JEE Main 2022 · June session

JEE Main 24 June 2022, Shift 2

90 questions: 30 numerical-answer and 60 multiple choice. 13 were rated hard and 36 easy. Its hard-question count was the joint 9th highest of the 24 2022 shifts Lumi analysed. The biggest chapter was Conic Sections, with 4 questions.

Questions: 90; none dropped
Numerical answer: 30; 33% of the shift
Hard: 13; 14% of the shift
Chapters asked: 69; 6 questions with a figure

Key findings · 24 Jun 2022, Shift 2

1
13 hard questions (14%): 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 11 of 30 rated hard; Chemistry had the most easy questions (17 of 30).
3
28 questions tested a new idea, one no earlier year had asked: 39% of the 72 questions tagged to an idea.
4
35 came back from 2021: the same idea had been asked in the 2021 shifts Lumi analysed.
5
33 shared an idea with another 2022 shift (46% of tagged questions). Shifts in the same year can test the same idea in a new setting.
6
Biggest chapters: Current Electricity in Physics (2), Aldehydes, Ketones and Carboxylic Acids in Chemistry (2) and Conic Sections in Maths (4).

The shift at a glance

Subjects, difficulty and chapters

Physics

30 Q · 10 numerical

Easy 16Medium 14Hard 0

Top chapters

Chemistry

30 Q · 10 numerical

Easy 17Medium 11Hard 2

Top chapters

Maths

30 Q · 10 numerical

Easy 3Medium 16Hard 11

Top chapters

Question format

Single correct55 Q
Numerical value30 Q
Two statements3 Q
Diagram based2 Q

What each question asks you to do

Multi-step32 Q
Calculation31 Q
Recall10 Q
Conceptual9 Q
Application6 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

16 easy14 medium0 hard

Q31
Identify the pair of physical quantities that have same dimensions :
Units and Measurements
EasySingle correct
Q32
The distance between Sun and Earth is R. The duration of year if the distance between Sun and Earth becomes 3R will be :
Gravitation Orbital Mechanics Kepler's Laws
EasySingle correctIdea asked in 4 yearsAsked the year before
Q33
A stone of mass m, tied to a string is being whirled in a vertical circle with a uniform speed. The tension in the string is
Laws of Motion Circular Motion Vertical Circular Motion
EasySingle correctIdea asked in 2 yearsAsked the year before
Q34
Two identical charged particles each having a mass 10 g and charge $2.0 \times 10^{-7}C$ are placed on a horizontal table with a separation of L between them such that they stay in limited equilibrium. If the coefficient of friction between each particle and…
Electric Charges and Fields Coulomb's Law Charge Equilibrium Analysis
MediumSingle correctNew ideaIdea asked in 2 years
Q35
A Carnot engine takes 5000 kcal of heat from a reservoir at $727^{\circ}C$ and gives heat to a sink at $127^{\circ}C$ . The work done by the engine is
Thermodynamics Heat Engines and Refrigerators Carnot Efficiency Limit
EasySingle correctIdea asked in 3 yearsAsked the year before
Q36
Two massless springs with spring constants 2 k and 9 k, carry 50 g and 100 g masses at their free ends. These two masses oscillate vertically such that their maximum velocities are equal. Then, the ratio of their respective amplitudes will be :
Oscillations SHM Equations Velocity and Accel vs Position
EasySingle correctIdea asked in 4 yearsAsked the year before
Q37
What will be the most suitable combination of three resistors A=2 $\Omega$ , B=4 $\Omega$ , C=6 $\Omega$ so that $\left(\frac{22}{3}\right)\Omega$ is equivalent resistance of combination?
Current Electricity
EasySingle correct
Q38
The soft-iron is a suitable material for making an electromagnet. This is because soft-iron has
Magnetism and Matter
EasySingle correct
Q39
A proton, a deuteron and an $\alpha$ -particle with same kinetic energy enter into a uniform magnetic field at right angle to magnetic field. The ratio of the radii of their respective circular paths is :
Moving Charges and Magnetism Lorentz Force Trajectory Radii Ratio for Same Kinetic Energy
EasySingle correctIdea asked in 3 yearsAsked the year before
Q40
Given below are two statements : Statement-I: The reactance of an ac circuit is zero. It is possible that the circuit contains a capacitor and an inductor. Statement-II: In ac circuit, the average power delivered by the source never becomes zero. In the…
Alternating Current Phasor Diagrams Series LC Phasor Subtraction
MediumTwo statementsNew idea
Q41
Potential energy as a function of $r$ is given by $U=\frac{A}{r^{10}}-\frac{B}{r^{5}}$ , where $r$ is the interatomic distance, A and B are positive constants. The equilibrium distance between the two atoms will be :
Work, Energy and Power Work and Kinetic Energy Potential Energy Curves and Equilibrium
MediumSingle correctIdea asked in 2 yearsAsked the year before
Q42
An object of mass 5 kg is thrown vertically upwards from the ground. The air resistance produces a constant retarding force of 10 N throughout the motion. The ratio of time of ascent to the time of descent will be equal to : [Use g = 10ms $^{-2}$ ].
Laws of Motion Kinematics in 1D Uniform Acceleration Equations
MediumSingle correctIdea asked in 4 yearsAsked the year before
Q43
A fly wheel is accelerated uniformly from rest and rotates through 5 rad in the first second. The angle rotated by the fly wheel in the next second, will be :
System of Particles and Rotational Motion Kinematics in 1D Displacement in the nth Second
EasySingle correctIdea asked in 4 yearsAsked the year before
Q44
A 100 g of iron nail is hit by a 1.5 kg hammer striking at a velocity of 60 $ms^{-1}$ . What will be the rise in the temperature of the nail if one fourth of energy of the hammer goes into heating the nail ? [Specific heat capacity of iron = 0.42…
Thermal Properties of Matter Calorimetry Mechanical Equivalent of Heat
EasySingle correctIdea asked in 3 years
Q45
If the charge on a capacitor is increased by 2 C, the energy stored in it increases by 44%. The original charge on the capacitor is (in C)
Electrostatic Potential and Capacitance Parallel Plate Capacitors Energy Stored in Capacitor
EasySingle correctNew ideaIdea asked in 2 years
Q46
A long cylindrical volume contains a uniformly distributed charge of density $\rho$ . The radius of cylindrical volume is $R$ . A charge particle ( $q$ ) revolves around the cylinder in a circular path. The kinetic energy of the particle is :
Electric Charges and Fields Gauss's Law Cylindrical Field
MediumSingle correctNew ideaIdea asked in 3 years
Q47
An electric bulb is rated as 200 W. What will be the peak magnetic field at 4 m distance produced by the radiations coming from this bulb? Consider this bulb as a point source with 3.5% efficiency.
Electromagnetic Waves
MediumSingle correct
Q48
The light of two different frequencies whose photons have energies 3.8 $eV$ and 1.4 $eV$ respectively, illuminate a metallic surface whose work function is 0.6 $eV$ successively. The ratio of maximum speeds of emitted electrons for the two frequencies…
Dual Nature of Radiation and Matter Photoelectric Effect Maximum Velocity Calculation
EasySingle correctNew idea
Q49
Two light beams of intensities in the ratio of 9 : 4 are allowed to interfere. The ratio of the intensity of maxima and minima will be :
Wave Optics Young's Double Slit Resultant Intensity Formula
EasySingle correctIdea asked in 5 yearsAsked the year before
Q50
In Bohr’s atomic model of hydrogen, let K, P and E are the kinetic energy, potential energy and total energy of the electron respectively. Choose the correct option when the electron undergoes transitions to a higher level :
Atoms Bohr's Atomic Model Energy Ratios in Bohr Orbits
EasySingle correctIdea asked in 3 yearsAsked the year before
Q51
A body is projected from the ground at an angle of $45^{\circ}$ with the horizontal. Its velocity after 2s is 20 ms $^{-1}$ . The maximum height reached by the body during its motion is ________ m. (use g = 10ms $^{-2}$ )
Motion in a Plane Projectile Motion Instantaneous Projectile Velocity
MediumNumerical valueNew ideaIdea asked in 3 years
Q52
An antenna is placed in a dielectric medium of dielectric constant 6.25. If the maximum size of that antenna is 5.0 mm, it can radiate a signal of minimum frequency of ______ GHz. (Given $\mu_r$ = 1 for dielectric medium)
Electromagnetic Waves
MediumNumerical value
Q53
A potentiometer wire of length 10 m and resistance 20 $\Omega$ is connected in series with a 25 V battery and an external resistance 30 $\Omega$ . A cell of emf E in secondary circuit is balanced by 250 cm long potentiometer wire. The value of E (in volt) is…
Current Electricity Potentiometer Potential Gradient Calculation
MediumNumerical valueIdea asked in 2 years
Q54
Two travelling waves of equal amplitudes and equal frequencies move in opposite directions along a string. They interfere to produce a stationary wave whose equation is given by $y = (10\cos \pi x \sin\frac{2\pi t}{T})$ cm The amplitude of the particle at…
Waves Waves on Strings Standing Wave Equation
EasyNumerical valueNew idea
Q55
In the given circuit, the value of current $I_L$ will be ________ mA. (When $R_L$ = 1k $\Omega$ )
Semiconductor Electronics: Materials, Devices and Simple Circuits Semiconductor Diodes Zener Diode Regulation
MediumNumerical value · figureIdea asked in 5 yearsAsked the year before
Q56
A sample contains $10^{-2}$ kg each of two substances A and B with half lives 4 s and 8 s respectively. The ratio of their atomic weights is 1 : 2. The ratio of the amounts of A and B after 16 s is $\frac{x}{100}$ . The value of $x$ is _______.
Nuclei Radioactive Decay Integer Half-Life Shortcuts
MediumNumerical valueIdea asked in 2 yearsAsked the year before
Q57
A ray of light is incident at an angle of incidence $60^{\circ}$ on the glass slab of refractive index $\sqrt{3}$ . After refraction, the light ray emerges out from other parallel faces and lateral shift between incident ray and emergent ray is $4\sqrt{3}$ …
Ray Optics and Optical Instruments Refraction and Snell's Law Lateral Shift in Slabs
MediumNumerical valueNew ideaIdea asked in 2 years
Q58
A circular coil of 1000 turns each with area 1m $^2$ is rotated about its vertical diameter at the rate of one revolution per second in a uniform horizontal magnetic field of 0.07T. The maximum voltage generation will be ______V.
Electromagnetic Induction Faraday's Law AC Generator Principle
EasyNumerical valueIdea asked in 4 yearsAsked the year before
Q59
A monoatomic gas performs a work of $\frac{Q}{4}$ where Q is the heat supplied to it. The molar heat capacity of the gas will be ________R during this transformation. Where R is the gas constant.
Thermodynamics First Law of Thermodynamics Molar Heat Capacity in Polytropic Processes
MediumNumerical valueIdea asked in 2 yearsAsked the year before
Q60
In an experiment to verify Newton’s law of cooling, a graph is plotted between, the temperature difference ( $\Delta$ T) of the water and surroundings and time as shown in figure. The initial temperature of water is taken as 80 $^{\circ}$ C. The value of t $_2$ …
Thermal Properties of Matter Thermal Radiation Logarithmic Graph of Newton's Law of Cooling
MediumNumerical value · figureIdea asked in 2 yearsAsked the year before

Chemistry30 questions

17 easy11 medium2 hard

Q61
120 g of an organic compound that contains only carbon and hydrogen gives 330 g of CO $_2$ and 270 g of water on complete combustion. The percentage of carbon and hydrogen, respectively are
Some Basic Concepts of Chemistry Mole Concept Empirical Formula from Mass Percentage
EasySingle correctIdea asked in 3 years
Q62
The energy of one mole of photons of radiation of wavelength 300 nm is (Given : h = 6.63 $\times$ 10 $^{-34}$ J s, N $_A$ = 6.02 $\times$ 10 $^{23}$ mol $^{-1}$ . c = 3 $\times$ 10 $^8$ m s $^{-1}$ )
Structure of Atom
EasySingle correct
Q63
The correct order of bond orders of C $_2^{2-}$ , N $_2^{2-}$ and O $_2^{2-}$ is, respectively
Chemical Bonding and Molecular Structure Molecular Orbital Theory Bond Order Calculation
MediumSingle correctIdea asked in 3 yearsAsked the year before
Q64
At 25°C and 1 atm pressure, the enthalpies of combustion are given below: Substance | H $_2$ | C (graphite) | C $_2$ H $_6$ (g) $\Delta_c H^{\ominus}$ / kJ mol $^{-1}$ | $-$ 286.0 | $-$ 394.0 | $-$ 1560.0 The enthalpy of formation of ethane is
Thermodynamics (Chemistry)Enthalpy and Hess's Law Formation and Combustion
MediumSingle correctIdea asked in 5 yearsAsked the year before
Q65
For a first order reaction, the time required for completion of 90% reaction is 'x' times the half life of the reaction. The value of 'x' is (Given: ln 10 = 2.303 and log 2 = 0.3010)
Chemical Kinetics Integrated Rate Equations First Order Fractional Completion Shortcuts
EasySingle correctIdea asked in 3 yearsAsked the year before
Q66
Metals generally melt at very high temperature. Amongst the following, the metal with the highest melting point will be
The d- and f-Block Elements d-Block Elements Melting Point Maxima and Anomalies
EasySingle correctNew idea
Q67
Which of the following chemical reactions represents Hall-Heroult Process?
General Principles and Processes of Isolation of Elements Metallurgical Extraction Hall-Heroult Process
EasySingle correctIdea asked in 3 yearsAsked the year before
Q68
In the industrial production of which of the following, molecular hydrogen is obtained as a byproduct ?
Hydrogen Hydrogen and Hydrides Dihydrogen from Brine Electrolysis
EasySingle correctNew ideaIdea asked in 2 years
Q69
Which one of the following compounds is used as a chemical in certain type of fire extinguishers?
The s-Block Elements
EasySingle correct
Q70
PCl $_5$ is well known, but NCl $_5$ is not. Because,
The p-Block Elements Group 15 & 16 Anomalous Properties of Nitrogen
EasySingle correctNew ideaIdea asked in 3 years
Q71
Transition metal complex with highest value of crystal field splitting ( $\Delta_o$ ) will be
Coordination Compounds Crystal Field Theory Effect of 3d vs 4d vs 5d Series on Splitting
MediumSingle correctNew idea
Q72
Some gases are responsible for heating of atmosphere (green house effect). Identify from the following the gaseous species which does not cause it.
Environmental Chemistry Greenhouse Effect
EasySingle correctIdea asked in 4 yearsAsked the year before
Q73
Arrange the following carbocations in decreasing order of stability.
Organic Chemistry: Some Basic Principles and Techniques Inductive and Electromeric Effects Carbocation Stability
HardDiagram based · figureNew ideaIdea asked in 4 years
Q74
Given below are two statements. Statement I: The presence of weaker $\pi$ -bonds make alkenes less stable than alkanes. Statement II: The strength of the double bond is greater than that of carbon-carbon single bond. In the light of the above statements,…
Hydrocarbons
MediumTwo statements
Q75
Which of the following reagents / reactions will convert 'A' to 'B'?
Aldehydes, Ketones and Carboxylic Acids Alkenes and Alkynes Hydroboration-Oxidation
MediumSingle correct · figureIdea asked in 4 years
Q76
Hex-4-ene-2-ol on treatment with PCC gives 'A'. 'A' on reaction with sodium hypoiodite gives 'B', which on further heating with soda lime gives 'C'. The compound 'C' is
Aldehydes, Ketones and Carboxylic Acids Carboxylic Acids and Derivatives Decarboxylation Reactions
MediumSingle correctIdea asked in 2 years
Q77
The conversion of propan-1-ol to n-butylamine involves the sequential addition of reagents. The correct sequential order of reagents is
Amines Reduction of Nitriles vs Isocyanides
MediumSingle correctIdea asked in 2 years
Q78
Which of the following is not an example of a condensation polymer?
Polymers Common Condensation Polymers
EasySingle correctNew ideaIdea asked in 2 years
Q79
The structure shown below is of which well-known drug molecule ?
Chemistry in Everyday Life Evolution of Antacids: Neutralizers vs Receptor Blockers
MediumDiagram based · figureIdea asked in 2 yearsAsked the year before
Q80
In the flame test of a mixture of salts, a green flame with blue centre was observed. Which one of the following cations may be present?
Practical Chemistry (Qualitative Analysis)Cation Analysis Cation Flame Colors
EasySingle correctNew ideaIdea asked in 3 years
Q81
At 300 K, a sample of 3.0 g of gas A occupies the same volume as 0.2 g of hydrogen at 200 K at the same pressure. The molar mass of gas A is ________ g mol $^{-1}$ . (nearest integer) Assume that the behaviour of gases is ideal. (Given: The molar mass of…
States of Matter Ideal Gas Laws Ideal Gas Equation
MediumNumerical valueIdea asked in 2 yearsAsked the year before
Q82
A company dissolves 'x' amount of CO $_2$ at 298 K in 1 litre of water to prepare soda water. X = ________ $\times$ 10 $^{-3}$ g. (nearest integer) (Given: partial pressure of CO $_2$ at 298 K = 0.835 bar. Henry's law constant for CO $_2$ at 298K = 1.67 kbar.…
Solutions
MediumNumerical value
Q83
PCl $_5$ dissociates as PCl $_5$ (g) $\rightleftharpoons$ PCl $_3$ (g)+Cl $_2$ (g) 5 moles of PCl $_5$ are placed in a 200 litre vessel which contains 2 moles of N $_2$ and is maintained at 600 K. The equilibrium pressure is 2.46 atm. The equilibrium constant K $_p$ for…
Equilibrium Chemical Equilibrium Kp Expression via Partial Pressures
HardNumerical valueNew ideaIdea asked in 2 years
Q84
The resistance of a conductivity cell containing 0.01 M KCl solution at 298 K is 1750 $\Omega$ . If the conductivity of 0.01M KCl solution at 298 K is 0.152 $\times$ 10 $^{-3}$ S cm $^{-1}$ , then the cell constant of the conductivity cell is ________ $\times$ …
Electrochemistry
EasyNumerical value
Q85
When 200 mL of 0.2 M acetic acid is shaken with 0.6 g of wood charcoal, the final concentration of acetic acid after adsorption is 0.1 M. The mass of acetic acid adsorbed per gram of carbon is ________ g.
Surface Chemistry
EasyNumerical value
Q86
(a) Baryte, (b) Galena, (c) Zinc blende and (d) Copper pyrites. How many of these minerals are sulphide based?
General Principles and Processes of Isolation of Elements
EasyNumerical value
Q87
Manganese (VI) has ability to disproportionate in acidic solution. The difference in oxidation states of two ions it forms in acidic solution is ________.
The d- and f-Block Elements d-Block Elements Disproportionation of Manganate(VI)
EasyNumerical valueNew ideaIdea asked in 2 years
Q88
0.2 g of an organic compound was subjected to estimation of nitrogen by Dumas method in which volume of N $_2$ evolved (at STP) was found to be 22.400 mL. The percentage of nitrogen in the compound is ________. [nearest integer] (Given: Molar mass of N $_2$ is…
Organic Chemistry: Some Basic Principles and Techniques Stoichiometry Stoichiometry of Dumas Method
EasyNumerical valueIdea asked in 4 yearsAsked the year before
Q89
Consider the above reaction. The number of $\pi$ electrons present in the product 'P' is ________.
Haloalkanes and Haloarenes SN1 and SN2 Mechanisms Allylic and Benzylic Reactivity in SN1
MediumNumerical value · figureIdea asked in 5 yearsAsked the year before
Q90
In alanylglycylleucylalanylvaline, the number of peptide linkages is ________.
Biomolecules
EasyNumerical value

Maths30 questions

3 easy16 medium11 hard

Q1
Let $x*y = x^2 + y^3$ and $(x*1)*1 = x*(1*1)$ . Then a value of $2\sin^{-1}\left(\frac{x^4+x^2-2}{x^4+x^2+2}\right)$ is
Inverse Trigonometric Functions Inverse Trig Identities Inverse Trig Simplification via tan(θ)
MediumSingle correctNew ideaIdea asked in 2 years
Q2
The sum of all the real roots of the equation $(e^{2x} - 4)(6e^{2x} - 5e^x + 1) = 0$ is
Complex Numbers and Quadratic Equations Quadratic Roots Exponential Reducible to Quadratic
EasySingle correctIdea asked in 4 yearsAsked the year before
Q3
Let the system of linear equations $x + y + \alpha z = 2$ $3x + y + z = 4$ $x + 2z = 1$ have a unique solution $(x^*, y^*, z^*)$ . If $(\alpha, x^*)$ , $(y^*, \alpha)$ and $(x^*, -y^*)$ are collinear points, then the sum of absolute values of all possible…
Determinants Cramer's Rule Condition for Unique Solution
MediumSingle correctIdea asked in 3 yearsAsked the year before
Q4
Let $x, y > 0$ . If $x^3y^2 = 2^{15}$ , then the least value of $3x + 2y$ is
Sequences and Series AP and GP Weighted AM-GM Inequality
MediumSingle correctNew ideaIdea asked in 2 years
Q5
Let $f(x) = \begin{cases} \frac{\sin(x-[x])}{x-[x]} & , x \in (-2,-1) \\ \max\{2x, 3[|x|]\} & , |x| < 1 \\ 1 & , \text{otherwise} \end{cases}$ where $[t]$ denotes greatest integer $\leq t$ . If $m$ is the number of points where $f$ is not continuous and $n$ …
Continuity and Differentiability Continuity involving GIF and Fractional Part
HardSingle correctIdea asked in 6 yearsAsked the year before
Q6
The value of the integral $\int_{-\pi/2}^{\pi/2} \frac{dx}{(1+e^x)(\sin^6 x + \cos^6 x)}$ is equal to
Integrals Definite Integral Properties Symmetric Limits with Exponential Denominator Trick
MediumSingle correctIdea asked in 5 yearsAsked the year before
Q7
$\lim_{n\to\infty}\left[\frac{n^2}{(n^2+1)(n+1)} + \frac{n^2}{(n^2+4)(n+2)} + \frac{n^2}{(n^2+9)(n+3)} + \ldots + \frac{n^2}{(n^2+n^2)(n+n)}\right]$ is equal to
Integrals Integral as Limit of Sum Summation to Riemann Form
HardSingle correctIdea asked in 4 yearsAsked the year before
Q8
A particle is moving in the $xy$ -plane along a curve $C$ passing through the point $(3, 3)$ . The tangent to the curve $C$ at the point $P$ meets the $x$ -axis at $Q$ . If the $y$ -axis bisects the segment $PQ$ , then $C$ is a parabola with
Differential Equations Variable Separable Form Geometric Applications
MediumSingle correctNew idea
Q9
Let the maximum area of the triangle that can be inscribed in the ellipse $\frac{x^2}{a^2} + \frac{y^2}{4} = 1$ , $a > 2$ , having one of its vertices at one end of the major axis of the ellipse and one of its sides parallel to the $y$ -axis, be $6\sqrt{3}$ .…
Conic Sections Ellipse Eccentricity Maximum Inscribed Rectangle Area via Eccentricity
HardSingle correctNew idea
Q10
Let the area of the triangle with vertices $A(1, \alpha)$ , $B(\alpha, 0)$ and $C(0, \alpha)$ be 4 sq. units. If the points $(\alpha, -\alpha)$ , $(-\alpha, \alpha)$ and $(\alpha^2, \beta)$ are collinear, then $\beta$ is equal to
Straight Lines Straight Line Distance Distance Formula via Determinant Area
MediumSingle correctNew idea
Q11
The number of distinct real roots of the equation $x^7 - 7x - 2 = 0$ is
Application of Derivatives Rolle's Theorem Polynomial Root Counting
MediumSingle correctIdea asked in 2 yearsAsked the year before
Q12
A random variable $X$ has the following probability distribution: | $X$ | 0 | 1 | 2 | 3 | 4 | | $P(X)$ | $k$ | $2k$ | $4k$ | $6k$ | $8k$ | The value of $P(1 < X < 4 \mid X \leq 2)$ is equal to :
Probability Conditional Probability Conditional Probability Definition
EasySingle correctIdea asked in 2 yearsAsked the year before
Q13
The number of solutions of the equation $\cos\left(x + \frac{\pi}{3}\right)\cos\left(\frac{\pi}{3} - x\right) = \frac{1}{4}\cos^2 2x$ , $x \in [-3\pi, 3\pi]$ is:
Trigonometric Functions General Solutions Product-to-Sum Transformation
MediumSingle correctNew ideaIdea asked in 2 years
Q14
If the shortest distance between the lines $\frac{x-1}{2} = \frac{y-2}{3} = \frac{z-3}{\lambda}$ and $\frac{x-2}{1} = \frac{y-4}{4} = \frac{z-5}{5}$ is $\frac{1}{\sqrt{3}}$ , then the sum of all possible values of $\lambda$ is:
Three Dimensional Geometry Skew Lines Cartesian Shortest Distance
MediumSingle correctIdea asked in 6 years
Q15
Let the points on the plane P be equidistant from the points $(-4, 2, 1)$ and $(2, -2, 3)$ . Then the acute angle between the plane P and the plane $2x + y + 3z = 1$ is
Three Dimensional Geometry Planes in 3D Perpendicular Bisector Plane
MediumSingle correctIdea asked in 3 yearsAsked the year before
Q16
Let $\hat{a}$ and $\hat{b}$ be two unit vectors such that $|(\hat{a} + \hat{b}) + 2(\hat{a} \times \hat{b})| = 2$ . If $\theta \in (0, \pi)$ is the angle between $\hat{a}$ and $\hat{b}$ , then among the statements: (S1) :…
Vector Algebra Dot Product Lagrange's Identity (Dot Product Form)
HardTwo statementsNew idea
Q17
If $y = \tan^{-1}\left(\sec x^3 - \tan x^3\right)$ , $\frac{\pi}{2} < x^3 < \frac{3\pi}{2}$ , then
Continuity and Differentiability Existence of Higher Order Derivatives
HardSingle correctIdea asked in 3 years
Q18
Consider the following statements: $A$ : Rishi is a judge. $B$ : Rishi is honest. $C$ : Rishi is not arrogant. The negation of the statement “if Rishi is a judge and he is not arrogant, then he is honest” is
Relations and Functions
MediumSingle correct
Q19
The slope of normal at any point $(x, y)$ , $x > 0$ , $y > 0$ on the curve $y = y(x)$ is given by $\frac{x^2}{xy - x^2y^2 - 1}$ . If the curve passes through the point $(1, 1)$ , then $e \cdot y(e)$ is equal to
Differential Equations Variable Separable Form Geometric Applications
HardSingle correctNew idea
Q20
Let $\lambda^*$ be the largest value of $\lambda$ for which the function $f_\lambda(x) = 4\lambda x^3 - 36\lambda x^2 + 36x + 48$ is increasing for all $x \in \mathbb{R}$ . Then $f_{\lambda^*}(1) + f_{\lambda^*}(-1)$ is equal to :
Application of Derivatives Monotonicity Parameter Range for Global Monotonicity
MediumSingle correctNew idea
Q21
Let $S = \{z \in \mathbb{C} : |z - 3| \leq 1$ and $z(4+3i) + \bar{z}(4-3i) \leq 24\}$ . If $\alpha + i\beta$ is the point in $S$ which is closest to $4i$ , then $25(\alpha + \beta)$ is equal to ____.
Complex Numbers and Quadratic Equations
HardNumerical value
Q22
Let $S = \left\{\begin{pmatrix} -1 & a \\ 0 & b \end{pmatrix} : a, b \in \{1, 2, 3, \ldots 100\}\right\}$ and let $T_n = \{A \in S : A^{n(n+1)} = I\}$ . Then the number of elements in $\bigcap_{n=1}^{100} T_n$ is ____.
Matrices Matrix Multiplication Powers of Square Matrices
HardNumerical valueIdea asked in 6 yearsAsked the year before
Q23
The number of 7-digit numbers which are multiples of 11 and are formed using all the digits 1, 2, 3, 4, 5, 7 and 9 is ____.
Permutations and Combinations Arrangements with Positional Parity Restrictions
HardNumerical valueNew idea
Q24
The sum of all the elements of the set $\{\alpha \in \{1, 2, \ldots\ldots, 100\} : HCF(\alpha, 24) = 1\}$ is ____.
Relations and Functions
MediumNumerical value
Q25
The remainder on dividing $1 + 3 + 3^2 + 3^3 + \ldots. + 3^{2021}$ by 50 is ____.
Binomial Theorem
HardNumerical value
Q26
The area (in sq. units) of the region enclosed between the parabola $y^2 = 2x$ and the line $x + y = 4$ is ____.
Application of Integrals Area Under Curves Area Between Parabola and Line
EasyNumerical valueIdea asked in 4 yearsAsked the year before
Q27
Let a circle $C : (x-h)^2 + (y-k)^2 = r^2$ , $k > 0$ , touch the $x$ -axis at $(1, 0)$ . If the line $x + y = 0$ intersects the circle $C$ at $P$ and $Q$ such that the length of the chord $PQ$ is 2, then the value of $h + k + r$ is equal to ____.
Conic Sections Circle Tangents Family of Circles Touching a Line
MediumNumerical valueIdea asked in 5 yearsAsked the year before
Q28
In an examination, there are 10 true-false type questions. Out of 10, a student can guess the answer of 4 questions correctly with probability $\frac{3}{4}$ and the remaining 6 questions correctly with probability $\frac{1}{4}$ . If the probability that the…
Probability
HardNumerical value
Q29
Let the hyperbola $H : \frac{x^2}{a^2} - y^2 = 1$ and the ellipse $E : 3x^2 + 4y^2 = 12$ be such that the length of latus rectum of $H$ is equal to the length of latus rectum of $E$ . If $e_H$ and $e_E$ are the eccentricities of $H$ and $E$ respectively, then…
Conic Sections Ellipse Eccentricity Length of Latus Rectum
MediumNumerical valueIdea asked in 6 years
Q30
Let $P_1$ be a parabola with vertex $(3, 2)$ and focus $(4, 4)$ and $P_2$ be its mirror image with respect to the line $x + 2y = 6$ . Then the directrix of $P_2$ is $x + 2y =$ ____.
Conic Sections Straight Line Distance Reflection of a Line in a Line
MediumNumerical valueNew ideaIdea asked in 2 years

Questions from the official paper published by NTA, answers from the final key. Lumi analyses 108 of the 174 JEE Main shifts held from 2017 to 2026; see the other 2022 shifts.