JEE Main 2021 · July session

JEE Main 25 July 2021, Shift 1

90 questions: 30 numerical-answer and 60 multiple choice. 12 were rated hard and 28 easy. Its hard-question count was the joint 3rd highest of the 24 2021 shifts Lumi analysed. The biggest chapter was Amines, with 3 questions.

Questions: 90; none dropped
Numerical answer: 30; 33% of the shift
Hard: 12; 13% of the shift
Chapters asked: 67; 16 questions with a figure

Key findings · 25 Jul 2021, Shift 1

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 7 of 30 rated hard; Physics had the most easy questions (11 of 30).
3
38 questions tested a new idea, one no earlier year had asked: 56% of the 68 questions tagged to an idea.
4
30 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 (53% of tagged questions). Shifts in the same year can test the same idea in a new setting.
6
Biggest chapters: System of Particles and Rotational Motion in Physics (3), Amines in Chemistry (3) and Binomial Theorem in Maths (3).

The shift at a glance

Subjects, difficulty and chapters

Physics

30 Q · 10 numerical

Easy 11Medium 17Hard 2

Top chapters

Chemistry

30 Q · 10 numerical

Easy 11Medium 16Hard 3

Top chapters

Maths

30 Q · 10 numerical

Easy 6Medium 17Hard 7

Top chapters

Question format

Single correct52 Q
Numerical value30 Q
Diagram based3 Q
Assertion–reason2 Q
Match the list1 Q
Two statements1 Q
Other1 Q

What each question asks you to do

Multi-step31 Q
Calculation24 Q
Application14 Q
Conceptual10 Q
Recall6 Q
Diagram reading3 Q
Data or graph2 Q

Most common: Multi-step.

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 easy17 medium2 hard

Chemistry30 questions

11 easy16 medium3 hard

Q31
For the following graphs, (a) Rate vs Time: horizontal line (b) $t_{1/2}$ vs Initial concentration: straight line through origin with positive slope (c) Concentration vs Time: decaying curve (d) Concentration vs Time: horizontal line (e) Rate vs…
Chemical Kinetics Integrated Rate Equations Kinetic Graphical Identification
MediumDiagram based · figureNew ideaIdea asked in 2 years
Q32
Sodium stearate $\mathrm{CH_3(CH_2)_{16}COO^-Na^+}$ is an anionic surfactant which forms micelles in oil. Choose the correct statement for it from the following :
Surface Chemistry Colloids Micelle Formation and CMC
MediumSingle correctIdea asked in 3 yearsAsked the year before
Q33
The ionic radii of $\mathrm{K^+}$ , $\mathrm{Na^+}$ , $\mathrm{Al^{3+}}$ and $\mathrm{Mg^{2+}}$ are in the order :
Classification of Elements and Periodicity in Properties Periodic Trends Isoelectronic Species Radii Comparison
EasySingle correctIdea asked in 5 yearsAsked the year before
Q34
In the leaching of alumina from bauxite, the ore expected to leach out in the process by reacting with NaOH is :
General Principles and Processes of Isolation of Elements Metallurgical Extraction Bayer Process Chemistry
EasySingle correctNew ideaIdea asked in 2 years
Q35
At 298.2 K the relationship between enthalpy of bond dissociation (in kJ mol $^{-1}$ ) for hydrogen ( $\mathrm{E_H}$ ) and its isotope, deuterium ( $\mathrm{E_D}$ ), is best described by :
Hydrogen Hydrogen and Hydrides H-H Bond Dissociation Enthalpy
MediumSingle correctNew idea
Q36
Given below are two statements : Statement I : None of the alkaline earth metal hydroxides dissolve in alkali. Statement II : Solubility of alkaline earth metal hydroxides in water increases down the group. In the light of the above statements, choose the…
The s-Block Elements s-Block Elements Amphoterism of Beryllium Compounds
MediumTwo statementsIdea asked in 4 yearsAsked the year before
Q37
Which one of the following compounds of Group-14 elements is not known ?
The p-Block Elements
MediumSingle correct
Q38
The correct order of following 3d metal oxides, according to their oxidation numbers is : (a) $\mathrm{CrO_3}$ (b) $\mathrm{Fe_2O_3}$ (c) $\mathrm{MnO_2}$ (d) $\mathrm{V_2O_5}$ (e) $\mathrm{Cu_2O}$
The d- and f-Block Elements d-Block Elements Oxidation State Trends
EasyOtherNew ideaIdea asked in 4 years
Q39
Which one of the following species responds to an external magnetic field ?
Coordination Compounds Crystal Field Theory High and Low Spin States
EasySingle correctIdea asked in 7 yearsAsked the year before
Q40
Which one of the following chemical agent is not being used for dry-cleaning of clothes ?
Chemistry in Everyday Life Environmental Chemistry Green Chemistry Alternatives in Daily Applications
MediumSingle correctNew ideaIdea asked in 2 years
Q41
Which one among the following resonating structures is not correct ?
Organic Chemistry: Some Basic Principles and Techniques Resonance and Hyperconjugation Resonance Structures
MediumDiagram based · figureNew idea
Q42
An Organic compound 'A' $\mathrm{C_4H_8}$ on treatment with $\mathrm{KMnO_4/H^+}$ yields compound 'B' $\mathrm{C_3H_6O}$ . Compound 'A' also yields compound 'B' an ozonolysis. Compound 'A' is :
Hydrocarbons Alkenes and Alkynes Reductive Ozonolysis Products
MediumSingle correctIdea asked in 7 yearsAsked the year before
Q43
[figure: but-3-en-2-one, CH2=CH-C(=O)-CH3, treated with (i) $\mathrm{C_2H_5MgBr}$ , dry ether (ii) $\mathrm{H_2O}$ , HCl giving P (Major Product)] Consider the above reaction, the major product 'P' is :
Aldehydes, Ketones and Carboxylic Acids Nucleophilic Addition Grignard Addition to Alpha,Beta-Unsaturated Carbonyls
HardSingle correct · figureNew ideaIdea asked in 2 years
Q44
[figure: 3,3-dimethylcyclopentanone (ketone C=O, gem-dimethyl on adjacent ring carbon) reacts with $\mathrm{CH_3CHO}$ , NaOH to give 'P' (Major Product); then (i) $\mathrm{I_2/NaOH}$ , Filter (ii) Filtrate + HCl gives 'X'] Consider the given reaction, the…
Aldehydes, Ketones and Carboxylic Acids Aldol and Cannizzaro Cross-Aldol Condensation
HardSingle correct · figureNew ideaIdea asked in 3 years
Q45
[figure: phenol converted to 4-bromophenol (Major Product)] The given reaction can occur in the presence of : (a) Bromine water (b) $\mathrm{Br_2}$ in $\mathrm{CS_2}$ , 273 K (c) $\mathrm{Br_2/FeBr_3}$ (d) $\mathrm{Br_2}$ in $\mathrm{CHCl_3}$ , 273 K Choose…
Haloalkanes and Haloarenes Alcohols and Phenols Phenol Bromination Solvent Effect
MediumSingle correct · figureNew ideaIdea asked in 3 years
Q46
Which one of the products of the following reactions does not react with Hinsberg reagent to form sulphonamide ?
Amines Hinsberg Test
MediumSingle correct · figureIdea asked in 5 yearsAsked the year before
Q47
Given below are two statements, one is labelled as Assertion (A) and other is labelled as Reason (R). Assertion (A) : Gabriel phthalimide synthesis cannot be used to prepare aromatic primary amines. Reason (R) : Aryl halides do not undergo nucleophilic…
Amines Gabriel Phthalimide Synthesis
EasyAssertion–reasonIdea asked in 6 yearsAsked the year before
Q48
[figure: polymer repeating unit $-(\mathrm{CH_2}$ -C6H3(OH)- $\mathrm{CH_2}$ -C6H3(OH)- $\mathrm{CH_2})-$ , phenol rings joined through CH2 groups at ortho and para positions relative to OH] is a repeating unit for :
Polymers Novolac Structure and Role
EasySingle correct · figureNew ideaIdea asked in 2 years
Q49
The water soluble protein is :
Biomolecules
EasySingle correct
Q50
Which one of the following compounds will liberate $\mathrm{CO_2}$ , when treated with $\mathrm{NaHCO_3}$ ?
Amines Organic Acidity and Basicity Sodium Bicarbonate Effervescence Test
MediumSingle correctNew ideaIdea asked in 2 years
Q51
When 10 mL of an aqueous solution of $\mathrm{Fe^{2+}}$ ions was titrated in the presence of dil $\mathrm{H_2SO_4}$ using diphenylamine indicator, 15 mL of 0.02 M solution of $\mathrm{K_2Cr_2O_7}$ was required to get the end point. The molarity of the…
Some Basic Concepts of Chemistry Redox Reactions Mohr's Salt Redox Titrations
MediumNumerical valueNew ideaIdea asked in 2 years
Q52
A home owner uses $4.00\times10^3\ \mathrm{m^3}$ of methane ( $\mathrm{CH_4}$ ) gas, (assume $\mathrm{CH_4}$ is an ideal gas) in a year to heat his home. Under the pressure of 1.0 atm and 300 K, mass of gas used is $x\times10^5$ g. The value of $x$ is…
States of Matter Ideal Gas Laws Ideal Gas Equation
EasyNumerical valueNew ideaIdea asked in 2 years
Q53
A source of monochromatic radiation of wavelength 400 nm provides 1000 J of energy in 10 seconds. When this radiation falls on the surface of sodium, $x\times10^{20}$ electrons are ejected per second. Assume that wavelength 400 nm is sufficient for ejection…
Structure of Atom
MediumNumerical value
Q54
The number of sigma bonds in $\mathrm{H_3C-C(H)=CH-C\equiv C-H}$ (the H on the second carbon drawn below it) is ____________.
Chemical Bonding and Molecular Structure
EasyNumerical value · figure
Q55
At 298 K, the enthalpy of fusion of a solid (X) is 2.8 kJ mol $^{-1}$ and the enthalpy of vaporisation of the liquid (X) is 98.2 kJ mol $^{-1}$ . The enthalpy of sublimation of the substance (X) in kJ mol $^{-1}$ is _________. (in nearest integer)
Thermodynamics (Chemistry)Enthalpy and Hess's Law Enthalpies of Phase Transitions
EasyNumerical valueNew idea
Q56
$\mathrm{CO_2}$ gas is bubbled through water during a soft drink manufacturing process at 298 K. If $\mathrm{CO_2}$ exerts a partial pressure of 0.835 bar then $x$ m mol of $\mathrm{CO_2}$ would dissolve in 0.9 L of water. The value of $x$ is _________.…
Solutions
MediumNumerical value
Q57
For the reaction $\mathrm{A+B\rightleftharpoons 2C}$ the value of equilibrium constant is 100 at 298 K. If the initial concentration of all the three species is 1 M each, then the equilibrium concentration of C is $x\times10^{-1}$ M. The value of $x$ is…
Equilibrium Chemical Equilibrium ICE Table Construction
MediumNumerical valueNew ideaIdea asked in 3 years
Q58
Consider the cell at 25 $^\circ$ C $\mathrm{Zn|Zn^{2+}(aq),(1\ M)\,||\,Fe^{3+}(aq),Fe^{2+}(aq)|Pt(s)}$ The fraction of total iron present as $\mathrm{Fe^{3+}}$ ion at the cell potential of 1.500 V is $x\times10^{-2}$ . The value of $x$ is _________. (Nearest…
Electrochemistry Nernst Equation Full Cell Nernst Equation
HardNumerical valueIdea asked in 6 yearsAsked the year before
Q59
Consider the complete combustion of butane, the amount of butane utilized to produce 72.0 g of water is ________ $\times10^{-1}$ g. (in nearest integer)
Some Basic Concepts of Chemistry Stoichiometry Hydrocarbon Combustion General Equation
EasyNumerical valueIdea asked in 4 yearsAsked the year before
Q60
Three moles of AgCl get precipitated when one mole of an octahedral co-ordination compound with empirical formula $\mathrm{CrCl_3.3NH_3.3H_2O}$ reacts with excess of silver nitrate. The number of chloride ions satisfying the secondary valency of the metal…
Coordination Compounds Coordination Isomerism Chemical Tests for Ionization Isomers
MediumNumerical valueNew idea

Maths30 questions

6 easy17 medium7 hard

Q61
Let an ellipse $E:\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$ , $a^2>b^2$ , passes through $\left(\sqrt{\frac{3}{2}},1\right)$ and has eccentricity $\frac{1}{\sqrt3}$ . If a circle, centered at focus $F(\alpha,0)$ , $\alpha>0$ , of E and radius $\frac{2}{\sqrt3}$ ,…
Conic Sections Ellipse Eccentricity Eccentricity calculation
HardSingle correctIdea asked in 6 yearsAsked the year before
Q62
Let $y=y(x)$ be the solution of the differential equation $\frac{dy}{dx}=1+xe^{y-x}$ , $-\sqrt2<x<\sqrt2$ , $y(0)=0$ then, the minimum value of $y(x)$ , $x\in(-\sqrt2,\sqrt2)$ is equal to :
Differential Equations Variable Separable Form Reducible to Separable
HardSingle correctIdea asked in 3 yearsAsked the year before
Q63
The area (in sq. units) of the region, given by the set $\{(x,y)\in\mathbf{R}\times\mathbf{R}\mid x\ge0,\ 2x^2\le y\le4-2x\}$ is :
Application of Integrals Area Under Curves Area of Inequalities
EasySingle correctIdea asked in 8 yearsAsked the year before
Q64
Let $f:[0,\infty)\to[0,\infty)$ be defined as $f(x)=\int_0^x[y]\,dy$ where $[x]$ is the greatest integer less than or equal to $x$ . Which of the following is true ?
Continuity and Differentiability Continuity and Differentiability of Integral Functions
MediumSingle correctNew idea
Q65
Let $f:\mathbf{R}\to\mathbf{R}$ be defined as $f(x)=\begin{cases}\frac{\lambda|x^2-5x+6|}{\mu(5x-x^2-6)}, & x<2\\ e^{\frac{\tan(x-2)}{x-[x]}}, & x>2\\ \mu, & x=2\end{cases}$ where $[x]$ is the greatest integer less than or equal to $x$ . If $f$ is continuous…
Continuity and Differentiability Continuity at a Point
MediumSingle correctIdea asked in 4 yearsAsked the year before
Q66
Let $g:\mathbf{N}\to\mathbf{N}$ be defined as $g(3n+1)=3n+2$ , $g(3n+2)=3n+3$ , $g(3n+3)=3n+1$ , for all $n\ge0$ . Then which of the following statements is true ?
Relations and Functions
HardSingle correct
Q67
Let $f(x)=3\sin^4x+10\sin^3x+6\sin^2x-3$ , $x\in\left[-\frac{\pi}{6},\frac{\pi}{2}\right]$ . Then, $f$ is :
Application of Derivatives Monotonicity Finding Intervals
MediumSingle correctIdea asked in 5 yearsAsked the year before
Q68
Let a parabola P be such that its vertex and focus lie on the positive $x$ -axis at a distance 2 and 4 units from the origin, respectively. If tangents are drawn from O(0, 0) to the parabola P which meet P at S and R, then the area (in sq. units) of…
Conic Sections
MediumSingle correct
Q69
Let the foot of perpendicular from a point P(1, 2, $-1$ ) to the straight line $L:\frac{x}{1}=\frac{y}{0}=\frac{z}{-1}$ be N. Let a line be drawn from P parallel to the plane $x+y+2z=0$ which meets L at point Q. If $\alpha$ is the acute angle between the…
Three Dimensional Geometry Lines in 3D Foot of the Perpendicular
MediumSingle correctIdea asked in 6 yearsAsked the year before
Q70
If b is very small as compared to the value of a, so that the cube and other higher powers of $\frac{b}{a}$ can be neglected in the identity $\frac{1}{a-b}+\frac{1}{a-2b}+\frac{1}{a-3b}+\ldots+\frac{1}{a-nb}=\alpha n+\beta n^2+\gamma n^3$ , then the value of…
Binomial Theorem
HardSingle correct
Q71
Let the vectors $(2+a+b)\hat{i}+(a+2b+c)\hat{j}-(b+c)\hat{k}$ , $(1+b)\hat{i}+2b\hat{j}-b\hat{k}$ and $(2+b)\hat{i}+2b\hat{j}+(1-b)\hat{k}$ , $a,b,c\in\mathbf{R}$ be co-planar. Then which of the following is true ?
Vector Algebra Triple Products Solving for Unknown in Coplanarity
MediumSingle correctIdea asked in 4 yearsAsked the year before
Q72
The number of real roots of the equation $e^{6x}-e^{4x}-2e^{3x}-12e^{2x}+e^x+1=0$ is :
Complex Numbers and Quadratic Equations Quadratic Roots Exponential Reducible to Quadratic
HardSingle correctIdea asked in 4 yearsAsked the year before
Q73
A spherical gas balloon of radius 16 meter subtends an angle $60^\circ$ at the eye of the observer A while the angle of elevation of its center from the eye of A is $75^\circ$ . Then the height (in meter) of the top most point of the balloon from the level of…
Trigonometric Functions
MediumSingle correct
Q74
Let 9 distinct balls be distributed among 4 boxes, $B_1$ , $B_2$ , $B_3$ and $B_4$ . If the probability that $B_3$ contains exactly 3 balls is $k\left(\frac{3}{4}\right)^9$ then k lies in the set :
Probability
MediumSingle correct
Q75
The locus of the centroid of the triangle formed by any point P on the hyperbola $16x^2-9y^2+32x+36y-164=0$ , and its foci is :
Conic Sections
HardSingle correct
Q76
Let $S_n$ be the sum of the first n terms of an arithmetic progression. If $S_{3n}=3S_{2n}$ , then the value of $\frac{S_{4n}}{S_{2n}}$ is :
Sequences and Series AP and GP AP Sum of n terms
EasySingle correctIdea asked in 7 yearsAsked the year before
Q77
The Boolean expression $(p\Rightarrow q)\wedge(q\Rightarrow\sim p)$ is equivalent to :
Mathematical Reasoning
EasySingle correct
Q78
The value of the definite integral $\int_{\pi/24}^{5\pi/24}\frac{dx}{1+\sqrt[3]{\tan2x}}$ is :
Integrals Definite Integral Properties King's Property Rational Shortcut
MediumSingle correctNew idea
Q79
The values of a and b, for which the system of equations $2x+3y+6z=8$ $x+2y+az=5$ $3x+5y+9z=b$ has no solution, are :
Determinants Cramer's Rule Condition for No Solution
EasySingle correctIdea asked in 6 yearsAsked the year before
Q80
The sum of all values of $x$ in $[0,2\pi]$ , for which $\sin x+\sin2x+\sin3x+\sin4x=0$ , is equal to :
Trigonometric Functions General Solutions Sum-to-Product Factorization
MediumSingle correctNew idea
Q81
The term independent of ' $x$ ' in the expansion of $\left(\frac{x+1}{x^{2/3}-x^{1/3}+1}-\frac{x-1}{x-x^{1/2}}\right)^{10}$ , where $x\ne0,1$ is equal to _________.
Binomial Theorem
MediumNumerical value
Q82
Let $\vec{p}=2\hat{i}+3\hat{j}+\hat{k}$ and $\vec{q}=\hat{i}+2\hat{j}+\hat{k}$ be two vectors. If a vector $\vec{r}=\left(\alpha\hat{i}+\beta\hat{j}+\gamma\hat{k}\right)$ is perpendicular to each of the vectors $\left(\vec{p}+\vec{q}\right)$ and…
Vector Algebra Cross Product Unit Normal Vector Calculation
EasyNumerical valueNew ideaIdea asked in 2 years
Q83
The ratio of the coefficient of the middle term in the expansion of $(1+x)^{20}$ and the sum of the coefficients of two middle terms in expansion of $(1+x)^{19}$ is _________.
Binomial Theorem Binomial Coefficients Maximum Coefficient for Odd n
EasyNumerical valueNew idea
Q84
If the value of $\left(1+\frac{2}{3}+\frac{6}{3^2}+\frac{10}{3^3}+\ldots\ \text{upto}\ \infty\right)^{\log_{(0.25)}\left(\frac{1}{3}+\frac{1}{3^2}+\frac{1}{3^3}+\ldots\ \text{upto}\ \infty\right)}$ is $l$ , then $l^2$ is equal to _________.
Sequences and Series Arithmetico-Geometric Series Sum of Infinite AGP
MediumNumerical valueNew ideaIdea asked in 4 years
Q85
Let $y=y(x)$ be solution of the following differential equation $e^y\frac{dy}{dx}-2e^y\sin x+\sin x\cos^2x=0$ , $y\left(\frac{\pi}{2}\right)=0$ . If $y(0)=\log_e(\alpha+\beta e^{-2})$ , then $4(\alpha+\beta)$ is equal to _________.
Differential Equations Linear Differential Equations Generalized Substitution for LDEs
MediumNumerical valueNew ideaIdea asked in 4 years
Q86
Consider the following frequency distribution : Class : 10 - 20 | 20 - 30 | 30 - 40 | 40 - 50 | 50 - 60 Frequency : $\alpha$ | 110 | 54 | 30 | $\beta$ If the sum of all frequencies is 584 and median is 45, then $|\alpha-\beta|$ is equal to _________.
Statistics
MediumNumerical value · figure
Q87
Let $S=\left\{n\in\mathbf{N}\ \middle|\ \begin{pmatrix}0&i\\1&0\end{pmatrix}^n\begin{pmatrix}a&b\\c&d\end{pmatrix}=\begin{pmatrix}a&b\\c&d\end{pmatrix}\ \forall\ a,b,c,d\in\mathbf{R}\right\}$ , where $i=\sqrt{-1}$ . Then the number of 2-digit numbers in the…
Matrices Matrix Multiplication Powers of Square Matrices
MediumNumerical valueIdea asked in 6 yearsAsked the year before
Q88
Let $M=\left\{A=\begin{pmatrix}a&b\\c&d\end{pmatrix}:a,b,c,d\in\{\pm3,\pm2,\pm1,0\}\right\}$ . Define $f:M\to\mathbf{Z}$ , as $f(A)=\det(A)$ , for all $A\in M$ , where $\mathbf{Z}$ is set of all integers. Then the number of $A\in M$ such that $f(A)=15$ is equal…
Determinants
MediumNumerical value
Q89
There are 5 students in class 10, 6 students in class 11 and 8 students in class 12. If the number of ways, in which 10 students can be selected from them so as to include at least 2 students from each class and at most 5 students from the total 11 students…
Permutations and Combinations Combinations of Distinct Objects
HardNumerical valueIdea asked in 8 yearsAsked the year before
Q90
If $\alpha,\beta$ are roots of the equation $x^2+5(\sqrt2)x+10=0$ , $\alpha>\beta$ and $P_n=\alpha^n-\beta^n$ for each positive integer n, then the value of $\left(\frac{P_{17}P_{20}+5\sqrt2P_{17}P_{19}}{P_{18}P_{19}+5\sqrt2P_{18}^2}\right)$ is equal to…
Complex Numbers and Quadratic Equations Quadratic Roots Newton's Sums
MediumNumerical valueIdea asked in 7 yearsAsked the year before

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.