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

17 easy12 medium1 hard

Q31
The dimensions of $\left(\frac{B^2}{\mu_0}\right)$ will be : (if $\mu_0$ : permeability of free space and B : magnetic field)
Units and Measurements Inductance Magnetic Energy Density
EasySingle correctNew ideaIdea asked in 2 years
Q32
A NCC parade is going at a uniform speed of 9 km/h under a mango tree on which a monkey is sitting at a height of 19.6 m. At any particular instant, the monkey drops a mango. A cadet will receive the mango whose distance from the tree at time of drop is :…
Motion in a Plane Projectile Motion Horizontal Projection from Height
EasySingle correctIdea asked in 4 yearsAsked the year before
Q33
In two different experiments, an object of mass 5 kg moving with a speed of $25\ \mathrm{ms^{-1}}$ hits two different walls and comes to rest within (i) 3 second, (ii) 5 seconds, respectively. Choose the correct option out of the following :
Laws of Motion Collisions Impulse-Momentum
EasySingle correctNew ideaIdea asked in 2 years
Q34
A balloon has mass of 10 g in air. The air escapes from the balloon at a uniform rate with velocity 4.5 cm/s. If the balloon shrinks in 5 s completely. Then, the average force acting on that balloon will be (in dyne).
Laws of Motion Newton's Laws of Motion Momentum Formulation (F = dp/dt)
MediumSingle correctIdea asked in 4 yearsAsked the year before
Q35
If the radius of earth shrinks by 2% while its mass remains same. The acceleration due to gravity on the earth's surface will approximately :
Gravitation
EasySingle correct
Q36
The force required to stretch a wire of cross-section $1\ \mathrm{cm^2}$ to double its length will be : (Given Yong's modulus of the wire $=2\times10^{11}\ \mathrm{N/m^2}$ )
Mechanical Properties of Solids Elasticity Elastic Moduli
EasySingle correctIdea asked in 7 yearsAsked the year before
Q37
A Carnot engine has efficiency of 50%. If the temperature of sink is reduced by $40^\circ$ C, its efficiency increases by 30%. The temperature of the source will be :
Thermodynamics Heat Engines and Refrigerators Carnot Efficiency Limit
MediumSingle correctIdea asked in 3 yearsAsked the year before
Q38
Given below are two statements : Statement I : The average momentum of a molecule in a sample of an ideal gas depends on temperature. Statement II : The rms speed of oxygen molecules in a gas is $v$ . If the temperature is doubled and the oxygen molecules…
Kinetic Theory Kinetic Theory of Gases Molecular Speed Types
MediumTwo statementsIdea asked in 3 yearsAsked the year before
Q39
In the wave equation $y=0.5\sin\frac{2\pi}{\lambda}(400\,t-x)\ \mathrm{m}$ the velocity of the wave will be :
Waves Waves on Strings Wave Speed on String
EasySingle correctIdea asked in 5 yearsAsked the year before
Q40
Two capacitors, each having capacitance 40 $\mu$ F are connected in series. The space between one of the capacitors is filled with dielectric material of dielectric constant K such that the equivalence capacitance of the system became 24 $\mu$ F. The value of…
Electrostatic Potential and Capacitance Dielectrics Capacitance with Full Slab
MediumSingle correctNew ideaIdea asked in 2 years
Q41
A wire of resistance $R_1$ is drawn out so that its length is increased by twice of its original length. The ratio of new resistance to original resistance is :
Current Electricity Ohm's Law and Resistance Resistance of Stretched Wires
EasySingle correctIdea asked in 4 yearsAsked the year before
Q42
The current sensitivity of a galvanometer can be increased by : (A) decreasing the number of turns (B) increasing the magnetic field (C) decreasing the area of the coil (D) decreasing the torsional constant of the spring Choose the most appropriate answer…
Moving Charges and Magnetism
MediumOther
Q43
As shown in the figure, a metallic rod of linear density $0.45\ \mathrm{kg\,m^{-1}}$ is lying horizontally on a smooth inclined plane which makes an angle of $45^\circ$ with the horizontal. The minimum current flowing in the rod required to keep it…
Moving Charges and Magnetism
MediumDiagram based · figure
Q44
The equation of current in a purely inductive circuit is $5\sin(49\pi t-30^\circ)$ . If the inductance is 30 mH then the equation for the voltage across the inductor, will be : {Let $\pi=\frac{22}{7}$ }
Alternating Current Phasor Diagrams Phase Relations in RLC
MediumSingle correctIdea asked in 5 yearsAsked the year before
Q45
As shown in the figure, after passing through the medium 1. The speed of light $v_2$ in medium 2 will be : (Given $c=3\times10^8\ \mathrm{ms^{-1}}$ )
Electromagnetic Waves
EasyDiagram based · figure
Q46
In normal adjustment, for a refracting telescope, the distance between objective and eye piece is 30 cm. The focal length of the objective, when the angular magnification of the telescope is 2, will be :
Ray Optics and Optical Instruments Optical Instruments Astronomical Telescope Magnification
EasySingle correctNew idea
Q47
The equation $\lambda=\frac{1.227}{x}$ nm can be used to find the de-Brogli wavelength of an electron. In this equation $x$ stands for : Where m = mass of electron P = momentum of electron K = Kinetic energy of electron V = Accelerating potential in volts…
Dual Nature of Radiation and Matter Matter Waves Wavelength from Accelerating Potential
EasySingle correctIdea asked in 4 yearsAsked the year before
Q48
The half life period of a radioactive substance is 60 days. The time taken for $\frac{7}{8}$ th of its original mass to disintegrate will be :
Nuclei Radioactive Decay Integer Half-Life Shortcuts
EasySingle correctIdea asked in 2 yearsAsked the year before
Q49
Identify the solar cell characteristics from the following options :
Semiconductor Electronics: Materials, Devices and Simple Circuits Semiconductor Diodes Solar Cell V-I Characteristics
EasyDiagram based · figureNew idea
Q50
In the case of amplitude modulation to avoid distortion the modulation index ( $\mu$ ) should be :
Communication Systems
EasySingle correct
Q51
If the projection of $2\hat{i}+4\hat{j}-2\hat{k}$ on $\hat{i}+2\hat{j}+\alpha\hat{k}$ is zero. Then, the value of $\alpha$ will be __________.
Motion in a Plane
EasyNumerical value
Q52
A freshly prepared radioactive source of half life 2 hours 30 minutes emits radiation which is 64 times the permissible safe level. The minimum time, after which it would be possible to work safely with source, will be __________ hours.
Nuclei Radioactive Decay Integer Half-Life Shortcuts
EasyNumerical valueIdea asked in 2 yearsAsked the year before
Q53
In a Young's double slit experiment, a laser light of 560 nm produces an interference pattern with consecutive bright fringes' separation of 7.2 mm. Now another light is used to produce an interference pattern with consecutive bright fringes' separation of…
Wave Optics Young's Double Slit YDSE Fringe Width
EasyNumerical valueIdea asked in 5 yearsAsked the year before
Q54
The frequencies at which the current amplitude in an LCR series circuit becomes $\frac{1}{\sqrt{2}}$ times its maximum value, are 212 rad s $^{-1}$ and 232 rad s $^{-1}$ . The value of resistance in the circuit is $R=5\ \Omega$ . The self inductance in the…
Alternating Current LCR Resonance Half-Power Frequencies
MediumNumerical valueNew idea
Q55
As shown in the figure, a potentiometer wire of resistance 20 $\Omega$ and length 300 cm is connected with resistance box (R.B.) and a standard cell of emf 4 V. For a resistance 'R' of resistance box introduced into the circuit, the null point for a cell of…
Current Electricity Potentiometer Potential Gradient Calculation
HardNumerical value · figureIdea asked in 2 years
Q56
Two electric dipoles of dipole moments $1.2\times10^{-30}$ Cm and $2.4\times10^{-30}$ Cm are placed in two different uniform electric fields of strengths $5\times10^4\ \mathrm{NC^{-1}}$ and $15\times10^4\ \mathrm{NC^{-1}}$ respectively. The ratio of maximum…
Electric Charges and Fields Electric Dipoles Torque in Uniform Field
EasyNumerical valueIdea asked in 4 years
Q57
The frequency of echo will be __________ Hz if the train blowing a whistle of frequency 320 Hz is moving with a velocity of 36 km/h towards a hill from which an echo is heard by the train driver. Velocity of sound in air is 330 m/s.
Waves Doppler Effect Moving Reflector Echo
MediumNumerical valueIdea asked in 3 years
Q58
The diameter of an air bubble which was initially 2 mm, rises steadily through a solution of density 1750 kg m $^{-3}$ at the rate of 0.35 cms $^{-1}$ . The coefficient of viscosity of the solution is __________ poise (in nearest integer). (the density of air…
Mechanical Properties of Fluids Viscosity Terminal Velocity of Rising Bubbles
MediumNumerical valueNew ideaIdea asked in 3 years
Q59
A block of mass 'm' (as shown in figure) moving with kinetic energy E compresses a spring through a distance 25 cm when, its speed is halved. The value of spring constant of used spring will be nE $\mathrm{Nm^{-1}}$ for n = __________.
Work, Energy and Power Work and Kinetic Energy Maximum Compression of a Spring
MediumNumerical value · figureIdea asked in 3 yearsAsked the year before
Q60
Four identical discs each of mass 'M' and diameter 'a' are arranged in a small plane as shown in figure. If the moment of inertia of the system about OO' is $\frac{x}{4}Ma^2$ . Then, the value of $x$ will be __________.
System of Particles and Rotational Motion Moment of Inertia MOI of Composite Bodies
MediumNumerical value · figureIdea asked in 5 yearsAsked the year before

Chemistry30 questions

12 easy16 medium2 hard

Q61
Identify the incorrect statement from the following.
Structure of Atom Quantum Numbers Magnetic Quantum Number
MediumSingle correctIdea asked in 4 yearsAsked the year before
Q62
Which of the following relation is not correct ?
Thermodynamics (Chemistry)Enthalpy and Hess's Law Relation Between ΔH and ΔU
EasySingle correctIdea asked in 4 yearsAsked the year before
Q63
Match List - I with List - II. List - I | List - II (A) $\mathrm{Cd(s)+2Ni(OH)_3(s)\to CdO(s)+2Ni(OH)_2(s)+H_2O(\mathit{l})}$ | (I) Primary battery (B) $\mathrm{Zn(Hg)+HgO(s)\to ZnO(s)+Hg(\mathit{l})}$ | (II) Discharging of secondary battery (C)…
Electrochemistry
MediumMatch the list
Q64
Match List - I with List - II. List - I (Reaction) | List - II (Catalyst) (A) $\mathrm{4NH_3(g)+5O_2(g)\to4NO(g)+6H_2O(g)}$ | (I) NO(g) (B) $\mathrm{N_2(g)+3H_2(g)\to2NH_3(g)}$ | (II) $\mathrm{H_2SO_4(\mathit{l})}$ (C)…
Surface Chemistry
EasyMatch the list
Q65
In which of the following pairs, electron gain enthalpies of constituent elements are nearly the same or identical ? (A) Rb and Cs (B) Na and K (C) Ar and Kr (D) I and At Choose the correct answer from the options given below :
Classification of Elements and Periodicity in Properties Periodic Trends Electron Gain Enthalpy
MediumOtherIdea asked in 5 yearsAsked the year before
Q66
Which of the reaction is suitable for concentrating ore by leaching process ?
General Principles and Processes of Isolation of Elements Metallurgical Extraction Bayer Process Chemistry
EasySingle correctIdea asked in 2 yearsAsked the year before
Q67
The metal salts formed during softening of hardwater using Clark's method are :
Hydrogen Hydrogen and Hydrides Clark's Method for Temporary Hardness
EasySingle correctNew idea
Q68
Which of the following statement is incorrect ?
The s-Block Elements s-Block Elements Anomalous Solubility of Fluorides
MediumSingle correctNew idea
Q69
Match List - I with List - II, match the gas evolved during each reaction. List - I | List - II (A) $\mathrm{(NH_4)_2Cr_2O_7\xrightarrow{\Delta}}$ | (I) $\mathrm{H_2}$ (B) $\mathrm{KMnO_4+HCl\to}$ | (II) $\mathrm{N_2}$ (C) $\mathrm{Al+NaOH+H_2O\to}$ | (III)…
The p-Block Elements Group 15 & 16 Preparation of Dinitrogen
MediumMatch the listIdea asked in 2 years
Q70
Which of the following has least tendency to liberate $\mathrm{H_2}$ from mineral acids ?
Electrochemistry Nernst Equation Standard Cell Potential
EasySingle correctIdea asked in 7 yearsAsked the year before
Q71
Given below are two statements : Statement I : In polluted water values of both dissolved oxygen and BOD are very low. Statement II : Eutrophication results in decrease in the amount of dissolved oxygen. In the light of the above statements, choose the most…
Environmental Chemistry BOD Values for Clean vs Polluted Water
EasyTwo statementsIdea asked in 2 yearsAsked the year before
Q72
Match List - I with List - II. List - I | List - II (A) [figure: five-membered saturated ring containing one O atom (tetrahydrofuran-type)] | (I) Spiro compound (B) [figure: bicyclic skeleton drawn as a bowtie/X shape with a vertical bond in the middle] |…
Organic Chemistry: Some Basic Principles and Techniques IUPAC Nomenclature Bicyclo and Spiro Nomenclature
MediumMatch the list · figureNew idea
Q73
Choose the correct option for the following reactions. $\mathrm{(CH_3)_3C-CH=CH_2}$ (3,3-dimethylbut-1-ene, drawn as $\mathrm{H_3C-C(CH_3)(CH_3)-CH=CH_2}$ ) $\xrightarrow{\mathrm{Hg(OAc)_2,\ H_2O\ /\ NaBH_4}}$ A and the same alkene…
Hydrocarbons Alkenes and Alkynes Oxymercuration-Demercuration
MediumSingle correct · figureNew idea
Q74
Among the following marked proton of which compound shows lowest $\mathrm{pK_a}$ value ?
Organic Chemistry: Some Basic Principles and Techniques Organic Acidity and Basicity Acidity of Alpha-Hydrogens in Carbonyls
MediumSingle correct · figureIdea asked in 2 years
Q75
Identify the major products A and B for the below given reaction sequence. Benzene $\xrightarrow{\text{(1) }\mathrm{CH_3CHClCH_3,\ AlCl_3};\ \text{(2) }\mathrm{O_2};\ \text{(3) }\mathrm{H^{\oplus},\ H_2O}}$ P; P $\xrightarrow{\mathrm{Na_2Cr_2O_7/H_2SO_4}}$ …
Alcohols, Phenols and Ethers Alcohols and Phenols Oxidation of Phenol to Benzoquinone
MediumSingle correct · figureNew idea
Q76
Identify the correct statement for the below given transformation. $\mathrm{CH_3-CH_2-CH_2-CH-CH_3}$ with $\mathrm{N^{\oplus}(CH_3)_3}$ on the CH carbon $\xrightarrow[\mathrm{C_2H_5OH}]{\mathrm{C_2H_5ONa}}$ A (Major) + B (Minor)
Amines Hofmann Elimination of Quaternary Salts
MediumSingle correctIdea asked in 2 yearsAsked the year before
Q77
Terylene polymer is obtained by condensation of :
Polymers Terylene (Dacron) Monomers
EasySingle correctNew idea
Q78
For the below given cyclic hemiacetal (X), the correct pyranose structure is :
Biomolecules Carbohydrates Haworth Projections
HardDiagram based · figureNew idea
Q79
Statements about Enzyme Inhibitor Drugs are given below : (A) There are Competitive and Non-competitive inhibitor drugs. (B) These can bind at the active sites and allosteric sites. (C) Competitive Drugs are allosteric site blocking drugs. (D)…
Chemistry in Everyday Life Competitive vs Allosteric Enzyme Inhibition
EasyOtherIdea asked in 2 yearsAsked the year before
Q80
For kinetic study of the reaction of iodide ion with $\mathrm{H_2O_2}$ at room temperature : (A) Always use freshly prepared starch solution. (B) Always keep the concentration of sodium thiosulphate solution less than that of KI solution. (C) Record the time…
Chemical Kinetics
MediumOther
Q81
In the given reaction, $\mathrm{X+Y+3Z\rightleftharpoons XYZ_3}$ if one mole of each of X and Y with 0.05 mol of Z gives compound $\mathrm{XYZ_3}$ . (Given : Atomic masses of X, Y and Z are 10, 20 and 30 amu, respectively.) The yield of $\mathrm{XYZ_3}$ is…
Some Basic Concepts of Chemistry Mole Concept Limiting Reagent and Reaction Yield
MediumNumerical valueIdea asked in 4 years
Q82
An element M crystallises in a body centred cubic unit cell with a cell edge of 300 pm. The density of the element is $6.0\ \mathrm{g\,cm^{-3}}$ . The number of atoms present in 180 g of the element is __________ $\times10^{23}$ . (Nearest integer)
The Solid State Crystal Lattices Unit Cell Density
MediumNumerical valueIdea asked in 4 yearsAsked the year before
Q83
The number of paramagnetic species among the following is __________. $\mathrm{B_2,\ Li_2,\ C_2,\ C_2^{-},\ O_2^{2-},\ O_2^{+}\ and\ He_2^{+}}$
Chemical Bonding and Molecular Structure Molecular Orbital Theory Magnetic Property Prediction
MediumNumerical valueIdea asked in 6 yearsAsked the year before
Q84
150 g of acetic acid was contaminated with 10.2 g ascorbic acid ( $\mathrm{C_6H_8O_6}$ ) to lower down its freezing point by $(x\times10^{-1})^\circ$ C. The value of $x$ is __________. (Nearest integer) [Given $K_f=3.9\ \mathrm{K\,kg\,mol^{-1}}$ ; molar mass of…
Solutions Colligative Properties Freezing Point Depression
EasyNumerical valueIdea asked in 4 yearsAsked the year before
Q85
$K_a$ for butyric acid ( $\mathrm{C_3H_7COOH}$ ) is $2\times10^{-5}$ . The pH of 0.2 M solution of butyric acid is __________ $\times10^{-1}$ . (Nearest integer) [Given $\log 2=0.30$ ]
Equilibrium pH and Buffers pH of Weak Acids and Bases
EasyNumerical valueIdea asked in 4 yearsAsked the year before
Q86
For the given first order reaction $\mathrm{A\to B}$ the half life of the reaction is 0.3010 min. The ratio of the initial concentration of reactant to the concentration of reactant at time 2.0 min will be equal to __________. (Nearest integer)
Chemical Kinetics Rate Laws First-Order Kinetics
EasyNumerical valueNew ideaIdea asked in 2 years
Q87
The number of interhalogens from the following having square pyramidal structure is : $\mathrm{ClF_3,\ IF_7,\ BrF_5,\ BrF_3,\ I_2Cl_6,\ IF_5,\ ClF,\ ClF_5}$
The p-Block Elements Group 17 & 18 Interhalogen Compounds
MediumNumerical valueIdea asked in 3 yearsAsked the year before
Q88
The disproportionation of $\mathrm{MnO_4^{2-}}$ in acidic medium resulted in the formation of two manganese compounds A and B. If the oxidation state of Mn in B is smaller than that of A, then the spin-only magnetic moment ( $\mu$ ) value of B in BM is…
The d- and f-Block Elements d-Block Elements Disproportionation of Manganate
MediumNumerical valueNew idea
Q89
Total number of relatively more stable isomer(s) possible for octahedral complex $\mathrm{[Cu(en)_2(SCN)_2]}$ will be __________.
Coordination Compounds Coordination Isomerism Ambidentate Ligands in Linkage Isomerism
HardNumerical valueNew ideaIdea asked in 3 years
Q90
On complete combustion of 0.492 g of an organic compound containing C, H and O, 0.7938 g of $\mathrm{CO_2}$ and 0.4428 g of $\mathrm{H_2O}$ was produced. The % composition of oxygen in the compound is __________.
Organic Chemistry: Some Basic Principles and Techniques Stoichiometry Empirical Formula from Combustion Data
EasyNumerical valueIdea asked in 4 yearsAsked the year before

Maths30 questions

3 easy15 medium12 hard

Q1
Let the solution curve of the differential equation $x\,dy=\left(\sqrt{x^2+y^2}+y\right)dx,\ x>0$ , intersect the line $x=1$ at $y=0$ and the line $x=2$ at $y=\alpha$ . Then the value of $\alpha$ is :
Differential Equations Homogeneous Equations Substitution y = vx
MediumSingle correctIdea asked in 6 yearsAsked the year before
Q2
Considering only the principal values of the inverse trigonometric functions, the domain of the function $f(x)=\cos^{-1}\left(\frac{x^2-4x+2}{x^2+3}\right)$ is :
Inverse Trigonometric Functions Inverse Trig Domains Domain of Rational Composite Inverse Trig
EasySingle correctNew idea
Q3
Let the vectors $\vec{a}=(1+t)\hat{i}+(1-t)\hat{j}+\hat{k},\ \vec{b}=(1-t)\hat{i}+(1+t)\hat{j}+2\hat{k}$ and $\vec{c}=t\hat{i}-t\hat{j}+\hat{k},\ t\in\mathbf{R}$ be such that for $\alpha,\beta,\gamma\in\mathbf{R}$ ,…
Vector Algebra Linear Independence Independence of Three Vectors
EasySingle correctNew idea
Q4
Considering the principal values of the inverse trigonometric functions, the sum of all the solutions of the equation $\cos^{-1}(x)-2\sin^{-1}(x)=\cos^{-1}(2x)$ is equal to :
Inverse Trigonometric Functions Inverse Trig Identities Cosine Inverse Sum and Difference
HardSingle correctNew idea
Q5
Let the operations $*,\odot\in\{\wedge,\vee\}$ . If $(p*q)\odot(p\odot\sim q)$ is a tautology, then the ordered pair $(*,\odot)$ is :
Mathematical Reasoning
MediumSingle correct
Q6
Let a vector $\vec{a}$ has magnitude 9. Let a vector $\vec{b}$ be such that for every $(x,y)\in\mathbf{R}\times\mathbf{R}-\{(0,0)\}$ , the vector $(x\vec{a}+y\vec{b})$ is perpendicular to the vector $(6y\vec{a}-18x\vec{b})$ . Then the value of…
Vector Algebra Dot Product Orthogonality Condition
MediumSingle correctNew idea
Q7
For $t\in(0,2\pi)$ , if ABC is an equilateral triangle with vertices $A(\sin t,-\cos t)$ , $B(\cos t,\sin t)$ and $C(a,b)$ such that its orthocentre lies on a circle with centre $\left(1,\frac{1}{3}\right)$ , then $(a^2-b^2)$ is equal to :
Straight Lines
HardSingle correct
Q8
For $\alpha\in\mathbf{N}$ , consider a relation $R$ on $\mathbf{N}$ given by $R=\{(x,y):3x+\alpha y\text{ is a multiple of }7\}$ . The relation $R$ is an equivalence relation if and only if :
Relations and Functions Sets and Relations Equivalence Relations
MediumSingle correctIdea asked in 4 yearsAsked the year before
Q9
Out of 60% female and 40% male candidates appearing in an exam, 60% candidates qualify it. The number of females qualifying the exam is twice the number of males qualifying it. A candidate is randomly chosen from the qualified candidates. The probability,…
Probability Conditional Probability Conditional Probability Definition
EasySingle correctIdea asked in 2 yearsAsked the year before
Q10
If $y=y(x),\ x\in(0,\pi/2)$ be the solution curve of the differential equation $(\sin^2 2x)\frac{dy}{dx}+(8\sin^2 2x+2\sin 4x)y=2e^{-4x}(2\sin 2x+\cos 2x)$ , with $y(\pi/4)=e^{-\pi}$ , then $y(\pi/6)$ is equal to :
Differential Equations Linear Differential Equations Integrating Factors with Trigonometric Functions
HardSingle correctNew ideaIdea asked in 4 years
Q11
If the tangents drawn at the points P and Q on the parabola $y^2=2x-3$ intersect at the point R(0, 1), then the orthocentre of the triangle PQR is :
Conic Sections General Conic Section Chord of Contact for General Conic
HardSingle correctIdea asked in 2 yearsAsked the year before
Q12
Let C be the centre of the circle $x^2+y^2-x+2y=\frac{11}{4}$ and P be a point on the circle. A line passes through the point C, makes an angle of $\frac{\pi}{4}$ with the line CP and intersects the circle at the points Q and R. Then the area of the triangle…
Conic Sections
MediumSingle correct
Q13
The remainder when $7^{2022}+3^{2022}$ is divided by 5 is :
Binomial Theorem
MediumSingle correct
Q14
Let the matrix $A=\begin{bmatrix}0&1&0\\0&0&1\\1&0&0\end{bmatrix}$ and the matrix $B_0=A^{49}+2A^{98}$ . If $B_n=\mathrm{Adj}(B_{n-1})$ for all $n\geq1$ , then $\det(B_4)$ is equal to :
Determinants Matrix Inverse Trap involving |adj(A)| vs |A^-1|
HardSingle correctIdea asked in 6 yearsAsked the year before
Q15
Let $S_1=\left\{z_1\in\mathbf{C}:|z_1-3|=\frac{1}{2}\right\}$ and $S_2=\{z_2\in\mathbf{C}:|z_2-|z_2+1||=|z_2+|z_2-1||\}$ . Then, for $z_1\in S_1$ and $z_2\in S_2$ , the least value of $|z_2-z_1|$ is :
Complex Numbers and Quadratic Equations Complex Numbers - Polar Form Polar Triangle Inequality Extrema
HardSingle correctIdea asked in 5 yearsAsked the year before
Q16
The foot of the perpendicular from a point on the circle $x^2+y^2=1,\ z=0$ to the plane $2x+3y+z=6$ lies on which one of the following curves ?
Three Dimensional Geometry Point-Plane Distance Foot of Perpendicular
HardSingle correctIdea asked in 3 yearsAsked the year before
Q17
If the minimum value of $f(x)=\frac{5x^2}{2}+\frac{\alpha}{x^5},\ x>0$ , is 14, then the value of $\alpha$ is equal to :
Application of Derivatives Maxima and Minima Optimization via AM-GM Inequality
MediumSingle correctIdea asked in 4 yearsAsked the year before
Q18
Let $\alpha,\beta$ and $\gamma$ be three positive real numbers. Let $f(x)=\alpha x^5+\beta x^3+\gamma x,\ x\in\mathbf{R}$ and $g:\mathbf{R}\to\mathbf{R}$ be such that $g(f(x))=x$ for all $x\in\mathbf{R}$ . If $a_1,a_2,a_3,\ldots,a_n$ be in arithmetic…
Relations and Functions
MediumSingle correct
Q19
Consider the sequence $a_1,a_2,a_3,\ldots$ such that $a_1=1,\ a_2=2$ and $a_{n+2}=\frac{2}{a_{n+1}}+a_n$ for $n=1,2,3,\ldots$ . If…
Sequences and Series
HardSingle correct
Q20
The minimum value of the twice differentiable function $f(x)=\int_0^x e^{x-t}f'(t)\,dt-(x^2-x+1)e^x,\ x\in\mathbf{R}$ , is :
Application of Derivatives Linear Differential Equations LDEs Formed via Leibniz Integral Rule
HardSingle correctIdea asked in 3 yearsAsked the year before
Q21
Let S be the set of all passwords which are six to eight characters long, where each character is either an alphabet from $\{A,B,C,D,E\}$ or a number from $\{1,2,3,4,5\}$ with the repetition of characters allowed. If the number of passwords in S whose at…
Permutations and Combinations Fundamental Counting Principle
MediumNumerical valueIdea asked in 5 yearsAsked the year before
Q22
Let $P(-2,-1,1)$ and $Q\left(\frac{56}{17},\frac{43}{17},\frac{111}{17}\right)$ be the vertices of the rhombus PRQS. If the direction ratios of the diagonal RS are $\alpha,-1,\beta$ , where both $\alpha$ and $\beta$ are integers of minimum absolute values,…
Three Dimensional Geometry Dot Product Perpendicular Diagonals of a Rhombus
MediumNumerical valueNew idea
Q23
Let $f:[0,1]\to\mathbf{R}$ be a twice differentiable function in $(0,1)$ such that $f(0)=3$ and $f(1)=5$ . If the line $y=2x+3$ intersects the graph of $f$ at only two distinct points in $(0,1)$ , then the least number of points $x\in(0,1)$ , at which…
Application of Derivatives Rolle's Theorem Higher-Order Derivative Roots
MediumNumerical valueIdea asked in 3 yearsAsked the year before
Q24
If $\int_0^{\sqrt{3}}\frac{15x^3}{\sqrt{1+x^2+\sqrt{(1+x^2)^3}}}\,dx=\alpha\sqrt{2}+\beta\sqrt{3}$ , where $\alpha,\beta$ are integers, then $\alpha+\beta$ is equal to __________.
Integrals Integration by Substitution Pure Quadratic Outside and Inside Root Substitution
HardNumerical valueNew idea
Q25
Let $A=\begin{bmatrix}1&-1\\2&\alpha\end{bmatrix}$ and $B=\begin{bmatrix}\beta&1\\1&0\end{bmatrix},\ \alpha,\beta\in\mathbf{R}$ . Let $\alpha_1$ be the value of $\alpha$ which satisfies $(A+B)^2=A^2+\begin{bmatrix}2&2\\2&2\end{bmatrix}$ and $\alpha_2$ be the…
Matrices Matrix Multiplication Binomial Expansion for Non-Commuting Matrices
MediumNumerical valueNew idea
Q26
For $p,q\in\mathbf{R}$ , consider the real valued function $f(x)=(x-p)^2-q,\ x\in\mathbf{R}$ and $q>0$ . Let $a_1,a_2,a_3$ and $a_4$ be in an arithmetic progression with mean $p$ and positive common difference. If $|f(a_i)|=500$ for all $i=1,2,3,4$ , then the…
Complex Numbers and Quadratic Equations Quadratic Roots Difference of Roots
MediumNumerical valueNew idea
Q27
For the hyperbola $H:x^2-y^2=1$ and the ellipse $E:\frac{x^2}{a^2}+\frac{y^2}{b^2}=1,\ a>b>0$ , let the (1) eccentricity of E be reciprocal of the eccentricity of H, and (2) the line $y=\sqrt{\frac{5}{2}}x+K$ be a common tangent of E and H. Then $4(a^2+b^2)$ …
Conic Sections General Conic Section Condition for Line to be Tangent
HardNumerical valueIdea asked in 3 yearsAsked the year before
Q28
Let $x_1,x_2,x_3,\ldots,x_{20}$ be in geometric progression with $x_1=3$ and the common ratio $\frac{1}{2}$ . A new data is constructed replacing each $x_i$ by $(x_i-i)^2$ . If $\bar{x}$ is the mean of new data, then the greatest integer less than or equal to…
Statistics AP and GP GP Sum of Squares
MediumNumerical valueNew ideaIdea asked in 2 years
Q29
$\lim_{x\to0}\left(\frac{(x+2\cos x)^3+2(x+2\cos x)^2+3\sin(x+2\cos x)}{(x+2)^3+2(x+2)^2+3\sin(x+2)}\right)^{\frac{100}{x}}$ is equal to __________.
Continuity and Differentiability L'Hopital's Rule e-Power Shortcut for 1^infinity Forms
HardNumerical valueIdea asked in 3 yearsAsked the year before
Q30
The sum of all real values of $x$ for which $\frac{3x^2-9x+17}{x^2+3x+10}=\frac{5x^2-7x+19}{3x^2+5x+12}$ is equal to __________.
Complex Numbers and Quadratic Equations Quadratic Roots Root-Coefficient Relation
MediumNumerical valueIdea asked in 6 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 2022 shifts.