JEE Main 2021 · July session

JEE Main 3 August 2021, Shift 2

90 questions: 30 numerical-answer and 60 multiple choice. 6 were rated hard and 29 easy. Its hard-question count was the joint 18th highest of the 24 2021 shifts Lumi analysed. The biggest chapter was Conic Sections, with 3 questions.

Questions: 90; none dropped
Numerical answer: 30; 33% of the shift
Hard: 6; 7% of the shift
Chapters asked: 60; 12 questions with a figure

Key findings · 3 Aug 2021, Shift 2

1
6 hard questions (7%): the joint 18th highest count of the 24 2021 shifts Lumi analysed. Across those shifts the hard share ran from 3% to 17%.
2
45 questions tested a new idea, one no earlier year had asked: 61% of the 74 questions tagged to an idea.
3
26 came back from 2020: the same idea had been asked in the 2020 shifts Lumi analysed.
4
41 shared an idea with another 2021 shift (55% of tagged questions). Shifts in the same year can test the same idea in a new setting.
5
Biggest chapters: Gravitation in Physics (3), Organic Chemistry: Some Basic Principles and Techniques in Chemistry (3) and Conic Sections in Maths (3).

The shift at a glance

Subjects, difficulty and chapters

Physics

30 Q · 10 numerical

Easy 14Medium 13Hard 3

Top chapters

Gravitation3 Q
Nuclei3 Q
Oscillations3 Q
Electromagnetic Induction2 Q
Electromagnetic Waves2 Q
Electrostatic Potential and Capacitance2 Q

Chemistry

30 Q · 10 numerical

Easy 9Medium 20Hard 1

Top chapters

Maths

30 Q · 10 numerical

Easy 6Medium 22Hard 2

Top chapters

Question format

Single correct52 Q
Numerical value30 Q
Assertion–reason3 Q
Diagram based2 Q
Two statements1 Q
Other1 Q
Match the list1 Q

What each question asks you to do

Calculation34 Q
Multi-step22 Q
Application14 Q
Conceptual9 Q
Recall7 Q
Diagram reading3 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

14 easy13 medium3 hard

Q1
A particle is executing simple harmonic motion with time period $T$ . What is the time taken by the particle to go directly to $\frac{1}{\sqrt{2}}$ of its amplitude from its mean position ?
Oscillations SHM Equations Phasor Diagram Technique
EasySingle correctNew ideaIdea asked in 3 years
Q2
A block of mass m rigidly attached with a spring of spring constant $K$ , is compressed through a small distance A. If the block is now released, the time taken by the block in going from P to Q will be
Oscillations SHM Equations Phasor Diagram Technique
MediumSingle correct · figureNew ideaIdea asked in 3 years
Q3
One mole Argon gas is sealed inside a thermally isolated chamber of 1 litre. A 100 W heater kept inside the chamber is switched on for 30 seconds. What will be the rise in temperature of the Argon gas? [R = 8.314 J mol $^{-1}$ K $^{-1}$ ]
Thermodynamics First Law of Thermodynamics Internal Energy Calculation
MediumSingle correctNew ideaIdea asked in 4 years
Q4
A solid sphere of radius $R$ has mass $M$ . A rod of mass $m$ is attached tangentially to the sphere. What would be the moment of inertia of the system about the axis of the rod, if its diameter is negligibly small compared to $R$ ?
System of Particles and Rotational Motion Moment of Inertia MOI of Spheres About a Tangent
MediumSingle correctNew ideaIdea asked in 2 years
Q5
One end of a metal wire is fixed at the centre of a uniform disc of radius 4.0 cm and mass 100 g. The other end of the wire is fixed with a clamp. The hanging disc is rotated about the wire through a small angle and is released. If the disc makes torsional…
Oscillations SHM Equations Torsional Pendulum Time Period
MediumSingle correct · figureNew idea
Q6
A particle is projected from the mid-point of the line joining two fixed particles each of mass $m$ in free space. If the separation between the fixed particles is $l$ , the minimum velocity of projection of the particle so as to escape to far away is equal to
Gravitation Orbital Mechanics Escape Velocity
HardSingle correctIdea asked in 5 yearsAsked the year before
Q7
A solid sphere of mass $M$ and radius $R$ lies on a horizontal rough surface. A horizontal force $F$ is applied at the centre of the sphere. The acceleration of the centre of the sphere will be : (the sphere rolls without slipping)
System of Particles and Rotational Motion
MediumSingle correct
Q8
A projectile is thrown with initial velocity of $(3\hat{i}+4\hat{j})\ \mathrm{m/s}$ . Here ' $\hat{i}$ ' is along the horizontal direction and ' $\hat{j}$ ' is assumed in vertical direction. The equation of the trajectory is - (Take $g = 10\ \mathrm{m/s^2}$ )
Motion in a Plane Projectile Motion Trajectory Equation Application
MediumSingle correctNew ideaIdea asked in 3 years
Q9
The velocity of a particle starting from origin which is set into motion at $t = 0$ varies as $V = V_0(2-t)$ , where $V_0$ is a positive constant. Find the distance covered and displacement by the particle in 4 second.
Motion in a Straight Line Kinematics in 1D Distance Calculation with Turning Points
MediumSingle correctNew idea
Q10
The following diagram shows the relation between stress and strain of two materials A and B. The ratio of Young's modulus of A and B is -
Mechanical Properties of Solids Elasticity Elastic Moduli
EasySingle correct · figureIdea asked in 7 yearsAsked the year before
Q11
An electric charge $10^{-2}\,\mu\mathrm{C}$ is placed at the origin (0, 0) of X–Y coordinate system. The coordinates of two points A and B are $(\sqrt{2},\sqrt{2})$ and (2, 0), respectively. The potential difference between the point A and B will be -
Electrostatic Potential and Capacitance Electric Potential Point Charge Potential
EasySingle correctNew ideaIdea asked in 3 years
Q12
If $A$ , $Z$ and $M$ represents the number of nucleons, protons and mass of the nucleus respectively, then mass defect $\Delta M$ is represented by ( $m_p$ and $m_n$ are mass of a proton and a neutron respectively)
Nuclei Nuclear Binding Energy Mass Defect Calculation
EasySingle correctNew ideaIdea asked in 3 years
Q13
A magnetic needle of length 12 cm, suspended at its middle point by a thread, stays at an angle of $45^\circ$ with the horizontal. If the pole strength of the needle is 2.4 Am, what vertical force should be applied to an end so as to keep it in horizontal…
Magnetism and Matter
HardSingle correct
Q14
Two concentric and coplanar circular coils have radius $a$ and $b\ (\gg a)$ as shown in figure. Resistance of the inner coil is $R$ . Current in the outer coil is increased from 0 to $i$ , then total charge circulating the inner coil is
Electromagnetic Induction Faraday's Law Induced Charge Flow
MediumSingle correct · figureIdea asked in 3 years
Q15
To view a small object with angular magnification 5, the power of the microscope needed is [The final image is formed at infinity] (Distance of distinct vision = 25 cm)
Ray Optics and Optical Instruments Optical Instruments Simple Microscope Magnification
EasySingle correctNew idea
Q16
If a man having the least distance of distinct vision of 50 cm, what should be the focal length of the spectacles for the man ?
Ray Optics and Optical Instruments Optical Instruments Hypermetropia Correction
EasySingle correctNew idea
Q17
Identify the resulting gate for the following circuit.
Semiconductor Electronics: Materials, Devices and Simple Circuits Logic Gates Combinational Logic Circuits
MediumSingle correct · figureIdea asked in 7 yearsAsked the year before
Q18
A radioactive isotope has a half life of T years. After how much time is its activity reduced to 6.25% of its original activity ?
Nuclei Radioactive Decay Half-Life Calculations
EasySingle correctIdea asked in 4 yearsAsked the year before
Q19
Given below are two statements : one is labeled as Assertion A and the other is labeled as Reason R. Assertion A : Communication techniques which are using much higher frequencies like TV signal broadcast, can not be received beyond the line-of-sight. Reason…
Electromagnetic Waves
EasyAssertion–reason
Q20
A block of mass 4.0 kg is pulled up on a smooth incline of angle $30^\circ$ with the horizontal. If the block moves with an acceleration of 1.0 m/s $^2$ , the power delivered by the pulling force at a time 3.0 s after the motion starts is
Work, Energy and Power Power Instantaneous Power
MediumSingle correctNew ideaIdea asked in 2 years
Q21
$\underline{\hspace{2em}}\times10^{-10}$ J (joule) energy is contained in a 120 m length of a laser beam operating at 4mW.
Electromagnetic Waves
MediumNumerical value
Q22
The magnetic flux linked with a coil at any instant 't' is given by $\phi = 5t^3 - 100t + 300$ (SI unit). The emf induced in the coil at t = 2 s is $\underline{\hspace{2em}}$ V.
Electromagnetic Induction Faraday's Law Faraday's Law Magnitude
EasyNumerical valueIdea asked in 6 yearsAsked the year before
Q23
The ratio of shortest wavelength to the largest wavelength in Brackett series of atomic spectra for hydrogen atom is $x : 25$ , where the value of $x$ is $\underline{\hspace{2em}}$ .
Atoms Bohr's Atomic Model Shortest and Longest Wavelengths
MediumNumerical valueIdea asked in 5 yearsAsked the year before
Q24
A train left the station at uniform acceleration of 2 m/s $^2$ . A man behind the train at a distance of 5 m is running with a constant speed of 6 m/s to catch the train. The time to catch the train will be $\underline{\hspace{2em}}$ s.
Motion in a Straight Line Kinematics in 1D 1D Overtaking and Meeting Time
EasyNumerical valueNew idea
Q25
A sample undergoing beta decay, reduced to $\frac{1}{16}^{\mathrm{th}}$ of its initial mass in 48 years. Its half life period is $\underline{\hspace{2em}}$ years.
Nuclei Radioactive Decay Half-Life Calculations
EasyNumerical valueIdea asked in 4 yearsAsked the year before
Q26
Two particles of equal mass 'm' go round a circle of radius $R$ under the action of their mutual gravitational attraction. The speed of each particle is $\sqrt{\frac{1}{x}\cdot\frac{Gm}{R}}$ where $x$ is $\underline{\hspace{2em}}$ .
Gravitation Orbital Mechanics Binary Star Systems
MediumNumerical valueNew ideaIdea asked in 2 years
Q27
In Young's double slit experiment, 58 fringes are observed in the field of view when a light of wavelength 6000 $\mathrm{\AA}$ was used. Number of fringes that can be viewed, if another light of wavelength 4000 $\mathrm{\AA}$ is used are…
Wave Optics Young's Double Slit YDSE Fringe Width
EasyNumerical valueIdea asked in 5 yearsAsked the year before
Q28
As the switch S is closed in the circuit shown in figure, current passed through it is $\underline{\hspace{2em}}$ A.
Current Electricity Kirchhoff's Laws Nodal Analysis Technique
EasyNumerical value · figureNew ideaIdea asked in 2 years
Q29
The electric potential at point $(x, 0, 0)$ is $V = \left(\frac{1000}{x}+\frac{1500}{x^2}+\frac{500}{x^3}\right)$ volt. The electric field strength at a point $x = 1\ m$ is $\underline{\hspace{2em}}\ \hat{i}\ V/m$ .
Electrostatic Potential and Capacitance Electric Potential Field-Potential Relation
EasyNumerical valueNew ideaIdea asked in 2 years
Q30
A body of mass 20 kg is placed at a latitude of $45^\circ$ on earth of radius $R$ and angular speed $\omega$ . The change in weight of the body (if earth stops rotating) is $x\,R\omega^2$ . The value of $x$ is $\underline{\hspace{2em}}$ .
Gravitation
HardNumerical value

Chemistry30 questions

9 easy20 medium1 hard

Q31
Given below are two statements : Statement I : In dichromate ion, all the Cr—O bonds are of equal length. Statement II : In dichromate ion, the Cr–O–Cr bond angle is less than the H—O—H bond angle in water In the light of the above statements, choose the…
The d- and f-Block Elements d-Block Elements Dichromate Reactions
MediumTwo statementsNew ideaIdea asked in 3 years
Q32
The lowest freezing point among the following solutions will be observed in [Atomic weight : C = 12, Mg = 24, Na = 23, Cl = 35.5, O = 16, N = 14]
Solutions Colligative Properties Freezing Point Depression
EasySingle correctIdea asked in 4 yearsAsked the year before
Q33
The correct order of the basic character for the following metal hydroxides is
The s-Block Elements f-Block Elements Basicity Trend of Lanthanide Hydroxides
MediumSingle correctNew idea
Q34
Given below are two statements: one is labelled as Assertion A and the other is labelled as Reason R Assertion A : Fluoride based compound is added during extraction of aluminum from bauxite. Reason R : Alumina is a poor conductor of electricity. In the…
General Principles and Processes of Isolation of Elements Metallurgical Extraction Hall-Heroult Process
EasyAssertion–reasonNew ideaIdea asked in 3 years
Q35
Which one of the following statements is incorrect? A. The correct order of increasing first ionization enthalpy is Li < H < F. B. Out of the three isotopes of hydrogen, two are radioactive C. Reactivity of halogens is much more than that of hydrogen. D. The…
Hydrogen Hydrogen and Hydrides Isotopes of Hydrogen
MediumOtherIdea asked in 4 yearsAsked the year before
Q36
A white solid (X) on heating forms a solid (Y) and a gas (Z). Another solid (B) forms (X) by reacting with (Z). (Y) can be converted into (X) and (B). What is (Y)?
The s-Block Elements s-Block Elements Thermal Stability Trends
MediumSingle correctNew ideaIdea asked in 2 years
Q37
Silicones are group of organosilicon polymers. Which one among the following acts as a chain terminating unit in silicone polymerization?
The p-Block Elements Group 13 & 14 Silicones and Zeolites
EasySingle correctNew ideaIdea asked in 2 years
Q38
Which one of the following set will give coloured aqueous solution?
The d- and f-Block Elements d-Block Elements Color and Catalysis
EasySingle correctNew ideaIdea asked in 4 years
Q39
Which one of the following statements is correct?
The d- and f-Block Elements f-Block Elements Cerium(IV) as an Oxidizing Agent
MediumSingle correctNew ideaIdea asked in 3 years
Q40
The oxidation states of Fe in $\mathrm{[Fe(NCS)(NH_3)_5]SO_4}$ , $\mathrm{Na_3[Fe(S_2O_3)_3]}$ and $\mathrm{[Fe(CO)_5]}$ respectively are
Coordination Compounds Coordination Nomenclature Oxidation State Calculation
EasySingle correctIdea asked in 4 yearsAsked the year before
Q41
Given below are two statements: one is labelled as Assertion A and the other is labelled as Reason R Assertion A : Fluoride ion concentration above 2 ppm causes brown mottling of teeth. Reason R : The presence of fluoride ions in drinking water converts…
Environmental Chemistry Fluoride Limits in Drinking Water
EasyAssertion–reasonIdea asked in 3 years
Q42
In the detection of nitrogen in an organic compound by Lassaigne's test, the iron compounds formed are
Organic Chemistry: Some Basic Principles and Techniques Anion Analysis Cyanide Prussian Blue Test
MediumSingle correctNew ideaIdea asked in 2 years
Q43
Order of reactivity for hydrolysis of substituted chlorobenzenes in the presence of aqueous NaOH is
Haloalkanes and Haloarenes Aromatic Hydrocarbons SNAr and Meisenheimer Complex
MediumSingle correct · figureNew ideaIdea asked in 3 years
Q44
The correct structure of $cis$ –2,4-dimethylhept-3-ene is,
Organic Chemistry: Some Basic Principles and Techniques Stereoisomerism Geometrical Isomerism
MediumDiagram based · figureNew idea
Q45
Arrange the following halides in the increasing order of their reactivity towards $\mathrm{S_N1}$ reaction mechanism.
Haloalkanes and Haloarenes SN1 and SN2 Mechanisms Allylic and Benzylic Reactivity in SN1
MediumSingle correct · figureIdea asked in 5 years
Q46
Which one of the following reagents are not suitable for the preparation of benzaldehyde from benzene?
Aldehydes, Ketones and Carboxylic Acids Aromatic Hydrocarbons Gattermann-Koch Formylation
MediumSingle correctNew ideaIdea asked in 3 years
Q47
A. $\xrightarrow[\text{2) Heat}]{\text{1) }\mathrm{NaNO_{2(aq.)}},\ \mathrm{HCl}}$ B. $\xrightarrow[\text{conc. }\mathrm{H_2SO_4}]{\mathrm{NO_2^+}}$ 2,4,6-trinitrophenol (picric acid) In the above reaction sequence the compounds A and B respectively are
Amines Diazonium Salts Replacement by H or OH
EasySingle correct · figureNew ideaIdea asked in 3 years
Q48
Match List I with II List I | List II (Ion detected) A. Borax Bead test | I. $\mathrm{As^{3+}}$ B. Charcoal cavity test | II. $\mathrm{Al^{3+}}$ C. Flame test | III. $\mathrm{Fe^{3+}}$ D. Lake test | IV. $\mathrm{Sr^{2+}}$ Choose the correct answer from the…
Practical Chemistry (Qualitative Analysis)Cation Analysis Confirmatory Tests
MediumMatch the listNew ideaIdea asked in 4 years
Q49
Select the odd group
Polymers Polymer Classification
MediumSingle correctNew ideaIdea asked in 3 years
Q50
Which one of the following represents the correct structure of a dinucleotide?
Biomolecules
MediumDiagram based · figure
Q51
Chlorine is prepared according to the following equation : $\mathrm{4HCl + MnO_{2(s)} \longrightarrow 2H_2O_{(l)} + MnCl_{2(aq)} + Cl_{2(g)}}$ . 10g sample of $\mathrm{MnO_2}$ produces 2.24L of chlorine under SATP, the percentage purity of the…
Some Basic Concepts of Chemistry Mole Concept Percentage Purity of Samples
MediumNumerical valueNew idea
Q52
The total kinetic energy of 10 moles of a monoatomic ideal gas at 25°C in kJ is $\underline{\hspace{2em}}$ . (Nearest integer) [R = 8.314 J K $^{-1}$ mol $^{-1}$ ]
States of Matter Ideal Gas Laws Translational Kinetic Energy
EasyNumerical valueNew idea
Q53
The ratio of radii for the first and third orbits of hydrogen atom is $1 : x$ . The value of $x$ is $\underline{\hspace{2em}}$ . (Integer answer)
Structure of Atom Bohr's Atomic Theory Bohr Radius Calculation
EasyNumerical valueIdea asked in 6 yearsAsked the year before
Q54
The work done by an ideal monoatomic gas when it is taken along the path ABCD as shown in the figure is $xP_0V_0$ . The value of $(-x)$ is $\underline{\hspace{2em}}$ . (Nearest integer) [ln 2 = 0.69]
Thermodynamics (Chemistry)
HardNumerical value · figure
Q55
The pH of the solution resulted when 150 mL of 0.1 M ammonia solution is titrated with 50 mL of 0.1M HCl [ $\mathrm{pK_b(NH_3)}$ = 4.7] is $\underline{\hspace{2em}}$ . (Nearest integer) [ $\log_{10} 2 = 0.30$ ]
Equilibrium pH and Buffers Buffer Formation via Partial Neutralization
MediumNumerical valueIdea asked in 2 yearsAsked the year before
Q56
The potential of a cell containing two hydrogen electrodes, one in contact with $10^{-8}$ M $\mathrm{H^+}$ concentration and the other in contact with 0.025 M $\mathrm{H^+}$ concentration is $x \times 10^{-4}$ V. The value of $x$ is…
Electrochemistry Nernst Equation Concentration Cells
MediumNumerical valueIdea asked in 3 yearsAsked the year before
Q57
The inactivation process of virus is first order with respect to virus concentration and 2% of the virus was inactivated in the first one minute. Time taken (in minutes) for the virus to become 75% inactivated is $\underline{\hspace{2em}}$ . (Nearest integer)…
Chemical Kinetics Integrated Rate Equations First Order Integrated Rate Law
MediumNumerical valueIdea asked in 6 yearsAsked the year before
Q58
In an adsorption isotherm, it is seen that the graph of $\log\left(\frac{x}{m}\right)$ vs $\log P$ , where $P$ is in atm, is a straight line inclined at $45^\circ$ and the intercept is 0.699. The amount of solute in grams adsorbed per gram of adsorbent at a…
Surface Chemistry Adsorption Freundlich Logarithmic Plot
MediumNumerical valueIdea asked in 4 yearsAsked the year before
Q59
The number of electrophiles in the list below is $\underline{\hspace{2em}}$ . (Integer answer) $\mathrm{\overset{\oplus}{N}O_2,\ CH_3-\overset{\oplus}{C}=O,\ H_2O,\ \overset{\oplus}{N}O}$
Organic Chemistry: Some Basic Principles and Techniques
MediumNumerical value
Q60
The number of isomers with molecular formula $\mathrm{C_3H_9N}$ , which will react with $\mathrm{CHCl_3 + KOH}$ is $\underline{\hspace{2em}}$ . (Integer answer)
Amines Carbylamine Reaction
MediumNumerical valueIdea asked in 4 yearsAsked the year before

Maths30 questions

6 easy22 medium2 hard

Q61
Let $\mathbb{R}$ be the set of real numbers and '*' be a binary operation on $\mathbb{R} - \{0\}$ defined by $a * b = a + 2b + \frac{a}{b}$ . Then the operation '*' is
Relations and Functions
MediumSingle correct
Q62
If the equations $2x^2 + kx - 5 = 0$ and $x^2 - 3x - 4 = 0$ have one root in common, then a value of ' $k$ ' is
Complex Numbers and Quadratic Equations Quadratic Roots Common Root Condition
EasySingle correctIdea asked in 3 yearsAsked the year before
Q63
Let $A = (a_{ij})$ be a $3\times3$ matrix, where $a_{ij} = 7^{\max\{i,j\}}$ . Then $\det A$ is equal to
Determinants
MediumSingle correct
Q64
Let $A$ be a $3\times3$ matrix and $|A| = -1$ . A matrix $B$ is obtained from the matrix $A$ by applying the following elementary row operations : $R_2 \to R_2 + 3R_1$ $R_3 \to 3R_3$ $R_1 \to R_1 - 5R_3$ . Then $|B|$ is equal to :
Determinants
EasySingle correct
Q65
For the system of linear equations $(k-3)x + y + z = 0$ $x + (k-3)y + z = 0$ $x + y + (k-3)z = 0$ the number of distinct values of $k$ , for which it has a non-trivial solution, is
Determinants Cramer's Rule Parameter Evaluation in Homogeneous Systems
MediumSingle correctIdea asked in 5 yearsAsked the year before
Q66
In the Binomial expansion of $\left(2x\sqrt{x} + 3y^{\frac{1}{16}}\right)^{33}$ , the number of terms, having positive integral powers of $x$ and $y$ is :
Binomial Theorem
MediumSingle correct
Q67
If ' $m$ ' is the arithmetic mean of two distinct real numbers $l$ and $n$ $(l, n > 1)$ and $G_1$ , $G_2$ and $G_3$ are three geometric means between $l$ and $n$ , then $G_1^4 + 2G_2^4 + G_3^4$ equals
Sequences and Series AP and GP Arithmetic and Geometric Means
MediumSingle correctIdea asked in 4 yearsAsked the year before
Q68
$\lim_{x\to0}(1+x)^{\frac{1}{e^x-1}}$ is equal to
Limits and Derivatives L'Hopital's Rule Indeterminate Power Forms
EasySingle correctNew ideaIdea asked in 3 years
Q69
If for non-zero distinct real numbers $a$ , $b$ and $c$ ,…
Continuity and Differentiability Finding Unknown Constants for Continuity
MediumSingle correctNew ideaIdea asked in 3 years
Q70
Let $f : \mathbb{R} \to \mathbb{R}$ be a differentiable function such that $f(0) = 2$ . If $\left|\frac{df}{dx}\right| \le 3$ , for all $x \in \mathbb{R}$ , then $f(1)$ lies in the interval
Application of Derivatives Mean Value Theorem Proving Lipschitz Condition via LMVT
MediumSingle correctNew idea
Q71
If the function $f : \mathbb{R} \to \mathbb{R}$ defined by $f(x) = \frac{x^3}{3} - (2a+3b)x^2 + 24abx + 5$ has a local maxima at $-4$ and a local minima at 4, then a value of $4a + 3b$ is
Application of Derivatives Maxima and Minima Finding Parameters from Given Extrema
MediumSingle correctNew ideaIdea asked in 3 years
Q72
The integral $\int \frac{dx}{\sqrt{2x+5}-\sqrt{2x+3}}$ is equal to (where C is a constant of integration)
Integrals
MediumSingle correct
Q73
A curve passes through the point $(-1, 4)$ . If the normal to it at any point $(x, y)$ on it passes through $(2, 0)$ , then its equation is
Differential Equations Tangents and Normals Finding Curve from Tangent/Normal Properties
MediumSingle correctNew idea
Q74
If an ellipse passes through the point $(4, 1)$ and has foci at $(\pm3, 0)$ , then its eccentricity is
Conic Sections Ellipse Eccentricity Focal distance property
MediumSingle correctIdea asked in 5 yearsAsked the year before
Q75
The tangents, to the parabola $y^2 = 2 - x$ at the points of its intersection with the line $y = x - 2$ , intersect at the point
Conic Sections General Conic Section Chord of Contact for General Conic
MediumSingle correctNew ideaIdea asked in 2 years
Q76
Let $L_1$ and $L_2$ be two lines passing through the points $P(b-c, c-a, a-b)$ and $Q\left(\frac{1}{l}, \frac{1}{m}, \frac{1}{n}\right)$ respectively, where $\frac{1}{l}, \frac{1}{m}$ and $\frac{1}{n}$ are the $a^{\mathrm{th}}$ , $b^{\mathrm{th}}$ and…
Three Dimensional Geometry Angle Between Lines Perpendicularity Condition
HardSingle correctNew idea
Q77
A plane intersects the $yz$ -plane at $x = 0,\ 2y - 3z = 5$ and it intersects the $xy$ -plane at $z = 0,\ 7x + 4y = 10$ . Then the distance of the point $(1, -2, -1)$ from this plane is
Three Dimensional Geometry Point-Plane Distance Distance from Point
MediumSingle correctNew ideaIdea asked in 2 years
Q78
A dice is thrown twice. If $A$ is the event that the sum of numbers appearing on them is 7 and $B$ is the event that number 5 appears at least once, then $P(B|A)$ is equal to
Probability Conditional Probability Reduced Sample Space Method
EasySingle correctNew idea
Q79
The maximum value of $(\cos\theta_1)\cdot(\cos\theta_2)\cdot(\cos\theta_3)\cdot\ldots\cdot(\cos\theta_{10})$ under the restrictions $0 < \theta_1, \theta_2, \ldots, \theta_{10} < \frac{\pi}{2}$ and…
Trigonometric Functions Trigonometric Identities Trigonometric Bounds and Inequalities
HardSingle correctNew idea
Q80
The statement $\sim(p \leftrightarrow \sim q)$ is equivalent to
Mathematical Reasoning
EasySingle correct
Q81
The number of elements in the set $\{x \in \mathbb{R} : (|x| - 2)\cdot|2x+3| = 1\}$ is $\underline{\hspace{2em}}$ .
Relations and Functions
MediumNumerical value
Q82
If six letter words (with or without meaning) are formed using A, B, C, D, E and F, then the number of words, in which exactly one of these alphabets is repeated three times and the other three are distinct, is $\underline{\hspace{2em}}$ .
Permutations and Combinations Permutations with Repetition
MediumNumerical valueIdea asked in 4 yearsAsked the year before
Q83
If the sum of all 3-digit numbers which are multiples of 9 is $41k$ , then ' $k$ ' is equal to $\underline{\hspace{2em}}$ .
Sequences and Series AP and GP AP Sum of n terms
EasyNumerical valueIdea asked in 7 yearsAsked the year before
Q84
If $[x]$ denotes the greatest integer $\le x$ , then the value of $\int_{-2}^{4}\left|[x] - x\right|dx$ is $\underline{\hspace{2em}}$ .
Integrals Definite Integral Properties Greatest Integer Function Integrals
MediumNumerical valueIdea asked in 7 yearsAsked the year before
Q85
If the area (in sq. units) bounded by the parabolas $5x^2 - y = 0$ and $2x^2 - y + b = 0$ $(b > 0)$ is $12\sqrt{3}$ , then ' $b$ ' is equal to $\underline{\hspace{2em}}$ .
Application of Integrals Area Under Curves Area Between Curves
MediumNumerical valueIdea asked in 7 yearsAsked the year before
Q86
The circles, $x^2 + y^2 + 2x + 4y - 4 = 0$ and $x^2 + y^2 - 4x + 4y + k = 0$ touch each other internally. If the point of contact of these circles is $(a, b)$ , then $5(a^2 + b^2)$ is equal to $\underline{\hspace{2em}}$ .
Conic Sections Circle Tangents Coordinates of Point of Contact
MediumNumerical valueNew idea
Q87
If the solution $y = y(x)$ of the differential equation $\left(y + 3x^4\right)\frac{dx}{dy} = x,\ x > 0$ , satisfies $y(1) = -1$ , then the value of $\frac{d^2y}{dx^2} - \frac{dy}{dx} + y$ at $x = 2$ is $\underline{\hspace{2em}}$ .
Differential Equations Linear Differential Equations Linear in x Form
MediumNumerical valueNew ideaIdea asked in 4 years
Q88
The mean and variance of 7 observations are 7 and 18 respectively. If 5 of the observations are 2, 4, 10, 11, 13 and the remaining observations are $x$ and $y$ , then $xy$ is equal to $\underline{\hspace{2em}}$ .
Statistics Variance and SD Finding Unknown Observations from Mean and Variance
MediumNumerical valueIdea asked in 7 yearsAsked the year before
Q89
A vertical pole of height $10\sqrt{3}$ meters is observed from three points, A, B and C in the same horizontal line passing through the foot 'O' of the pole. The angles of elevation of the top, 'P', of the pole from A, B and C are in A.P. If…
Trigonometric Functions
MediumNumerical value
Q90
Let $\vec{a} = 2\hat{i} + \alpha\hat{j} + \hat{k}$ and $\vec{b} = \beta\hat{i} - 5\hat{j} + \gamma\hat{k}$ , where $\alpha$ , $\beta$ and $\gamma$ are real numbers. If $\vec{a}\times\vec{b} = 26\hat{i} - 11\hat{j} - 19\hat{k}$ , then $\alpha - \beta + \gamma$ …
Vector Algebra Cross Product Determinant Calculation Method
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.