JEE Main 2021 · March session

JEE Main 18 March 2021, Shift 2

90 questions: 30 numerical-answer and 60 multiple choice. 8 were rated hard and 31 easy. Its hard-question count was the joint 12th 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: 8; 9% of the shift
Chapters asked: 68; 14 questions with a figure

Key findings · 18 Mar 2021, Shift 2

1
8 hard questions (9%): the joint 12th 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 (14 of 30).
3
45 questions tested a new idea, one no earlier year had asked: 63% of the 72 questions tagged to an idea.
4
24 came back from 2020: the same idea had been asked in the 2020 shifts Lumi analysed.
5
44 shared an idea with another 2021 shift (61% of tagged questions). Shifts in the same year can test the same idea in a new setting.
6
Biggest chapters: Gravitation in Physics (3), Amines in Chemistry (3) and Application of Derivatives in Maths (2).

The shift at a glance

Subjects, difficulty and chapters

Physics

30 Q · 10 numerical

Easy 14Medium 15Hard 1

Top chapters

Chemistry

30 Q · 10 numerical

Easy 14Medium 16Hard 0

Top chapters

Amines3 Q
Aldehydes, Ketones and Carboxylic Acids2 Q
Equilibrium2 Q
Some Basic Concepts of Chemistry2 Q
Alcohols, Phenols and Ethers1 Q
Biomolecules1 Q

Maths

30 Q · 10 numerical

Easy 3Medium 20Hard 7

Top chapters

Question format

Single correct45 Q
Numerical value30 Q
Diagram based7 Q
Two statements3 Q
Match the list3 Q
Other2 Q

What each question asks you to do

Multi-step25 Q
Calculation22 Q
Conceptual17 Q
Application12 Q
Recall9 Q
Data or graph3 Q
Diagram reading2 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

14 easy15 medium1 hard

Q1
Consider a sample of oxygen behaving like an ideal gas. At 300 K, the ratio of root mean square (rms) velocity to the average velocity of gas molecule would be : (Molecular weight of oxygen is 32 g/mol; R $=8.3$ J K $^{-1}$ mol $^{-1}$ )
Kinetic Theory Kinetic Theory of Gases Molecular Speed Ratios
EasySingle correctNew ideaIdea asked in 4 years
Q2
An object of mass $m_1$ collides with another object of mass $m_2$ , which is at rest. After the collision the objects move with equal speeds in opposite direction. The ratio of the masses $m_2 : m_1$ is :
Work, Energy and Power Collisions 1D Elastic Collisions
MediumSingle correctNew idea
Q3
A particle of mass m moves in a circular orbit under the central potential field, $U(r) = -\frac{C}{r}$ , where C is a positive constant. The correct radius –velocity graph of the particle’s motion is :
Gravitation Orbital Mechanics Orbital Velocity
MediumDiagram based · figureIdea asked in 5 yearsAsked the year before
Q4
A solid cylinder of mass m is wrapped with an inextensible light string and, is placed on a rough inclined plane as shown in the figure. The frictional force acting between the cylinder and the inclined plane is : [The coefficient of static friction,…
System of Particles and Rotational Motion Friction Direction of Static Friction
HardDiagram based · figureNew idea
Q5
If the angular velocity of earth’s spin is increased such that the bodies at the equator start floating, the duration of the day would be approximately : [Take g $=10$ ms $^{-2}$ , the radius of earth, R $=6400\times10^3$ m, Take $\pi=3.14$ ]
Gravitation Orbital Mechanics Orbital Time Period
MediumSingle correctIdea asked in 6 yearsAsked the year before
Q6
Consider a uniform wire of mass M and length L. It is bent into a semicircle. Its moment of inertia about a line perpendicular to the plane of the wire passing through the centre is :
System of Particles and Rotational Motion Moment of Inertia MOI Change by Bending a Rod
EasySingle correctNew ideaIdea asked in 2 years
Q7
The speed of electrons in a scanning electron microscope is $1\times10^7$ ms $^{-1}$ . If the protons having the same speed are used instead of electrons, then the resolving power of scanning proton microscope will be changed by a factor of :
Dual Nature of Radiation and Matter Matter Waves Wavelength Ratios of Different Particles
MediumSingle correctIdea asked in 6 yearsAsked the year before
Q8
The velocity - displacement graph of a particle is shown in the figure. The acceleration - displacement graph of the same particle is represented by :
Motion in a Straight Line Kinematics in 1D Velocity-Position Graph Subnormal
MediumDiagram based · figureNew idea
Q9
For an adiabatic expansion of an ideal gas, the fractional change in its pressure is equal to (where $\gamma$ is the ratio of specific heats) :
Thermodynamics Thermodynamic Processes Adiabatic Process Analysis
EasySingle correctIdea asked in 7 yearsAsked the year before
Q10
The angular momentum of a planet of mass M moving around the sun in an elliptical orbit is $\vec{L}$ . The magnitude of the areal velocity of the planet is :
Gravitation Orbital Mechanics Areal Velocity and Angular Momentum
EasySingle correctNew ideaIdea asked in 2 years
Q11
An ideal gas in a cylinder is separated by a piston in such a way that the entropy of one part is $S_1$ and that of the other part is $S_2$ . Given that $S_1 > S_2$ . If the piston is removed then the total entropy of the system will be :
Thermodynamics Second Law and Entropy Extensive Nature of Entropy
EasySingle correctNew idea
Q12
The function of time representing a simple harmonic motion with a period of $\frac{\pi}{\omega}$ is :
Oscillations SHM Equations SHM Definition and Force Law
MediumSingle correctNew ideaIdea asked in 4 years
Q13
In a series LCR circuit, the inductive reactance ( $X_L$ ) is 10 $\Omega$ and the capacitive reactance ( $X_C$ ) is 4 $\Omega$ . The resistance (R) in the circuit is 6 $\Omega$ . The power factor of the circuit is :
Alternating Current Phasor Diagrams Power Factor from Phasors
EasySingle correctIdea asked in 5 yearsAsked the year before
Q14
Which of the following statements are correct ? (A) Electric monopoles do not exist whereas magnetic monopoles exist. (B) Magnetic field lines due to a solenoid at its ends and outside cannot be completely straight and confined. (C) Magnetic field lines are…
Magnetism and Matter Ampere's Circuital Law Toroidal Magnetic Field
MediumOtherNew idea
Q15
The correct relation between $\alpha$ (ratio of collector current to emitter current) and $\beta$ (ratio of collector current to base current) of a transistor is :
Semiconductor Electronics: Materials, Devices and Simple Circuits Transistors Current Gains Alpha and Beta
EasySingle correctNew ideaIdea asked in 2 years
Q16
A proton and an $\alpha$ -particle, having kinetic energies $K_p$ and $K_\alpha$ respectively, enter into a magnetic field at right angles. The ratio of the radii of trajectory of proton to that of $\alpha$ -particle is 2 : 1. The ratio of $K_p : K_\alpha$ is :
Moving Charges and Magnetism Lorentz Force Circular Motion in Uniform B-Field
MediumSingle correctIdea asked in 4 yearsAsked the year before
Q17
Three rays of light, namely red (R), green (G) and blue (B) are incident on the face PQ of a right angled prism PQR as shown in the figure. The refractive indices of the material of the prism for red, green and blue wavelength are 1.27, 1.42 and 1.49…
Ray Optics and Optical Instruments Refraction and Snell's Law Wavelength Dependence of TIR
MediumDiagram based · figureNew idea
Q18
The time taken for the magnetic energy to reach 25% of its maximum value, when a solenoid of resistance R, inductance L is connected to a battery, is :
Electromagnetic Induction
MediumSingle correct
Q19
A plane electromagnetic wave propagating along $y$ -direction can have the following pair of electric field ( $\vec{E}$ ) and magnetic field ( $\vec{B}$ ) components.
Electromagnetic Waves
EasySingle correct
Q20
The decay of a proton to neutron is :
Nuclei Radioactive Decay Beta Decay Mechanisms
MediumSingle correctNew idea
Q21
The projectile motion of a particle of mass 5 g is shown in the figure. The initial velocity of the particle is $5\sqrt{2}$ ms $^{-1}$ and the air resistance is assumed to be negligible. The magnitude of the change in momentum between the points A and B is…
Motion in a Plane Projectile Motion Change in Momentum During Flight
EasyNumerical value · figureNew idea
Q22
An infinite number of point charges, each carrying 1 $\mu$ C charge, are placed along the $y$ -axis at $y=1$ m, 2 m, 4 m, 8 m ................. . The total force on a 1 C point charge, placed at the origin, is $x\times10^3$ N. The value of $x$ , to the nearest…
Electric Charges and Fields Coulomb's Law Superposition of Forces
MediumNumerical valueNew ideaIdea asked in 3 years
Q23
Two wires of same length and thickness having specific resistances 6 $\Omega$ cm and 3 $\Omega$ cm respectively are connected in parallel. The effective resistivity is $\rho$ $\Omega$ cm. The value of $\rho$ , to the nearest integer, is _________.
Current Electricity Ohm's Law and Resistance Resistivity and Geometry
EasyNumerical valueIdea asked in 4 yearsAsked the year before
Q24
A ball of mass 4 kg, moving with a velocity of 10 ms $^{-1}$ , collides with a spring of length 8 m and force constant 100 Nm $^{-1}$ . The length of the compressed spring is $x$ m. The value of $x$ , to the nearest integer, is _________.
Work, Energy and Power Work and Kinetic Energy Maximum Compression of a Spring
EasyNumerical valueNew ideaIdea asked in 3 years
Q25
The typical output characteristics curve for a transistor working in the common-emitter configuration is shown in the figure. The estimated current gain from the figure is _________.
Semiconductor Electronics: Materials, Devices and Simple Circuits Transistors CE Configuration Characteristics
MediumNumerical value · figureNew idea
Q26
Consider a 72 cm long wire AB as shown in the figure. The galvanometer jockey is placed at P on AB at a distance $x$ cm from A. The galvanometer shows zero deflection. The value of $x$ , to the nearest integer, is _________.
Current Electricity Wheatstone Bridge Meter Bridge Principles
EasyNumerical value · figureIdea asked in 4 yearsAsked the year before
Q27
A galaxy is moving away from the earth at a speed of 286 kms $^{-1}$ . The shift in the wavelength of a redline at 630 nm is $x\times10^{-10}$ m. The value of $x$ , to the nearest integer, is _________. [Take the value of speed of light c, as $3\times10^8$ …
Waves Doppler Effect Light Doppler Approximation
EasyNumerical valueNew ideaIdea asked in 2 years
Q28
Consider a water tank as shown in the figure. It’s cross-sectional area is 0.4 m $^2$ . The tank has an opening B near the bottom whose cross-section area is 1 cm $^2$ . A load of 24 kg is applied on the water at the top when the height of the water level is 40…
Mechanical Properties of Fluids Fluid Dynamics Torricelli's Law of Efflux
MediumNumerical value · figureNew ideaIdea asked in 2 years
Q29
A TV transmission tower antenna is at a height of 20 m. Suppose that the receiving antenna is at. (i) ground level (ii) a height of 5 m. The increase in antenna range in case (ii) relative to case (i) is n%. The value of n, to the nearest integer, is…
Communication Systems
MediumNumerical value
Q30
The radius of a sphere is measured to be $(7.50\pm0.85)$ cm. Suppose the percentage error in its volume is $x$ . The value of $x$ , to the nearest $x$ , is _________.
Units and Measurements
EasyNumerical value

Chemistry30 questions

14 easy16 medium0 hard

Q31
A hard substance melts at high temperature and is an insulator in both solid and in molten state. This solid is most likely to be a/an :
The Solid State
EasySingle correct
Q32
Given below are two statements : Statement I : Bohr’s theory accounts for the stability and line spectrum of $\mathrm{Li^+}$ ion. Statement II : Bohr’s theory was unable to explain the splitting of spectral lines in the presence of a magnetic field. In the…
Structure of Atom Bohr's Atomic Theory Limitations of Bohr's Model
EasyTwo statementsNew idea
Q33
The charges on the colloidal CdS sol and $\mathrm{TiO_2}$ sol are, respectively :
Surface Chemistry
MediumSingle correct
Q34
The first ionization energy of magnesium is smaller as compared to that of elements X and Y, but higher than that of Z. The elements X, Y and Z, respectively, are :
Classification of Elements and Periodicity in Properties Periodic Trends Ionization Enthalpy Trends
EasySingle correctIdea asked in 6 yearsAsked the year before
Q35
Match List - I with List - II : List - I List - II (a) Mercury (i) Vapour phase refining (b) Copper (ii) Distillation Refining (c) Silicon (iii) Electrolytic Refining (d) Nickel (iv) Zone Refining Choose the most appropriate answer from the option given…
General Principles and Processes of Isolation of Elements Metallurgical Extraction Metal Refining Techniques
EasyMatch the listIdea asked in 4 yearsAsked the year before
Q36
In basic medium, $\mathrm{H_2O_2}$ exhibits which of the following reactions ? (A) $\mathrm{Mn^{2+}} \rightarrow \mathrm{Mn^{4+}}$ (B) $\mathrm{I_2} \rightarrow \mathrm{I^-}$ (C) $\mathrm{PbS} \rightarrow \mathrm{PbSO_4}$ Choose the most appropriate answer…
Hydrogen Hydrogen and Hydrides Oxidizing Action of H2O2
MediumOtherNew ideaIdea asked in 2 years
Q37
Match List - I with List - II : List - I List - II (a) Be (i) treatment of cancer (b) Mg (ii) extraction of metals (c) Ca (iii) incendiary bombs and signals (d) Ra (iv) windows of X-ray tubes (v) bearings for motor engines. Choose the most appropriate answer…
The s-Block Elements
MediumMatch the list
Q38
The oxidation states of nitrogen in NO, $\mathrm{NO_2}$ , $\mathrm{N_2O}$ and $\mathrm{NO_3^-}$ are in the order of :
Redox Reactions Oxidation Number Assignment
EasySingle correctNew ideaIdea asked in 4 years
Q39
The oxide that shows magnetic property is :
The d- and f-Block Elements d-Block Elements Spin-only Magnetism
MediumSingle correctIdea asked in 6 yearsAsked the year before
Q40
The secondary valency and the number of hydrogen bonded water molecule(s) in $\mathrm{CuSO_4 \cdot 5H_2O}$ , respectively, are :
Coordination Compounds
MediumSingle correct
Q41
Given below are two statements : Statement I : Non-biodegradable wastes are generated by the thermal power plants. Statement II : Bio-degradable detergents leads to eutrophication. In the light of the above statements, choose the most appropriate answer from…
Environmental Chemistry Eutrophication and Dissolved Oxygen Limits
EasyTwo statementsIdea asked in 3 yearsAsked the year before
Q42
Given below are two statements : Statement I : $\mathrm{C_2H_5OH}$ and AgCN both can generate nucleophile. Statement II : KCN and AgCN both will generate nitrile nucleophile with all reaction conditions. Choose the most appropriate option :
Haloalkanes and Haloarenes SN1 and SN2 Mechanisms Ambident Nucleophiles (Cyanide/Nitrite)
MediumTwo statementsNew ideaIdea asked in 3 years
Q43
In the following molecule, [figure] Hybridisation of Carbon a, b and c respectively are :
Organic Chemistry: Some Basic Principles and Techniques
EasyDiagram based · figure
Q44
Main Products formed during a reaction of 1-methoxy naphthalene with hydroiodic acid are :
Alcohols, Phenols and Ethers Ethers Cleavage of Alkyl-Aryl Ethers
EasySingle correct · figureNew ideaIdea asked in 3 years
Q45
In the reaction of hypobromite with amide, the carbonyl carbon is lost as :
Amines Hofmann Bromamide Degradation
EasySingle correctNew ideaIdea asked in 5 years
Q46
[figure: reaction scheme] Consider the above reaction, the product ‘X’ and ‘Y’ respectively are :
Aldehydes, Ketones and Carboxylic Acids Aldol and Cannizzaro Self-Aldol Condensation
MediumDiagram based · figureNew ideaIdea asked in 3 years
Q47
An organic compound “A” on treatment with benzene sulphonyl chloride gives compound B. B is soluble in dil. NaOH solution. Compound A is :
Amines Hinsberg Test
MediumSingle correctIdea asked in 5 yearsAsked the year before
Q48
[figure: reaction scheme] Consider the given reaction, percentage yield of :
Amines Aromatic Hydrocarbons Ortho-Para vs Meta Directing
MediumDiagram based · figureNew ideaIdea asked in 5 years
Q49
Match List - I with List - II : List - I (Class of Chemicals) List - II (Example) (a) Antifertility drug (i) Meprobamate (b) Antibiotic (ii) Alitame (c) Tranquilizer (iii) Norethindrone (d) Artificial Sweetener (iv) Salvarsan Choose the most appropriate…
Chemistry in Everyday Life Drug Classification
MediumMatch the listNew ideaIdea asked in 2 years
Q50
Deficiency of vitamin K causes :
Biomolecules
EasySingle correct
Q51
10.0 mL of $\mathrm{Na_2CO_3}$ solution is titrated against 0.2 M HCl solution. The following titre values were obtained in 5 readings : 4.8 mL, 4.9 mL, 5.0 mL, 5.0 mL and 5.0 mL. Based on these readings, and convention of titrimetric estimation the…
Some Basic Concepts of Chemistry Mole Concept Molarity Calculation
EasyNumerical valueIdea asked in 5 yearsAsked the year before
Q52
The number of species below that have two lone pairs of electrons in their central atom is __________. (Round off to the Nearest Integer). $\mathrm{SF_4}$ , $\mathrm{BF_4^-}$ , $\mathrm{ClF_3}$ , $\mathrm{AsF_3}$ , $\mathrm{PCl_5}$ , $\mathrm{BrF_5}$ ,…
Chemical Bonding and Molecular Structure VSEPR Theory Trigonal Bipyramidal Positions
MediumNumerical valueIdea asked in 5 yearsAsked the year before
Q53
The gas phase reaction $\mathrm{2A(g) \rightleftharpoons A_2(g)}$ at 400 K has $\Delta G^\circ = +25.2$ kJ mol $^{-1}$ . The equilibrium constant $\mathrm{K_C}$ for this reaction is __________ $\times 10^{-2}$ . (Round off to the Nearest Integer). [Use : R =…
Equilibrium Chemical Equilibrium Standard Gibbs Free Energy and Equilibrium Constant
MediumNumerical valueNew ideaIdea asked in 3 years
Q54
A solute A dimerizes in water. The boiling point of a 2 molal solution of A is 100.52℃. The percentage association of A is __________. (Round off to the Nearest Integer). [Use : $\mathrm{K_b}$ for water = 0.52 K kg mol $^{-1}$ Boiling point of water = 100℃]
Solutions Colligative Properties Degree of Association from Van't Hoff Factor
MediumNumerical valueIdea asked in 3 years
Q55
The solubility of $\mathrm{CdSO_4}$ in water is $8.0 \times 10^{-4}$ mol L $^{-1}$ . Its solubility in 0.01 M $\mathrm{H_2SO_4}$ solution is __________ $\times 10^{-6}$ mol L $^{-1}$ . (Round off to the Nearest Integer). (Assume that solubility is much less than…
Equilibrium Solubility Product Common Ion Effect
MediumNumerical valueNew ideaIdea asked in 4 years
Q56
The molar conductivities at infinite dilution of barium chloride, sulphuric acid and hydrochloric acid are 280, 860 and 426 S cm $^2$ mol $^{-1}$ respectively. The molar conductivity at infinite dilution of barium sulphate is __________ S cm $^2$ mol $^{-1}$ .…
Electrochemistry
EasyNumerical value
Q57
A reaction has a half life of 1 min. The time required for 99.9% completion of the reaction is __________ min. (Round off to the Nearest Integer). [Use : ln 2 = 0.69; ln 10 = 2.3]
Chemical Kinetics Integrated Rate Equations First Order Fractional Completion Shortcuts
EasyNumerical valueNew ideaIdea asked in 3 years
Q58
A xenon compound ‘A’ upon partial hydrolysis gives $\mathrm{XeO_2F_2}$ . The number of lone pair of electrons present in compound A is __________. (Round off to the Nearest Integer)
The p-Block Elements Group 17 & 18 Hydrolysis Reactions of Xenon Fluorides
MediumNumerical valueNew idea
Q59
[figure: reaction scheme] Consider the above reaction where 6.1 g of Benzoic acid is used to get 7.8 g of m-bromo benzoic acid. The percentage yield of the product is __________. (Round off to the Nearest Integer). [Given : Atomic masses : C : 12.0 u, H :…
Some Basic Concepts of Chemistry Stoichiometry Theoretical and Percent Yield
EasyNumerical value · figureNew ideaIdea asked in 2 years
Q60
In Tollen’s test for aldehyde, the overall number of electron(s) transferred to the Tollen’s reagent formula $\mathrm{[Ag(NH_3)_2]^+}$ per aldehyde group to form silver mirror is __________. (Round off to the Nearest Integer).
Aldehydes, Ketones and Carboxylic Acids
MediumNumerical value

Maths30 questions

3 easy20 medium7 hard

Q61
If $15\sin^4\alpha + 10\cos^4\alpha = 6$ , for some $\alpha\in\mathbb{R}$ , then the value of $27\sec^6\alpha + 8\mathrm{cosec}^6\alpha$ is equal to :
Trigonometric Functions Trigonometric Identities High Power Reduction Identities
MediumSingle correctNew ideaIdea asked in 2 years
Q62
If P and Q are two statements, then which of the following compound statement is a tautology ?
Mathematical Reasoning
EasySingle correct
Q63
A pole stands vertically inside a triangular park ABC. Let the angle of elevation of the top of the pole from each corner of the park be $\frac{\pi}{3}$ . If the radius of the circumcircle of $\Delta$ ABC is 2, then the height of the pole is equal to :
Trigonometric Functions Properties of Triangles Inradius and Circumradius
MediumSingle correctNew ideaIdea asked in 3 years
Q64
Let in a series of $2n$ observations, half of them are equal to a and remaining half are equal to $-$ a. Also by adding a constant b in each of these observations, the mean and standard deviation of new set become 5 and 20, respectively. Then the value of…
Statistics Variance and SD Effect of Linear Transformations
MediumSingle correctIdea asked in 2 yearsAsked the year before
Q65
Let in a Binomial distribution, consisting of 5 independent trials, probabilities of exactly 1 and 2 successes be 0.4096 and 0.2048 respectively. Then the probability of getting exactly 3 successes is equal to :
Probability
MediumSingle correct
Q66
In a triangle ABC, if $\left|\overrightarrow{BC}\right|=8$ , $\left|\overrightarrow{CA}\right|=7$ , $\left|\overrightarrow{AB}\right|=10$ , then the projection of the vector $\overrightarrow{AB}$ on $\overrightarrow{AC}$ is equal to :
Vector Algebra Dot Product Scalar and Vector Projection
MediumSingle correctIdea asked in 5 yearsAsked the year before
Q67
Let the centroid of an equilateral triangle ABC be at the origin. Let one of the sides of the equilateral triangle be along the straight line $x+y=3$ . If R and r be the radius of circumcircle and incircle respectively of $\Delta$ ABC, then $(R+r)$ is equal to :
Straight Lines Straight Line Distance Distance from Origin to a Line
MediumSingle correctNew idea
Q68
Let a tangent be drawn to the ellipse $\frac{x^2}{27}+y^2=1$ at $(3\sqrt{3}\cos\theta,\sin\theta)$ where $\theta\in\left(0,\frac{\pi}{2}\right)$ . Then the value of $\theta$ such that the sum of intercepts on axes made by this tangent is minimum is equal to :
Application of Derivatives Maxima and Minima Optimization Problems
HardSingle correctIdea asked in 5 years
Q69
Let $y=y(x)$ be the solution of the differential equation $\frac{dy}{dx}=(y+1)\left((y+1)e^{x^2/2}-x\right)$ , $0<x<2.1$ , with $y(2)=0$ . Then the value of $\frac{dy}{dx}$ at $x=1$ is equal to :
Differential Equations Linear Differential Equations Bernoulli Equation Reduction
HardSingle correctNew ideaIdea asked in 2 years
Q70
The area bounded by the curve $4y^2=x^2(4-x)(x-2)$ is equal to :
Application of Integrals Area Under Curves Area of Algebraic Curve Loops
HardSingle correctNew idea
Q71
Let $g(x)=\int_0^x f(t)\,dt$ , where $f$ is continuous function in $[0, 3]$ such that $\frac{1}{3}\le f(t)\le 1$ for all $t\in[0, 1]$ and $0\le f(t)\le\frac{1}{2}$ for all $t\in(1, 3]$ . The largest possible interval in which $g(3)$ lies is :
Integrals Definite Integral Properties Bounding and Estimation Property
MediumSingle correctIdea asked in 3 yearsAsked the year before
Q72
Let $f:\mathbb{R}\to\mathbb{R}$ be a function defined as $f(x)=\begin{cases}\frac{\sin(a+1)x+\sin 2x}{2x}, & \text{if } x<0\\ b, & \text{if } x=0\\ \frac{\sqrt{x+bx^3}-\sqrt{x}}{bx^{5/2}}, & \text{if } x>0\end{cases}$ If $f$ is continuous at $x=0$ , then the…
Continuity and Differentiability Finding Unknown Constants for Continuity
MediumSingle correct · figureNew ideaIdea asked in 3 years
Q73
Let $S_1$ be the sum of first $2n$ terms of an arithmetic progression. Let $S_2$ be the sum of first $4n$ terms of the same arithmetic progression. If $(S_2-S_1)$ is 1000, then the sum of the first $6n$ terms of the arithmetic progression is equal to :
Sequences and Series AP and GP AP Sum of n terms
MediumSingle correctIdea asked in 7 yearsAsked the year before
Q74
Let $S_1 : x^2+y^2=9$ and $S_2 : (x-2)^2+y^2=1$ . Then the locus of center of a variable circle S which touches $S_1$ internally and $S_2$ externally always passes through the points :
Conic Sections
HardSingle correct
Q75
Let the system of linear equations $4x+\lambda y+2z=0$ $2x-y+z=0$ $\mu x+2y+3z=0$ , $\lambda, \mu\in\mathbb{R}$ . has a non-trivial solution. Then which of the following is true ?
Determinants Cramer's Rule Parameter Evaluation in Homogeneous Systems
EasySingle correctIdea asked in 5 yearsAsked the year before
Q76
Let $f:\mathbb{R}-\{3\}\to\mathbb{R}-\{1\}$ be defined by $f(x)=\frac{x-2}{x-3}$ . Let $g:\mathbb{R}\to\mathbb{R}$ be given as $g(x)=2x-3$ . Then, the sum of all the values of $x$ for which $f^{-1}(x)+g^{-1}(x)=\frac{13}{2}$ is equal to.
Relations and Functions Functions and Domain Domain and Range of Inverse Functions
MediumSingle correctNew ideaIdea asked in 2 years
Q77
Let a complex number be $w=1-\sqrt{3}\,i$ . Let another complex number $z$ be such that $|zw|=1$ and $\arg(z)-\arg(w)=\frac{\pi}{2}$ . Then the area of the triangle with vertices origin, $z$ and $w$ is equal to :
Complex Numbers and Quadratic Equations Complex Numbers - Polar Form Area of Rotated Complex Triangles
MediumSingle correctNew idea
Q78
Define a relation R over a class of $n\times n$ real matrices A and B as "ARB iff there exists a non-singular matrix P such that $PAP^{-1}=B$ ". Then which of the following is true ?
Relations and Functions Sets and Relations Equivalence Relations
EasySingle correctNew ideaIdea asked in 4 years
Q79
Consider a hyperbola H : $x^2-2y^2=4$ . Let the tangent at a point P $(4, \sqrt{6})$ meet the $x$ -axis at Q and latus rectum at R $(x_1, y_1)$ , $x_1>0$ . If F is a focus of H which is nearer to the point P, then the area of $\Delta$ QFR is equal to .
Conic Sections
HardSingle correct
Q80
Let $\vec{a}$ and $\vec{b}$ be two non-zero vectors perpendicular to each other and $\left|\vec{a}\right|=\left|\vec{b}\right|$ . If $\left|\vec{a}\times\vec{b}\right|=\left|\vec{a}\right|$ , then the angle between the vectors…
Vector Algebra Cross Product Magnitude and Sine Formula
MediumSingle correctIdea asked in 7 yearsAsked the year before
Q81
If $f(x)$ and $g(x)$ are two polynomials such that the polynomial $P(x)=f(x^3)+x\,g(x^3)$ is divisible by $x^2+x+1$ , then $P(1)$ is equal to ___________.
Complex Numbers and Quadratic Equations Roots of Unity Polynomial Remainder via Roots
MediumNumerical valueNew idea
Q82
Let I be an identity matrix of order $2\times2$ and $P=\begin{bmatrix}2&-1\\5&-3\end{bmatrix}$ . Then the value of $n\in\mathbb{N}$ for which $P^n=5I-8P$ is equal to ___________.
Matrices Matrix Multiplication Cayley-Hamilton Theorem Application
MediumNumerical valueNew ideaIdea asked in 5 years
Q83
The term independent of $x$ in the expansion of $\left[\frac{x+1}{x^{2/3}-x^{1/3}+1}-\frac{x-1}{x-x^{1/2}}\right]^{10}$ , $x\neq1$ , is equal to ___________.
Binomial Theorem
MediumNumerical value
Q84
If $\sum_{r=1}^{10} r!\,(r^3+6r^2+2r+5)=\alpha\,(11!)$ , then the value of $\alpha$ is equal to ___________.
Permutations and Combinations
HardNumerical value
Q85
Let $P(x)$ be a real polynomial of degree 3 which vanishes at $x=-3$ . Let $P(x)$ have local minima at $x=1$ , local maxima at $x=-1$ and $\int_{-1}^{1}P(x)\,dx=18$ , then the sum of all the coefficients of the polynomial $P(x)$ is equal to ___________.
Application of Derivatives Maxima and Minima Determining Coefficients from Extrema
MediumNumerical valueIdea asked in 3 yearsAsked the year before
Q86
Let $y=y(x)$ be the solution of the differential equation $x\,dy-y\,dx=\sqrt{(x^2-y^2)}\,dx$ , $x\ge1$ , with $y(1)=0$ . If the area bounded by the line $x=1$ , $x=e^{\pi}$ , $y=0$ and $y=y(x)$ is $\alpha e^{2\pi}+\beta$ , then the value of $10(\alpha+\beta)$ is…
Differential Equations Homogeneous Equations Substitution y = vx
HardNumerical valueIdea asked in 6 yearsAsked the year before
Q87
Let P be a plane containing the line $\frac{x-1}{3}=\frac{y+6}{4}=\frac{z+5}{2}$ and parallel to the line $\frac{x-3}{4}=\frac{y-2}{-3}=\frac{z+5}{7}$ . If the point $(1, -1, \alpha)$ lies on the plane P, then the value of $|5\alpha|$ is equal to ___________.
Three Dimensional Geometry Skew Lines Plane Containing One Line and Parallel to Other
MediumNumerical valueNew idea
Q88
Let the mirror image of the point $(1, 3, a)$ with respect to the plane $\vec{r}\cdot\left(2\hat{i}-\hat{j}+\hat{k}\right)-b=0$ be $(-3, 5, 2)$ . Then, the value of $|a+b|$ is equal to ___________.
Three Dimensional Geometry Point-Plane Distance Image of a Point
MediumNumerical valueIdea asked in 4 years
Q89
Let ${}^nC_r$ denote the binomial coefficient of $x^r$ in the expansion of $(1+x)^n$ . If $\sum_{k=0}^{10}(2^2+3k)\,{}^nC_k=\alpha\cdot3^{10}+\beta\cdot2^{10}$ , $\alpha, \beta\in\mathbb{R}$ , then $\alpha+\beta$ is equal to ___________.
Binomial Theorem Binomial Coefficients Binomial Coefficients with AP Multiplier
MediumNumerical valueNew ideaIdea asked in 2 years
Q90
Let $f:\mathbb{R}\to\mathbb{R}$ satisfy the equation $f(x+y)=f(x)\cdot f(y)$ for all $x, y\in\mathbb{R}$ and $f(x)\neq0$ for any $x\in\mathbb{R}$ . If the function $f$ is differentiable at $x=0$ and $f'(0)=3$ , then $\lim_{h\to0}\frac{1}{h}\left(f(h)-1\right)$ …
Continuity and Differentiability Differentiability via Functional Equations
MediumNumerical valueIdea asked in 6 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.