JEE Main 2022 · June session

JEE Main 26 June 2022, Shift 1

89 questions (1 more dropped by NTA): 30 numerical-answer and 59 multiple choice. 14 were rated hard and 30 easy. Its hard-question count was the joint 7th highest of the 24 2022 shifts Lumi analysed. The biggest chapter was Organic Chemistry: Some Basic Principles and Techniques, with 4 questions.

Questions: 89; 1 dropped by NTA
Numerical answer: 30; 34% of the shift
Hard: 14; 16% of the shift
Chapters asked: 65; 13 questions with a figure

Key findings · 26 Jun 2022, Shift 1

1
14 hard questions (16%): the joint 7th 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 8 of 29 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 74 questions tagged to an idea.
4
41 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: Electromagnetic Induction in Physics (2), Organic Chemistry: Some Basic Principles and Techniques in Chemistry (4) and Conic Sections 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 13Hard 5

Top chapters

Maths

29 Q · 10 numerical

Easy 1Medium 20Hard 8

Top chapters

Conic Sections3 Q
Application of Derivatives2 Q
Complex Numbers and Quadratic Equations2 Q
Determinants2 Q
Differential Equations2 Q
Integrals2 Q

Question format

Single correct47 Q
Numerical value30 Q
Diagram based6 Q
Other3 Q
Two statements2 Q
Assertion–reason1 Q

What each question asks you to do

Calculation31 Q
Multi-step28 Q
Conceptual12 Q
Recall11 Q
Data or graph4 Q
Application3 Q

Most common: Calculation.

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
Open29 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 89 questions of this shift

The opening words of each question, with its chapter, topic, idea and how hard it was. Open one for the full text, options and the official answer. “New idea” means no earlier year had tested it.

Subject

Difficulty

Idea

Find

89 of 89 questions

Physics30 questions

17 easy12 medium1 hard

Q31
An expression for a dimensionless quantity P is given by $P=\frac{\alpha}{\beta}\log_e\left(\frac{kt}{\beta x}\right)$ ; where $\alpha$ and $\beta$ are constants, $x$ is distance; k is Boltzmann constant and t is the temperature. Then the dimensions of…
Units and Measurements
MediumSingle correct
Q32
A person is standing in an elevator. In which situation, he experiences weight loss ?
Laws of Motion Newton's Laws of Motion Apparent Weight in Accelerating Frames
EasySingle correctIdea asked in 3 yearsAsked the year before
Q33
An object is thrown vertically upwards. At its maximum height, which of the following quantity becomes zero ?
Motion in a Straight Line Kinematics in 1D Free Fall Sign Conventions
EasySingle correctIdea asked in 5 years
Q34
A ball is released from rest from point P of a smooth semi-spherical vessel as shown in figure. The ratio of the centripetal force and normal reaction on the ball at point Q is A while angular position of point Q is $\alpha$ with respect to point P. Which of…
Laws of Motion Circular Motion Vertical Circular Motion
HardDiagram based · figureIdea asked in 2 yearsAsked the year before
Q35
A thin circular ring of mass M and radius R is rotating with a constant angular velocity 2 $\mathrm{rads^{-1}}$ in a horizontal plane about an axis vertical to its plane and passing through the center of the ring. If two objects each of mass m be attached…
System of Particles and Rotational Motion Angular Momentum Conservation of Angular Momentum
EasySingle correctIdea asked in 5 yearsAsked the year before
Q36
The variation of acceleration due to gravity (g) with distance (r) from the center of the earth is correctly represented by : (Given R=radius of earth)
Gravitation
EasyDiagram based · figure
Q37
The efficiency of a Carnot's engine, working between steam point and ice point, will be :
Thermodynamics Heat Engines and Refrigerators Carnot Efficiency Limit
EasySingle correctIdea asked in 3 yearsAsked the year before
Q38
Time period of a simple pendulum in a stationary lift is 'T'. If the lift accelerates with $\frac{g}{6}$ vertically upwards then the time period will be : (Where g=acceleration due to gravity)
Oscillations Pendulums Effective Gravity Effects
EasySingle correctNew idea
Q39
A thermally insulated vessel contains an ideal gas of molecular mass M and ratio of specific heats 1.4. Vessel is moving with speed $v$ and is suddenly brought to rest. Assuming no heat is lost to the surrounding and vessel temperature of the gas increases…
Kinetic Theory Kinetic Theory of Gases Gamma and Degrees of Freedom
MediumSingle correctIdea asked in 5 yearsAsked the year before
Q40
Two capacitors having capacitance $C_1$ and $C_2$ respectively are connected as shown in figure. Initially, capacitor $C_1$ is charged to a potential difference V volt by a battery. The battery is then removed and the charged capacitor $C_1$ is now connected…
Electrostatic Potential and Capacitance Parallel Plate Capacitors Charge Sharing and Heat Loss
EasyDiagram based · figureIdea asked in 6 yearsAsked the year before
Q41
Given below two statements : One is labelled as Assertion (A) and other is labelled as Reason (R). Assertion (A) : Non-polar materials do not have any permanent dipole moment. Reason (R) : When a non-polar material is placed in an electric field, the centre…
Electrostatic Potential and Capacitance Dielectrics Polar vs. Non-Polar Dielectrics
EasyAssertion–reasonNew ideaIdea asked in 2 years
Q42
The magnetic flux through a coil perpendicular to its plane is varying according to the relation $\phi=(5t^3+4t^2+2t-5)$ Weber. If the resistance of the coil is 5 ohm, then the induced current through the coil at t=2 s will be,
Electromagnetic Induction Faraday's Law Magnetic Flux Calculation
EasySingle correctNew ideaIdea asked in 2 years
Q43
An aluminium wire is stretched to make its length, 0.4% larger. The percentage change in resistance is :
Current Electricity Ohm's Law and Resistance Small Stretch Approximation for Resistance
EasySingle correctNew idea
Q44
A proton and an alpha particle of the same velocity enter in a uniform magnetic field which is acting perpendicular to their direction of motion. The ratio of the radii of the circular paths described by the alpha particle and proton is :
Moving Charges and Magnetism Lorentz Force Trajectory Radii Ratio for Same Velocity
EasySingle correctNew idea
Q45
If Electric field intensity of a uniform plane electro magnetic wave is given as $E=-301.6\,\sin(kz-\omega t)\,\hat{a}_x+452.4\,\sin(kz-\omega t)\,\hat{a}_y\ \frac{V}{m}$ . Then, magnetic intensity 'H' of this wave in $\mathrm{Am^{-1}}$ will be : [Given :…
Electromagnetic Waves
MediumSingle correct
Q46
In free space, an electromagnetic wave of 3 GHz frequency strikes over the edge of an object of size $\frac{\lambda}{100}$ , where $\lambda$ is the wavelength of the wave in free space. The phenomenon, which happens there will be :
Electromagnetic Waves
MediumSingle correct
Q47
An electron with speed $v$ and a photon with speed c have the same de-Broglie wavelength. If the kinetic energy and momentum of electron are $E_e$ and $p_e$ and that of photon are $E_{ph}$ and $p_{ph}$ respectively. Which of the following is correct ?
Dual Nature of Radiation and Matter Matter Waves Electron vs Photon (Same Wavelength)
MediumSingle correctIdea asked in 3 yearsAsked the year before
Q48
How many alpha and beta particles are emitted when Uranium $_{92}\mathrm{U}^{238}$ decays to lead $_{82}\mathrm{Pb}^{206}$ ?
Nuclei Radioactive Decay Counting Alpha and Beta Emissions
EasySingle correctIdea asked in 2 yearsAsked the year before
Q49
The I-V characteristics of a p-n junction diode in forward bias is shown in the figure. The ratio of dynamic resistance, corresponding to forward bias voltage of 2 V and 4 V respectively, is :
Semiconductor Electronics: Materials, Devices and Simple Circuits Semiconductor Diodes Dynamic Resistance Calculation
MediumDiagram based · figureNew idea
Q50
Choose the correct statement for amplitude modulation :
Communication Systems
EasySingle correct
Q51
A fighter jet is flying horizontally at a certain altitude with a speed of 200 $\mathrm{ms^{-1}}$ . When it passes directly overhead an anti-aircraft gun, a bullet is fired from the gun, at an angle $\theta$ with the horizontal, to hit the jet. If the bullet…
Motion in a Plane Projectile Motion Relative Motion of Two Projectiles
MediumNumerical valueNew idea
Q52
A ball of mass 0.5 kg is dropped from the height of 10 m. The height, at which the magnitude of velocity becomes equal to the magnitude of acceleration due to gravity, is __________ m. [Use $g=10\ \mathrm{m/s^2}$ ]
Motion in a Straight Line Kinematics in 1D Free Fall Sign Conventions
EasyNumerical valueIdea asked in 5 years
Q53
The elastic behaviour of material for linear stress and linear strain, is shown in the figure. The energy density for a linear strain of $5\times10^{-4}$ is __________ $\mathrm{kJ/m^3}$ . Assume that material is elastic upto the linear strain of…
Mechanical Properties of Solids Elasticity Elastic Energy Density
MediumNumerical value · figureIdea asked in 4 years
Q54
The elongation of a wire on the surface of the earth is $10^{-4}$ m. The same wire of same dimensions is elongated by $6\times10^{-5}$ m on another planet. The acceleration due to gravity on the planet will be __________ $\mathrm{ms^{-2}}$ . (Take…
Mechanical Properties of Solids Elasticity Elongation Due to Self-Weight
EasyNumerical valueIdea asked in 3 yearsAsked the year before
Q55
A 10 $\Omega$ , 20 mH coil carrying constant current is connected to a battery of 20 V through a switch. Now after switch is opened current becomes zero in 100 $\mu$ s. The average e.m.f. induced in the coil is __________ V.
Electromagnetic Induction Inductance Self-Inductance and Back EMF
EasyNumerical valueIdea asked in 5 yearsAsked the year before
Q56
A light ray is incident, at an incident angle $\theta_1$ , on the system of two plane mirrors $M_1$ and $M_2$ having an inclination angle $75^\circ$ between them (as shown in figure). After reflecting from mirror $M_1$ it gets reflected back by the mirror…
Ray Optics and Optical Instruments Spherical Mirrors Multiple Reflections in Two-Mirror Systems
MediumNumerical value · figureNew idea
Q57
In a vernier callipers, each cm on the main scale is divided into 20 equal parts. If tenth vernier scale division coincides with nineth main scale division. Then the value of vernier contant will be __________ $\times10^{-2}$ mm.
Units and Measurements
EasyNumerical value
Q58
As per the given circuit, the value of current through the battery will be __________ A.
Semiconductor Electronics: Materials, Devices and Simple Circuits Semiconductor Diodes Multi-Diode State Determination
MediumNumerical value · figureNew idea
Q59
A 110 V, 50 Hz, AC source is connected in the circuit (as shown in figure). The current through the resistance 55 $\Omega$ , at resonance in the circuit, will be __________ A.
Alternating Current LCR Resonance Parallel Resonance (Anti-resonance)
MediumNumerical value · figureNew idea
Q60
An ideal fluid of density 800 $\mathrm{kgm^{-3}}$ , flows smoothly through a bent pipe (as shown in figure) that tapers in cross-sectional area from a to $\frac{a}{2}$ . The pressure difference between the wide and narrow sections of pipe is 4100 Pa. At wider…
Mechanical Properties of Fluids Fluid Dynamics Bernoulli's Equation
MediumNumerical value · figureIdea asked in 5 years

Chemistry30 questions

12 easy13 medium5 hard

Q61
A commercially sold conc. HCl is 35% HCl by mass. If the density of this commercial acid is 1.46 g/mL, the molarity of this solution is : (Atomic mass : Cl=35.5 amu, H=1 amu)
Some Basic Concepts of Chemistry Mole Concept Molarity Calculation
EasySingle correctIdea asked in 5 yearsAsked the year before
Q62
An evacuated glass vessel weighs 40.0 g when empty, 135.0 g when filled with a liquid of density 0.95 g mL $^{-1}$ and 40.5 g when filled with an ideal gas at 0.82 atm at 250 K. The molar mass of the gas in g mol $^{-1}$ is : (Given : R=0.082 L atm K $^{-1}$ …
States of Matter Ideal Gas Laws Ideal Gas Equation
MediumSingle correctIdea asked in 2 yearsAsked the year before
Q63
If the radius of the 3 $^{rd}$ Bohr's orbit of hydrogen atom is r $_3$ and the radius of 4 $^{th}$ Bohr's orbit is r $_4$ . Then :
Structure of Atom Bohr's Atomic Theory Bohr Radius Calculation
EasySingle correctIdea asked in 6 yearsAsked the year before
Q64
Consider the ions/molecule $\mathrm{O_2^+, O_2, O_2^-, O_2^{2-}}$ For increasing bond order the correct option is :
Chemical Bonding and Molecular Structure Molecular Orbital Theory Oxygen Ion Series Comparison
EasySingle correctIdea asked in 3 yearsAsked the year before
Q65
The $\left(\frac{\partial E}{\partial T}\right)_P$ of different types of half cells are as follows : A: $1\times10^{-4}$ B: $2\times10^{-4}$ C: $0.1\times10^{-4}$ D: $0.2\times10^{-4}$ (Where E is the electromotive force) Which of the above half cells would…
Electrochemistry Nernst Equation Temperature Coefficient of Cell EMF
MediumSingle correctIdea asked in 2 yearsAsked the year before
Q66
Choose the correct stability order of group 13 elements in their +1 oxidation state.
The p-Block Elements Group 13 & 14 Inert Pair Effect
EasySingle correctIdea asked in 5 yearsAsked the year before
Q67
Given below are two statements : Statement I : According to the Ellingham diagram, any metal oxide with higher $\Delta G^\circ$ is more stable than the one with lower $\Delta G^\circ$ . Statement II : The metal involved in the formation of oxide placed lower…
General Principles and Processes of Isolation of Elements Metallurgical Extraction Ellingham Diagram Basics
MediumTwo statementsIdea asked in 3 yearsAsked the year before
Q68
Consider the following reaction : $\mathrm{2HSO_4^-(aq)\xrightarrow[(2)\ Hydrolysis]{(1)\ Electrolysis}2HSO_4^-+2H^++\textbf{A}}$ The dihedral angle in product A in its solid phase at 110 K is :
Hydrogen Hydrogen and Hydrides Non-Planar Structure of H2O2
MediumSingle correctNew idea
Q69
The correct order of melting point is :
The s-Block Elements s-Block Elements Melting and Boiling Point Trends
EasySingle correctNew ideaIdea asked in 2 years
Q70
The correct order of melting points of hydrides of group 16 elements is :
The p-Block Elements Group 15 & 16 Group 16 Hydrides Trends
EasySingle correctIdea asked in 4 yearsAsked the year before
Q71
Consider the following reaction : A + alkali $\rightarrow$ B (Major Product) If B is an oxoacid of phosphorus with no P–H bond, then A is :
The p-Block Elements Group 15 & 16 Phosphorus Oxoacids
MediumSingle correctIdea asked in 4 yearsAsked the year before
Q72
Polar stratospheric clouds facilitate the formation of :
Environmental Chemistry Role of Polar Stratospheric Clouds (PSCs)
EasySingle correctNew idea
Q73
Given below are two statements : Statement I : In 'Lassaigne's Test', when both nitrogen and sulphur are present in an organic compound, sodium thiocyanate is formed. Statement II : If both nitrogen and sulphur are present in an organic compound, then the…
Organic Chemistry: Some Basic Principles and Techniques
MediumTwo statements
Q74
$(\mathrm{C_7H_5O_2})_2\xrightarrow{h\nu}[\mathrm{X}]\rightarrow2\dot{\mathrm{C}}_6\mathrm{H}_5+2\mathrm{CO_2}$ Consider the above reaction and identify the intermidiate 'X'
Organic Chemistry: Some Basic Principles and Techniques
MediumOther · figure
Q75
[figure: drawn structure with the label MgBr at its upper end and the label MgBr at its lower end, the two joined by a bracket-shaped line] + $\mathrm{CH_3-C(=O)-CH_2-C(=O)-CH_3}$ (each C=O drawn with O above) $\rightarrow$ 'A' $\xrightarrow{\mathrm{H_2O}}$ …
Aldehydes, Ketones and Carboxylic Acids Nucleophilic Addition Grignard Reagent Addition
HardOther · figureIdea asked in 4 yearsAsked the year before
Q76
Which will have the highest enol content ?
Aldehydes, Ketones and Carboxylic Acids Structural Isomerism Enol Content Stability Factors
MediumDiagram based · figureNew idea
Q77
Among the following structures, which will show the most stable enamine formation ? (Where Me is $-\mathrm{CH_3}$ )
Amines Nucleophilic Addition Enamine Formation
HardDiagram based · figureNew idea
Q78
Which of the following sets are correct regarding polymer ? (A) Copolymer : Buna–S (B) Condensation polymer : Nylon–6,6 (C) Fibres : Nylon–6,6 (D) Thermosetting polymer : Terylene (E) Homopolymer : Buna–N Choose the correct answer from given options below :
Polymers Polymer Classification
EasyOtherIdea asked in 3 yearsAsked the year before
Q79
A chemical which stimulates the secretion of pepsin is :
Chemistry in Everyday Life Antihistamine Receptor Specificity
EasySingle correctIdea asked in 2 years
Q80
Which statement is not true with respect to nitrate ion test ?
Organic Chemistry: Some Basic Principles and Techniques Anion Analysis Nitrate Brown Ring Test
EasySingle correctNew idea
Q81
For complete combustion of methanol $\mathrm{CH_3OH(l)}+\frac{3}{2}\mathrm{O_2(g)}\rightarrow\mathrm{CO_2(g)}+2\mathrm{H_2O(l)}$ the amount of heat produced as measured by bomb calorimeter is 726 kJ mol $^{-1}$ at 27°C. The enthalpy of combustion for the…
Thermodynamics (Chemistry)Enthalpy and Hess's Law Relation Between ΔH and ΔU
MediumNumerical valueIdea asked in 4 yearsAsked the year before
Q82
A 0.5 percent solution of potassium chloride was found to freeze at $-0.24$ °C. The percentage dissociation of potassium chloride is __________. (Nearest integer) (Molal depression constant for water is 1.80 K kg mol $^{-1}$ and molar mass of KCl is 74.6 g…
Solutions Colligative Properties Degree of Dissociation from Van't Hoff Factor
MediumNumerical valueIdea asked in 5 yearsAsked the year before
Q83
50 mL of 0.1 M $\mathrm{CH_3COOH}$ is being titrated against 0.1 M NaOH. When 25 mL of NaOH has been added, the pH of the solution will be __________ $\times10^{-2}$ . (Nearest integer) (Given : $\mathrm{pK_a}$ ( $\mathrm{CH_3COOH}$ )=4.76) log 2=0.30 log…
Equilibrium pH and Buffers pH at Titration Half-Equivalence Point
EasyNumerical valueNew ideaIdea asked in 2 years
Q84
A flask is filled with equal moles of A and B. The half lives of A and B are 100 s and 50 s respectively and are independent of the initial concentration. The time required for the concentration of A to be four times that of B is __________s. (Given : ln…
Chemical Kinetics Integrated Rate Equations First Order Half-Life
HardNumerical valueNew ideaIdea asked in 3 years
Q85
2.0 g of $\mathrm{H_2}$ gas is adsorbed on 2.5 g of platinum powder at 300 K and 1 bar pressure. The volume of the gas adsorbed per gram of the adsorbent is __________ mL. (Given : R=0.083 L bar K $^{-1}$ mol $^{-1}$ )
States of Matter Ideal Gas Laws Ideal Gas Equation
MediumNumerical valueIdea asked in 2 yearsAsked the year before
Q86
The spin-only magnetic moment value of the most basic oxide of vanadium among $\mathrm{V_2O_3}$ , $\mathrm{V_2O_4}$ and $\mathrm{V_2O_5}$ is __________ B.M. (Nearest integer)
The d- and f-Block Elements d-Block Elements Spin-only Magnetism
MediumNumerical valueIdea asked in 6 yearsAsked the year before
Q87
The spin-only magnetic moment value of an octahedral complex among $\mathrm{CoCl_3\cdot4NH_3}$ , $\mathrm{NiCl_2\cdot6H_2O}$ and $\mathrm{PtCl_4\cdot2HCl}$ , which upon reaction with excess of $\mathrm{AgNO_3}$ gives 2 moles of AgCl is __________ B.M. (Nearest…
Coordination Compounds Crystal Field Theory Spin-only Magnetic Moment
HardNumerical valueIdea asked in 7 yearsAsked the year before
Q88
On complete combustion 0.30 g of an organic compound gave 0.20 g of carbon dioxide and 0.10 g of water. The percentage of carbon in the given organic compound is __________. (Nearest Integer)
Organic Chemistry: Some Basic Principles and Techniques Stoichiometry Empirical Formula from Combustion Data
EasyNumerical valueIdea asked in 4 yearsAsked the year before
Q89
Compound 'P' on nitration with dil. $\mathrm{HNO_3}$ yields two isomers (A) and (B). These isomers can be separated by steam distillation. Isomers (A) and (B) show the intramolecular and intermolecular hydrogen bonding respectively. Compound (P) on reaction…
Alcohols, Phenols and Ethers Alcohols and Phenols Phenol Nitration Reagent Effects
HardNumerical valueNew ideaIdea asked in 2 years
Q90
The number of oxygens present in a nucleotide formed from a base, that is present only in RNA is __________.
Biomolecules
MediumNumerical value

Maths29 questions

1 easy20 medium8 hard

Q1
Let $f(x) = \frac{x-1}{x+1}$ , $x \in \mathbf{R} - \{0, -1, 1\}$ . If $f^{n+1}(x) = f(f^{n}(x))$ for all $n \in \mathbf{N}$ , then $f^{6}(6) + f^{7}(7)$ is equal to :
Relations and Functions
MediumSingle correct
Q2
Let $A = \left\{z \in \mathbf{C} : \left|\frac{z+1}{z-1}\right| < 1\right\}$ and $B = \left\{z \in \mathbf{C} : \arg\left(\frac{z-1}{z+1}\right) = \frac{2\pi}{3}\right\}$ . Then $A \cap B$ is :
Complex Numbers and Quadratic Equations Complex Numbers - Polar Form Locus of Argument Ratio (Circular Arc)
HardSingle correctIdea asked in 3 yearsAsked the year before
Q3
Let A be a $3 \times 3$ invertible matrix. If $|\mathrm{adj}\,(24A)| = |\mathrm{adj}\,(3\,\mathrm{adj}\,(2A))|$ , then $|A|^2$ is equal to :
Determinants Matrix Inverse Trap involving |adj(A)| vs |A^-1|
MediumSingle correctIdea asked in 6 yearsAsked the year before
Q4
The ordered pair (a, b), for which the system of linear equations $3x - 2y + z = b$ $5x - 8y + 9z = 3$ $2x + y + az = -1$ has no solution, is :
Determinants Cramer's Rule Condition for No Solution
MediumSingle correctIdea asked in 6 yearsAsked the year before
Q5
The remainder when $(2021)^{2023}$ is divided by 7 is :
Binomial Theorem
MediumSingle correct
Q6
$\lim_{x \to \frac{1}{\sqrt{2}}} \frac{\sin(\cos^{-1}x) - x}{1 - \tan(\cos^{-1}x)}$ is equal to :
Limits and Derivatives L'Hopital's Rule L'Hopital with Inverse Trigonometric Functions
MediumSingle correctNew ideaIdea asked in 2 years
Q7
Let $f, g : \mathbf{R} \to \mathbf{R}$ be two real valued functions defined as $f(x) = \begin{cases} -|x+3|, & x < 0 \\ e^{x}, & x \geq 0 \end{cases}$ and $g(x) = \begin{cases} x^2 + k_1 x, & x < 0 \\ 4x + k_2, & x \geq 0 \end{cases}$ , where $k_1$ and $k_2$ …
Continuity and Differentiability Continuity and Differentiability of Composite Functions
HardSingle correctIdea asked in 5 yearsAsked the year before
Q8
The sum of the absolute minimum and the absolute maximum values of the function $f(x) = |3x - x^2 + 2| - x$ in the interval $[-1, 2]$ is :
Application of Derivatives Maxima and Minima Extrema of Absolute Value Functions
MediumSingle correctNew idea
Q9
Let S be the set of all natural numbers, for which the line $\frac{x}{a} + \frac{y}{b} = 2$ is a tangent to the curve $\left(\frac{x}{a}\right)^n + \left(\frac{y}{b}\right)^n = 2$ at the point $(a, b)$ , $ab \neq 0$ . Then :
Application of Derivatives Tangents and Normals Tangents to Implicit Curves
MediumSingle correctIdea asked in 2 years
Q11
Let R be the point $(3, 7)$ and let P and Q be two points on the line $x + y = 5$ such that PQR is an equilateral triangle. Then the area of $\Delta$ PQR is :
Straight Lines Straight Line Distance Altitude Length of a Triangle
MediumSingle correctNew ideaIdea asked in 2 years
Q12
Let C be a circle passing through the points A $(2, -1)$ and B $(3, 4)$ . The line segment AB is not a diameter of C. If r is the radius of C and its centre lies on the circle $(x-5)^2 + (y-1)^2 = \frac{13}{2}$ , then $r^2$ is equal to :
Conic Sections
MediumSingle correct
Q13
Let the normal at the point P on the parabola $y^2 = 6x$ pass through the point $(5, -8)$ . If the tangent at P to the parabola intersects its directrix at the point Q, then the ordinate of the point Q is :
Conic Sections Normals to Conics Parabola Normal Parametric Form
MediumSingle correctNew idea
Q14
If the two lines $l_1 : \frac{x-2}{3} = \frac{y+1}{-2}, z = 2$ and $l_2 : \frac{x-1}{1} = \frac{2y+3}{\alpha} = \frac{z+5}{2}$ are perpendicular, then an angle between the lines $l_2$ and $l_3 : \frac{1-x}{3} = \frac{2y-1}{-4} = \frac{z}{4}$ is :
Three Dimensional Geometry Angle Between Lines Angle from Symmetric Equations
MediumSingle correctNew idea
Q15
Let the plane $2x + 3y + z + 20 = 0$ be rotated through a right angle about its line of intersection with the plane $x - 3y + 5z = 8$ . If the mirror image of the point $\left(2, -\frac{1}{2}, 2\right)$ in the rotated plane is B $(a, b, c)$ , then :
Three Dimensional Geometry Planes in 3D Family of Planes Perpendicular to a Third Plane
HardSingle correctIdea asked in 3 yearsAsked the year before
Q16
If $\vec{a} \cdot \vec{b} = 1$ , $\vec{b} \cdot \vec{c} = 2$ and $\vec{c} \cdot \vec{a} = 3$ , then the value of $\left[\vec{a} \times (\vec{b} \times \vec{c}),\ \vec{b} \times (\vec{c} \times \vec{a}),\ \vec{c} \times (\vec{b} \times \vec{a})\right]$ is :
Vector Algebra Triple Products VTP BAC-CAB Expansion
MediumSingle correctIdea asked in 2 yearsAsked the year before
Q17
Let a biased coin be tossed 5 times. If the probability of getting 4 heads is equal to the probability of getting 5 heads, then the probability of getting atmost two heads is :
Probability
MediumSingle correct
Q18
The mean of the numbers a, b, 8, 5, 10 is 6 and their variance is 6.8. If M is the mean deviation of the numbers about the mean, then 25 M is equal to :
Statistics Variance and SD Finding Unknown Observations from Mean and Variance
MediumSingle correctIdea asked in 7 yearsAsked the year before
Q19
Let $f(x) = 2\cos^{-1}x + 4\cot^{-1}x - 3x^2 - 2x + 10$ , $x \in [-1, 1]$ . If $[a, b]$ is the range of the function $f$ , then $4a - b$ is equal to :
Inverse Trigonometric Functions Inverse Trig Identities Range of Combined Inverse Trig Functions
MediumSingle correctNew idea
Q20
Let $\Delta, \nabla \in \{\wedge, \vee\}$ be such that $p \,\nabla\, q \Rightarrow ((p \,\Delta\, q) \,\nabla\, r)$ is a tautology. Then $(p \,\nabla\, q) \,\Delta\, r$ is logically equivalent to :
Mathematical Reasoning
MediumSingle correct
Q21
The sum of the cubes of all the roots of the equation $x^4 - 3x^3 - 2x^2 + 3x + 1 = 0$ is ________.
Complex Numbers and Quadratic Equations Quadratic Roots Newton's Sums
MediumNumerical valueIdea asked in 7 yearsAsked the year before
Q22
There are ten boys $B_1, B_2, \ldots, B_{10}$ and five girls $G_1, G_2, \ldots, G_5$ in a class. Then the number of ways of forming a group consisting of three boys and three girls, if both $B_1$ and $B_2$ together should not be the members of a group, is…
Permutations and Combinations Combinations of Distinct Objects
EasyNumerical valueIdea asked in 8 yearsAsked the year before
Q23
Let the common tangents to the curves $4(x^2 + y^2) = 9$ and $y^2 = 4x$ intersect at the point Q. Let an ellipse, centered at the origin O, has lengths of semi-minor and semi-major axes equal to OQ and 6, respectively. If e and $l$ respectively denote the…
Conic Sections Circle Tangents Common Tangent to Circle and Conic
HardNumerical valueIdea asked in 3 yearsAsked the year before
Q24
Let $f(x) = \max\{|x+1|, |x+2|, \ldots, |x+5|\}$ . Then $\int_{-6}^{0} f(x)\,dx$ is equal to ________.
Integrals Definite Integral Properties Integrals of Min and Max Functions
MediumNumerical valueNew idea
Q25
Let the solution curve $y = y(x)$ of the differential equation $(4 + x^2)\,dy - 2x(x^2 + 3y + 4)\,dx = 0$ pass through the origin. Then $y(2)$ is equal to ________.
Differential Equations Linear Differential Equations Particular Solution Solving
MediumNumerical valueIdea asked in 4 yearsAsked the year before
Q26
If $\sin^2(10^\circ)\,\sin(20^\circ)\,\sin(40^\circ)\,\sin(50^\circ)\,\sin(70^\circ) = \alpha - \frac{1}{16}\sin(10^\circ)$ , then $16 + \alpha^{-1}$ is equal to ________.
Trigonometric Functions Trigonometric Identities Sine Product Identity with 60 Degrees
HardNumerical valueNew ideaIdea asked in 2 years
Q27
Let $A = \{n \in \mathbf{N} : \mathrm{H.C.F.}\,(n, 45) = 1\}$ and Let $B = \{2k : k \in \{1, 2, \ldots, 100\}\}$ . Then the sum of all the elements of $A \cap B$ is ________.
Relations and Functions Sets and Relations Inclusion-Exclusion for Three Sets
MediumNumerical valueIdea asked in 4 yearsAsked the year before
Q28
The value of the integral $\frac{48}{\pi^4}\int_0^{\pi}\left(\frac{3\pi x^2}{2} - x^3\right)\frac{\sin x}{1 + \cos^2 x}\,dx$ is equal to ________.
Integrals Definite Integral Properties King's Property
HardNumerical valueIdea asked in 6 yearsAsked the year before
Q29
Let $A = \sum_{i=1}^{10}\sum_{j=1}^{10}\min\{i, j\}$ and $B = \sum_{i=1}^{10}\sum_{j=1}^{10}\max\{i, j\}$ . Then $A + B$ is equal to ________.
Sequences and Series
HardNumerical value
Q30
Let $S = (0, 2\pi) - \left\{\frac{\pi}{2}, \frac{3\pi}{4}, \frac{3\pi}{2}, \frac{7\pi}{4}\right\}$ . Let $y = y(x)$ , $x \in S$ , be the solution curve of the differential equation $\frac{dy}{dx} = \frac{1}{1 + \sin 2x}$ ,…
Differential Equations Variable Separable Form Initial Value Problem
HardNumerical valueIdea asked in 7 yearsAsked the year before

Dropped from the final key and left out of every count: Q10.

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.