JEE Main 2021 · February session

JEE Main 24 February 2021, Shift 1

90 questions: 30 numerical-answer and 60 multiple choice. 6 were rated hard and 32 easy. Its hard-question count was the joint 18th highest of the 24 2021 shifts Lumi analysed. The biggest chapter was Gravitation, with 3 questions.

Questions: 90; none dropped
Numerical answer: 30; 33% of the shift
Hard: 6; 7% of the shift
Chapters asked: 63; 15 questions with a figure

Key findings · 24 Feb 2021, Shift 1

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
Maths was the hardest section, with 5 of 30 rated hard; Physics had the most easy questions (17 of 30).
3
51 questions tested a new idea, one no earlier year had asked: 68% of the 75 questions tagged to an idea.
4
24 came back from 2020: the same idea had been asked in the 2020 shifts Lumi analysed.
5
35 shared an idea with another 2021 shift (47% of tagged questions). Shifts in the same year can test the same idea in a new setting.
6
Biggest chapters: Gravitation in Physics (3), Hydrocarbons in Chemistry (3) and Integrals in Maths (3).

The shift at a glance

Subjects, difficulty and chapters

Physics

30 Q · 10 numerical

Easy 17Medium 13Hard 0

Top chapters

Chemistry

30 Q · 10 numerical

Easy 12Medium 17Hard 1

Top chapters

Hydrocarbons3 Q
Alcohols, Phenols and Ethers2 Q
Amines2 Q
Equilibrium2 Q
General Principles and Processes of Isolation of Elements2 Q
Redox Reactions2 Q

Maths

30 Q · 10 numerical

Easy 3Medium 22Hard 5

Top chapters

Integrals3 Q
Trigonometric Functions3 Q
Application of Derivatives2 Q
Complex Numbers and Quadratic Equations2 Q
Conic Sections2 Q
Matrices2 Q

Question format

Single correct45 Q
Numerical value30 Q
Diagram based11 Q
Match the list2 Q
Two statements2 Q

What each question asks you to do

Calculation26 Q
Multi-step22 Q
Application18 Q
Conceptual11 Q
Recall11 Q
Data or graph1 Q
Diagram reading1 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

17 easy13 medium0 hard

Q1
The workdone by a gas molecule in an isolated system is given by, $W = \alpha\beta^{2} e^{-\frac{x^{2}}{\alpha kT}}$ , where $x$ is the displacement, $k$ is the Boltzmann constant and T is the temperature. $\alpha$ and $\beta$ are constants. Then the…
Units and Measurements
MediumSingle correct
Q2
Two stars of masses m and 2m at a distance d rotate about their common centre of mass in free space. The period of revolution is :
Gravitation Orbital Mechanics Binary Star Systems
MediumSingle correctNew ideaIdea asked in 2 years
Q3
Four identical particles of equal masses 1 kg made to move along the circumference of a circle of radius 1 m under the action of their own mutual gravitational attraction. The speed of each particle will be :
Gravitation Orbital Mechanics Multi-Star Polygonal Orbits
MediumSingle correctNew idea
Q4
Moment of inertia (M.I.) of four bodies, having same mass and radius, are reported as ; $I_1$ = M.I. of thin circular ring about its diameter, $I_2$ = M.I. of circular disc about an axis perpendicular to disc and going through the centre, $I_3$ = M.I. of…
System of Particles and Rotational Motion Moment of Inertia Comparing MOI of Standard Shapes
MediumSingle correctNew idea
Q5
Consider two satellites $S_1$ and $S_2$ with periods of revolution 1 hr. and 8 hr. respectively revolving around a planet in circular orbits. The ratio of angular velocity of satellite $S_1$ to the angular velocity of satellite $S_2$ is :
Gravitation Orbital Mechanics Orbital Time Period
EasySingle correctIdea asked in 6 yearsAsked the year before
Q6
Each side of a box made of metal sheet in cubic shape is ‘a’ at room temperature ‘T’, the coefficient of linear expansion of the metal sheet is ‘ $\alpha$ ’. The metal sheet is heated uniformly, by a small temperature $\Delta$ T, so that its new temperature is…
Thermal Properties of Matter Thermal Expansion Volume Expansion
EasySingle correctNew ideaIdea asked in 2 years
Q7
If Y, K and $\eta$ are the values of Young’s modulus, bulk modulus and modulus of rigidity of any material respectively. Choose the correct relation for these parameters.
Mechanical Properties of Solids Elasticity Interrelations Between Elastic Constants
MediumSingle correctNew idea
Q8
If the velocity-time graph has the shape AMB, what would be the shape of the corresponding acceleration-time graph ?
Motion in a Straight Line Kinematics in 1D Kinematic Graph Conversion
EasyDiagram based · figureNew ideaIdea asked in 2 years
Q9
n mole of a perfect gas undergoes a cyclic process ABCA (see figure) consisting of the following processes. A → B : Isothermal expansion at temperature T so that the volume is doubled from $V_1$ to $V_2=2V_1$ and pressure changes from $P_1$ to $P_2$ . B → C :…
Thermodynamics Thermodynamic Processes Cyclic Process Work
MediumDiagram based · figureNew ideaIdea asked in 4 years
Q10
Match List I with List II. List I | List II (a) Isothermal | (i) Pressure constant (b) Isochoric | (ii) Temperature constant (c) Adiabatic | (iii) Volume constant (d) Isobaric | (iv) Heat content is constant Choose the correct answer from the options given…
Thermodynamics Thermodynamic Processes Isothermal Process Analysis
EasyMatch the listNew ideaIdea asked in 3 years
Q11
In the given figure, a mass M is attached to a horizontal spring which is fixed on one side to a rigid support. The spring constant of the spring is k. The mass oscillates on a frictionless surface with time period T and amplitude A. When the mass is in…
Oscillations Energy in SHM Mass Dropped at Mean Position
MediumSingle correct · figureIdea asked in 3 yearsAsked the year before
Q12
A cell $E_1$ of emf 6 V and internal resistance 2 $\Omega$ is connected with another cell $E_2$ of emf 4 V and internal resistance 8 $\Omega$ (as shown in the figure). The potential difference across points X and Y is :
Current Electricity Kirchhoff's Laws Reversed Cells in Series
MediumSingle correct · figureNew ideaIdea asked in 2 years
Q13
A current through a wire depends on time as $i=\alpha_0 t+\beta t^{2}$ where $\alpha_0=20$ A/s and $\beta=8$ As $^{-2}$ . Find the charge crossed through a section of the wire in 15 s.
Current Electricity RC Circuits Total Charge Flown Integration
EasySingle correctNew idea
Q14
Two equal capacitors are first connected in series and then in parallel. The ratio of the equivalent capacities in the two cases will be :
Electrostatic Potential and Capacitance Parallel Plate Capacitors Series and Parallel Combinations
EasySingle correctIdea asked in 6 yearsAsked the year before
Q15
A cube of side ‘a’ has point charges +Q located at each of its vertices except at the origin where the charge is $-$ Q. The electric field at the centre of cube is :
Electric Charges and Fields Electric Field Missing Charge Superposition Trick
MediumDiagram based · figureNew idea
Q16
If an emitter current is changed by 4 mA, the collector current changes by 3.5 mA. The value of $\beta$ will be :
Semiconductor Electronics: Materials, Devices and Simple Circuits Transistors Current Gains Alpha and Beta
EasySingle correctNew ideaIdea asked in 2 years
Q17
In the given figure, the energy levels of hydrogen atom have been shown along with some transitions marked A, B, C, D and E. The transitions A, B and C respectively represent :
Atoms Bohr's Atomic Model Shortest and Longest Wavelengths
MediumDiagram based · figureIdea asked in 5 yearsAsked the year before
Q18
Given below are two statements : Statement I : Two photons having equal linear momenta have equal wavelengths. Statement II : If the wavelength of photon is decreased, then the momentum and energy of a photon will also decrease. In the light of the above…
Dual Nature of Radiation and Matter Photoelectric Effect Photon Energy and Momentum
EasyTwo statementsNew ideaIdea asked in 4 years
Q19
The focal length $f$ is related to the radius of curvature r of the spherical convex mirror by :
Ray Optics and Optical Instruments Spherical Mirrors Focal Length Relation
EasySingle correctNew idea
Q20
In a Young’s double slit experiment, the width of the one of the slit is three times the other slit. The amplitude of the light coming from a slit is proportional to the slit-width. Find the ratio of the maximum to the minimum intensity in the interference…
Wave Optics Young's Double Slit Intensity Ratio from Slit Widths
EasySingle correctNew ideaIdea asked in 4 years
Q21
The coefficient of static friction between a wooden block of mass 0.5 kg and a vertical rough wall is 0.2. The magnitude of horizontal force that should be applied on the block to keep it adhere to the wall will be __________ N. [ g $=10$ ms $^{-2}$ ]
Laws of Motion Friction Block on a Vertical Wall
EasyNumerical valueNew ideaIdea asked in 2 years
Q22
An unpolarized light beam is incident on the polarizer of a polarization experiment and the intensity of light beam emerging from the analyzer is measured as 100 Lumens. Now, if the analyzer is rotated around the horizontal axis (direction of light) by…
Wave Optics Polarization Malus's Law Calculation
EasyNumerical valueNew ideaIdea asked in 5 years
Q23
A ball with a speed of 9 m/s collides with another identical ball at rest. After the collision, the direction of each ball makes an angle of $30^\circ$ with the original direction. The ratio of velocities of the balls after collision is $x:y$ , where $x$ is…
Work, Energy and Power Collisions Oblique Collisions
MediumNumerical valueIdea asked in 2 yearsAsked the year before
Q24
A hydraulic press can lift 100 kg when a mass ‘m’ is placed on the smaller piston. It can lift __________ kg when the diameter of the larger piston is increased by 4 times and that of the smaller piston is decreased by 4 times keeping the same mass ‘m’ on…
Mechanical Properties of Fluids Fluid Statics Pascal's Law
EasyNumerical valueNew ideaIdea asked in 3 years
Q25
An inclined plane is bent in such a way that the vertical cross-section is given by $y=\frac{x^{2}}{4}$ where $y$ is in vertical and $x$ in horizontal direction. If the upper surface of this curved plane is rough with coefficient of friction $\mu=0.5$ , the…
Laws of Motion Friction Friction on Inclines
MediumNumerical valueNew ideaIdea asked in 3 years
Q26
A resonance circuit having inductance and resistance $2\times10^{-4}$ H and 6.28 $\Omega$ respectively oscillates at 10 MHz frequency. The value of quality factor of this resonator is __________. [ $\pi=3.14$ ]
Alternating Current LCR Resonance Quality Factor Definition
EasyNumerical valueNew idea
Q27
An audio signal $\nu_{\mathrm{m}}=20\ \sin\ 2\pi(1500t)$ amplitude modulates a carrier $\nu_{\mathrm{c}}=80\ \sin\ 2\pi(100{,}000t)$ . The value of percent modulation is __________.
Communication Systems
EasyNumerical value
Q28
In connection with the circuit drawn below, the value of current flowing through 2 k $\Omega$ resistor is __________ $\times10^{-4}$ A.
Semiconductor Electronics: Materials, Devices and Simple Circuits Semiconductor Diodes Zener Diode Regulation
MediumNumerical value · figureIdea asked in 5 yearsAsked the year before
Q29
An electromagnetic wave of frequency 5 GHz, is travelling in a medium whose relative electric permittivity and relative magnetic permeability both are 2. Its velocity in this medium is __________ $\times10^{7}$ m/s.
Electromagnetic Waves
EasyNumerical value
Q30
A common transistor radio set requires 12 V (D.C.) for its operation. The D.C. source is constructed by using a transformer and a rectifier circuit, which are operated at 220 V (A.C.) on standard domestic A.C. supply. The number of turns of secondary coil…
Alternating Current
EasyNumerical value

Chemistry30 questions

12 easy17 medium1 hard

Q31
Which of the following are isostructural pairs ? A. $\mathrm{SO_4^{2-}}$ and $\mathrm{CrO_4^{2-}}$ B. $\mathrm{SiCl_4}$ and $\mathrm{TiCl_4}$ C. $\mathrm{NH_3}$ and $\mathrm{NO_3^{-}}$ D. $\mathrm{BCl_3}$ and $\mathrm{BrCl_3}$
Chemical Bonding and Molecular Structure VSEPR Theory Isostructural and Isoelectronic Identification
MediumSingle correctNew ideaIdea asked in 2 years
Q32
In Freundlich adsorption isotherm, slope of AB line is :
Surface Chemistry Adsorption Freundlich Logarithmic Plot
MediumDiagram based · figureIdea asked in 4 yearsAsked the year before
Q33
Consider the elements Mg, Al, S, P and Si, the correct increasing order of their first ionization enthalpy is :
Classification of Elements and Periodicity in Properties Periodic Trends Penetration Effect on Ionization Enthalpy
MediumSingle correctIdea asked in 2 yearsAsked the year before
Q34
Which of the following ore is concentrated using group 1 cyanide salt ?
General Principles and Processes of Isolation of Elements
EasySingle correct
Q35
(A) $\mathrm{HOCl + H_2O_2 \rightarrow H_3O^+ + Cl^- + O_2}$ (B) $\mathrm{I_2 + H_2O_2 + 2OH^- \rightarrow 2I^- + 2H_2O + O_2}$ Choose the correct option.
Redox Reactions Hydrogen and Hydrides Redox Behavior of Hydrogen Peroxide
EasySingle correctNew ideaIdea asked in 2 years
Q36
$\mathrm{Al_2O_3}$ was leached with alkali to get X. The solution of X on passing of gas Y, forms Z. X, Y and Z respectively are :
General Principles and Processes of Isolation of Elements Metallurgical Extraction Bayer Process Chemistry
EasySingle correctNew ideaIdea asked in 2 years
Q37
The electrode potential of $\mathrm{M^{2+}/M}$ of 3d-series elements shows positive value for :
The d- and f-Block Elements d-Block Elements Standard Electrode Potential Trends
EasySingle correctNew ideaIdea asked in 3 years
Q38
The major components in “Gun Metal” are :
The d- and f-Block Elements
EasySingle correct
Q39
The gas released during anaerobic degradation of vegetation may lead to :
Environmental Chemistry Greenhouse Effect
EasySingle correctIdea asked in 4 yearsAsked the year before
Q40
Which of the following compound gives pink colour on reaction with phthalic anhydride in conc. $\mathrm{H_2SO_4}$ followed by treatment with NaOH ?
Alcohols, Phenols and Ethers Alcohols and Phenols Phthalein Dye Test
EasySingle correct · figureNew ideaIdea asked in 3 years
Q41
What is the major product formed by HI on reaction with [figure] ?
Hydrocarbons Alkenes and Alkynes Carbocation Rearrangements
MediumDiagram based · figureIdea asked in 3 yearsAsked the year before
Q42
Which of the following reagent is used for the following reaction ? $\mathrm{CH_3CH_2CH_3 \xrightarrow{\ ?\ } CH_3CH_2CHO}$
Hydrocarbons Alkanes and Free Radicals Controlled Oxidation Catalysts
EasySingle correctNew idea
Q43
The product formed in the first step of the reaction of [figure] with excess $\mathrm{Mg/Et_2O(Et=C_2H_5)}$ is :
Haloalkanes and Haloarenes
MediumDiagram based · figure
Q44
What is the final product (major) ‘A’ in the given reaction ? [figure] $\xrightarrow{\mathrm{HCl}}$ ‘A’ (major product)
Alcohols, Phenols and Ethers Alkenes and Alkynes Carbocation Rearrangements
MediumDiagram based · figureIdea asked in 3 yearsAsked the year before
Q45
Identify products A and B. [figure] $\xrightarrow[273\ \mathrm{K}]{\mathrm{dil.\ KMnO_4}} \mathrm{A} \xrightarrow{\mathrm{CrO_3}} \mathrm{B}$
Hydrocarbons Alcohols and Phenols Selective Alcohol Oxidation
MediumDiagram based · figureNew ideaIdea asked in 2 years
Q46
In the following reaction the reason why meta-nitro product also formed is :
Amines Direct Nitration of Aniline
MediumDiagram based · figureNew ideaIdea asked in 2 years
Q47
‘A’ and ‘B’ in the following reactions are : [figure] $\xrightarrow[\mathrm{KCN}]{\mathrm{NaNO_2/HCl}}$ (A) $\xrightarrow{\mathrm{SnCl_2/HCl/H_3O^+}}$ (B)
Amines Diazonium Salts Cyanide Replacement via Sandmeyer
MediumDiagram based · figureNew idea
Q48
Match List I with List II. List I (Monomer Unit) List II (Polymer) (a) Caprolactum (i) Natural rubber (b) 2-Chloro-1,3-butadiene (ii) Buna-N (c) Isoprene (iii) Nylon 6 (d) Acrylonitrile (iv) Neoprene Choose the correct answer from the options given below :
Polymers Homopolymers vs Copolymers (Buna-S and Buna-N)
EasyMatch the listIdea asked in 2 yearsAsked the year before
Q49
Out of the following, which type of interaction is responsible for the stabilisation of $\alpha$ -helix structure of proteins ?
Biomolecules
EasySingle correct
Q50
Given below are two statements : Statement I : Colourless cupric metaborate is reduced to cuprous metaborate in a luminous flame. Statement II : Cuprous metaborate is obtained by heating boric anhydride and copper sulphate in a non-luminous flame. In the…
The p-Block Elements Group 13 & 14 Borax Bead Test Chemistry
MediumTwo statementsNew ideaIdea asked in 2 years
Q51
4.5 g of compound A (MW=90) was used to make 250 mL of its aqueous solution. The molarity of the solution in M is $x \times 10^{-1}$ . The value of $x$ is __________. (Rounded off to the nearest integer)
Some Basic Concepts of Chemistry Mole Concept Molarity Calculation
EasyNumerical valueIdea asked in 5 yearsAsked the year before
Q52
The coordination number of an atom in a body-centered cubic structure is __________. [Assume that the lattice is made up of atoms.]
The Solid State Crystal Lattices BCC Neighbor Distances
EasyNumerical valueNew idea
Q53
A proton and a $\mathrm{Li^{3+}}$ nucleus are accelerated by the same potential. If $\lambda_{\mathrm{Li}}$ and $\lambda_{\mathrm{p}}$ denote the de Broglie wavelengths of $\mathrm{Li^{3+}}$ and proton respectively, then the value of…
Structure of Atom Bohr's Atomic Theory Circumference and De Broglie Wavelength
MediumNumerical valueNew ideaIdea asked in 3 years
Q54
For the reaction $\mathrm{A_{(g)} \rightarrow B_{(g)}}$ , the value of the equilibrium constant at 300 K and 1 atm is equal to 100.0. The value of $\Delta_r G$ for the reaction at 300 K and 1 atm in $\mathrm{J\ mol^{-1}}$ is $-x\mathrm{R}$ , where $x$ is…
Equilibrium Gibbs Free Energy Delta G vs Delta G-degree at Equilibrium
MediumNumerical valueNew idea
Q55
When 9.45 g of $\mathrm{ClCH_2COOH}$ is added to 500 mL of water, its freezing point drops by $0.5^\circ\mathrm{C}$ . The dissociation constant of $\mathrm{ClCH_2COOH}$ is $x \times 10^{-3}$ . The value of $x$ is __________. (Rounded off to the nearest…
Solutions Colligative Properties Calculating Ka from Freezing Point Depression
HardNumerical valueNew ideaIdea asked in 4 years
Q56
At 1990 K and 1 atm pressure, there are equal number of $\mathrm{Cl_2}$ molecules and Cl atoms in the reaction mixture. The value of $K_p$ for the reaction $\mathrm{Cl_{2(g)} \rightleftharpoons 2Cl_{(g)}}$ under the above conditions is $x \times 10^{-1}$ .…
Equilibrium Chemical Equilibrium Kp for PCl5 Dissociation Type Reactions
MediumNumerical valueNew ideaIdea asked in 4 years
Q57
The reaction of sulphur in alkaline medium is given below : $\mathrm{S_{8(s)} + a\ OH^-_{(aq)} \longrightarrow b\ S^{2-}_{(aq)} + c\ S_2O_3^{2-}{}_{(aq)} + d\ H_2O_{(l)}}$ The values of ‘a’ is __________. (Integer answer)
Redox Reactions Ion-Electron Basic
MediumNumerical valueNew idea
Q58
Gaseous cyclobutene isomerizes to butadiene in a first order process which has a ‘k’ value of $3.3\times10^{-4}\ \mathrm{s^{-1}}$ at 153°C. The time in minutes it takes for the isomerization to proceed 40% to completion at this temperature is __________.…
Chemical Kinetics Integrated Rate Equations First Order Fractional Completion Shortcuts
MediumNumerical valueNew ideaIdea asked in 3 years
Q59
Number of amphoteric compounds among the following is __________. (A) BeO (B) BaO (C) $\mathrm{Be(OH)_2}$ (D) $\mathrm{Sr(OH)_2}$
The s-Block Elements s-Block Elements Amphoterism of Beryllium Compounds
MediumNumerical valueIdea asked in 4 yearsAsked the year before
Q60
The stepwise formation of $\mathrm{[Cu(NH_3)_4]^{2+}}$ is given below : $\mathrm{Cu^{2+} + NH_3 \overset{K_1}{\rightleftharpoons} [Cu(NH_3)]^{2+}}$ $\mathrm{[Cu(NH_3)]^{2+} + NH_3 \overset{K_2}{\rightleftharpoons} [Cu(NH_3)_2]^{2+}}$ …
Coordination Compounds
MediumNumerical value

Maths30 questions

3 easy22 medium5 hard

Q61
Let $f : \mathbb{R} \to \mathbb{R}$ be defined as $f(x) = 2x - 1$ and $g : \mathbb{R} - \{1\} \to \mathbb{R}$ be defined as $g(x) = \frac{x - \frac{1}{2}}{x - 1}$ . Then the composition function $f(g(x))$ is :
Relations and Functions
MediumSingle correct
Q62
Let p and q be two positive numbers such that $p + q = 2$ and $p^4 + q^4 = 272$ . Then p and q are roots of the equation :
Complex Numbers and Quadratic Equations Quadratic Roots Symmetric Functions
MediumSingle correctNew ideaIdea asked in 5 years
Q63
The system of linear equations $3x - 2y - kz = 10$ $2x - 4y - 2z = 6$ $x + 2y - z = 5\mathrm{m}$ is inconsistent if :
Determinants Cramer's Rule Condition for No Solution
MediumSingle correctIdea asked in 6 yearsAsked the year before
Q64
The value of $-{}^{15}C_1 + 2 \cdot {}^{15}C_2 - 3 \cdot {}^{15}C_3 + \dots - 15 \cdot {}^{15}C_{15} + {}^{14}C_1 + {}^{14}C_3 + {}^{14}C_5 + \dots + {}^{14}C_{11}$ is :
Binomial Theorem Binomial Coefficients Zero Alternating Moments
MediumSingle correctNew idea
Q65
If $e^{(\cos^2 x + \cos^4 x + \cos^6 x + \dots \infty)\log_e 2}$ satisfies the equation $t^2 - 9t + 8 = 0$ , then the value of $\frac{2\sin x}{\sin x + \sqrt{3}\cos x}\left(0 < x < \frac{\pi}{2}\right)$ is :
Trigonometric Functions AP and GP Infinite GP Sum
MediumSingle correctIdea asked in 6 yearsAsked the year before
Q66
$\lim_{x \to 0} \frac{\int_0^{x^2} (\sin\sqrt{t})\,dt}{x^3}$ is equal to :
Integrals Leibniz Rule Limit Evaluation
MediumSingle correctNew ideaIdea asked in 2 years
Q67
The function $f(x) = \frac{4x^3 - 3x^2}{6} - 2\sin x + (2x - 1)\cos x$ :
Application of Derivatives Monotonicity First Derivative Test
MediumSingle correctIdea asked in 3 yearsAsked the year before
Q68
A scientific committee is to be formed from 6 Indians and 8 foreigners, which includes at least 2 Indians and double the number of foreigners as Indians. Then the number of ways, the committee can be formed, is :
Permutations and Combinations Combinations of Distinct Objects
MediumSingle correctIdea asked in 8 yearsAsked the year before
Q69
If $f : \mathbb{R} \to \mathbb{R}$ is a function defined by $f(x) = [x - 1]\cos\left(\frac{2x - 1}{2}\right)\pi$ , where $[\cdot]$ denotes the greatest integer function, then $f$ is :
Continuity and Differentiability Continuity involving GIF and Fractional Part
MediumSingle correctIdea asked in 6 yearsAsked the year before
Q70
If $\int \frac{\cos x - \sin x}{\sqrt{8 - \sin 2x}}dx = a\sin^{-1}\left(\frac{\sin x + \cos x}{b}\right) + c$ , where c is a constant of integration, then the ordered pair (a, b) is equal to :
Integrals Integration by Substitution sin(2x) Derivative Substitution
MediumSingle correctNew idea
Q71
The area (in sq. units) of the part of the circle $x^2 + y^2 = 36$ , which is outside the parabola $y^2 = 9x$ , is :
Application of Integrals Area Under Curves Area Between Curves
MediumSingle correctIdea asked in 7 yearsAsked the year before
Q72
The population $P = P(t)$ at time 't' of a certain species follows the differential equation $\frac{dP}{dt} = 0.5P - 450$ . If $P(0) = 850$ , then the time at which population becomes zero is :
Differential Equations Variable Separable Form Physical Modeling Applications
MediumSingle correctNew ideaIdea asked in 2 years
Q73
A man is walking on a straight line. The arithmetic mean of the reciprocals of the intercepts of this line on the coordinate axes is $\frac{1}{4}$ . Three stones A, B and C are placed at the points $(1, 1)$ , $(2, 2)$ and $(4, 4)$ respectively. Then which of…
Straight Lines
MediumSingle correct
Q74
The locus of the mid-point of the line segment joining the focus of the parabola $y^2 = 4ax$ to a moving point of the parabola, is another parabola whose directrix is :
Conic Sections General Conic Section Focus-Directrix Form Extraction
MediumSingle correctNew ideaIdea asked in 3 years
Q75
If the tangent to the curve $y = x^3$ at the point $P(t, t^3)$ meets the curve again at Q, then the ordinate of the point which divides PQ internally in the ratio $1 : 2$ is :
Application of Derivatives Tangents and Normals Tangent Intersecting Curve Again
MediumSingle correctNew idea
Q76
The equation of the plane passing through the point $(1, 2, -3)$ and perpendicular to the planes $3x + y - 2z = 5$ and $2x - 5y - z = 7$ , is :
Three Dimensional Geometry Planes in 3D Plane Perpendicular to Two Planes
EasySingle correctNew ideaIdea asked in 3 years
Q77
The distance of the point $(1, 1, 9)$ from the point of intersection of the line $\frac{x - 3}{1} = \frac{y - 4}{2} = \frac{z - 5}{2}$ and the plane $x + y + z = 17$ is :
Three Dimensional Geometry Planes in 3D Intersection Point of Line and Plane
EasySingle correctNew ideaIdea asked in 2 years
Q78
An ordinary dice is rolled for a certain number of times. If the probability of getting an odd number 2 times is equal to the probability of getting an even number 3 times, then the probability of getting an odd number for odd number of times is :
Probability
MediumSingle correct
Q79
Two vertical poles are 150 m apart and the height of one is three times that of the other. If from the middle point of the line joining their feet, an observer finds the angles of elevation of their tops to be complementary, then the height of the shorter…
Trigonometric Functions
MediumSingle correct
Q80
The statement among the following that is a tautology is :
Mathematical Reasoning
EasySingle correct
Q81
If the least and the largest real values of $\alpha$ , for which the equation $z + \alpha|z - 1| + 2i = 0$ ( $z \in C$ and $i = \sqrt{-1}$ ) has a solution, are p and q respectively; then $4(p^2 + q^2)$ is equal to ___________.
Complex Numbers and Quadratic Equations Quadratic Roots Nature of Roots
HardNumerical valueIdea asked in 3 yearsAsked the year before
Q82
Let $B_i$ ( $i = 1, 2, 3$ ) be three independent events in a sample space. The probability that only $B_1$ occur is $\alpha$ , only $B_2$ occurs is $\beta$ and only $B_3$ occurs is $\gamma$ . Let p be the probability that none of the events $B_i$ occurs and…
Probability Conditional Probability Independence Criterion
HardNumerical valueNew ideaIdea asked in 3 years
Q83
Let $P = \begin{bmatrix} 3 & -1 & -2 \\ 2 & 0 & \alpha \\ 3 & -5 & 0 \end{bmatrix}$ , where $\alpha \in \mathbf{R}$ . Suppose $Q = [q_{ij}]$ is a matrix satisfying $PQ = kI_3$ for some non-zero $k \in \mathbf{R}$ . If $q_{23} = -\frac{k}{8}$ and…
Matrices Matrix Inverse Inverse via Adjoint Formula
HardNumerical valueNew idea
Q84
Let M be any $3 \times 3$ matrix with entries from the set $\{0, 1, 2\}$ . The maximum number of such matrices, for which the sum of diagonal elements of $M^T M$ is seven, is ___________.
Matrices
HardNumerical value
Q85
Let $A = \{n \in N : n \text{ is a 3-digit number}\}$ $B = \{9k + 2 : k \in N\}$ and $C = \{9k + l : k \in N\}$ for some $l$ ( $0 < l < 9$ ) If the sum of all the elements of the set $A \cap (B \cup C)$ is $274 \times 400$ , then $l$ is equal to ___________.
Relations and Functions Sets and Relations Set Operations
HardNumerical valueNew idea
Q86
The minimum value of $\alpha$ for which the equation $\frac{4}{\sin x} + \frac{1}{1 - \sin x} = \alpha$ has at least one solution in $\left(0, \frac{\pi}{2}\right)$ is ___________.
Trigonometric Functions Maxima and Minima Optimization via AM-GM Inequality
MediumNumerical valueIdea asked in 4 yearsAsked the year before
Q87
If $\int_{-a}^{a}(|x| + |x - 2|)dx = 22$ , ( $a > 2$ ) and $[x]$ denotes the greatest integer $\le x$ , then $\int_{a}^{-a}(x + [x])dx$ is equal to ___________.
Integrals Definite Integral Properties Greatest Integer Function Integrals
MediumNumerical valueIdea asked in 7 yearsAsked the year before
Q88
If one of the diameters of the circle $x^2 + y^2 - 2x - 6y + 6 = 0$ is a chord of another circle 'C', whose center is at $(2, 1)$ , then its radius is ___________.
Conic Sections General Conic Section Condition for a Circle
MediumNumerical valueNew ideaIdea asked in 3 years
Q89
Let three vectors $\vec{a}, \vec{b}$ and $\vec{c}$ be such that $\vec{c}$ is coplanar with $\vec{a}$ and $\vec{b}$ , $\vec{a} \cdot \vec{c} = 7$ and $\vec{b}$ is perpendicular to $\vec{c}$ , where $\vec{a} = -\hat{i} + \hat{j} + \hat{k}$ and…
Vector Algebra Triple Products Constructing Orthogonal Coplanar Vectors
MediumNumerical valueNew idea
Q90
$\lim_{n \to \infty} \tan\left\{\sum_{r=1}^{n} \tan^{-1}\left(\frac{1}{1 + r + r^2}\right)\right\}$ is equal to ___________.
Inverse Trigonometric Functions Inverse Trig Identities Telescoping Inverse Trig Series
MediumNumerical valueIdea asked in 4 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.