JEE Main 2021 · February session

JEE Main 26 February 2021, Shift 2

90 questions: 30 numerical-answer and 60 multiple choice. 11 were rated hard and 24 easy. Its hard-question count was the joint 6th highest of the 24 2021 shifts Lumi analysed. The biggest chapter was Aldehydes, Ketones and Carboxylic Acids, with 4 questions.

Questions: 90; none dropped
Numerical answer: 30; 33% of the shift
Hard: 11; 12% of the shift
Chapters asked: 63; 12 questions with a figure

Key findings · 26 Feb 2021, Shift 2

1
11 hard questions (12%): the joint 6th 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 5 of 30 rated hard; Physics had the most easy questions (11 of 30).
3
62 questions tested a new idea, one no earlier year had asked: 79% of the 78 questions tagged to an idea.
4
12 came back from 2020: the same idea had been asked in the 2020 shifts Lumi analysed.
5
39 shared an idea with another 2021 shift (50% of tagged questions). Shifts in the same year can test the same idea in a new setting.
6
Biggest chapters: Oscillations in Physics (4), Aldehydes, Ketones and Carboxylic Acids in Chemistry (4) and Application of Derivatives 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 6Medium 20Hard 4

Top chapters

Maths

30 Q · 10 numerical

Easy 7Medium 18Hard 5

Top chapters

Question format

Single correct42 Q
Numerical value30 Q
Diagram based7 Q
Match the list5 Q
Assertion–reason2 Q
Two statements2 Q
Other2 Q

What each question asks you to do

Calculation29 Q
Multi-step26 Q
Conceptual16 Q
Application9 Q
Recall8 Q
Data or graph2 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 easy17 medium2 hard

Q1
A radioactive sample is undergoing $\alpha$ decay. At any time $t_1$ , its activity is A and another time $t_2$ , the activity is $\frac{A}{5}$ . What is the average life time for the sample ?
Nuclei Radioactive Decay Mean Life Calculation
EasySingle correctNew ideaIdea asked in 2 years
Q2
Given below are two statements : one is labeled as Assertion A and the other is labeled as Reason R. Assertion A : For a simple microscope, the angular size of the object equals the angular size of the image. Reason R : Magnification is achieved as the small…
Ray Optics and Optical Instruments Optical Instruments Visual Angle and Angular Magnification
MediumAssertion–reasonNew idea
Q3
A tuning fork A of unknown frequency produces 5 beats/s with a fork of known frequency 340 Hz. When fork A is filed, the beat frequency decreases to 2 beats/s. What is the frequency of fork A ?
Waves Beats Effect of Filing Fork
EasySingle correctNew idea
Q4
Draw the output signal Y in the given combination of gates.
Semiconductor Electronics: Materials, Devices and Simple Circuits Logic Gates Timing Diagrams and Waveform Analysis
MediumDiagram based · figureNew ideaIdea asked in 4 years
Q5
Given below are two statements : Statement I : A second's pendulum has a time period of 1 second. Statement II : It takes precisely one second to move between the two extreme positions. In the light of the above statements, choose the correct answer from the…
Oscillations Pendulums Seconds Pendulum
EasyTwo statementsNew ideaIdea asked in 2 years
Q6
If ‘C’ and ‘V’ represent capacity and voltage respectively then what are the dimensions of $\lambda$ where C/V = $\lambda$ ?
Units and Measurements
MediumSingle correct
Q7
An aeroplane, with its wings spread 10 m, is flying at a speed of 180 km/h in a horizontal direction. The total intensity of earth’s field at that part is $2.5 \times 10^{-4}$ Wb/m $^2$ and the angle of dip is $60^\circ$ . The emf induced between the tips of…
Electromagnetic Induction Faraday's Law Linear Motional EMF
MediumSingle correctNew ideaIdea asked in 3 years
Q8
A cord is wound round the circumference of wheel of radius r. The axis of the wheel is horizontal and the moment of inertia about it is I. A weight mg is attached to the cord at the end. The weight falls from rest. After falling through a distance ‘h’, the…
System of Particles and Rotational Motion Work and Kinetic Energy Conservation of Mechanical Energy
MediumSingle correctNew ideaIdea asked in 4 years
Q9
The trajectory of a projectile in a vertical plane is $y = \alpha x - \beta x^2$ , where $\alpha$ and $\beta$ are constants and x & y are respectively the horizontal and vertical distances of the projectile from the point of projection. The angle of…
Motion in a Plane Projectile Motion Trajectory Equation Application
EasySingle correctNew ideaIdea asked in 3 years
Q10
Two masses A and B, each of mass M are fixed together by a massless spring. A force acts on the mass B as shown in figure. If the mass A starts moving away from mass B with acceleration ‘a’, then the acceleration of mass B will be :
Laws of Motion Newton's Laws of Motion F=ma for Single Body
MediumDiagram based · figureNew ideaIdea asked in 6 years
Q11
Given below are two statements : Statement I : An electric dipole is placed at the centre of a hollow sphere. The flux of electric field through the sphere is zero but the electric field is not zero anywhere in the sphere. Statement II : If R is the radius…
Electric Charges and Fields Gauss's Law Flux of Enclosed Dipoles
MediumTwo statementsNew idea
Q12
A scooter accelerates from rest for time $t_1$ at constant rate $a_1$ and then retards at constant rate $a_2$ for time $t_2$ and comes to rest. The correct value of $\frac{t_1}{t_2}$ will be :
Motion in a Straight Line Kinematics in 1D Uniform Acceleration Equations
EasySingle correctNew ideaIdea asked in 4 years
Q13
The internal energy (U), pressure (P) and volume (V) of an ideal gas are related as U = 3PV + 4. The gas is :
Kinetic Theory Kinetic Theory of Gases Law of Equipartition
MediumSingle correctNew ideaIdea asked in 4 years
Q14
The recoil speed of a hydrogen atom after it emits a photon in going from n = 5 state to n = 1 state will be :
Atoms Bohr's Atomic Model Atom Recoil During Emission
MediumSingle correctNew ideaIdea asked in 2 years
Q15
The length of metallic wire is $l_1$ when tension in it is $T_1$ . It is $l_2$ when the tension is $T_2$ . The original length of the wire will be :
Mechanical Properties of Solids Elasticity Finding Natural Length from Two Tensions
EasySingle correctNew ideaIdea asked in 3 years
Q16
A particle executes S.H.M., the graph of velocity as a function of displacement is :
Oscillations SHM Equations Phase Space Trajectory (v vs x)
EasySingle correctNew idea
Q17
The incident ray, reflected ray and the outward drawn normal are denoted by the unit vectors $\vec{a}$ , $\vec{b}$ and $\vec{c}$ respectively. Then choose the correct relation for these vectors.
Ray Optics and Optical Instruments
MediumSingle correct
Q18
Find the peak current and resonant frequency of the following circuit (as shown in figure).
Alternating Current LCR Resonance Resonant Frequency Calculation
MediumDiagram based · figureNew ideaIdea asked in 4 years
Q19
An inclined plane making an angle of $30^\circ$ with the horizontal is placed in a uniform horizontal electric field $200\ \frac{N}{C}$ as shown in the figure. A body of mass 1 kg and charge 5 mC is allowed to slide down from rest at a height of 1 m. If the…
Electric Charges and Fields Friction Motion Down a Rough Incline
HardDiagram based · figureNew ideaIdea asked in 3 years
Q20
A wire of 1 $\Omega$ has a length of 1 m. It is stretched till its length increases by 25%. The percentage change in resistance to the nearest integer is :
Current Electricity Ohm's Law and Resistance Resistance of Stretched Wires
EasySingle correctNew ideaIdea asked in 4 years
Q21
The zener diode has a $V_z$ = 30 V. The current passing through the diode for the following circuit is ________ mA.
Semiconductor Electronics: Materials, Devices and Simple Circuits Semiconductor Diodes Zener Diode Regulation
EasyNumerical value · figureIdea asked in 5 yearsAsked the year before
Q22
Time period of a simple pendulum is T. The time taken to complete $\frac{5}{8}$ oscillations starting from mean position is $\frac{\alpha}{\beta}$ T. The value of $\alpha$ is ________.
Oscillations SHM Equations General Solution of SHM
MediumNumerical valueNew ideaIdea asked in 2 years
Q23
The volume V of a given mass of monoatomic gas changes with temperature T according to the relation V = K $T^{\frac{2}{3}}$ . The workdone when temperature changes by 90 K will be xR. The value of $x$ is ________. [R = universal gas constant]
Thermodynamics Thermodynamic Processes Polytropic Process Analysis
MediumNumerical valueNew ideaIdea asked in 2 years
Q24
Two stream of photons, possessing energies equal to twice and ten times the work function of metal are incident on the metal surface successively. The value of ratio of maximum velocities of the photoelectrons emitted in the two respective cases is $x : y$ .…
Dual Nature of Radiation and Matter Photoelectric Effect Einstein's Photoelectric Equation
MediumNumerical valueIdea asked in 5 yearsAsked the year before
Q25
If the highest frequency modulating a carrier is 5 kHz, then the number of AM broadcast stations accommodated in a 90 kHz bandwidth are ________.
Communication Systems
EasyNumerical value
Q26
In the reported figure of earth, the value of acceleration due to gravity is same at point A and C but it is smaller than that of its value at point B (surface of the earth). The value of OA : AB will be $x : y$ . The value of $x$ is ________.
Gravitation
MediumNumerical value · figure
Q27
1 mole of rigid diatomic gas performs a work of $\frac{Q}{5}$ when heat Q is supplied to it. The molar heat capacity of the gas during this transformation is $\frac{xR}{8}$ . The value of $x$ is ________. [R = universal gas constant]
Thermodynamics First Law of Thermodynamics Molar Heat Capacity in Polytropic Processes
MediumNumerical valueNew ideaIdea asked in 2 years
Q28
A point source of light S, placed at a distance 60 cm infront of the centre of a plane mirror of width 50 cm, hangs vertically on a wall. A man walks infront of the mirror along a line parallel to the mirror at a distance 1.2 m from it (see in the figure).…
Ray Optics and Optical Instruments Spherical Mirrors Field of View of Spherical Mirrors
MediumNumerical value · figureNew idea
Q29
A particle executes S.H.M. with amplitude ‘a’ and time period ‘T’. The displacement of the particle when its speed is half of maximum speed is $\frac{\sqrt{x}\,a}{2}$ . The value of $x$ is ________.
Oscillations SHM Equations Velocity and Accel vs Position
EasyNumerical valueNew ideaIdea asked in 4 years
Q30
27 similar drops of mercury are maintained at 10 V each. All these spherical drops combine into a single big drop. The potential energy of the bigger drop is ________ times that of a smaller drop.
Electrostatic Potential and Capacitance Electric Potential Self-Energy of a Spherical Shell
HardNumerical valueNew ideaIdea asked in 2 years

Chemistry30 questions

6 easy20 medium4 hard

Q31
Match List-I with List-II. List-I (Molecule) | List-II (Bond order) (a) $\mathrm{Ne_2}$ | (i) 1 (b) $\mathrm{N_2}$ | (ii) 2 (c) $\mathrm{F_2}$ | (iii) 0 (d) $\mathrm{O_2}$ | (iv) 3 Choose the correct answer from the options given below :
Chemical Bonding and Molecular Structure Molecular Orbital Theory Bond Order Calculation
MediumMatch the listNew ideaIdea asked in 3 years
Q32
The nature of charge on resulting colloidal particles when $\mathrm{FeCl_3}$ is added to excess of hot water is :
Surface Chemistry Colloids Chemical Methods of Preparation
MediumSingle correctNew idea
Q33
The correct order of electron gain enthalpy is :
Classification of Elements and Periodicity in Properties Periodic Trends Electron Gain Enthalpy
MediumSingle correctNew ideaIdea asked in 5 years
Q34
Match List-I with List-II. List-I | List-II (a) Siderite | (i) Cu (b) Calamine | (ii) Ca (c) Malachite | (iii) Fe (d) Cryolite | (iv) Al | (v) Zn Choose the correct answer from the options given below :
General Principles and Processes of Isolation of Elements
EasyMatch the list
Q35
Which of the following forms of hydrogen emits low energy $\beta^{-}$ particles ?
Hydrogen Hydrogen and Hydrides Radioactive Decay of Tritium
EasySingle correctNew idea
Q36
Match List-I with List-II. List-I | List-II (a) Sodium Carbonate | (i) Deacon (b) Titanium | (ii) Castner-Kellner (c) Chlorine | (iii) van-Arkel (d) Sodium hydroxide | (iv) Solvay Choose the correct answer from the options given below :
General Principles and Processes of Isolation of Elements
MediumMatch the list
Q37
Which pair of oxides is acidic in nature ?
The p-Block Elements Group 13 & 14 Acid-Base Nature of Group 13 Oxides
MediumSingle correctNew ideaIdea asked in 2 years
Q38
Given below are two statements : one is labelled as Assertion A and the other is labelled as Reason R. Assertion A : In $\mathrm{TlI_3}$ , isomorphous to $\mathrm{CsI_3}$ , the metal is present in $+1$ oxidation state. Reason R : Tl metal has fourteen $f$ …
The p-Block Elements Group 13 & 14 Inert Pair Effect
MediumAssertion–reasonNew ideaIdea asked in 5 years
Q39
Calgon is used for water treatment. Which of the following statement is NOT true about Calgon ?
The s-Block Elements Hydrogen and Hydrides Calgon's Method for Permanent Hardness
EasySingle correctNew idea
Q40
Ceric ammonium nitrate and $\mathrm{CHCl_3}$ /alc. KOH are used for the identification of functional groups present in __________ and __________ respectively.
Alcohols, Phenols and Ethers
MediumSingle correct
Q41
In $\overset{1}{\mathrm{CH_2}}=\overset{2}{\mathrm{C}}=\overset{3}{\mathrm{CH}}-\overset{4}{\mathrm{CH_3}}$ molecule, the hybridization of carbon 1, 2, 3 and 4 respectively, are :
Organic Chemistry: Some Basic Principles and Techniques
MediumSingle correct
Q42
[reaction scheme: structure drawn in line-bond form] $\xrightarrow[\text{2) } \mathrm{Cr_2O_3}\text{ , 773 K, 10 - 20 atm}]{\text{1) Zn/HCl}}$ Considering the above reaction, the major product among the following is :
Aldehydes, Ketones and Carboxylic Acids Alkanes and Free Radicals Aromatization (Reforming) of Alkanes
MediumOther · figureNew idea
Q43
Identify A in the given reaction. [structure drawn] $\xrightarrow{\mathrm{SOCl_2}}$ A (Major Product)
Alcohols, Phenols and Ethers Alcohols and Phenols Darzens Halogenation (SOCl2)
MediumDiagram based · figureNew idea
Q44
Identify A in the given chemical reaction. [structure drawn: benzene ring with two adjacent $\mathrm{CH_2CH_2CHO}$ substituents] $\xrightarrow[\mathrm{C_2H_5OH,\ H_2O},\ \Delta]{\mathrm{NaOH}}$ A (Major Product)
Aldehydes, Ketones and Carboxylic Acids Aldol and Cannizzaro Intramolecular Aldol
MediumDiagram based · figureIdea asked in 5 yearsAsked the year before
Q45
Identify A in the following chemical reaction. [structure drawn: benzene ring with CHO at one position and $\mathrm{CH_3O}$ at the opposite (para) position] $\xrightarrow{\text{i) HCHO, NaOH; ii) }\mathrm{CH_3CH_2Br}\text{, NaH, DMF; iii) HI, }\Delta}$ A
Aldehydes, Ketones and Carboxylic Acids Aldol and Cannizzaro Cross-Cannizzaro Reaction
HardDiagram based · figureNew idea
Q46
Match List-I with List-II. List-I | List-II (a) [figure: benzene ring with $\mathrm{N_2^+Cl^-}$ ] $\xrightarrow{\mathrm{Cu_2Cl_2}}$ [figure: benzene ring with Cl] $+\ \mathrm{N_2}$ | (i) Wurtz reaction (b) [figure: benzene ring with $\mathrm{N_2^+Cl^-}$ ]…
Haloalkanes and Haloarenes Diazonium Salts Sandmeyer Reaction
MediumMatch the list · figureIdea asked in 4 yearsAsked the year before
Q47
Seliwanoff test and Xanthoproteic test are used for the identification of __________ and __________ respectively.
Biomolecules Carbohydrates Seliwanoff's Test
MediumSingle correctIdea asked in 2 yearsAsked the year before
Q48
Match List-I with List-II. List-I | List-II (a) Sucrose | (i) $\beta$ -D-Galactose and $\beta$ -D-Glucose (b) Lactose | (ii) $\alpha$ -D-Glucose and $\beta$ -D-Fructose (c) Maltose | (iii) $\alpha$ -D-Glucose and $\alpha$ -D-Glucose Choose the correct answer from…
Biomolecules Carbohydrates Sucrose Structure and Linkage
EasyMatch the listNew ideaIdea asked in 2 years
Q49
2,4-DNP test can be used to identify :
Aldehydes, Ketones and Carboxylic Acids Nucleophilic Addition 2,4-DNP (Brady's Reagent) Test
EasySingle correctNew ideaIdea asked in 2 years
Q50
A. Phenyl methanamine B. N,N-Dimethylaniline C. N-Methyl aniline D. Benzenamine Choose the correct order of basic nature of the above amines.
Amines Aromatic Amine Basicity
MediumOtherNew ideaIdea asked in 3 years
Q51
The $\mathrm{NaNO_3}$ weighed out to make 50 mL of an aqueous solution containing 70.0 mg $\mathrm{Na^+}$ per mL is __________ g. (Rounded off to the nearest integer) [Given : Atomic weight in g $\mathrm{mol^{-1}}$ - Na : 23 ; N : 14 ; O : 16]
Some Basic Concepts of Chemistry Mole Concept Molarity Calculation
MediumNumerical valueIdea asked in 5 yearsAsked the year before
Q52
The number of octahedral voids per lattice site in a lattice is __________. (Rounded off to the nearest integer)
The Solid State Crystal Lattices Tetrahedral and Octahedral
EasyNumerical valueNew idea
Q53
A ball weighing 10 g is moving with a velocity of 90 $\mathrm{ms^{-1}}$ . If the uncertainty in its velocity is 5%, then the uncertainty in its position is __________ $\times 10^{-33}$ m. (Rounded off to the nearest integer) [Given : h…
Structure of Atom
MediumNumerical value
Q54
The average S $-$ F bond energy in kJ $\mathrm{mol^{-1}}$ of $\mathrm{SF_6}$ is __________. (Rounded off to the nearest integer) [Given : The values of standard enthalpy of formation of $\mathrm{SF_6}$ (g), S(g) and F(g) are - 1100, 275 and 80 kJ…
Thermodynamics (Chemistry)Enthalpy and Hess's Law Bond Enthalpy Calculations
MediumNumerical valueNew ideaIdea asked in 4 years
Q55
When 12.2 g of benzoic acid is dissolved in 100 g of water, the freezing point of solution was found to be $-0.93^\circ$ C ( $\mathrm{K_f}$ ( $\mathrm{H_2O}$ ) $= 1.86$ K kg $\mathrm{mol^{-1}}$ ). The number (n) of benzoic acid molecules associated (assuming 100%…
Solutions Colligative Properties Degree of Association from Van't Hoff Factor
MediumNumerical valueIdea asked in 3 years
Q56
The pH of ammonium phosphate solution, if $pk_a$ of phosphoric acid and $pk_b$ of ammonium hydroxide are 5.23 and 4.75 respectively, is __________.
Equilibrium pH and Buffers pH of Weak Acid-Weak Base Salts
HardNumerical valueIdea asked in 3 years
Q57
Emf of the following cell at 298 K in V is $x \times 10^{-2}$ . $\mathrm{Zn|Zn^{2+}}$ (0.1 M) $\|$ $\mathrm{Ag^+}$ (0.01 M) $|$ Ag The value of $x$ is __________. (Rounded off to the nearest integer) [Given : $\mathrm{E^{\theta}_{Zn^{2+}/Zn}} = -0.76$ V ;…
Electrochemistry Nernst Equation Full Cell Nernst Equation
MediumNumerical valueIdea asked in 6 yearsAsked the year before
Q58
If the activation energy of a reaction is 80.9 kJ $\mathrm{mol^{-1}}$ , the fraction of molecules at 700 K, having enough energy to react to form products is $e^{-x}$ . The value of $x$ is __________. (Rounded off to the nearest integer) [Use R $= 8.31$ J…
Chemical Kinetics Arrhenius Equation Arrhenius Exponential Form
MediumNumerical valueIdea asked in 4 years
Q59
In mildly alkaline medium, thiosulphate ion is oxidized by $\mathrm{MnO_4^-}$ to "A". The oxidation state of sulphur in "A" is __________.
Redox Reactions Permanganate n-Factor Across Media
HardNumerical valueNew ideaIdea asked in 3 years
Q60
The number of stereoisomers possible for $[\mathrm{Co(ox)_2(Br)(NH_3)}]^{2-}$ is __________. [ox = oxalate]
Coordination Compounds Coordination Isomerism Total Isomer Calculation
HardNumerical valueNew ideaIdea asked in 3 years

Maths30 questions

7 easy18 medium5 hard

Q61
If the mirror image of the point $(1, 3, 5)$ with respect to the plane $4x-5y+2z=8$ is $(\alpha, \beta, \gamma)$ , then $5(\alpha+\beta+\gamma)$ equals :
Three Dimensional Geometry Point-Plane Distance Image of a Point
EasySingle correctIdea asked in 4 years
Q62
Let $A=\{1, 2, 3, \ldots, 10\}$ and $f : A \to A$ be defined as $f(k)=\begin{cases} k+1 & \text{if } k \text{ is odd} \\ k & \text{if } k \text{ is even} \end{cases}$ Then the number of possible functions $g : A \to A$ such that $g \circ f = f$ is :
Relations and Functions Permutations and Combinations Total Number of Functions
MediumSingle correctNew ideaIdea asked in 4 years
Q63
Let $A_1$ be the area of the region bounded by the curves $y=\sin x$ , $y=\cos x$ and $y$ -axis in the first quadrant. Also, let $A_2$ be the area of the region bounded by the curves $y=\sin x$ , $y=\cos x$ , $x$ -axis and $x=\frac{\pi}{2}$ in the first quadrant.…
Application of Integrals Area Under Curves Area Between Trigonometric Curves
MediumSingle correctNew idea
Q64
If $0 < a, b < 1$ , and $\tan^{-1}a + \tan^{-1}b = \frac{\pi}{4}$ , then the value of $(a+b) - \left(\frac{a^2+b^2}{2}\right) + \left(\frac{a^3+b^3}{3}\right) - \left(\frac{a^4+b^4}{4}\right) + \ldots$ is :
Inverse Trigonometric Functions Inverse Trig Identities Tan-1 Sum and Difference
MediumSingle correctNew ideaIdea asked in 3 years
Q65
Let slope of the tangent line to a curve at any point $P(x, y)$ be given by $\frac{xy^2+y}{x}$ . If the curve intersects the line $x+2y=4$ at $x=-2$ , then the value of $y$ , for which the point $(3, y)$ lies on the curve, is :
Differential Equations Linear Differential Equations Bernoulli Equation Reduction
MediumSingle correctNew ideaIdea asked in 2 years
Q66
The sum of the series $\sum_{n=1}^{\infty} \frac{n^2+6n+10}{(2n+1)!}$ is equal to :
Sequences and Series
HardSingle correct
Q67
Let $f(x)=\int_0^x e^t f(t)\,dt + e^x$ be a differentiable function for all $x \in \mathbf{R}$ . Then $f(x)$ equals :
Differential Equations Linear Differential Equations LDEs Formed via Leibniz Integral Rule
MediumSingle correctNew ideaIdea asked in 3 years
Q68
Let $f(x)$ be a differentiable function at $x=a$ with $f'(a)=2$ and $f(a)=4$ . Then $\lim_{x \to a} \frac{xf(a)-af(x)}{x-a}$ equals :
Continuity and Differentiability LHD and RHD Calculation
EasySingle correctNew idea
Q69
Let $f(x)=\sin^{-1}x$ and $g(x)=\frac{x^2-x-2}{2x^2-x-6}$ . If $g(2)=\lim_{x \to 2} g(x)$ , then the domain of the function $f \circ g$ is :
Relations and Functions Functions and Domain Domain of Composite Functions
MediumSingle correctNew idea
Q70
Let $A(1, 4)$ and $B(1, -5)$ be two points. Let $P$ be a point on the circle $(x-1)^2+(y-1)^2=1$ such that $(PA)^2+(PB)^2$ have maximum value, then the points, $P$ , $A$ and $B$ lie on :
Conic Sections Circle Tangents Maximum and Minimum Distance to a Circle
MediumSingle correctNew ideaIdea asked in 2 years
Q71
If vectors $\vec{a}_1 = x\hat{i} - \hat{j} + \hat{k}$ and $\vec{a}_2 = \hat{i} + y\hat{j} + z\hat{k}$ are collinear, then a possible unit vector parallel to the vector $x\hat{i} + y\hat{j} + z\hat{k}$ is :
Vector Algebra Linear Independence Proportionality Test for Collinearity
EasySingle correctNew idea
Q72
Let $F_1(A, B, C) = (A \wedge \sim B) \vee [\sim C \wedge (A \vee B)] \vee \sim A$ and $F_2(A, B) = (A \vee B) \vee (B \to \sim A)$ be two logical expressions. Then :
Mathematical Reasoning
MediumSingle correct
Q73
A seven digit number is formed using digits $3, 3, 4, 4, 4, 5, 5$ . The probability, that number so formed is divisible by $2$ , is :
Probability Permutations and Combinations Permutations with Repetition
MediumSingle correctIdea asked in 4 yearsAsked the year before
Q74
Consider the following system of equations : $x+2y-3z=a$ $2x+6y-11z=b$ $x-2y+7z=c$ , where a, b and c are real constants. Then the system of equations :
Determinants Cramer's Rule Condition for Infinite Solutions
MediumSingle correctIdea asked in 7 yearsAsked the year before
Q75
The triangle of maximum area that can be inscribed in a given circle of radius ‘r’ is :
Application of Derivatives Maxima and Minima Standard Geometric Optimization Results
EasySingle correctNew idea
Q76
Let L be a line obtained from the intersection of two planes $x+2y+z=6$ and $y+2z=4$ . If point $P(\alpha, \beta, \gamma)$ is the foot of perpendicular from $(3, 2, 1)$ on L, then the value of $21(\alpha+\beta+\gamma)$ equals :
Three Dimensional Geometry Lines in 3D Foot of the Perpendicular
MediumSingle correctIdea asked in 6 yearsAsked the year before
Q77
Let $f : \mathbf{R} \to \mathbf{R}$ be defined as $f(x)=\begin{cases} 2\sin\left(-\frac{\pi x}{2}\right), & \text{if } x<-1 \\ |ax^2+x+b|, & \text{if } -1 \le x \le 1 \\ \sin(\pi x), & \text{if } x>1 \end{cases}$ If $f(x)$ is continuous on $\mathbf{R}$ , then…
Continuity and Differentiability Finding Unknown Constants for Continuity
MediumSingle correctNew ideaIdea asked in 3 years
Q78
If the locus of the mid-point of the line segment from the point $(3, 2)$ to a point on the circle, $x^2+y^2=1$ is a circle of radius r, then r is equal to :
Conic Sections
EasySingle correct
Q79
A natural number has prime factorization given by $n=2^x 3^y 5^z$ , where $y$ and $z$ are such that $y+z=5$ and $y^{-1}+z^{-1}=\frac{5}{6}$ , $y>z$ . Then the number of odd divisors of n, including 1, is :
Permutations and Combinations Number of Divisors with Specific Properties
EasySingle correctNew idea
Q80
For $x>0$ , if $f(x)=\int_1^x \frac{\log_e t}{(1+t)}dt$ , then $f(e)+f\left(\frac{1}{e}\right)$ is equal to :
Integrals Definite Integral Properties Reciprocal Substitution (x = 1/t)
MediumSingle correctNew ideaIdea asked in 2 years
Q81
If $I_{m,n}=\int_0^1 x^{m-1}(1-x)^{n-1}\,dx$ , for $m, n \ge 1$ , and $\int_0^1 \frac{x^{m-1}+x^{n-1}}{(1+x)^{m+n}}dx=\alpha I_{m,n}$ , $\alpha \in \mathbf{R}$ , then $\alpha$ equals __________.
Integrals Definite Integral Properties Beta Function Integrals
HardNumerical valueNew ideaIdea asked in 2 years
Q82
Let $z$ be those complex numbers which satisfy $|z+5| \le 4$ and $z(1+i)+\overline{z}(1-i) \ge -10$ , $i=\sqrt{-1}$ . If the maximum value of $|z+1|^2$ is $\alpha+\beta\sqrt{2}$ , then the value of $(\alpha+\beta)$ is __________.
Complex Numbers and Quadratic Equations Complex Numbers - Polar Form Polar Triangle Inequality Extrema
HardNumerical valueNew ideaIdea asked in 5 years
Q83
Let the normals at all the points on a given curve pass through a fixed point $(a, b)$ . If the curve passes through $(3, -3)$ and $(4, -2\sqrt{2})$ , and given that $a-2\sqrt{2}\,b=3$ , then $(a^2+b^2+ab)$ is equal to __________.
Application of Derivatives Tangents and Normals Equation of Normal
MediumNumerical valueIdea asked in 3 yearsAsked the year before
Q84
Let a be an integer such that all the real roots of the polynomial $2x^5+5x^4+10x^3+10x^2+10x+10$ lie in the interval $(a, a+1)$ . Then, $|a|$ is equal to __________.
Application of Derivatives Rolle's Theorem Polynomial Root Counting
MediumNumerical valueNew ideaIdea asked in 2 years
Q85
Let $X_1, X_2, \ldots, X_{18}$ be eighteen observations such that $\sum_{i=1}^{18}(X_i-\alpha)=36$ and $\sum_{i=1}^{18}(X_i-\beta)^2=90$ , where $\alpha$ and $\beta$ are distinct real numbers. If the standard deviation of these observations is $1$ , then the…
Statistics Variance and SD Minimum Sum of Squared Deviations
MediumNumerical valueNew idea
Q86
If the matrix $A=\begin{bmatrix} 1 & 0 & 0 \\ 0 & 2 & 0 \\ 3 & 0 & -1 \end{bmatrix}$ satisfies the equation $A^{20}+\alpha A^{19}+\beta A=\begin{bmatrix} 1 & 0 & 0 \\ 0 & 4 & 0 \\ 0 & 0 & 1 \end{bmatrix}$ for some real numbers $\alpha$ and $\beta$ , then…
Matrices Matrix Multiplication Cayley-Hamilton Theorem Application
HardNumerical valueNew ideaIdea asked in 5 years
Q87
Let $\alpha$ and $\beta$ be two real numbers such that $\alpha+\beta=1$ and $\alpha\beta=-1$ . Let $p_n=(\alpha)^n+(\beta)^n$ , $p_{n-1}=11$ and $p_{n+1}=29$ for some integer $n \ge 1$ . Then, the value of $p_n^2$ is __________.
Complex Numbers and Quadratic Equations Quadratic Roots Newton's Sums
EasyNumerical valueIdea asked in 7 yearsAsked the year before
Q88
The total number of 4-digit numbers whose greatest common divisor with $18$ is $3$ , is __________.
Permutations and Combinations Principle of Inclusion-Exclusion in Counting
MediumNumerical valueNew ideaIdea asked in 3 years
Q89
If the arithmetic mean and geometric mean of the $p^{th}$ and $q^{th}$ terms of the sequence $-16, 8, -4, 2, \ldots$ satisfy the equation $4x^2-9x+5=0$ , then $p+q$ is equal to __________.
Sequences and Series AP and GP GP General Term
MediumNumerical valueNew ideaIdea asked in 5 years
Q90
Let L be a common tangent line to the curves $4x^2+9y^2=36$ and $(2x)^2+(2y)^2=31$ . Then the square of the slope of the line L is __________.
Conic Sections Circle Tangents Common Tangent to Circle and Conic
HardNumerical valueNew ideaIdea asked in 3 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.