JEE Main 2021 · August session

JEE Main 27 August 2021, Shift 1

90 questions: 30 numerical-answer and 60 multiple choice. 12 were rated hard and 24 easy. Its hard-question count was the joint 3rd highest of the 24 2021 shifts Lumi analysed. The biggest chapter was Conic Sections, with 3 questions.

Questions: 90; none dropped
Numerical answer: 30; 33% of the shift
Hard: 12; 13% of the shift
Chapters asked: 70; 15 questions with a figure

Key findings · 27 Aug 2021, Shift 1

1
12 hard questions (13%): the joint 3rd 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 6 of 30 rated hard; Chemistry had the most easy questions (12 of 30).
3
50 questions tested a new idea, one no earlier year had asked: 63% of the 80 questions tagged to an idea.
4
25 came back from 2020: the same idea had been asked in the 2020 shifts Lumi analysed.
5
40 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: Moving Charges and Magnetism in Physics (3), Alcohols, Phenols and Ethers in Chemistry (2) and Conic Sections in Maths (3).

The shift at a glance

Subjects, difficulty and chapters

Physics

30 Q · 10 numerical

Easy 10Medium 18Hard 2

Top chapters

Chemistry

30 Q · 10 numerical

Easy 12Medium 14Hard 4

Top chapters

Maths

30 Q · 10 numerical

Easy 2Medium 22Hard 6

Top chapters

Question format

Single correct50 Q
Numerical value30 Q
Diagram based7 Q
Match the list2 Q
Assertion–reason1 Q

What each question asks you to do

Multi-step31 Q
Calculation26 Q
Conceptual11 Q
Recall11 Q
Application7 Q
Data or graph2 Q
Diagram reading2 Q

Most common: Multi-step.

Against the other 2021 shifts

This shift next to the 23 other 2021 shifts Lumi analysed

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

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

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

Drill down

What this shift asked, and its history

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

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

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

Squares are the 8 years with analysed shifts, 2017 to 2026. A number is the questions that year when there is more than one; the shade is scaled to one shift, so a year with more analysed shifts (1 to 24 a year) does not look darker. Share is of the level above.

Every question

All 90 questions of this shift

The opening words of each question, with its chapter, topic, idea and how hard it was. Open one for the full text, options and the official answer. “New idea” means no earlier year had tested it.

Subject

Difficulty

Idea

Find

90 of 90 questions

Physics30 questions

10 easy18 medium2 hard

Q1
In Millikan’s oil drop experiment, what is viscous force acting on an uncharged drop of radius $2.0\times10^{-5}$ m and density $1.2\times10^{3}$ kgm $^{-3}$ ? Take viscosity of liquid $=1.8\times10^{-5}$ Nsm $^{-2}$ . (Neglect buoyancy due to air).
Moving Charges and Magnetism Viscosity Stokes' Law
MediumSingle correctNew idea
Q2
Moment of inertia of a square plate of side $l$ about the axis passing through one of the corner and perpendicular to the plane of square plate is given by :
System of Particles and Rotational Motion Moment of Inertia Parallel Axis Theorem
EasySingle correctIdea asked in 4 yearsAsked the year before
Q3
A huge circular arc of length 4.4 ly subtends an angle ‘4s’ at the centre of the circle. How long it would take for a body to complete 4 revolution if its speed is 8 AU per second ? Given : 1 ly $=9.46\times10^{15}$ m 1 AU $=1.5\times10^{11}$ m
Units and Measurements
MediumSingle correct
Q4
The resultant of these forces $\overrightarrow{OP}$ , $\overrightarrow{OQ}$ , $\overrightarrow{OR}$ , $\overrightarrow{OS}$ and $\overrightarrow{OT}$ is approximately __________ N. [Take $\sqrt{3}=1.7$ , $\sqrt{2}=1.4$ Given $\hat{i}$ and $\hat{j}$ unit vectors…
Motion in a Plane Kinematics in 2D Vector Component Resolution
MediumSingle correct · figureIdea asked in 4 yearsAsked the year before
Q5
Which of the following is not a dimensionless quantity ?
Units and Measurements Biot-Savart Law Dimensions of Permeability
EasySingle correctNew ideaIdea asked in 2 years
Q6
If E and H represents the intensity of electric field and magnetising field respectively, then the unit of E/H will be :
Magnetism and Matter
MediumSingle correct
Q7
An ideal gas is expanding such that $\mathrm{PT}^3=$ constant. The coefficient of volume expansion of the gas is :
Thermodynamics Thermal Expansion Volume Expansion Coefficient of Ideal Gas
MediumSingle correctNew ideaIdea asked in 2 years
Q8
A balloon carries a total load of 185 kg at normal pressure and temperature of $27^\circ$ C. What load will the balloon carry on rising to a height at which the barometric pressure is 45 cm of Hg and the temperature is $-7^\circ$ C. Assuming the volume…
Kinetic Theory Fluid Statics Gas Balloon Payload Capacity
MediumSingle correctNew idea
Q9
The variation of displacement with time of a particle executing free simple harmonic motion is shown in the figure. The potential energy U( $x$ ) versus time (t) plot of the particle is correctly shown in figure :
Oscillations Energy in SHM Energy-Time Graphs in SHM
MediumSingle correct · figureIdea asked in 2 years
Q10
Calculate the amount of charge on capacitor of 4 $\mu$ F. The internal resistance of battery is 1 $\Omega$ :
Electrostatic Potential and Capacitance RC Circuits Initial and Steady State Behavior
MediumSingle correct · figureIdea asked in 5 years
Q11
A uniformly charged disc of radius R having surface charge density $\sigma$ is placed in the $xy$ plane with its center at the origin. Find the electric field intensity along the $z$ -axis at a distance Z from origin :
Electric Charges and Fields Electric Field Axial Field of a Charged Disc
MediumSingle correctNew idea
Q12
Five identical cells each of internal resistance 1 $\Omega$ and emf 5 V are connected in series and in parallel with an external resistance ‘R’. For what value of ‘R’, current in series and parallel combination will remain the same ?
Current Electricity Kirchhoff's Laws Series Combination of Cells
MediumSingle correctIdea asked in 4 yearsAsked the year before
Q13
Two ions of masses 4 amu and 16 amu have charges $+2$ e and $+3$ e respectively. These ions pass through the region of constant perpendicular magnetic field. The kinetic energy of both ions is same. Then :
Moving Charges and Magnetism Lorentz Force Trajectory Radii Ratio for Same Kinetic Energy
MediumSingle correctNew ideaIdea asked in 3 years
Q14
A bar magnet is passing through a conducting loop of radius R with velocity $v$ . The radius of the bar magnet is such that it just passes through the loop. The induced e.m.f. in the loop can be represented by the approximate curve :
Electromagnetic Induction Lenz's Law Current Reversal During Magnet Pass-Through
HardSingle correct · figureNew idea
Q15
Electric field in a plane electromagnetic wave is given by $\mathrm{E}=50\ \sin(500x-10\times10^{10}\mathrm{t})$ V/m The velocity of electromagnetic wave in this medium is : (Given C $=$ speed of light in vacuum)
Electromagnetic Waves
EasySingle correct
Q16
An object is placed beyond the centre of curvature C of the given concave mirror. If the distance of the object is $\mathrm{d}_1$ from C and the distance of the image formed is $\mathrm{d}_2$ from C, the radius of curvature of this mirror is :
Ray Optics and Optical Instruments Spherical Mirrors Mirror Formula Application
MediumSingle correctNew ideaIdea asked in 3 years
Q17
Find the distance of the image from object O, formed by the combination of lenses in the figure :
Ray Optics and Optical Instruments Lens Maker's Formula Lenses Separated by a Distance
MediumDiagram based · figureIdea asked in 6 years
Q18
In a photoelectric experiment, increasing the intensity of incident light :
Dual Nature of Radiation and Matter Photoelectric Effect Intensity vs Frequency Effects
EasySingle correctNew ideaIdea asked in 3 years
Q19
There are $10^{10}$ radioactive nuclei in a given radioactive element. Its half-life time is 1 minute. How many nuclei will remain after 30 seconds ? ( $\sqrt{2}=1.414$ )
Nuclei Radioactive Decay Half-Life Calculations
EasySingle correctIdea asked in 4 yearsAsked the year before
Q20
For a transistor in CE mode to be used as an amplifier, it must be operated in :
Semiconductor Electronics: Materials, Devices and Simple Circuits Transistors Transistor Biasing and Structure
EasySingle correctNew idea
Q21
A circuit is arranged as shown in figure. The output voltage $\mathrm{V_o}$ is equal to __________ V.
Semiconductor Electronics: Materials, Devices and Simple Circuits Semiconductor Diodes Ideal vs. Practical Diode Circuit Analysis
MediumNumerical value · figureIdea asked in 7 yearsAsked the year before
Q22
Two persons A and B perform same amount of work in moving a body through a certain distance d with application of forces acting at angles $45^\circ$ and $60^\circ$ with the direction of displacement respectively. The ratio of force applied by person A to the…
Work, Energy and Power Work and Kinetic Energy Work via Dot Product
EasyNumerical valueNew ideaIdea asked in 2 years
Q23
If the velocity of a body related to displacement $x$ is given by $v=\sqrt{5000+24x}$ m/s, then the acceleration of the body is __________ m/s $^2$ .
Motion in a Straight Line Kinematics in 1D Position-Dependent Acceleration (v dv/dx)
EasyNumerical valueNew ideaIdea asked in 4 years
Q24
A body of mass (2M) splits into four masses {m, M $-$ m, m, M $-$ m}, which are rearranged to form a square as shown in the figure. The ratio of $\frac{\mathrm{M}}{\mathrm{m}}$ for which, the gravitational potential energy of the system becomes maximum is…
Gravitation Gravitational Potential Potential Energy of Systems
HardNumerical value · figureNew ideaIdea asked in 2 years
Q25
Two cars X and Y are approaching each other with velocities 36 km/h and 72 km/h respectively. The frequency of a whistle sound as emitted by a passenger in car X, heard by the passenger in car Y is 1320 Hz. If the velocity of sound in air is 340 m/s, the…
Waves Doppler Effect General Doppler Formula
EasyNumerical valueNew idea
Q26
First, a set of n equal resistors of 10 $\Omega$ each are connected in series to a battery of emf 20 V and internal resistance 10 $\Omega$ . A current I is observed to flow. Then, the n resistors are connected in parallel to the same battery. It is observed…
Current Electricity Kirchhoff's Laws Cell EMF and Internal Resistance
MediumNumerical valueIdea asked in 6 yearsAsked the year before
Q27
A uniform conducting wire of length is 24a, and resistance R is wound up as a current carrying coil in the shape of an equilateral triangle of side ‘a’ and then in the form of a square of side ‘a’. The coil is connected to a voltage source $\mathrm{V_0}$ .…
Moving Charges and Magnetism Magnetic Dipole Moment Magnetic Moment of Polygons from Fixed Wire
MediumNumerical valueNew idea
Q28
The alternating current is given by $i=\left\{\sqrt{42}\ \sin\left(\frac{2\pi}{\mathrm{T}}\mathrm{t}\right)+10\right\}$ A The r.m.s. value of this current is __________ A.
Alternating Current RMS and Peak Values RMS of DC and AC Superposition
EasyNumerical valueNew ideaIdea asked in 2 years
Q29
A transmitting antenna has a height of 320 m and that of receiving antenna is 2000 m. The maximum distance between them for satisfactory communication in line of sight mode is ‘d’. The value of ‘d’ is __________ km.
Communication Systems
MediumNumerical value
Q30
A rod CD of thermal resistance 10.0 KW $^{-1}$ is joined at the middle of an identical rod AB as shown in figure. The ends A, B and D are maintained at $200^\circ$ C, $100^\circ$ C and $125^\circ$ C respectively. The heat current in CD is P watt. The value of P…
Thermal Properties of Matter Heat Conduction Junction Temperature
MediumNumerical value · figureNew ideaIdea asked in 4 years

Chemistry30 questions

12 easy14 medium4 hard

Q31
The unit of the van der Waals gas equation parameter ‘a’ in $\left(\mathrm{P}+\frac{\mathrm{an^2}}{\mathrm{V^2}}\right)(\mathrm{V}-\mathrm{nb})=\mathrm{nRT}$ is :
States of Matter Real Gases Units of VdW Constants
EasySingle correctNew ideaIdea asked in 2 years
Q32
Match items of List - I with those of List - II : List - I (Property) List - II (Example) (a) Diamagnetism (i) MnO (b) Ferrimagnetism (ii) $\mathrm{O_2}$ (c) Paramagnetism (iii) NaCl (d) Antiferromagnetism (iv) $\mathrm{Fe_3O_4}$ Choose the most appropriate…
The Solid State Solid State Defects Magnetic Properties
EasyMatch the listNew idea
Q33
Match List - I with List - II : List - I (Species) List - II (No. of lone pairs of electrons on the central atom) (a) $\mathrm{XeF_2}$ (i) 0 (b) $\mathrm{XeO_2F_2}$ (ii) 1 (c) $\mathrm{XeO_3F_2}$ (iii) 2 (d) $\mathrm{XeF_4}$ (iv) 3 Choose the most…
Chemical Bonding and Molecular Structure VSEPR Theory Equatorial Oxygen in TBP Xenon Compounds
MediumMatch the listNew idea
Q34
Tyndall effect is more effectively shown by :
Surface Chemistry Colloids Optical and Kinetic Properties
EasySingle correctIdea asked in 5 yearsAsked the year before
Q35
In which one of the following molecules strongest back donation of an electron pair from halide to boron is expected ?
The p-Block Elements Group 13 & 14 Back-Bonding and Lewis Acidity in Boron Trihalides
EasySingle correctNew idea
Q36
Which refining process is generally used in the purification of low melting metals ?
General Principles and Processes of Isolation of Elements Metallurgical Extraction Liquation Refining Principle
EasySingle correctNew ideaIdea asked in 2 years
Q37
Deuterium resembles hydrogen in properties but :
Hydrogen Hydrogen and Hydrides Isotopes of Hydrogen
EasySingle correctIdea asked in 4 yearsAsked the year before
Q38
The number of water molecules in gypsum, dead burnt plaster and plaster of Paris, respectively are :
The s-Block Elements s-Block Elements Gypsum and Plaster of Paris
EasySingle correctNew idea
Q39
In polythionic acid, $\mathrm{H_2S_xO_6}$ ( $x$ = 3 to 5) the oxidation state(s) of sulphur is/are :
Redox Reactions Oxidation Number Assignment
MediumSingle correctNew ideaIdea asked in 4 years
Q40
The nature of oxides $\mathrm{V_2O_3}$ and CrO is indexed as ‘X’ and ‘Y’ type respectively. The correct set of X and Y is :
The d- and f-Block Elements d-Block Elements Acid-Base Character of Oxides
EasySingle correctIdea asked in 4 years
Q41
The gas ‘A’ is having very low reactivity reaches to stratosphere. It is non-toxic and non-flammable but dissociated by UV-radiations in stratosphere. The intermediates formed initially from the gas ‘A’ are :
Environmental Chemistry Ozone Layer Depletion
EasySingle correct · figureIdea asked in 4 yearsAsked the year before
Q42
In the following sequence of reactions, the final product D is : $\mathrm{CH_3-C\equiv C-H + NaNH_2} \rightarrow \mathrm{A}$ $\mathrm{A}$ + [reagent drawn above the arrow] $\rightarrow \mathrm{B}$ $\mathrm{B}$ $\xrightarrow{\mathrm{H_2/Pd\text{-}C}}$ …
Alcohols, Phenols and Ethers Alkenes and Alkynes Acetylide Ion Alkylation
MediumDiagram based · figureNew ideaIdea asked in 2 years
Q43
In the following sequence of reactions the P is : [ring structure with Cl drawn] + Mg $\xrightarrow[\text{ether}]{\text{dry}}$ [ A ] $\xrightarrow{\text{ethanol}}$ P (Major Product)
Haloalkanes and Haloarenes Alkanes and Free Radicals Alkane Synthesis via Grignard Protonation
EasyDiagram based · figureNew ideaIdea asked in 2 years
Q44
The structure of the starting compound P used in the reaction given below is : P $\xrightarrow[\text{2. }\mathrm{H_3O^+}]{\text{1. NaOCl}}$ [product drawn]
Aldehydes, Ketones and Carboxylic Acids Aldol and Cannizzaro Haloform Interference with Aldol
MediumDiagram based · figureNew idea
Q45
Given below are two statements : one is labelled as Assertion (A) and the other is labelled as Reason (R). Assertion (A) : Synthesis of ethyl phenyl ether may be achieved by Williamson synthesis. Reason (R) : Reaction of bromobenzene with sodium ethoxide…
Alcohols, Phenols and Ethers Ethers Williamson Synthesis Limits
MediumAssertion–reasonNew ideaIdea asked in 3 years
Q46
The major product of the following reaction is : $\mathrm{CH_3-CH-CH_2-CH_2-C(=O)-Cl}$ [CH3 drawn above the CH, O drawn above the C]…
Amines Hofmann Bromamide Degradation
MediumDiagram based · figureNew ideaIdea asked in 5 years
Q47
Which of the following is not a correct statement for primary aliphatic amines ?
Amines Aliphatic Amine Basicity
MediumSingle correctNew ideaIdea asked in 2 years
Q48
[Four structures labelled (A), (B), (C) and (D) are drawn.] The correct statement about (A), (B), (C) and (D) is :
Chemistry in Everyday Life Classes of Tranquilizers
MediumDiagram based · figureNew ideaIdea asked in 2 years
Q49
Out of following isomeric forms of uracil, which one is present in RNA ?
Biomolecules
MediumDiagram based · figure
Q50
Acidic ferric chloride solution on treatment with excess of potassium ferrocyanide gives a Prussian blue coloured colloidal species. It is :
Coordination Compounds
MediumSingle correct
Q51
The kinetic energy of an electron in the second Bohr orbit of a hydrogen atom is equal to $\frac{h^2}{x\,m\,a_0^2}$ . The value of $10x$ is __________. ( $a_0$ is radius of Bohr’s orbit) (Nearest integer) [ Given : $\pi = 3.14$ ]
Structure of Atom Bohr's Atomic Theory Angular Momentum Quantization
HardNumerical valueIdea asked in 3 yearsAsked the year before
Q52
200 mL of 0.2 M HCl is mixed with 300 mL of 0.1 M NaOH. The molar heat of neutralization of this reaction is $-57.1$ kJ. The increase in temperature in °C of the system on mixing is $x \times 10^{-2}$ . The value of $x$ is __________. (Nearest integer) [Given…
Thermodynamics (Chemistry)Enthalpy and Hess's Law Constant Enthalpy of Neutralization
MediumNumerical valueNew ideaIdea asked in 3 years
Q53
1 kg of 0.75 molal aqueous solution of sucrose can be cooled up to $-4℃$ before freezing. The amount of ice (in g) that will be separated out is __________. (Nearest integer) [Given : $\mathrm{K_f}$ ( $\mathrm{H_2O}$ ) = 1.86 K kg $\mathrm{mol^{-1}}$ ]
Solutions Colligative Properties Freezing Point Depression
HardNumerical valueIdea asked in 4 yearsAsked the year before
Q54
The number of moles of $\mathrm{NH_3}$ , that must be added to 2 L of 0.80 M $\mathrm{AgNO_3}$ in order to reduce the concentration of $\mathrm{Ag^+}$ ions to $5.0 \times 10^{-8}$ M ( $\mathrm{K_{formation}}$ for $[\mathrm{Ag(NH_3)_2}]^+ = 1.0 \times 10^8$ ) is…
Equilibrium Solubility Product Complexation Effect on Solubility
HardNumerical valueNew idea
Q55
When 10 mL of an aqueous solution of $\mathrm{KMnO_4}$ was titrated in acidic medium, equal volume of 0.1 M of an aqueous solution of ferrous sulphate was required for complete discharge of colour. The strength of $\mathrm{KMnO_4}$ in grams per litre is…
Some Basic Concepts of Chemistry Redox Reactions Mohr's Salt Redox Titrations
MediumNumerical valueNew ideaIdea asked in 2 years
Q56
The reaction that occurs in a breath analyser, a device used to determine the alcohol level in a person’s blood stream is $\mathrm{2K_2Cr_2O_7 + 8H_2SO_4 + 3C_2H_6O \rightarrow 2Cr_2(SO_4)_3 + 3C_2H_4O_2 + 2K_2SO_4 + 11H_2O}$ If the rate of appearance of…
Chemical Kinetics Rate Laws Rate and Stoichiometry
EasyNumerical valueIdea asked in 5 yearsAsked the year before
Q57
The number of $f$ electrons in the ground state electronic configuration of Np (Z = 93) is __________. (Integer answer)
Structure of Atom Quantum Numbers Anomalous Configurations of Heavy Metals
MediumNumerical valueNew idea
Q58
1 mol of an octahedral metal complex with formula $\mathrm{MCl_3{\cdot}2L}$ on reaction with excess of $\mathrm{AgNO_3}$ gives 1 mol of AgCl. The denticity of Ligand L is __________. (Integer answer)
Coordination Compounds
MediumNumerical value
Q59
In Carius method for estimation of halogens, 0.2 g of an organic compound gave 0.188 g of AgBr. The percentage of bromine in the compound is __________. (Nearest integer) [Atomic mass : Ag = 108, Br = 80]
Organic Chemistry: Some Basic Principles and Techniques
EasyNumerical value
Q60
The number of moles of CuO, that will be utilized in Dumas method for estimating nitrogen in a sample of 57.5 g of N,N-dimethylaminopentane is __________ $\times 10^{-2}$ . (Nearest integer)
Organic Chemistry: Some Basic Principles and Techniques Stoichiometry Stoichiometry of Dumas Method
HardNumerical valueIdea asked in 4 yearsAsked the year before

Maths30 questions

2 easy22 medium6 hard

Q61
If $x^2+9y^2-4x+3=0$ , $x, y \in \mathbb{R}$ , then $x$ and $y$ respectively lie in the intervals :
Relations and Functions Quadratic Roots Nature of Roots
MediumSingle correctIdea asked in 3 yearsAsked the year before
Q62
If $S=\left\{z \in \mathbb{C} : \frac{z-i}{z+2i} \in \mathbb{R}\right\}$ , then :
Complex Numbers and Quadratic Equations Complex Numbers - Polar Form Collinearity via Argument
MediumSingle correctNew idea
Q63
If the matrix $A=\begin{pmatrix}0 & 2\\ K & -1\end{pmatrix}$ satisfies $A(A^3+3I)=2I$ , then the value of $K$ is :
Matrices Matrix Multiplication Cayley-Hamilton Theorem Application
MediumSingle correctNew ideaIdea asked in 5 years
Q64
If for $x, y \in \mathbf{R}$ , $x>0$ , $y=\log_{10}x+\log_{10}x^{1/3}+\log_{10}x^{1/9}+\dots$ upto $\infty$ terms and $\frac{2+4+6+\dots+2y}{3+6+9+\dots+3y}=\frac{4}{\log_{10}x}$ , then the ordered pair $(x, y)$ is equal to :
Sequences and Series AP and GP Infinite GP Sum
MediumSingle correctIdea asked in 6 yearsAsked the year before
Q65
If $\alpha$ , $\beta$ are the distinct roots of $x^2+bx+c=0$ , then $\lim_{x\to\beta}\frac{e^{2(x^2+bx+c)}-1-2(x^2+bx+c)}{(x-\beta)^2}$ is equal to :
Limits and Derivatives L'Hopital's Rule Choosing Series Expansion over L'Hopital
MediumSingle correctIdea asked in 7 yearsAsked the year before
Q66
A wire of length 20 m is to be cut into two pieces. One of the pieces is to be made into a square and the other into a regular hexagon. Then the length of the side (in meters) of the hexagon, so that the combined area of the square and the hexagon is…
Application of Derivatives Maxima and Minima Optimization Problems
MediumSingle correctIdea asked in 5 years
Q67
Let A be a fixed point $(0, 6)$ and B be a moving point $(2t, 0)$ . Let M be the mid-point of AB and the perpendicular bisector of AB meets the $y$ -axis at C. The locus of the mid-point P of MC is :
Straight Lines
MediumSingle correct
Q68
If $U_n=\left(1+\frac{1}{n^2}\right)\left(1+\frac{2^2}{n^2}\right)^2 \dots \left(1+\frac{n^2}{n^2}\right)^n$ , then $\lim_{n\to\infty}(U_n)^{\frac{-4}{n^2}}$ is equal to :
Integrals Integral as Limit of Sum Product Series with Variable Exponents
HardSingle correctNew idea
Q69
Let $y=y(x)$ be the solution of the differential equation $\frac{dy}{dx}=2(y+2\sin x-5)x-2\cos x$ such that $y(0)=7$ . Then $y(\pi)$ is equal to :
Differential Equations Linear Differential Equations Integrating Factor Calculation
MediumSingle correctIdea asked in 7 yearsAsked the year before
Q70
A tangent and a normal are drawn at the point $P(2, -4)$ on the parabola $y^2=8x$ , which meet the directrix of the parabola at the points A and B respectively. If $Q(a, b)$ is a point such that AQBP is a square, then $2a+b$ is equal to :
Conic Sections Normals to Conics Parabola Normal Point Form
MediumSingle correctNew ideaIdea asked in 2 years
Q71
Let us consider a curve, $y=f(x)$ passing through the point $(-2, 2)$ and the slope of the tangent to the curve at any point $(x, f(x))$ is given by $f(x)+xf'(x)=x^2$ . Then :
Differential Equations Linear Differential Equations Exact Differential Recognition in LDEs
MediumSingle correctNew ideaIdea asked in 3 years
Q72
Equation of a plane at a distance $\sqrt{\frac{2}{21}}$ from the origin, which contains the line of intersection of the planes $x-y-z-1=0$ and $2x+y-3z+4=0$ , is :
Three Dimensional Geometry Planes in 3D Family of Planes
MediumSingle correctIdea asked in 4 yearsAsked the year before
Q73
The distance of the point $(1, -2, 3)$ from the plane $x-y+z=5$ measured parallel to a line, whose direction ratios are $2, 3, -6$ is :
Three Dimensional Geometry Point-Plane Distance Distance Along a Line
MediumSingle correctIdea asked in 5 yearsAsked the year before
Q74
Let $\frac{\sin A}{\sin B}=\frac{\sin(A-C)}{\sin(C-B)}$ , where A, B, C are angles of a triangle ABC. If the lengths of the sides opposite these angles are a, b, c respectively, then :
Trigonometric Functions Properties of Triangles Sine Rule Application
MediumSingle correctNew ideaIdea asked in 2 years
Q75
When a certain biased die is rolled, a particular face occurs with probability $\frac{1}{6}-x$ and its opposite face occurs with probability $\frac{1}{6}+x$ . All other faces occur with probability $\frac{1}{6}$ . Note that opposite faces sum to 7 in any die.…
Probability Conditional Probability Conditioning on Dice Sums
MediumSingle correctNew idea
Q76
If $(\sin^{-1}x)^2-(\cos^{-1}x)^2=a$ ; $0<x<1$ , $a\neq 0$ , then the value of $2x^2-1$ is :
Inverse Trigonometric Functions Inverse Trig Identities Complementary Identities
MediumSingle correctNew ideaIdea asked in 3 years
Q77
$\int_{6}^{16}\frac{\log_e x^2}{\log_e x^2+\log_e(x^2-44x+484)}dx$ is equal to :
Integrals Definite Integral Properties King's Property
MediumSingle correctIdea asked in 6 yearsAsked the year before
Q78
If $0<x<1$ , then $\frac{3}{2}x^2+\frac{5}{3}x^3+\frac{7}{4}x^4+\dots$ , is equal to :
Sequences and Series Arithmetico-Geometric Series Integration Method for Harmonic-Geometric Series
HardSingle correctNew idea
Q79
The statement $(p \wedge (p \rightarrow q) \wedge (q \rightarrow r)) \rightarrow r$ is :
Mathematical Reasoning
MediumSingle correct
Q80
$\sum_{k=0}^{20}\left({}^{20}C_k\right)^2$ is equal to :
Binomial Theorem Binomial Coefficients Sum of Squares of Binomial Coefficients
EasySingle correctNew idea
Q81
If $A=\{x \in \mathbf{R} : |x-2|>1\}$ , $B=\left\{x \in \mathbf{R} : \sqrt{x^2-3}>1\right\}$ , $C=\{x \in \mathbf{R} : |x-4|\geq 2\}$ and $\mathbf{Z}$ is the set of all integers, then the number of subsets of the set $(A \cap B \cap C)^c \cap \mathbf{Z}$ is…
Relations and Functions Sets and Relations Intervals as Subsets of R
MediumNumerical valueIdea asked in 3 yearsAsked the year before
Q82
If the system of linear equations $2x+y-z=3$ $x-y-z=\alpha$ $3x+3y+\beta z=3$ has infinitely many solution, then $\alpha+\beta-\alpha\beta$ is equal to __________.
Determinants Cramer's Rule Condition for Infinite Solutions
MediumNumerical valueIdea asked in 7 yearsAsked the year before
Q83
A number is called a palindrome if it reads the same backward as well as forward. For example 285582 is a six digit palindrome. The number of six digit palindromes, which are divisible by 55, is _________.
Permutations and Combinations Fundamental Counting Principle
HardNumerical valueNew ideaIdea asked in 5 years
Q84
If $y^{1/4}+y^{-1/4}=2x$ , and $(x^2-1)\frac{d^2y}{dx^2}+\alpha x\frac{dy}{dx}+\beta y=0$ , then $|\alpha-\beta|$ is equal to __________.
Continuity and Differentiability Chain Rule Second Order Implicit Chain Rule
HardNumerical valueIdea asked in 4 yearsAsked the year before
Q85
The number of distinct real roots of the equation $3x^4+4x^3-12x^2+4=0$ is __________.
Application of Derivatives Maxima and Minima Analyzing Polynomial Roots via Extrema
MediumNumerical valueIdea asked in 4 yearsAsked the year before
Q86
Let the equation $x^2+y^2+px+(1-p)y+5=0$ represent circles of varying radius $r \in (0, 5]$ . Then the number of elements in the set $S=\{q : q=p^2 \text{ and } q \text{ is an integer}\}$ is __________.
Conic Sections General Conic Section Condition for a Circle
HardNumerical valueNew ideaIdea asked in 3 years
Q87
If the minimum area of the triangle formed by a tangent to the ellipse $\frac{x^2}{b^2}+\frac{y^2}{4a^2}=1$ and the co-ordinate axis is $kab$ , then $k$ is equal to __________.
Conic Sections Tangents and Normals Optimizing Tangent-Bounded Area
MediumNumerical valueNew idea
Q88
Let n be an odd natural number such that the variance of $1, 2, 3, 4, \dots, n$ is 14. Then n is equal to ___________.
Statistics Variance and SD Variance of First n Naturals
EasyNumerical valueNew idea
Q89
Let $\vec{a}=\hat{i}+5\hat{j}+\alpha\hat{k}$ , $\vec{b}=\hat{i}+3\hat{j}+\beta\hat{k}$ and $\vec{c}=-\hat{i}+2\hat{j}-3\hat{k}$ be three vectors such that, $|\vec{b}\times\vec{c}|=5\sqrt{3}$ and $\vec{a}$ is perpendicular to $\vec{b}$ . Then the greatest…
Vector Algebra Cross Product Magnitude and Sine Formula
MediumNumerical valueIdea asked in 7 yearsAsked the year before
Q90
If $\int\frac{dx}{(x^2+x+1)^2}=a\tan^{-1}\left(\frac{2x+1}{\sqrt{3}}\right)+b\left(\frac{2x+1}{x^2+x+1}\right)+C$ , $x>0$ where C is the constant of integration, then the value of $9(\sqrt{3}a+b)$ is equal to __________.
Integrals Integration by Substitution Tangent Trigonometric Substitution
HardNumerical valueNew idea

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.