JEE Main 2021 · August session

JEE Main 31 August 2021, Shift 2

90 questions: 30 numerical-answer and 60 multiple choice. 12 were rated hard and 27 easy. Its hard-question count was the joint 3rd highest of the 24 2021 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: 12; 13% of the shift
Chapters asked: 71; 13 questions with a figure

Key findings · 31 Aug 2021, Shift 2

1
12 hard questions (13%): the joint 3rd highest count of the 24 2021 shifts Lumi analysed. Across those shifts the hard share ran from 3% to 17%.
2
Maths was the hardest section, with 8 of 30 rated hard; Chemistry had the most easy questions (14 of 30).
3
56 questions tested a new idea, one no earlier year had asked: 72% of the 78 questions tagged to an idea.
4
22 came back from 2020: the same idea had been asked in the 2020 shifts Lumi analysed.
5
36 shared an idea with another 2021 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 Application of Derivatives in Maths (3).

The shift at a glance

Subjects, difficulty and chapters

Physics

30 Q · 10 numerical

Easy 11Medium 16Hard 3

Top chapters

Chemistry

30 Q · 10 numerical

Easy 14Medium 15Hard 1

Top chapters

Aldehydes, Ketones and Carboxylic Acids2 Q
Amines2 Q
Coordination Compounds2 Q
Haloalkanes and Haloarenes2 Q
The p-Block Elements2 Q
Biomolecules1 Q

Maths

30 Q · 10 numerical

Easy 2Medium 20Hard 8

Top chapters

Application of Derivatives3 Q
Conic Sections3 Q
Complex Numbers and Quadratic Equations2 Q
Differential Equations2 Q
Integrals2 Q
Sequences and Series2 Q

Question format

Single correct42 Q
Numerical value30 Q
Diagram based10 Q
Two statements3 Q
Other2 Q
Match the list2 Q
Assertion–reason1 Q

What each question asks you to do

Calculation29 Q
Multi-step28 Q
Conceptual12 Q
Application9 Q
Recall9 Q
Diagram reading2 Q
Data or graph1 Q

Most common: Calculation.

Against the other 2021 shifts

This shift next to the 23 other 2021 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 Feb, Shift 1	90	36%	7%	33%	68%	24%
24 Feb, Shift 2	88	43%	5%	32%	63%	21%
25 Feb, Shift 1	90	23%	11%	33%	65%	30%
25 Feb, Shift 2	89	35%	10%	34%	67%	24%
26 Feb, Shift 1	88	50%	3%	33%	78%	17%
26 Feb, Shift 2	90	27%	12%	33%	80%	29%
16 Mar, Shift 2	90	31%	7%	33%	69%	33%
17 Mar, Shift 1	89	39%	8%	34%	61%	26%
17 Mar, Shift 2	89	36%	17%	34%	66%	32%
18 Mar, Shift 1	90	40%	9%	33%	74%	24%
18 Mar, Shift 2	90	34%	9%	33%	63%	28%
22 Jul, Shift 2	90	40%	12%	33%	67%	28%
25 Jul, Shift 1	90	31%	13%	33%	56%	34%
25 Jul, Shift 2	90	48%	3%	33%	63%	27%
3 Aug, Shift 1	90	32%	11%	33%	68%	21%
3 Aug, Shift 2	90	32%	7%	33%	61%	24%
4 Aug, Shift 1	90	40%	8%	33%	62%	29%
4 Aug, Shift 2	90	32%	7%	33%	65%	41%
26 Aug, Shift 1	89	27%	15%	34%	66%	30%
26 Aug, Shift 2	89	21%	9%	34%	60%	28%
27 Aug, Shift 1	90	27%	13%	33%	63%	34%
27 Aug, Shift 2	90	29%	9%	33%	72%	29%
31 Aug, Shift 1	90	27%	11%	33%	70%	24%
31 Aug, Shift 2	90	30%	13%	33%	72%	31%

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

11 easy16 medium3 hard

Chemistry30 questions

14 easy15 medium1 hard

Q31
The incorrect expression among the following is :
Thermodynamics (Chemistry)Gibbs Free Energy Gibbs Energy and Equil
MediumSingle correctNew ideaIdea asked in 5 years
Q32
Match List - I with List - II : List - I (Parameter) | List - II (Unit) (a) Cell constant | (i) $\mathrm{S\ cm^2\ mol^{-1}}$ (b) Molar conductivity | (ii) Dimensionless (c) Conductivity | (iii) $\mathrm{m^{-1}}$ (d) Degree of dissociation of electrolyte |…
Electrochemistry
EasyMatch the list
Q33
Which one of the following correctly represents the order of stability of oxides, $\mathrm{X_2O}$ ; (X=halogen) ?
The p-Block Elements
MediumSingle correct
Q34
Which one of the following statements is incorrect ?
Hydrogen Hydrogen and Hydrides H-H Bond Dissociation Enthalpy
MediumSingle correctNew idea
Q35
Given below are two statements : one is labelled as Assertion (A) and the other is labelled as Reason (R). Assertion (A) : Lithium salts are hydrated. Reason (R) : Lithium has higher polarising power than other alkali metal group members. In the light of the…
The s-Block Elements s-Block Elements Hydration Enthalpy Trends
EasyAssertion–reasonNew ideaIdea asked in 3 years
Q36
The number of S=O bonds present in sulphurous acid, peroxodisulphuric acid and pyrosulphuric acid, respectively are :
The p-Block Elements Group 15 & 16 Sulfur Oxoacids
MediumSingle correctIdea asked in 4 yearsAsked the year before
Q37
The $\mathrm{Eu^{2+}}$ ion is a strong reducing agent in spite of its ground state electronic configuration (outermost) : [Atomic number of Eu = 63]
The d- and f-Block Elements f-Block Elements Europium(II) as a Reducing Agent
EasySingle correctNew idea
Q38
In which one of the following sets all species show disproportionation reaction ?
Redox Reactions Disproportionation Balancing
MediumSingle correctNew ideaIdea asked in 3 years
Q39
Spin only magnetic moment in BM of $[\mathrm{Fe(CO)_4(C_2O_4)}]^+$ is :
Coordination Compounds Crystal Field Theory Spin-only Magnetic Moment
MediumSingle correctIdea asked in 7 yearsAsked the year before
Q40
The deposition of X and Y on ground surfaces is referred as wet and dry depositions, respectively. X and Y are :
Environmental Chemistry Acid Rain Mechanism
EasySingle correctIdea asked in 4 yearsAsked the year before
Q41
The major product of the following reaction is : [reaction: starting material drawn as a five-membered ring; reagent above/below arrow: NaOH / C2H5OH; Major Product]
Haloalkanes and Haloarenes Alkenes and Alkynes Saytzeff vs. Hofmann Elimination
MediumDiagram based · figureIdea asked in 6 yearsAsked the year before
Q42
Arrange the following conformational isomers of n-butane in order of their increasing potential energy : [four Newman projections labelled I, II, III, IV]
Hydrocarbons Alkanes and Free Radicals Ethane and Butane Conformers
MediumDiagram based · figureNew ideaIdea asked in 2 years
Q43
For the following sequence of reactions, the correct products are : [benzene ring drawn with alternating double bonds] 1. Br2/Fe/ $\Delta$ 2. Mg/dry ether 3. $\mathrm{CH_3OH}$ -> Products
Haloalkanes and Haloarenes Alkanes and Free Radicals Alkane Synthesis via Grignard Protonation
MediumDiagram based · figureNew ideaIdea asked in 2 years
Q44
For the reaction given below : [benzene ring drawn with an inner circle; CHO attached at top, $\mathrm{CH_2OH}$ attached at bottom] 1. NaOH, $\Delta$ 2. $\mathrm{H_3O^+}$ -> Product The compound which is not formed as a product in the reaction is a :
Aldehydes, Ketones and Carboxylic Acids Aldol and Cannizzaro Cannizzaro Reaction
MediumDiagram based · figureIdea asked in 4 yearsAsked the year before
Q45
The structures of A and B formed in the following reaction are : [Ph = $-\mathrm{C_6H_5}$ ] [benzene ring (with inner circle) + a five-membered cyclic anhydride drawn as a ring containing O with two C=O groups] -> (AlCl $_3$ (2 eq)) A -> (Zn/Hg, HCl) B
Aldehydes, Ketones and Carboxylic Acids Aromatic Hydrocarbons Friedel-Crafts Acylation and Acylium Ion
MediumDiagram based · figureNew ideaIdea asked in 2 years
Q46
Identify correct A, B and C in the reaction sequence given below : [benzene ring] conc. $\mathrm{HNO_3}$ + conc. $\mathrm{H_2SO_4}$ , $\Delta$ -> A -> ( $\mathrm{Cl_2}$ , Anhyd. $\mathrm{AlCl_3}$ ) B -> (Fe/HCl) C
Amines Aromatic Hydrocarbons Ortho-Para vs Meta Directing
MediumDiagram based · figureNew ideaIdea asked in 5 years
Q47
The major products A and B formed in the following reaction sequence are : [benzene ring (inner circle) bearing $\mathrm{NH_2}$ ; acetic anhydride drawn above the arrow] -> A -> ( $\mathrm{Br_2}$ , $\mathrm{CH_3COOH}$ , Room Temperature) B
Amines Protection of Aniline for Mono-Substitution
EasyDiagram based · figureNew ideaIdea asked in 3 years
Q48
Which among the following is not a polyester ?
Polymers Polyesters: Terylene vs Glyptal
EasySingle correctNew idea
Q49
Which of the following is NOT an example of fibrous protein ?
Biomolecules
EasySingle correct
Q50
Match List - I with List - II : List - I (Metal Ion) | List - II (Group in Qualitative analysis) (a) $\mathrm{Mn^{2+}}$ | (i) Group - III (b) $\mathrm{As^{3+}}$ | (ii) Group – IIA (c) $\mathrm{Cu^{2+}}$ | (iii) Group - IV (d) $\mathrm{Al^{3+}}$ | (iv) Group…
Coordination Compounds Cation Analysis Group IIA and IIB Sulfide Separation
MediumMatch the listNew idea
Q51
Sodium oxide reacts with water to produce sodium hydroxide. 20.0 g of sodium oxide is dissolved in 500 mL of water. Neglecting the change in volume, the concentration of the resulting NaOH solution is __________ $\times 10^{-1}$ M. (Nearest integer) [Atomic…
Some Basic Concepts of Chemistry Mole Concept Molarity Calculation
EasyNumerical valueIdea asked in 5 yearsAsked the year before
Q52
The empirical formula for a compound with a cubic close packed arrangement of anions and with cations occupying all the octahedral sites in $\mathrm{A_xB}$ . The value of $x$ is __________. (Integer answer)
The Solid State Crystal Lattices Empirical Formula from Void Occupancy
EasyNumerical valueIdea asked in 3 yearsAsked the year before
Q53
The value of magnetic quantum number of the outermost electron of $\mathrm{Zn^+}$ ion is __________. (Integer answer)
Structure of Atom Quantum Numbers Magnetic Quantum Number
EasyNumerical valueIdea asked in 4 yearsAsked the year before
Q54
According to molecular orbital theory, the number of unpaired electron(s) in $\mathrm{O_2^{2-}}$ is __________.
Chemical Bonding and Molecular Structure Molecular Orbital Theory Magnetic Property Prediction
EasyNumerical valueIdea asked in 6 yearsAsked the year before
Q55
1.22 g of an organic acid is separately dissolved in 100 g of benzene ( $\mathrm{K_b = 2.6\ K\ kg\ mol^{-1}}$ ) and 100 g of acetone ( $\mathrm{K_b = 1.7\ K\ kg\ mol^{-1}}$ ). The acid is known to dimerize in benzene but remain as a monomer in acetone. The…
Solutions Colligative Properties Carboxylic Acid Dimerization in Non-Polar Solvents
HardNumerical valueNew idea
Q56
The pH of a solution obtained by mixing 50 mL of 1 M HCl and 30 mL of 1 M NaOH is $x \times 10^{-4}$ . The value of $x$ is __________. (Nearest integer) [log 2.5 = 0.3979]
Equilibrium pH and Buffers pH of Strong Acids and Bases
EasyNumerical valueNew ideaIdea asked in 4 years
Q57
For the reaction A → B, the rate constant k (in $\mathrm{s^{-1}}$ ) is given by $\log_{10} k = 20.35 - \frac{(2.47 \times 10^3)}{T}$ The energy of activation in $\mathrm{kJ\ mol^{-1}}$ is __________. (Nearest integer) [Given : R = 8.314 J…
Chemical Kinetics Arrhenius Equation Activation Energy Calculation
EasyNumerical valueNew ideaIdea asked in 3 years
Q58
$\mathrm{CH_4}$ is adsorbed on 1 g charcoal at 0°C following the Freundlich adsorption isotherm. 10.0 mL of $\mathrm{CH_4}$ is adsorbed at 100 mm of Hg, whereas 15.0 mL is adsorbed at 200 mm of Hg. The volume of $\mathrm{CH_4}$ adsorbed at 300 mm of Hg is…
Surface Chemistry Adsorption Freundlich Isotherm Equation
MediumNumerical valueNew idea
Q59
In the electrolytic refining of blister copper, the total number of main impurities, from the following, removed as anode mud is __________. Pb, Sb, Se, Te, Ru, Ag, Au and Pt
General Principles and Processes of Isolation of Elements Metallurgical Extraction Anode Mud Composition
MediumNumerical valueNew idea
Q60
The transformation occurring in Duma’s method is given below $\mathrm{C_2H_7N} + \left(2x + \frac{y}{2}\right)\mathrm{CuO} \rightarrow x\ \mathrm{CO_2} + \frac{y}{2}\mathrm{H_2O} + \frac{z}{2}\mathrm{N_2} + \left(2x + \frac{y}{2}\right)\mathrm{Cu}$ The value…
Organic Chemistry: Some Basic Principles and Techniques Stoichiometry Stoichiometry of Dumas Method
EasyNumerical valueIdea asked in 4 yearsAsked the year before

Maths30 questions

2 easy20 medium8 hard

Q61
Let $f : \mathbf{N} \to \mathbf{N}$ be a function such that $f(\mathrm{m}+\mathrm{n}) = f(\mathrm{m}) + f(\mathrm{n})$ for every $\mathrm{m}, \mathrm{n} \in \mathbf{N}$ . If $f(6) = 18$ , then $f(2) \cdot f(3)$ is equal to :
Relations and Functions
EasySingle correct
Q62
If $z$ is a complex number such that $\frac{z-i}{z-1}$ is purely imaginary, then the minimum value of $|z-(3+3i)|$ is :
Complex Numbers and Quadratic Equations Complex Numbers - Polar Form Locus of Argument Ratio (Circular Arc)
MediumSingle correctNew ideaIdea asked in 3 years
Q63
The sum of the roots of the equation, $x+1-2\log_2(3+2^x)+2\log_4(10-2^{-x}) = 0$ , is :
Complex Numbers and Quadratic Equations Quadratic Roots Exponential Reducible to Quadratic
MediumSingle correctIdea asked in 4 yearsAsked the year before
Q64
If $\alpha+\beta+\gamma = 2\pi$ , then the system of equations $x+(\cos\gamma)y+(\cos\beta)z = 0$ $(\cos\gamma)x+y+(\cos\alpha)z = 0$ $(\cos\beta)x+(\cos\alpha)y+z = 0$ has :
Determinants Cramer's Rule Homogeneous System Solutions
MediumSingle correctIdea asked in 4 yearsAsked the year before
Q65
Let $\mathrm{a_1, a_2, a_3, ...}$ be an A.P. If $\frac{\mathrm{a_1+a_2+...+a_{10}}}{\mathrm{a_1+a_2+...+a_p}} = \frac{100}{\mathrm{p}^2}$ , $\mathrm{p} \neq 10$ , then $\frac{\mathrm{a_{11}}}{\mathrm{a_{10}}}$ is equal to :
Sequences and Series AP and GP AP Recognition from Quadratic Sum
MediumSingle correctNew ideaIdea asked in 2 years
Q66
An angle of intersection of the curves, $\frac{x^2}{\mathrm{a}^2}+\frac{y^2}{\mathrm{b}^2} = 1$ and $x^2+y^2 = \mathrm{ab}$ , $\mathrm{a} > \mathrm{b}$ , is :
Application of Derivatives Tangents and Normals Angle Between Curves
MediumSingle correctNew idea
Q67
If $\alpha = \lim_{x\to\pi/4}\frac{\tan^3 x-\tan x}{\cos\left(x+\frac{\pi}{4}\right)}$ and $\beta = \lim_{x\to 0}(\cos x)^{\cot x}$ are the roots of the equation, $\mathrm{a}x^2+\mathrm{b}x-4 = 0$ , then the ordered pair $(\mathrm{a}, \mathrm{b})$ is :
Limits and Derivatives L'Hopital's Rule Indeterminate Power Forms
MediumSingle correctNew ideaIdea asked in 3 years
Q68
Let $f$ be any continuous function on $[0, 2]$ and twice differentiable on $(0, 2)$ . If $f(0) = 0$ , $f(1) = 1$ and $f(2) = 2$ , then :
Application of Derivatives Rolle's Theorem Higher-Order Derivative Roots
MediumSingle correctNew ideaIdea asked in 3 years
Q69
If $[x]$ is the greatest integer $\le x$ , then $\pi^2 \int_0^2 \left(\sin\frac{\pi x}{2}\right)(x-[x])^{[x]}\,\mathrm{d}x$ is equal to :
Integrals Definite Integral Properties Greatest Integer Function Integrals
HardSingle correctIdea asked in 7 yearsAsked the year before
Q70
If $\frac{\mathrm{d}y}{\mathrm{d}x} = \frac{2^x y + 2^y \cdot 2^x}{2^x + 2^{x+y}\log_e 2}$ , $y(0) = 0$ , then for $y = 1$ , the value of $x$ lies in the interval :
Differential Equations Variable Separable Form Direct Integration
HardSingle correctNew ideaIdea asked in 3 years
Q71
If $y\frac{\mathrm{d}y}{\mathrm{d}x} = x\left[\frac{y^2}{x^2} + \frac{\phi\left(\frac{y^2}{x^2}\right)}{\phi'\left(\frac{y^2}{x^2}\right)}\right]$ , $x > 0$ , $\phi > 0$ , and $y(1) = -1$ , then $\phi\left(\frac{y^2}{4}\right)$ is equal to :
Differential Equations Homogeneous Equations Homogeneous Equations with Transcendental Functions
HardSingle correctNew ideaIdea asked in 2 years
Q72
Let A be the set of all points $(\alpha, \beta)$ such that the area of triangle formed by the points $(5, 6)$ , $(3, 2)$ and $(\alpha, \beta)$ is 12 square units. Then the least possible length of a line segment joining the origin to a point in A, is :
Straight Lines Straight Line Distance Locus of Points at a Fixed Distance
MediumSingle correctNew ideaIdea asked in 2 years
Q73
The locus of mid-points of the line segments joining $(-3, -5)$ and the points on the ellipse $\frac{x^2}{4}+\frac{y^2}{9} = 1$ is :
Conic Sections
MediumSingle correct
Q74
The distance of the point $(-1, 2, -2)$ from the line of intersection of the planes $2x+3y+2z = 0$ and $x-2y+z = 0$ is :
Three Dimensional Geometry Lines in 3D Perpendicular Distance from Point to Line
MediumSingle correctNew ideaIdea asked in 6 years
Q75
Let $\vec{a}, \vec{b}, \vec{c}$ be three vectors mutually perpendicular to each other and have same magnitude. If a vector $\vec{r}$ satisfies…
Vector Algebra Triple Products VTP BAC-CAB Expansion
HardSingle correctNew ideaIdea asked in 2 years
Q76
Let $\mathrm{S} = \{1, 2, 3, 4, 5, 6\}$ . Then the probability that a randomly chosen onto function $\mathrm{g}$ from S to S satisfies $\mathrm{g}(3) = 2\mathrm{g}(1)$ is :
Probability Permutations and Combinations Number of Surjective (Onto) Functions
MediumSingle correctNew ideaIdea asked in 4 years
Q77
The mean and variance of 7 observations are 8 and 16 respetively. If two observations are 6 and 8, then the variance of the remaining 5 observations is :
Statistics Variance and SD Finding Unknown Observations from Mean and Variance
MediumSingle correctIdea asked in 7 yearsAsked the year before
Q78
The number of solutions of the equation $32^{\tan^2 x}+32^{\sec^2 x} = 81$ , $0 \le x \le \frac{\pi}{4}$ is :
Trigonometric Functions General Solutions Quadratic Substitution in Trigonometry
MediumSingle correctNew ideaIdea asked in 5 years
Q79
The domain of the function $f(x) = \sin^{-1}\left(\frac{3x^2+x-1}{(x-1)^2}\right)+\cos^{-1}\left(\frac{x-1}{x+1}\right)$ is :
Inverse Trigonometric Functions Inverse Trig Domains Domain Intersection of Multiple Inverse Trig Functions
HardSingle correctNew idea
Q80
Negation of the statement $(\mathrm{p} \vee \mathrm{r}) \Rightarrow (\mathrm{q} \vee \mathrm{r})$ is :
Mathematical Reasoning
EasySingle correct
Q81
The number of elements in the set $\left\{\mathrm{A} = \begin{pmatrix}\mathrm{a} & \mathrm{b}\\ 0 & \mathrm{d}\end{pmatrix} : \mathrm{a}, \mathrm{b}, \mathrm{d} \in \{-1, 0, 1\} \text{ and } (\mathrm{I}-\mathrm{A})^3 = \mathrm{I}-\mathrm{A}^3\right\}$ , where…
Matrices Matrix Multiplication Binomial Expansion for Commuting Matrices
MediumNumerical valueNew ideaIdea asked in 3 years
Q82
The number of 4-digit numbers which are neither multiple of 7 nor multiple of 3 is ______________.
Permutations and Combinations Principle of Inclusion-Exclusion in Counting
MediumNumerical valueNew ideaIdea asked in 3 years
Q83
If the coefficient of $\mathrm{a}^7\mathrm{b}^8$ in the expansion of $(\mathrm{a}+2\mathrm{b}+4\mathrm{ab})^{10}$ is $\mathrm{K}\cdot 2^{16}$ , then $\mathrm{K}$ is equal to ______________.
Binomial Theorem Permutations and Combinations Multinomial Theorem in Counting
MediumNumerical valueNew ideaIdea asked in 2 years
Q84
If $\mathrm{S} = \frac{7}{5}+\frac{9}{5^2}+\frac{13}{5^3}+\frac{19}{5^4}+...$ , then $160\,\mathrm{S}$ is equal to ______________.
Sequences and Series Arithmetico-Geometric Series Second-Order AGP Summation
HardNumerical valueNew ideaIdea asked in 3 years
Q85
Let $f(x)$ be a cubic polynomial with $f(1) = -10$ , $f(-1) = 6$ , and has a local minima at $x = 1$ , and $f'(x)$ has a local minima at $x = -1$ . Then $f(3)$ is equal to ______________.
Application of Derivatives Maxima and Minima Determining Coefficients from Extrema
MediumNumerical valueIdea asked in 3 yearsAsked the year before
Q86
If $\int\frac{\sin x}{\sin^3 x+\cos^3 x}\,\mathrm{d}x = \alpha\log_e|1+\tan x|+\beta\log_e|1-\tan x+\tan^2 x|+\gamma\tan^{-1}\left(\frac{2\tan x-1}{\sqrt{3}}\right)+\mathrm{C}$ , when C is constant of integration, then the value of $18(\alpha+\beta+\gamma^2)$ …
Integrals Integration by Substitution Negative Even Sum of Trig Powers Substitution
HardNumerical valueNew idea
Q87
If the line $y = \mathrm{m}x$ bisects the area enclosed by the lines $x = 0$ , $y = 0$ , $x = \frac{3}{2}$ and the curve $y = 1+4x-x^2$ , then $12\,\mathrm{m}$ is equal to ______________.
Application of Integrals Area Under Curves Bisection of Bounded Areas
MediumNumerical valueIdea asked in 2 yearsAsked the year before
Q88
Let B be the centre of the circle $x^2+y^2-2x+4y+1 = 0$ . Let the tangents at two points P and Q on the circle intersect at the point $\mathrm{A}(3, 1)$ . Then $8\cdot\left(\frac{\text{area }\Delta\mathrm{APQ}}{\text{area }\Delta\mathrm{BPQ}}\right)$ is equal…
Conic Sections Circle Tangents Area of Tangent-Chord Triangle
HardNumerical valueNew ideaIdea asked in 2 years
Q89
A tangent line L is drawn at the point $(2, -4)$ on the parabola $y^2 = 8x$ . If the line L is also tangent to the circle $x^2+y^2 = \mathrm{a}$ , then 'a' is equal to ______________.
Conic Sections Circle Tangents Common Tangent to Circle and Conic
MediumNumerical valueNew ideaIdea asked in 3 years
Q90
Suppose the line $\frac{x-2}{\alpha} = \frac{y-2}{-5} = \frac{z+2}{2}$ lies on the plane $x+3y-2z+\beta = 0$ . Then $(\alpha+\beta)$ is equal to ______________.
Three Dimensional Geometry Planes in 3D Condition for Line to Lie in Plane
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 2021 shifts.