JEE Main 2021 · February session

JEE Main 25 February 2021, Shift 2

89 questions (1 more dropped by NTA): 30 numerical-answer and 59 multiple choice. 9 were rated hard and 31 easy. Its hard-question count was the 11th highest of the 24 2021 shifts Lumi analysed. The biggest chapter was Conic Sections, with 4 questions.

Questions: 89; 1 dropped by NTA
Numerical answer: 30; 34% of the shift
Hard: 9; 10% of the shift
Chapters asked: 60; 13 questions with a figure

Key findings · 25 Feb 2021, Shift 2

1
9 hard questions (10%): the 11th 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 (12 of 29).
3
48 questions tested a new idea, one no earlier year had asked: 67% 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
42 shared an idea with another 2021 shift (58% of tagged questions). Shifts in the same year can test the same idea in a new setting.
6
Biggest chapters: Dual Nature of Radiation and Matter in Physics (3), Alcohols, Phenols and Ethers in Chemistry (2) and Conic Sections in Maths (4).

The shift at a glance

Subjects, difficulty and chapters

Physics

29 Q · 10 numerical

Easy 12Medium 17Hard 0

Top chapters

Chemistry

30 Q · 10 numerical

Easy 12Medium 16Hard 2

Top chapters

Maths

30 Q · 10 numerical

Easy 7Medium 16Hard 7

Top chapters

Conic Sections4 Q
Integrals4 Q
Relations and Functions4 Q
Application of Derivatives2 Q
Complex Numbers and Quadratic Equations2 Q
Determinants2 Q

Question format

Single correct45 Q
Numerical value30 Q
Diagram based8 Q
Two statements4 Q
Match the list1 Q
Other1 Q

What each question asks you to do

Calculation30 Q
Multi-step21 Q
Conceptual13 Q
Recall11 Q
Application10 Q
Diagram reading3 Q
Data or graph1 Q

Most common: Calculation.

Against the other 2021 shifts

This shift next to the 23 other 2021 shifts Lumi analysed

Shares are of each shift’s scored questions. The highlighted row is this shift.

Shift	Questions	Easy	Hard	Numerical	New ideas	Multi-step
24 Feb, Shift 1	90	36%	7%	33%	68%	24%
24 Feb, Shift 2	88	43%	5%	32%	63%	21%
25 Feb, Shift 1	90	23%	11%	33%	65%	30%
25 Feb, Shift 2	89	35%	10%	34%	67%	24%
26 Feb, Shift 1	88	50%	3%	33%	78%	17%
26 Feb, Shift 2	90	27%	12%	33%	80%	29%
16 Mar, Shift 2	90	31%	7%	33%	69%	33%
17 Mar, Shift 1	89	39%	8%	34%	61%	26%
17 Mar, Shift 2	89	36%	17%	34%	66%	32%
18 Mar, Shift 1	90	40%	9%	33%	74%	24%
18 Mar, Shift 2	90	34%	9%	33%	63%	28%
22 Jul, Shift 2	90	40%	12%	33%	67%	28%
25 Jul, Shift 1	90	31%	13%	33%	56%	34%
25 Jul, Shift 2	90	48%	3%	33%	63%	27%
3 Aug, Shift 1	90	32%	11%	33%	68%	21%
3 Aug, Shift 2	90	32%	7%	33%	61%	24%
4 Aug, Shift 1	90	40%	8%	33%	62%	29%
4 Aug, Shift 2	90	32%	7%	33%	65%	41%
26 Aug, Shift 1	89	27%	15%	34%	66%	30%
26 Aug, Shift 2	89	21%	9%	34%	60%	28%
27 Aug, Shift 1	90	27%	13%	33%	63%	34%
27 Aug, Shift 2	90	29%	9%	33%	72%	29%
31 Aug, Shift 1	90	27%	11%	33%	70%	24%
31 Aug, Shift 2	90	30%	13%	33%	72%	31%

New ideas: share of a shift’s questions testing an idea no earlier year had asked. Lumi analyses 108 of the 174 JEE Main shifts held from 2017 to 2026.

Drill down

What this shift asked, and its history

Every chapter this shift asked, with all its topics and ideas. Counts and squares cover every analysed shift from 2017 to 2026; rows this shift did not ask are faded, and each row lists this shift’s questions with the most recent from other shifts.

subjectAskedShareYears’17’20’21’22’23’24’25’26

Open30 this shift3092 · 33%33%8/830299720720358540200225
Open30 this shift3065 · 33%33%8/829299715710354533200225
Open29 this shift3084 · 33%33%8/830298716720359538199224

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 89 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

89 of 89 questions

Physics29 questions

12 easy17 medium0 hard

Q1
If e is the electronic charge, c is the speed of light in free space and h is Planck’s constant, the quantity $\frac{1}{4\pi\varepsilon_0}\frac{|e|^2}{\hbar c}$ has dimensions of :
Units and Measurements
MediumSingle correct
Q2
A stone is dropped from the top of a building. When it crosses a point 5 m below the top, another stone starts to fall from a point 25 m below the top. Both stones reach the bottom of building simultaneously. The height of the building is :
Motion in a Straight Line Kinematics in 1D Relative Motion in Free Fall
MediumSingle correctNew idea
Q4
The point A moves with a uniform speed along the circumference of a circle of radius 0.36 m and covers $30^{\circ}$ in 0.1 s. The perpendicular projection ‘P’ from ‘A’ on the diameter MN represents the simple harmonic motion of ‘P’. The restoration force per…
Oscillations SHM Equations SHM Definition and Force Law
MediumDiagram based · figureNew ideaIdea asked in 4 years
Q5
Thermodynamic process is shown below on a P-V diagram for one mole of an ideal gas. If $V_2 = 2V_1$ then the ratio of temperature $T_2/T_1$ is :
Thermodynamics Thermodynamic Processes Polytropic Process Analysis
EasyDiagram based · figureNew ideaIdea asked in 2 years
Q6
Given below are two statements : Statement I : In a diatomic molecule, the rotational energy at a given temperature obeys Maxwell’s distribution. Statement II : In a diatomic molecule, the rotational energy at a given temperature equals the translational…
Kinetic Theory Kinetic Theory of Gases Law of Equipartition
MediumTwo statementsNew ideaIdea asked in 4 years
Q7
Two identical springs of spring constant ‘2k’ are attached to a block of mass m and to fixed support (see figure). When the mass is displaced from equilibrium position on either side, it executes simple harmonic motion. The time period of oscillations of…
Oscillations SHM Equations Springs in Parallel
EasyDiagram based · figureNew ideaIdea asked in 3 years
Q8
$Y = A\sin(\omega t + \phi_0)$ is the time-displacement equation of a SHM. At $t = 0$ the displacement of the particle is $Y = \frac{A}{2}$ and it is moving along negative $x$ -direction. Then the initial phase angle $\phi_0$ will be :
Oscillations SHM Equations Phase and Initial Conditions
MediumSingle correctNew ideaIdea asked in 3 years
Q9
A charge ‘q’ is placed at one corner of a cube as shown in figure. The flux of electrostatic field $\vec{E}$ through the shaded area is :
Electric Charges and Fields Gauss's Law Flux via Cube Symmetry
MediumDiagram based · figureNew ideaIdea asked in 2 years
Q10
An electron with kinetic energy $K_1$ enters between parallel plates of a capacitor at an angle ‘ $\alpha$ ’ with the plates. It leaves the plates at angle ‘ $\beta$ ’ with kinetic energy $K_2$ . Then the ratio of kinetic energies $K_1 : K_2$ will be :
Electrostatic Potential and Capacitance Electric Field Charged Particle Trajectory in Uniform Field
MediumSingle correctIdea asked in 3 yearsAsked the year before
Q11
In a ferromagnetic material, below the curie temperature, a domain is defined as :
Magnetism and Matter
EasySingle correct
Q12
An LCR circuit contains resistance of $110\ \Omega$ and a supply of 220 V at 300 rad/s angular frequency. If only capacitance is removed from the circuit, current lags behind the voltage by $45^{\circ}$ . If on the other hand, only inductor is removed the…
Alternating Current LCR Resonance Impedance at Resonance
MediumSingle correctNew ideaIdea asked in 3 years
Q13
The stopping potential for electrons emitted from a photosensitive surface illuminated by light of wavelength 491 nm is 0.710 V. When the incident wavelength is changed to a new value, the stopping potential is 1.43 V. The new wavelength is :
Dual Nature of Radiation and Matter Photoelectric Effect Two-Wavelength System of Equations
MediumSingle correctIdea asked in 5 yearsAsked the year before
Q14
Consider the diffraction pattern obtained from the sunlight incident on a pinhole of diameter 0.1 µm. If the diameter of the pinhole is slightly increased, it will affect the diffraction pattern such that :
Wave Optics Single Slit Diffraction Angular Width of Central Max
MediumSingle correctNew ideaIdea asked in 3 years
Q15
An electron of mass $m_e$ and a proton of mass $m_p = 1836\ m_e$ are moving with the same speed. The ratio of their de Broglie wavelength $\frac{\lambda_{electron}}{\lambda_{proton}}$ will be :
Dual Nature of Radiation and Matter Matter Waves Wavelength Ratios of Different Particles
EasySingle correctIdea asked in 6 yearsAsked the year before
Q16
The wavelength of the photon emitted by a hydrogen atom when an electron makes a transition from n = 2 to n = 1 state is :
Atoms Bohr's Atomic Model Rydberg Formula and Spectra
EasySingle correctIdea asked in 6 yearsAsked the year before
Q17
If a message signal of frequency ‘ $f_m$ ’ is amplitude modulated with a carrier signal of frequency ‘ $f_c$ ’ and radiated through an antenna, the wavelength of the corresponding signal in air is :
Communication Systems
EasySingle correct
Q18
For extrinsic semiconductors; when doping level is increased;
Semiconductor Electronics: Materials, Devices and Simple Circuits
MediumSingle correct
Q19
Match List I with List II. List I List II (a) Rectifier (i) Used either for stepping up or stepping down the a.c. voltage (b) Stabilizer (ii) Used to convert a.c. voltage into d.c. voltage (c) Transformer (iii) Used to remove any ripple in the rectified…
Semiconductor Electronics: Materials, Devices and Simple Circuits Semiconductor Diodes Diode as a Rectifier
EasyMatch the listNew idea
Q20
The truth table for the following logic circuit is :
Semiconductor Electronics: Materials, Devices and Simple Circuits Logic Gates Combinational Logic Circuits
MediumDiagram based · figureIdea asked in 7 yearsAsked the year before
Q21
Two particles having masses 4 g and 16 g respectively are moving with equal kinetic energies. The ratio of the magnitudes of their linear momentum is n : 2. The value of n will be ____________.
Work, Energy and Power Work and Kinetic Energy Momentum and Kinetic Energy Relation
EasyNumerical valueNew ideaIdea asked in 4 years
Q22
The initial velocity $v_i$ required to project a body vertically upward from the surface of the earth to reach a height of 10R, where R is the radius of the earth, may be described in terms of escape velocity $v_e$ such that…
Gravitation Gravitational Potential Maximum Height Using Exact Potential
MediumNumerical valueNew ideaIdea asked in 2 years
Q23
The percentage increase in the speed of transverse waves produced in a stretched string if the tension is increased by 4%, will be ____________%.
Waves Waves on Strings Wave Speed on String
EasyNumerical valueIdea asked in 5 yearsAsked the year before
Q24
If $\vec{P} \times \vec{Q} = \vec{Q} \times \vec{P}$ , the angle between $\vec{P}$ and $\vec{Q}$ is $\theta$ ( $0^{\circ} < \theta < 360^{\circ}$ ). The value of ‘ $\theta$ ’ will be ____________ $^{\circ}$ .
Motion in a Plane
EasyNumerical value
Q25
A reversible heat engine converts one-fourth of the heat input into work. When the temperature of the sink is reduced by 52 K, its efficiency is doubled. The temperature in Kelvin of the source will be ____________.
Thermodynamics Heat Engines and Refrigerators Carnot Efficiency Limit
MediumNumerical valueNew ideaIdea asked in 3 years
Q26
Two small spheres each of mass 10 mg are suspended from a point by threads 0.5 m long. They are equally charged and repel each other to a distance of 0.20 m. The charge on each of the sphere is $\frac{a}{21} \times 10^{-8}$ C. The value of ‘a’ will be…
Electric Charges and Fields Coulomb's Law Suspended Charge Mechanics
MediumNumerical valueNew idea
Q27
Two identical conducting spheres with negligible volume have 2.1 nC and $-$ 0.1 nC charges, respectively. They are brought into contact and then separated by a distance of 0.5 m. The electrostatic force acting between the spheres is ____________…
Electric Charges and Fields Coulomb's Law Charge Redistribution via Conduction
EasyNumerical valueNew ideaIdea asked in 3 years
Q28
The peak electric field produced by the radiation coming from the 8 W bulb at a distance of 10 m is $\frac{x}{10}\sqrt{\frac{\mu_0 c}{\pi}}\ \frac{V}{m}$ . The efficiency of the bulb is 10% and it is a point source. The value of $x$ is ____________.
Electromagnetic Waves
MediumNumerical value
Q29
A current of 6 A enters one corner P of an equilateral triangle PQR having 3 wires of resistance $2\ \Omega$ each and leaves by the corner R. The currents $i_1$ in ampere is ____________ .
Current Electricity Kirchhoff's Laws Current Division Rule
EasyNumerical value · figureNew ideaIdea asked in 3 years
Q30
The wavelength of an X-ray beam is 10 Å. The mass of a fictitious particle having the same energy as that of the X-ray photons is $\frac{x}{3}h$ kg. The value of $x$ is ____________. (h = Planck’s constant)
Dual Nature of Radiation and Matter Matter Waves Matter Wave vs Photon Energy Scaling
MediumNumerical valueNew idea

Chemistry30 questions

12 easy16 medium2 hard

Q31
Which among the following species has unequal bond lengths ?
Chemical Bonding and Molecular Structure VSEPR Theory See-Saw Geometry (SN=5, 1 LP)
EasySingle correctNew ideaIdea asked in 2 years
Q32
The solubility of $\mathrm{Ca(OH)_2}$ in water is : [Given : The solubility product of $\mathrm{Ca(OH)_2}$ in water $=5.5\times10^{-6}$ ]
Equilibrium Solubility Product Solubility-Ksp Relationship
MediumSingle correctIdea asked in 5 yearsAsked the year before
Q33
Which one of the following statements is FALSE for hydrophilic sols ?
Surface Chemistry Colloids Lyophilic and Lyophobic Sols
EasySingle correctNew ideaIdea asked in 3 years
Q34
The correct order of bond dissociation enthalpy of halogens is :
The p-Block Elements Group 17 & 18 Halogen Reactivity Trends
MediumSingle correctNew ideaIdea asked in 2 years
Q35
The method used for the purification of Indium is :
General Principles and Processes of Isolation of Elements Metallurgical Extraction Metal Refining Techniques
EasySingle correctIdea asked in 4 yearsAsked the year before
Q36
Water does not produce CO on reacting with :
Hydrogen Hydrogen and Hydrides Syngas Production via Reforming
EasySingle correctNew ideaIdea asked in 2 years
Q37
Given below are two statements : Statement I : $\alpha$ and $\beta$ forms of sulphur can change reversibly between themselves with slow heating or slow cooling. Statement II : At room temperature the stable crystalline form of sulphur is monoclinic sulphur.…
The p-Block Elements Group 15 & 16 Sulfur Allotropes
EasyTwo statementsNew idea
Q38
The major components of German Silver are :
The d- and f-Block Elements
EasySingle correct
Q39
In which of the following order the given complex ions are arranged correctly with respect to their decreasing spin only magnetic moment ? (i) $\mathrm{[FeF_6]^{3-}}$ (ii) $\mathrm{[Co(NH_3)_6]^{3+}}$ (iii) $\mathrm{[NiCl_4]^{2-}}$ (iv)…
Coordination Compounds Crystal Field Theory Spin-only Magnetic Moment
MediumSingle correctIdea asked in 7 yearsAsked the year before
Q40
Given below are two statements : Statement I : The pH of rain water is normally ~5.6. Statement II : If the pH of rain water drops below 5.6, it is called acid rain. In the light of the above statements, choose the correct answer from the options given below :
Environmental Chemistry Acid Rain pH Threshold
EasyTwo statementsNew idea
Q41
Which of the following compound is added to the sodium extract before addition of silver nitrate for testing of halogens ?
Organic Chemistry: Some Basic Principles and Techniques
EasySingle correct
Q42
The major product of the following reaction is : [figure: benzene ring (circle inside hexagon), then an arrow with a skeletal reagent drawn above it (see figure_note) and $\mathrm{H_2SO_4}$ below]
Haloalkanes and Haloarenes Aromatic Hydrocarbons Friedel-Crafts Alkylation
MediumDiagram based · figureIdea asked in 5 yearsAsked the year before
Q43
The major product of the following reaction is : $\mathrm{CH_3CH_2CH{=}CH_2}\xrightarrow[\text{Rh catalyst}]{\mathrm{H_2/CO}}$
Aldehydes, Ketones and Carboxylic Acids Hydrogen and Hydrides Hydroformylation of Olefins
EasySingle correct · figureNew idea
Q44
The correct sequence of reagents used in the preparation of 4-bromo-2-nitroethyl benzene from benzene is :
Haloalkanes and Haloarenes Aromatic Hydrocarbons Bypassing FC Alkylation Rearrangements
HardSingle correctNew idea
Q45
The correct order of acid character of the following compounds is : [figure: four benzene-ring compounds labelled I, II, III, IV]
Alcohols, Phenols and Ethers Organic Acidity and Basicity Carboxylic Acid Acidity
MediumDiagram based · figureIdea asked in 4 yearsAsked the year before
Q46
What is ‘X’ in the given reaction ? [figure: $\mathrm{CH_2OH}$ over $\mathrm{CH_2OH}$ joined by a vertical line] + oxalic acid $\xrightarrow{210^\circ\mathrm{C}}$ X (major product)
Alcohols, Phenols and Ethers
MediumSingle correct · figure
Q47
[figure: benzene ring with $\mathrm{NH_2}$ $\xrightarrow[288\ \mathrm{K}]{\mathrm{HNO_3,\ H_2SO_4}}$ (A) + (B) + (C)] Correct statement about the given chemical reaction is :
Amines Direct Nitration of Aniline
MediumOther · figureNew ideaIdea asked in 2 years
Q48
Carbylamine test is used to detect the presence of primary amino group in an organic compound. Which of the following compound is formed when this test is performed with aniline ?
Amines Carbylamine Reaction
EasySingle correct · figureIdea asked in 4 yearsAsked the year before
Q49
Which of the following is correct structure of $\alpha$ -anomer of maltose ?
Biomolecules Carbohydrates Maltose Structure and Linkage
MediumDiagram based · figureIdea asked in 3 yearsAsked the year before
Q50
Given below are two statements : Statement I : The identification of $\mathrm{Ni^{2+}}$ is carried out by dimethyl glyoxime in the presence of $\mathrm{NH_4OH}$ . Statement II : The dimethyl glyoxime is a bidentate neutral ligand. In the light of the above…
Coordination Compounds
MediumTwo statements
Q51
Consider titration of NaOH solution versus 1.25 M oxalic acid solution. At the end point following burette readings were obtained. (i) 4.5 mL (ii) 4.5 mL (iii) 4.4 mL (iv) 4.4 mL (v) 4.4 mL If the volume of oxalic acid taken was 10.0 mL then the molarity of…
Some Basic Concepts of Chemistry Mole Concept Normality and n-factor Calculation
MediumNumerical valueIdea asked in 4 yearsAsked the year before
Q52
The unit cell of copper corresponds to a face centered cube of edge length 3.596 Å with one copper atom at each lattice point. The calculated density of copper in kg/m $^3$ is ________. [Molar mass of Cu : 63.54 g ; Avogadro Number $=6.022\times10^{23}$ ]
The Solid State Crystal Lattices Unit Cell Density
MediumNumerical valueIdea asked in 4 yearsAsked the year before
Q53
Electromagnetic radiation of wavelength 663 nm is just sufficient to ionise the atom of metal A. The ionization energy of metal A in kJ mol $^{-1}$ is ________. (Rounded-off to the nearest integer) [h $=6.63\times10^{-34}$ Js, c $=3.00\times10^{8}$ ms $^{-1}$ ,…
Structure of Atom Bohr's Atomic Theory Ionization and Excitation Energy
MediumNumerical valueNew ideaIdea asked in 5 years
Q54
Five moles of an ideal gas at 293 K is expanded isothermally from an initial pressure of 2.1 MPa to 1.3 MPa against at constant external pressure 4.3 MPa. The heat transferred in this process is ________ kJ mol $^{-1}$ . (Rounded-off to the nearest integer)…
Thermodynamics (Chemistry)
MediumNumerical value
Q55
If a compound AB dissociates to the extent of 75% in an aqueous solution, the molality of the solution which shows a 2.5 K rise in the boiling point of the solution is ________ molal. (Rounded-off to the nearest integer) [ $\mathrm{K_b}=0.52$ K kg mol $^{-1}$ ]
Solutions Colligative Properties Degree of Dissociation from Van't Hoff Factor
MediumNumerical valueNew ideaIdea asked in 5 years
Q56
Copper reduces $\mathrm{NO_3^-}$ into NO and $\mathrm{NO_2}$ depending upon the concentration of $\mathrm{HNO_3}$ in solution. (Assuming fixed $\mathrm{[Cu^{2+}]}$ and $\mathrm{P_{NO}=P_{NO_2}}$ ), the $\mathrm{HNO_3}$ concentration at which the thermodynamic…
Electrochemistry Nernst Equation pH Dependence of Electrode Potential
HardNumerical valueIdea asked in 4 yearsAsked the year before
Q57
The rate constant of a reaction increases by five times on increase in temperature from 27 $^\circ$ C to 52 $^\circ$ C. The value of activation energy in kJ mol $^{-1}$ is ________. (Rounded-off to the nearest integer) [R $=8.314$ J K $^{-1}$ mol $^{-1}$ ]
Chemical Kinetics Arrhenius Equation Two-Point Arrhenius Equation
MediumNumerical valueIdea asked in 4 yearsAsked the year before
Q58
Among the following, number of metal/s which can be used as electrodes in the photoelectric cell is ________. (Integer answer) (A) Li (B) Na (C) Rb (D) Cs
The s-Block Elements s-Block Elements Photoelectric Application of K and Cs
EasyNumerical valueIdea asked in 2 yearsAsked the year before
Q59
The spin only magnetic moment of a divalent ion in aqueous solution (atomic number 29) is ________ BM.
The d- and f-Block Elements d-Block Elements Spin-only Magnetism
MediumNumerical valueIdea asked in 6 yearsAsked the year before
Q60
The number of compound/s given below which contain/s $-$ COOH group is ________. (Integer answer) (A) Sulphanilic acid (B) Picric acid (C) Aspirin (D) Ascorbic acid
Biomolecules
EasyNumerical value

Maths30 questions

7 easy16 medium7 hard

Q61
If for the matrix, $A = \begin{bmatrix} 1 & -\alpha \\ \alpha & \beta \end{bmatrix}$ , $AA^T = I_2$ , then the value of $\alpha^4 + \beta^4$ is :
Matrices Matrix Multiplication Orthogonal Matrix Multiplication Definition
EasySingle correctNew ideaIdea asked in 2 years
Q62
Let A be a $3 \times 3$ matrix with det(A) $= 4$ . Let $R_i$ denote the $i^{th}$ row of A. If a matrix B is obtained by performing the operation $R_2 \to 2R_2 + 5R_3$ on 2A, then det(B) is equal to :
Determinants
EasySingle correct
Q63
The following system of linear equations $2x + 3y + 2z = 9$ $3x + 2y + 2z = 9$ $x - y + 4z = 8$
Determinants Cramer's Rule Condition for Unique Solution
EasySingle correctNew ideaIdea asked in 3 years
Q64
If $I_n = \int_{\pi/4}^{\pi/2} \cot^n x \, dx$ , then :
Integrals Definite Integral Properties Reduction Formulae in Definite Integrals
HardSingle correctNew ideaIdea asked in 4 years
Q65
A function $f(x)$ is given by $f(x) = \frac{5^x}{5^x + 5}$ , then the sum of the series $f\left(\frac{1}{20}\right) + f\left(\frac{2}{20}\right) + f\left(\frac{3}{20}\right) + \ldots\ldots + f\left(\frac{39}{20}\right)$ is equal to :
Relations and Functions
MediumSingle correct
Q66
Let $\alpha$ and $\beta$ be the roots of $x^2 - 6x - 2 = 0$ . If $a_n = \alpha^n - \beta^n$ for $n \geq 1$ , then the value of $\frac{a_{10} - 2a_8}{3a_9}$ is :
Complex Numbers and Quadratic Equations Quadratic Roots Newton's Sums
MediumSingle correctIdea asked in 7 yearsAsked the year before
Q67
The minimum value of $f(x) = a^{a^x} + a^{1-a^x}$ , where $a, x \in \mathbf{R}$ and $a > 0$ , is equal to :
Application of Derivatives Maxima and Minima Optimization via AM-GM Inequality
MediumSingle correctIdea asked in 4 yearsAsked the year before
Q68
The integral $\int \frac{e^{3\log_e 2x} + 5e^{2\log_e 2x}}{e^{4\log_e x} + 5e^{3\log_e x} - 7e^{2\log_e x}} \, dx$ , $x > 0$ , is equal to : (where c is a constant of integration)
Integrals Integration by Substitution Logarithmic Derivative Integration Rule
MediumSingle correctNew idea
Q69
If $\alpha, \beta \in \mathbf{R}$ are such that $1 - 2i$ (here $i^2 = -1$ ) is a root of $z^2 + \alpha z + \beta = 0$ , then $(\alpha - \beta)$ is equal to :
Complex Numbers and Quadratic Equations Quadratic Roots Complex Conjugate Roots
EasySingle correctIdea asked in 4 yearsAsked the year before
Q70
If the curve $x^2 + 2y^2 = 2$ intersects the line $x + y = 1$ at two points P and Q, then the angle subtended by the line segment PQ at the origin is :
Conic Sections Pair of Straight Lines Homogenization of curves
HardSingle correctNew idea
Q71
The shortest distance between the line $x - y = 1$ and the curve $x^2 = 2y$ is :
Conic Sections Tangents and Normals Shortest Distance via Common Normal
MediumSingle correctNew idea
Q72
A hyperbola passes through the foci of the ellipse $\frac{x^2}{25} + \frac{y^2}{16} = 1$ and its transverse and conjugate axes coincide with major and minor axes of the ellipse, respectively. If the product of their eccentricities is one, then the equation…
Conic Sections Ellipse Eccentricity Coordinates of foci
MediumSingle correctNew ideaIdea asked in 4 years
Q73
A plane passes through the points A(1, 2, 3), B(2, 3, 1) and C(2, 4, 2). If O is the origin and P is $(2, -1, 1)$ , then the projection of $\overrightarrow{OP}$ on this plane is of length :
Three Dimensional Geometry Planes in 3D Projection of Line Segment on Plane
MediumSingle correctNew idea
Q74
$\lim_{n\to\infty}\left[\frac{1}{n} + \frac{n}{(n+1)^2} + \frac{n}{(n+2)^2} + \ldots\ldots\ldots + \frac{n}{(2n-1)^2}\right]$ is equal to :
Integrals Integral as Limit of Sum Summation to Riemann Form
MediumSingle correctNew ideaIdea asked in 4 years
Q75
In a group of 400 people, 160 are smokers and non-vegetarian; 100 are smokers and vegetarian and the remaining 140 are non-smokers and vegetarian. Their chances of getting a particular chest disorder are 35%, 20% and 10% respectively. A person is chosen from…
Probability Bayes' Theorem Bayes' Theorem Application
MediumSingle correctNew ideaIdea asked in 6 years
Q76
Let A be a set of all 4-digit natural numbers whose exactly one digit is 7. Then the probability that a randomly chosen element of A leaves remainder 2 when divided by 5 is :
Permutations and Combinations Fundamental Counting Principle
HardSingle correctNew ideaIdea asked in 5 years
Q77
If $0 < x, y < \pi$ and $\cos x + \cos y - \cos(x+y) = \frac{3}{2}$ , then $\sin x + \cos y$ is equal to :
Trigonometric Functions Trigonometric Identities Trigonometric Bounds and Inequalities
MediumSingle correctNew idea
Q78
Let $x$ denote the total number of one-one functions from a set A with 3 elements to a set B with 5 elements and $y$ denote the total number of one-one functions from the set A to the set $A \times B$ . Then :
Relations and Functions Permutations and Combinations Number of Injective Functions
EasySingle correctNew ideaIdea asked in 4 years
Q79
$\mathrm{cosec}\left[2\cot^{-1}(5) + \cos^{-1}\left(\frac{4}{5}\right)\right]$ is equal to :
Inverse Trigonometric Functions Inverse Trig Identities Trig of Sum of Inverse Trig Functions
MediumSingle correctNew ideaIdea asked in 3 years
Q80
The contrapositive of the statement “If you will work, you will earn money” is :
Relations and Functions
EasySingle correct
Q81
A function $f$ is defined on $[-3, 3]$ as $f(x) = \begin{cases} \min\{|x|, 2 - x^2\}, & -2 \leq x \leq 2 \\ [|x|], & 2 < |x| \leq 3 \end{cases}$ where $[x]$ denotes the greatest integer $\leq x$ . The number of points, where $f$ is not differentiable in…
Continuity and Differentiability Differentiability of Max/Min Functions
HardNumerical valueIdea asked in 5 yearsAsked the year before
Q82
If the curve, $y = y(x)$ represented by the solution of the differential equation $(2xy^2 - y)dx + x\,dy = 0$ , passes through the intersection of the lines, $2x - 3y = 1$ and $3x + 2y = 8$ , then $|y(1)|$ is equal to ________.
Differential Equations Linear Differential Equations Bernoulli Equation Reduction
HardNumerical valueNew ideaIdea asked in 2 years
Q83
The total number of two digit numbers ‘n’, such that $3^n + 7^n$ is a multiple of 10, is ________.
Relations and Functions
MediumNumerical value
Q84
If $\lim_{x\to 0} \frac{ax - (e^{4x} - 1)}{ax(e^{4x} - 1)}$ exists and is equal to b, then the value of $a - 2b$ is ________.
Limits and Derivatives L'Hopital's Rule Determining Unknown Constants from Finite Limits
MediumNumerical valueNew ideaIdea asked in 4 years
Q85
If the curves $x = y^4$ and $xy = k$ cut at right angles, then $(4k)^6$ is equal to ________.
Application of Derivatives Tangents and Normals Orthogonal Intersection of Curves
MediumNumerical valueNew idea
Q86
The value of $\int_{-2}^{2} |3x^2 - 3x - 6|\,dx$ is ________.
Integrals Definite Integral Properties Modulus Function Interval Splitting
MediumNumerical valueIdea asked in 3 yearsAsked the year before
Q87
If the remainder when $x$ is divided by 4 is 3, then the remainder when $(2020 + x)^{2022}$ is divided by 8 is ________.
Binomial Theorem
MediumNumerical value
Q88
A line ‘ $l$ ’ passing through origin is perpendicular to the lines $l_1: \vec{r} = (3+t)\hat{i} + (-1+2t)\hat{j} + (4+2t)\hat{k}$ $l_2: \vec{r} = (3+2s)\hat{i} + (3+2s)\hat{j} + (2+s)\hat{k}$ If the co-ordinates of the point in the first octant on ‘ $l_2$ ’ at…
Three Dimensional Geometry Lines in 3D Line Perpendicular to Two Given Lines
HardNumerical valueNew ideaIdea asked in 5 years
Q89
A line is a common tangent to the circle $(x-3)^2 + y^2 = 9$ and the parabola $y^2 = 4x$ . If the two points of contact (a, b) and (c, d) are distinct and lie in the first quadrant, then $2(a + c)$ is equal to ________.
Conic Sections Circle Tangents Common Tangent to Circle and Conic
HardNumerical valueNew ideaIdea asked in 3 years
Q90
Let $\vec{a} = \hat{i} + \alpha\hat{j} + 3\hat{k}$ and $\vec{b} = 3\hat{i} - \alpha\hat{j} + \hat{k}$ . If the area of the parallelogram whose adjacent sides are represented by the vectors $\vec{a}$ and $\vec{b}$ is $8\sqrt{3}$ square units, then…
Vector Algebra Cross Product Area of Triangles and Parallelograms
EasyNumerical valueNew ideaIdea asked in 6 years

Dropped from the final key and left out of every count: Q3.

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.