JEE Main 2021 · March session

JEE Main 16 March 2021, Shift 2

90 questions: 30 numerical-answer and 60 multiple choice. 6 were rated hard and 28 easy. Its hard-question count was the joint 18th 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: 6; 7% of the shift
Chapters asked: 67; 11 questions with a figure

Key findings · 16 Mar 2021, Shift 2

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 6 of 30 rated hard; Physics had the most easy questions (14 of 30).
3
50 questions tested a new idea, one no earlier year had asked: 69% of the 72 questions tagged to an idea.
4
22 came back from 2020: the same idea had been asked in the 2020 shifts Lumi analysed.
5
37 shared an idea with another 2021 shift (51% of tagged questions). Shifts in the same year can test the same idea in a new setting.
6
Biggest chapters: Current Electricity in Physics (2), Aldehydes, Ketones and Carboxylic Acids in Chemistry (2) and Conic Sections in Maths (3).

The shift at a glance

Subjects, difficulty and chapters

Physics

30 Q · 10 numerical

Easy 14Medium 16Hard 0

Top chapters

Chemistry

30 Q · 10 numerical

Easy 12Medium 18Hard 0

Top chapters

Maths

30 Q · 10 numerical

Easy 2Medium 22Hard 6

Top chapters

Question format

Single correct47 Q
Numerical value30 Q
Diagram based8 Q
Two statements2 Q
Other2 Q
Match the list1 Q

What each question asks you to do

Calculation30 Q
Multi-step30 Q
Application13 Q
Recall10 Q
Conceptual6 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 shift3084 · 33%33%8/830298716720359538199224
Open30 this shift3065 · 33%33%8/829299715710354533200225

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

Every question

All 90 questions of this shift

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

Subject

Difficulty

Idea

Find

90 of 90 questions

Physics30 questions

14 easy16 medium0 hard

Q1
Two identical antennas mounted on identical towers are separated from each other by a distance of 45 km. What should nearly be the minimum height of receiving antenna to receive the signals in line of sight ? (Assume radius of earth is 6400 km)
Communication Systems
MediumSingle correct
Q2
The de-Broglie wavelength associated with an electron and a proton were calculated by accelerating them through same potential of 100 V. What should nearly be the ratio of their wavelengths ? ( $\mathrm{m_p}=1.00727\,u$ , $\mathrm{m_e}=0.00055\,u$ )
Dual Nature of Radiation and Matter Matter Waves Wavelength Ratios of Different Particles
MediumSingle correctIdea asked in 6 yearsAsked the year before
Q3
The refractive index of a converging lens is 1.4. What will be the focal length of this lens if it is placed in a medium of same refractive index ? Assume the radii of curvature of the faces of lens are $R_1$ and $R_2$ respectively.
Ray Optics and Optical Instruments Lens Maker's Formula Lens Immersed in a Liquid
EasySingle correctNew ideaIdea asked in 5 years
Q4
Red light differs from blue light as they have :
Wave Optics
EasySingle correct
Q5
The magnetic field in a region is given by $\vec{B}=B_o\left(\frac{x}{a}\right)\hat{k}$ . A square loop of side d is placed with its edges along the $x$ and $y$ axes. The loop is moved with a constant velocity $\vec{v}=v_o\hat{i}$ . The emf induced in the loop…
Electromagnetic Induction Faraday's Law EMF in Spatially Varying Fields
MediumDiagram based · figureNew idea
Q6
Amplitude of a mass-spring system, which is executing simple harmonic motion decreases with time. If mass = 500 g, Decay constant = 20 g/s then how much time is required for the amplitude of the system to drop to half of its initial value ? ( $\ln 2=0.693$ )
Oscillations SHM Equations Damped Amplitude Decay
MediumSingle correctNew idea
Q7
Calculate the value of mean free path ( $\lambda$ ) for oxygen molecules at temperature $27^\circ\mathrm{C}$ and pressure $1.01\times10^5$ Pa. Assume the molecular diameter 0.3 nm and the gas is ideal. ( $k=1.38\times10^{-23}\ \mathrm{JK^{-1}}$ )
Kinetic Theory Kinetic Theory of Gases Mean Free Path
EasySingle correctIdea asked in 4 yearsAsked the year before
Q8
What will be the nature of flow of water from a circular tap, when its flow rate increased from 0.18 L/min to 0.48 L/min ? The radius of the tap and viscosity of water are 0.5 cm and $10^{-3}$ Pa s, respectively. (Density of water : $10^3\ \mathrm{kg/m^3}$ )
Mechanical Properties of Fluids Fluid Dynamics Reynolds Number and Flow Regime
MediumSingle correctNew idea
Q9
A charge Q is moving $\vec{dl}$ distance in the magnetic field $\vec{B}$ . Find the value of work done by $\vec{B}$ .
Moving Charges and Magnetism Lorentz Force Work Done by Magnetic Force
EasySingle correctNew ideaIdea asked in 2 years
Q10
Calculate the time interval between 33% decay and 67% decay if half-life of a substance is 20 minutes.
Nuclei Radioactive Decay Half-Life Calculations
EasySingle correctIdea asked in 4 yearsAsked the year before
Q11
For the given circuit, comment on the type of transformer used.
Alternating Current
EasyDiagram based · figure
Q12
The half-life of $\mathrm{Au^{198}}$ is 2.7 days. The activity of 1.50 mg of $\mathrm{Au^{198}}$ if its atomic weight is 198 g $\mathrm{mol^{-1}}$ is, ( $N_A=6\times10^{23}$ /mol).
Nuclei Radioactive Decay Activity and Rate of Decay
MediumSingle correctNew ideaIdea asked in 3 years
Q13
A bimetallic strip consists of metals A and B. It is mounted rigidly as shown. The metal A has higher coefficient of expansion compared to that of metal B. When the bimetallic strip is placed in a cold bath, it will :
Thermal Properties of Matter Thermal Expansion Bimetallic Strips
EasyDiagram based · figureNew idea
Q14
A large block of wood of mass M = 5.99 kg is hanging from two long massless cords. A bullet of mass m = 10 g is fired into the block and gets embedded in it. The (block + bullet) then swing upwards, their centre of mass rising a vertical distance h = 9.8 cm…
Work, Energy and Power Collisions Ballistic Pendulum
MediumDiagram based · figureNew ideaIdea asked in 3 years
Q15
Statement I : A cyclist is moving on an unbanked road with a speed of 7 $\mathrm{kmh^{-1}}$ and takes a sharp circular turn along a path of radius of 2m without reducing the speed. The static friction coefficient is 0.2. The cyclist will not slip and pass…
Laws of Motion Circular Motion Friction on Unbanked Flat Curves
MediumTwo statementsNew ideaIdea asked in 2 years
Q16
A mosquito is moving with a velocity $\vec{v}=0.5\,t^2\hat{i}+3t\hat{j}+9\hat{k}$ m/s and accelerating in uniform conditions. What will be the direction of mosquito after 2 s ?
Motion in a Plane Kinematics in 2D Vector Calculus for 2D Kinematics
MediumSingle correctNew ideaIdea asked in 2 years
Q17
In order to determine the Young's Modulus of a wire of radius 0.2 cm (measured using a scale of least count = 0.001 cm) and length 1m (measured using a scale of least count = 1 mm), a weight of mass 1 kg (measured using a scale of least count = 1 g) was…
Units and Measurements Elasticity Error Analysis in Searle's Experiment
MediumSingle correctNew ideaIdea asked in 4 years
Q18
A resistor develops 500 J of thermal energy in 20 s when a current of 1.5A is passed through it. If the current is increased from 1.5 A to 3 A, what will be the energy developed in 20 s.
Current Electricity Ohm's Law and Resistance Joule Heating and Power
EasySingle correctIdea asked in 6 yearsAsked the year before
Q19
Find out the surface charge density at the intersection of point $x=3$ m plane and $x$ -axis, in the region of uniform line charge of 8 nC/m lying along the $z$ -axis in free space.
Electric Charges and Fields Electric Field Finite/Infinite Line Field
MediumSingle correctNew idea
Q20
The following logic gate is equivalent to :
Semiconductor Electronics: Materials, Devices and Simple Circuits Logic Gates Combinational Logic Circuits
MediumDiagram based · figureIdea asked in 7 yearsAsked the year before
Q21
If one wants to remove all the mass of the earth to infinity in order to break it up completely. The amount of energy that needs to be supplied will be $\frac{x}{5}\frac{GM^2}{R}$ where $x$ is ________. (Round off to the Nearest Integer) (M is the mass of…
Gravitation Gravitational Potential Gravitational Binding Energy
MediumNumerical valueNew idea
Q22
A swimmer can swim with velocity of 12 km/h in still water. Water flowing in a river has velocity 6 km/h. The direction with respect to the direction of flow of river water he should swim in order to reach the point on the other bank just opposite to his…
Motion in a Plane Relative Velocity River-Boat Drift Analysis
EasyNumerical valueNew idea
Q23
A body of mass 2 kg moves under a force of $(2\hat{i}+3\hat{j}+5\hat{k})$ N. It starts from rest and was at the origin initially. After 4 s, its new coordinates are (8, b, 20). The value of b is ________. (Round off to the Nearest Integer)
Motion in a Straight Line Newton's Laws of Motion F=ma for Single Body
EasyNumerical valueNew ideaIdea asked in 6 years
Q24
A force $\vec{F}=4\hat{i}+3\hat{j}+4\hat{k}$ is applied on an intersection point of $x=2$ plane and $x$ -axis. The magnitude of torque of this force about a point (2, 3, 4) is ________. (Round off to the Nearest Integer)
System of Particles and Rotational Motion Torque and Equilibrium Torque Calculation
EasyNumerical valueIdea asked in 5 yearsAsked the year before
Q25
A solid disc of radius 'a' and mass 'm' rolls down without slipping on an inclined plane making an angle $\theta$ with the horizontal. The acceleration of the disc will be $\frac{2}{b}g\sin\theta$ where b is ________. (Round off to the Nearest Integer) (g =…
System of Particles and Rotational Motion
MediumNumerical value · figure
Q26
For an ideal heat engine, the temperature of the source is $127^\circ$ C. In order to have 60% efficiency the temperature of the sink should be ________ $^\circ$ C. (Round off to the Nearest Integer)
Thermodynamics Heat Engines and Refrigerators Carnot Efficiency Limit
EasyNumerical valueNew ideaIdea asked in 3 years
Q27
In a parallel plate capacitor set up, the plate area of capacitor is $2\ \mathrm{m^2}$ and the plates are separated by 1 m. If the space between the plates are filled with a dielectric material of thickness 0.5 m and area $2\ \mathrm{m^2}$ (see fig) the…
Electrostatic Potential and Capacitance Parallel Plate Capacitors Partially Filled Dielectric Slab
EasyNumerical value · figureNew ideaIdea asked in 2 years
Q28
The energy dissipated by a resistor is 10 mJ in 1 s when an electric current of 2 mA flows through it. The resistance is ________ $\Omega$ . (Round off to the Nearest Integer)
Current Electricity Ohm's Law and Resistance Joule Heating and Power
EasyNumerical valueIdea asked in 6 yearsAsked the year before
Q29
A deviation of $2^\circ$ is produced in the yellow ray when prism of crown and flint glass are achromatically combined. Taking dispersive powers of crown and flint glass as 0.02 and 0.03 respectively and refractive index for yellow light for these glasses…
Ray Optics and Optical Instruments Prism Deviation Achromatic Prism Combination
MediumNumerical valueNew ideaIdea asked in 2 years
Q30
A closed organ pipe of length L and an open organ pipe contain gases of densities $\rho_1$ and $\rho_2$ respectively. The compressibility of gases are equal in both the pipes. Both the pipes are vibrating in their first overtone with same frequency. The…
Waves Resonance in Air Columns Harmonic Frequency Ratios
MediumNumerical valueNew idea

Chemistry30 questions

12 easy18 medium0 hard

Q31
The INCORRECT statement regarding the structure of $\mathrm{C_{60}}$ is :
The p-Block Elements Group 13 & 14 Carbon Allotropes
MediumSingle correctNew idea
Q32
The INCORRECT statements below regarding colloidal solutions is :
Surface Chemistry Colloids Colligative Properties of Colloids
EasySingle correctNew idea
Q33
The characteristics of elements X, Y and Z with atomic numbers, respectively, 33, 53 and 83 are :
Classification of Elements and Periodicity in Properties Periodic Trends Metallic and Non-Metallic Character Trends
MediumSingle correctNew ideaIdea asked in 3 years
Q34
Which of the following reduction reaction CANNOT be carried out with coke ?
General Principles and Processes of Isolation of Elements Metallurgical Extraction Gibbs Energy Equation in Metallurgy
EasySingle correctNew idea
Q35
The correct statements about $\mathrm{H_2O_2}$ are : (A) used in the treatment of effluents. (B) used as both oxidising and reducing agents. (C) the two hydroxyl groups lie in the same plane. (D) miscible with water. Choose the correct answer from the…
Hydrogen Hydrogen and Hydrides Redox Behavior of Hydrogen Peroxide
MediumOtherNew ideaIdea asked in 2 years
Q36
Identify the elements X and Y using the ionisation energy values given below : Ionization energy (kJ/mol) 1st 2nd X 495 4563 Y 731 1450
Classification of Elements and Periodicity in Properties Periodic Trends Successive Ionization Enthalpy Jumps
EasySingle correctIdea asked in 3 yearsAsked the year before
Q37
The exact volumes of 1 M NaOH solution required to neutralise 50 mL of 1 M $\mathrm{H_3PO_3}$ solution and 100 mL of 2 M $\mathrm{H_3PO_2}$ solution, respectively, are :
The p-Block Elements Group 15 & 16 Phosphorus Oxoacids
MediumSingle correctIdea asked in 4 yearsAsked the year before
Q38
Arrange the following metal complex/compounds in the increasing order of spin only magnetic moment. Presume all the three, high spin system. (Atomic numbers Ce = 58, Gd = 64 and Eu = 63.) (a) $\mathrm{(NH_4)_2[Ce(NO_3)_6]}$ (b) $\mathrm{Gd(NO_3)_3}$ and (c)…
The d- and f-Block Elements f-Block Elements Magnetic and Spectral Properties
MediumSingle correctNew ideaIdea asked in 2 years
Q39
$\mathrm{FeX_2}$ and $\mathrm{FeY_3}$ are known when $x$ and $y$ are :
The d- and f-Block Elements d-Block Elements Stability of Transition Metal Halides
MediumSingle correctNew idea
Q40
The green house gas/es is (are ) : (A) Carbon dioxide (B) Oxygen (C) Water vapour (D) Methane Choose the most appropriate answer from the options given below :
Environmental Chemistry Greenhouse Effect
EasyOtherIdea asked in 4 yearsAsked the year before
Q41
Match List-I with List-II : List-I (Test/Reagents/Observation(s)) (a) Lassaigne's Test (b) Cu(II) oxide (c) Silver nitrate (d) The sodium fusion extract gives black precipitate with acetic acid and lead acetate List-II (Species detected) (i) Carbon (ii)…
Organic Chemistry: Some Basic Principles and Techniques
MediumMatch the list
Q42
Statement I : Sodium hydride can be used as an oxidising agent. Statement II : The lone pair of electrons on nitrogen in pyridine makes it basic. Choose the CORRECT answer from the options given below :
Hydrogen Hydrogen and Hydrides Ionic or Saline Hydrides
EasyTwo statementsNew ideaIdea asked in 2 years
Q43
An unsaturated hydrocarbon X on ozonolysis gives A. Compound A when warmed with ammonical silver nitrate forms a bright silver mirror along the sides of the test tube. The unsaturated hydrocarbon X is :
Hydrocarbons Alkenes and Alkynes Alkyne Ozonolysis
MediumSingle correct · figureNew idea
Q44
[figure: cyclohexene reacts with reagent A to give 1,4-dichlorocyclohex-2-ene (a cyclohexene ring with Cl on the two ring carbons para to each other, flanking the double bond)] Identify the reagent(s) 'A' and condition(s) for the reaction
Hydrocarbons Alkenes and Alkynes Allylic Halogenation with NBS
MediumDiagram based · figureNew ideaIdea asked in 3 years
Q45
[figure: 3-methylanisole (benzene ring with CH3 and OCH3 meta to each other) --"A"--> 3-methoxybenzoic acid (benzene ring with COOH and OCH3 meta to each other)] In the above reaction, the reagent "A" is :
Aldehydes, Ketones and Carboxylic Acids Carboxylic Acids and Derivatives Side-Chain Oxidation of Alkylbenzenes
EasyDiagram based · figureNew ideaIdea asked in 2 years
Q46
[figure: 2-substituted benzonitrile (benzene ring bearing CN and a CH(CH3)(OCH3) group ortho to each other)] reacts with (i) $\mathrm{C_6H_5MgBr}$ (1.0 equivalent), dry Ether; (ii) $\mathrm{H_3O^+}$ to give X, Major Product. The structure of X is :
Aldehydes, Ketones and Carboxylic Acids
MediumDiagram based · figure
Q47
Which of the following is least basic ?
Amines Basicity of Amides vs Amines
EasySingle correctNew ideaIdea asked in 2 years
Q48
Ammonolysis of Alkyl halides followed by the treatment with NaOH solution can be used to prepare primary, secondary and tertiary amines. The purpose of NaOH in the reaction is :
Amines Ammonolysis of Alkyl Halides
MediumSingle correctNew ideaIdea asked in 2 years
Q49
Which of the following polymer is used in the manufacture of wood laminates ?
Polymers Urea-Formaldehyde Resin
EasySingle correctNew idea
Q50
The secondary structure of protein is stabilised by :
Biomolecules
EasySingle correct
Q51
When 35 mL of 0.15 M lead nitrate solution is mixed with 20 mL of 0.12 M chromic sulphate solution, ________ $\times10^{-5}$ moles of lead sulphate precipitate out. (Round off to the Nearest Integer).
Some Basic Concepts of Chemistry Stoichiometry Limiting Reagent Identification
MediumNumerical valueNew ideaIdea asked in 4 years
Q52
Ga (atomic mass 70 u) crystallizes in a hexagonal close packed structure. The total number of voids in 0.581 g of Ga is ________ $\times10^{21}$ . (Round off to the Nearest Integer). [Given : $N_A=6.023\times10^{23}$ ]
The Solid State Crystal Lattices Tetrahedral and Octahedral
MediumNumerical valueNew idea
Q53
The number of orbitals with $n=5$ , $m_l=+2$ is ________. (Round off to the Nearest Integer).
Structure of Atom Quantum Numbers Magnetic Quantum Number
EasyNumerical valueIdea asked in 4 yearsAsked the year before
Q54
At $25^\circ$ C, 50 g of iron reacts with HCl to form $\mathrm{FeCl_2}$ . The evolved hydrogen gas expands against a constant pressure of 1 bar. The work done by the gas during this expansion is ________ J. (Round off to the Nearest Integer). [Given :…
Thermodynamics (Chemistry)
MediumNumerical value
Q55
At 363 K, the vapour pressure of A is 21 kPa and that of B is 18 kPa. One mole of A and 2 moles of B are mixed. Assuming that this solution is ideal, the vapour pressure of the mixture is ________ kPa. (Round off to the Nearest Integer).
Solutions
EasyNumerical value
Q56
Sulphurous acid ( $\mathrm{H_2SO_3}$ ) has $K_{a_1}=1.7\times10^{-2}$ and $K_{a_2}=6.4\times10^{-8}$ . The pH of 0.588 M $\mathrm{H_2SO_3}$ is ________. (Round off to the Nearest Integer).
Equilibrium pH and Buffers pH of Polyprotic Acids
MediumNumerical valueNew ideaIdea asked in 2 years
Q57
A 5.0 m mol $\mathrm{dm^{-3}}$ aqueous solution of KCl has a conductance of 0.55 mS when measured in a cell of cell constant 1.3 $\mathrm{cm^{-1}}$ . The molar conductivity of this solution is ________ $\mathrm{mSm^2\,mol^{-1}}$ . (Round off to the Nearest…
Electrochemistry
MediumNumerical value
Q58
A and B decompose via first order kinetics with half-lives 54.0 min and 18.0 min respectively. Starting from an equimolar non reactive mixture of A and B, the time taken for the concentration of A to become 16 times that of B is ________ min. (Round off to…
Chemical Kinetics Integrated Rate Equations First Order Integrated Rate Law
MediumNumerical valueIdea asked in 6 yearsAsked the year before
Q59
$\mathrm{[Ti(H_2O)_6]^{3+}}$ absorbs light of wavelength 498 nm during a d-d transition. The octahedral splitting energy for the above complex is ________ $\times10^{-19}$ J. (Round off to the Nearest Integer). $h=6.626\times10^{-34}$ Js;…
Coordination Compounds Crystal Field Theory Color and d-d Transitions
EasyNumerical valueNew ideaIdea asked in 5 years
Q60
In Duma's method of estimation of nitrogen, 0.1840 g of an organic compound gave 30 mL of nitrogen collected at 287 K and 758 mm of Hg pressure. The percentage composition of nitrogen in the compound is ________. (Round off to the Nearest Integer). [Given :…
Organic Chemistry: Some Basic Principles and Techniques Stoichiometry Stoichiometry of Dumas Method
MediumNumerical valueIdea asked in 4 yearsAsked the year before

Maths30 questions

2 easy22 medium6 hard

Q61
If the foot of the perpendicular from point (4, 3, 8) on the line $L_1:\frac{x-a}{l}=\frac{y-2}{3}=\frac{z-b}{4}$ , $l\neq0$ is (3, 5, 7), then the shortest distance between the line $L_1$ and line $L_2:\frac{x-2}{3}=\frac{y-4}{4}=\frac{z-5}{5}$ is equal to :
Three Dimensional Geometry Lines in 3D Foot of the Perpendicular
MediumSingle correctIdea asked in 6 yearsAsked the year before
Q62
Let the lengths of intercepts on $x$ -axis and $y$ -axis made by the circle $x^2+y^2+ax+2ay+c=0$ , ( $a<0$ ) be $2\sqrt2$ and $2\sqrt5$ , respectively. Then the shortest distance from origin to a tangent to this circle which is perpendicular to the line $x+2y=0$ ,…
Conic Sections Circle Tangents Tangent Parallel or Perpendicular to a Line
MediumSingle correctNew idea
Q63
Let $\vec a=\hat i+2\hat j-3\hat k$ and $\vec b=2\hat i-3\hat j+5\hat k$ . If $\vec r\times\vec a=\vec b\times\vec r$ , $\vec r\cdot(\alpha\hat i+2\hat j+\hat k)=3$ and $\vec r\cdot(2\hat i+5\hat j-\alpha\hat k)=-1$ , $\alpha\in\mathbf R$ , then the value of…
Vector Algebra Cross Product Solving Vector Equations with Cross Products
HardSingle correctIdea asked in 6 yearsAsked the year before
Q64
Let $f$ be a real valued function, defined on $\mathbf R-\{-1,1\}$ and given by $f(x)=3\log_e\left|\frac{x-1}{x+1}\right|-\frac{2}{x-1}$ . Then in which of the following intervals, function $f(x)$ is increasing ?
Application of Derivatives Monotonicity Finding Intervals
MediumSingle correctIdea asked in 5 yearsAsked the year before
Q65
If the points of intersections of the ellipse $\frac{x^2}{16}+\frac{y^2}{b^2}=1$ and the circle $x^2+y^2=4b$ , $b>4$ lie on the curve $y^2=3x^2$ , then b is equal to :
Conic Sections
MediumSingle correct
Q66
Let C be the locus of the mirror image of a point on the parabola $y^2=4x$ with respect to the line $y=x$ . Then the equation of tangent to C at P(2, 1) is :
Conic Sections
MediumSingle correct
Q67
Let A denote the event that a 6-digit integer formed by 0, 1, 2, 3, 4, 5, 6 without repetitions, be divisible by 3. Then probability of event A is equal to :
Probability
MediumSingle correct
Q68
If $y=y(x)$ is the solution of the differential equation $\frac{dy}{dx}+(\tan x)\,y=\sin x$ , $0\le x\le\frac{\pi}{3}$ , with $y(0)=0$ , then $y\left(\frac{\pi}{4}\right)$ equal to :
Differential Equations Linear Differential Equations Particular Solution Solving
MediumSingle correctNew ideaIdea asked in 4 years
Q69
Let $\alpha\in\mathbf R$ be such that the function $f(x)=\begin{cases}\frac{\cos^{-1}(1-\{x\}^2)\sin^{-1}(1-\{x\})}{\{x\}-\{x\}^3}, & x\neq0\\ \alpha, & x=0\end{cases}$ is continuous at $x=0$ , where $\{x\}=x-[x]$ , $[x]$ is the greatest integer less than or…
Continuity and Differentiability Continuity involving GIF and Fractional Part
HardSingle correctIdea asked in 6 yearsAsked the year before
Q70
If $(x,y,z)$ be an arbitrary point lying on a plane P which passes through the points (42, 0, 0), (0, 42, 0) and (0, 0, 42), then the value of the expression…
Three Dimensional Geometry
MediumSingle correct
Q71
Let $A=\{2,3,4,5,\ldots,30\}$ and ' $\simeq$ ' be an equivalence relation on $A\times A$ , defined by $(a,b)\simeq(c,d)$ , if and only if $ad=bc$ . Then the number of ordered pairs which satisfy this equivalence relation with ordered pair (4, 3) is equal to :
Relations and Functions Sets and Relations Equivalence Relations
MediumSingle correctNew ideaIdea asked in 4 years
Q72
Let $P(x)=x^2+bx+c$ be a quadratic polynomial with real coefficients such that $\int_0^1P(x)\,dx=1$ and $P(x)$ leaves remainder 5 when it is divided by $(x-2)$ . Then the value of $9(b+c)$ is equal to :
Integrals
EasySingle correct
Q73
Consider a rectangle ABCD having 5, 7, 6, 9 points in the interior of the line segments AB, CD, BC, DA respectively. Let $\alpha$ be the number of triangles having these points from different sides as vertices and $\beta$ be the number of quadrilaterals…
Permutations and Combinations Number of Triangles from Points
MediumSingle correctNew ideaIdea asked in 3 years
Q74
Consider the integral $I=\int_0^{10}\frac{[x]\,e^{[x]}}{e^{x-1}}\,dx$ , where $[x]$ denotes the greatest integer less than or equal to $x$ . Then the value of I is equal to :
Integrals Definite Integral Properties Greatest Integer Function Integrals
MediumSingle correctIdea asked in 7 yearsAsked the year before
Q75
Let A(-1, 1), B(3, 4) and C(2, 0) be given three points. A line $y=mx$ , $m>0$ , intersects lines AC and BC at point P and Q respectively. Let $A_1$ and $A_2$ be the areas of $\triangle$ ABC and $\triangle$ PQC respectively, such that $A_1=3A_2$ , then the value…
Straight Lines
MediumSingle correct
Q76
The least value of $|z|$ where $z$ is complex number which satisfies the inequality $\exp\left(\frac{(|z|+3)(|z|-1)}{||z|+1|}\log_e 2\right)\ge\log_{\sqrt2}|5\sqrt7+9i|$ , $i=\sqrt{-1}$ , is equal to :
Complex Numbers and Quadratic Equations
MediumSingle correct
Q77
The maximum value of $f(x)=\begin{vmatrix}\sin^2x & 1+\cos^2x & \cos2x\\ 1+\sin^2x & \cos^2x & \cos2x\\ \sin^2x & \cos^2x & \sin2x\end{vmatrix}$ , $x\in\mathbf R$ is :
Determinants Trigonometric Identities Extrema of Linear Trigonometric Forms
MediumSingle correctNew ideaIdea asked in 2 years
Q78
Given that the inverse trigonometric functions take principal values only. Then, the number of real values of $x$ which satisfy $\sin^{-1}\left(\frac{3x}{5}\right)+\sin^{-1}\left(\frac{4x}{5}\right)=\sin^{-1}x$ is equal to :
Inverse Trigonometric Functions Inverse Trig Identities Sine Inverse Sum and Difference
MediumSingle correctIdea asked in 3 yearsAsked the year before
Q79
Let $f:S\to S$ where $S=(0,\infty)$ be a twice differentiable function such that $f(x+1)=xf(x)$ . If $g:S\to\mathbf R$ be defined as $g(x)=\log_e f(x)$ , then the value of $|g''(5)-g''(1)|$ is equal to :
Application of Derivatives Chain Rule Differentiating Functional Equations
HardSingle correctNew ideaIdea asked in 2 years
Q80
Let $C_1$ be the curve obtained by the solution of differential equation $2xy\frac{dy}{dx}=y^2-x^2$ , $x>0$ . Let the curve $C_2$ be the solution of $\frac{2xy}{x^2-y^2}=\frac{dy}{dx}$ . If both the curves pass through (1, 1), then the area enclosed by the…
Application of Integrals Area Under Curves Area Between Curves
HardSingle correctIdea asked in 7 yearsAsked the year before
Q81
For real numbers $\alpha,\beta,\gamma$ and $\delta$ , if…
Integrals Integration by Substitution Composite Function Substitution
HardNumerical valueNew ideaIdea asked in 3 years
Q82
In $\triangle$ ABC, the lengths of sides AC and AB are 12 cm and 5 cm, respectively. If the area of $\triangle$ ABC is 30 $\mathrm{cm^2}$ and R and r are respectively the radii of circumcircle and incircle of $\triangle$ ABC, then the value of $2R+r$ (in cm) is…
Trigonometric Functions Properties of Triangles Inradius and Circumradius
MediumNumerical valueNew ideaIdea asked in 3 years
Q83
If the distance of the point (1, $-2$ , 3) from the plane $x+2y-3z+10=0$ measured parallel to the line, $\frac{x-1}{3}=\frac{2-y}{m}=\frac{z+3}{1}$ is $\sqrt{\frac72}$ , then the value of $|m|$ is equal to ________.
Three Dimensional Geometry Point-Plane Distance Distance Along a Line
MediumNumerical valueIdea asked in 5 yearsAsked the year before
Q84
Let $\vec c$ be a vector perpendicular to the vectors $\vec a=\hat i+\hat j-\hat k$ and $\vec b=\hat i+2\hat j+\hat k$ . If $\vec c\cdot(\hat i+\hat j+3\hat k)=8$ then the value of $\vec c\cdot(\vec a\times\vec b)$ is equal to ________.
Vector Algebra Triple Products STP Definition and Determinant
MediumNumerical valueIdea asked in 2 yearsAsked the year before
Q85
Let $f:\mathbf R\to\mathbf R$ and $g:\mathbf R\to\mathbf R$ be defined as $f(x)=\begin{cases}x+a, & x<0\\ |x-1|, & x\ge0\end{cases}$ and $g(x)=\begin{cases}x+1, & x<0\\ (x-1)^2+b, & x\ge0\end{cases}$ , where a, b are non-negative real numbers. If…
Continuity and Differentiability Continuity and Differentiability of Composite Functions
MediumNumerical valueNew ideaIdea asked in 5 years
Q86
Consider the statistics of two sets of observations as follows : Size Mean Variance Observation I 10 2 2 Observation II n 3 1 If the variance of the combined set of these two observations is $\frac{17}{9}$ , then the value of n is equal to ________.
Statistics Variance and SD Combined Variance of Two Sets
EasyNumerical valueNew ideaIdea asked in 3 years
Q87
Let n be a positive integer. Let $A=\sum_{k=0}^{n}(-1)^k\,{}^nC_k\left[\left(\frac12\right)^k+\left(\frac34\right)^k+\left(\frac78\right)^k+\left(\frac{15}{16}\right)^k+\left(\frac{31}{32}\right)^k\right]$ If $63A=1-\frac{1}{2^{30}}$ , then n is equal to…
Binomial Theorem Binomial Coefficients Alternating Binomial Series
MediumNumerical valueNew ideaIdea asked in 2 years
Q88
Let $A=\begin{bmatrix}a_1\\a_2\end{bmatrix}$ and $B=\begin{bmatrix}b_1\\b_2\end{bmatrix}$ be two $2\times1$ matrices with real entries such that $A=XB$ , where $X=\frac{1}{\sqrt3}\begin{bmatrix}1&-1\\1&k\end{bmatrix}$ , and $k\in\mathbf R$ . If…
Matrices Matrix Multiplication Row-by-Column Multiplication
MediumNumerical valueNew ideaIdea asked in 2 years
Q89
Let $\frac{1}{16}$ , a and b be in G.P. and $\frac1a,\frac1b$ , 6 be in A.P., where $a,b>0$ . Then $72(a+b)$ is equal to ________.
Sequences and Series AP and GP GP General Term
MediumNumerical valueNew ideaIdea asked in 5 years
Q90
Let $S_n(x)=\log_{a^{1/2}}x+\log_{a^{1/3}}x+\log_{a^{1/6}}x+\log_{a^{1/11}}x+\log_{a^{1/18}}x+\log_{a^{1/27}}x+\ldots$ up to n-terms, where $a>1$ . If $S_{24}(x)=1093$ and $S_{12}(2x)=265$ , then value of a is equal to ________.
Sequences and Series
HardNumerical value

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.