JEE Main 2023 · April session

JEE Main 11 April 2023, Shift 2

88 questions (2 more dropped by NTA): 30 numerical-answer and 58 multiple choice. 16 were rated hard and 21 easy. Its hard-question count was the 3rd highest of the 12 2023 shifts Lumi analysed. The biggest chapter was Haloalkanes and Haloarenes, with 5 questions.

Questions: 88; 2 dropped by NTA
Numerical answer: 30; 34% of the shift
Hard: 16; 18% of the shift
Chapters asked: 61; 11 questions with a figure

Key findings · 11 Apr 2023, Shift 2

1
16 hard questions (18%): the 3rd highest count of the 12 2023 shifts Lumi analysed. Across those shifts the hard share ran from 9% to 24%.
2
Maths was the hardest section, with 12 of 29 rated hard; Chemistry had the most easy questions (10 of 30).
3
21 questions tested a new idea, one no earlier year had asked: 28% of the 74 questions tagged to an idea.
4
44 came back from 2022: the same idea had been asked in the 2022 shifts Lumi analysed.
5
31 shared an idea with another 2023 shift (42% of tagged questions). Shifts in the same year can test the same idea in a new setting.
6
Biggest chapters: Electromagnetic Induction in Physics (3), Haloalkanes and Haloarenes in Chemistry (5) and Three Dimensional Geometry in Maths (3).

The shift at a glance

Subjects, difficulty and chapters

Physics

29 Q · 10 numerical

Easy 9Medium 18Hard 2

Top chapters

Chemistry

30 Q · 10 numerical

Easy 10Medium 18Hard 2

Top chapters

Haloalkanes and Haloarenes5 Q
Coordination Compounds4 Q
Aldehydes, Ketones and Carboxylic Acids2 Q
Solutions2 Q
The s-Block Elements2 Q
Amines1 Q

Maths

29 Q · 10 numerical

Easy 2Medium 15Hard 12

Top chapters

Three Dimensional Geometry3 Q
Complex Numbers and Quadratic Equations2 Q
Conic Sections2 Q
Determinants2 Q
Integrals2 Q
Permutations and Combinations2 Q

Question format

Single correct43 Q
Numerical value30 Q
Diagram based4 Q
Assertion–reason4 Q
Two statements3 Q
Other3 Q
Match the list1 Q

What each question asks you to do

Multi-step34 Q
Calculation22 Q
Conceptual16 Q
Recall11 Q
Application5 Q

Most common: Multi-step.

Against the other 2023 shifts

This shift next to the 11 other 2023 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
6 Apr, Shift 1	90	31%	11%	33%	30%	27%
6 Apr, Shift 2	89	33%	17%	34%	25%	20%
8 Apr, Shift 1	89	38%	9%	34%	29%	21%
8 Apr, Shift 2	89	42%	16%	34%	23%	30%
10 Apr, Shift 1	88	39%	9%	33%	12%	25%
10 Apr, Shift 2	90	38%	12%	33%	20%	21%
11 Apr, Shift 1	90	34%	16%	33%	25%	33%
11 Apr, Shift 2	88	24%	18%	34%	28%	39%
12 Apr, Shift 1	89	30%	24%	34%	29%	35%
13 Apr, Shift 1	89	24%	19%	34%	21%	37%
13 Apr, Shift 2	90	33%	13%	33%	35%	32%
15 Apr, Shift 1	90	43%	14%	33%	18%	30%

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
Open29 this shift3084 · 33%33%8/830298716720359538199224
Open29 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 88 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

88 of 88 questions

Physics29 questions

9 easy18 medium2 hard

Q31
A car P travelling at $20\ \mathrm{ms^{-1}}$ sounds its horn at a frequency of $400\ \mathrm{Hz}$ . Another car Q is travelling behind the first car in the same direction with a velocity $40\ \mathrm{ms^{-1}}$ . The frequency heard by the passenger of the car…
Waves Doppler Effect Moving Source Cases
MediumSingle correctIdea asked in 2 yearsAsked the year before
Q32
The root mean square speed of molecules of nitrogen gas at $27^\circ\mathrm{C}$ is approximately : (Given mass of a nitrogen molecule = $4.6\times10^{-26}\ \mathrm{kg}$ and take Boltzmann constant $\mathrm{k_B} = 1.4\times10^{-23}\ \mathrm{JK^{-1}}$ )
Kinetic Theory Kinetic Theory of Gases Molecular Speed Types
EasySingle correctIdea asked in 3 yearsAsked the year before
Q33
A space ship of mass $2\times10^{4}\ \mathrm{kg}$ is launched into a circular orbit close to the earth surface. The additional velocity to be imparted to the space ship in the orbit to overcome the gravitational pull will be (if $g = 10\ \mathrm{m/s^2}$ and…
Gravitation Orbital Mechanics Escape from an Orbiting State
MediumSingle correctNew idea
Q34
The Thermodynamic process, in which internal energy of the system remains constant is
Thermodynamics Thermodynamic Processes Isothermal Process Analysis
EasySingle correctIdea asked in 3 years
Q35
A body of mass $500\ \mathrm{g}$ moves along x-axis such that it's velocity varies with displacement x according to the relation $v = 10\sqrt{x}\ \mathrm{m/s}$ the force acting on the body is:-
Laws of Motion Newton's Laws of Motion Position-Dependent Force Integration
MediumSingle correctNew idea
Q36
Eight equal drops of water are falling through air with a steady speed of $10\ \mathrm{cm/s}$ . If the drops coalesce, the new velocity is:-
Mechanical Properties of Fluids Viscosity Terminal Velocity of Coalescing Drops
MediumSingle correctNew ideaIdea asked in 2 years
Q37
If V is the gravitational potential due to sphere of uniform density on it's surface, then it's value at the center of sphere will be:-
Gravitation Gravitational Potential Potential of Solid Sphere
MediumSingle correctNew idea
Q39
A projectile is projected at $30^\circ$ from horizontal with initial velocity $40\ \mathrm{ms^{-1}}$ . The velocity of the projectile at t = 2 s from the start will be : (Given $g = 10\ \mathrm{m/s^2}$ )
Motion in a Plane Projectile Motion Instantaneous Projectile Velocity
EasySingle correctIdea asked in 3 yearsAsked the year before
Q40
If force (F), velocity (V) and time (T) are considered as fundamental physical quantity, then dimensional formula of density will be :
Units and Measurements
MediumSingle correct
Q41
In satellite communication, the uplink frequency band used is :
Communication Systems
EasySingle correct
Q42
The logic operations performed by the given digital circuit 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
Q43
When one light ray is reflected from a plane mirror with $30^\circ$ angle of reflection, the angle of deviation of the ray after reflection is:
Ray Optics and Optical Instruments
EasySingle correct
Q44
The energy of $\mathrm{He^+}$ ion in its first excited state is, (The ground state energy for the Hydrogen atom is $-13.6\ \mathrm{eV}$ ):
Atoms Bohr's Atomic Model Energy Level Calculations
EasySingle correctIdea asked in 6 yearsAsked the year before
Q45
The ratio of the de-Broglie wavelengths of proton and electron having same Kinetic energy : (Assume $\mathrm{m_p = m_e\times1849}$ )
Dual Nature of Radiation and Matter Matter Waves Wavelength Ratios of Different Particles
EasySingle correctIdea asked in 6 yearsAsked the year before
Q46
A plane electromagnetic wave of frequency $20\ \mathrm{MHz}$ propagates in free space along x-direction. At a particular space and time, $\vec{E} = 6.6\,\hat{j}\ \mathrm{V/m}$ . What is $\vec{B}$ at this point?
Electromagnetic Waves
MediumSingle correct
Q47
Given below are two statements: one is labelled as Assertion A and the other is labelled as Reason R Assertion A: A bar magnet dropped through a metallic cylindrical pipe takes more time to come down compared to a non-magnetic bar with same geometry and…
Electromagnetic Induction Lenz's Law Magnet Falling in a Conducting Tube
EasyAssertion–reasonNew idea
Q48
An electron is allowed to move with constant velocity along the axis of current carrying straight solenoid. A. The electron will experience magnetic force along the axis of the solenoid. B. The electron will not experience magnetic force. C. The electron…
Moving Charges and Magnetism Lorentz Force Helical Motion and Pitch
MediumOtherIdea asked in 4 years
Q49
The current flowing through $\mathrm{R_2}$ is :
Current Electricity Kirchhoff's Laws Current Divider Rule
MediumDiagram based · figureNew idea
Q50
A capacitor of capacitance C is charged to a potential V. The flux of the electric field through a closed surface enclosing the positive plate of the capacitor is :
Electric Charges and Fields Gauss's Law Gauss Law Application
MediumSingle correctNew ideaIdea asked in 4 years
Q51
A wire of density $8\times10^{3}\ \mathrm{kg/m^3}$ is stretched between two clamps $0.5\ \mathrm{m}$ apart. The extension developed in the wire is $3.2\times10^{-4}\ \mathrm{m}$ . If $\mathrm{Y} = 8\times10^{10}\ \mathrm{N/m^2}$ , the fundamental frequency of…
Waves Waves on Strings Wave Speed on String
MediumNumerical valueIdea asked in 5 yearsAsked the year before
Q52
The surface tension of soap solution is $3.5\times10^{-2}\ \mathrm{Nm^{-1}}$ . The amount of work done required to increase the radius of soap bubble from $10\ \mathrm{cm}$ to $20\ \mathrm{cm}$ is ____________ $\times10^{-4}\ \mathrm{J}$ . (take $\pi = 22/7$ )
Mechanical Properties of Fluids Surface Tension Surface Energy and Work
MediumNumerical valueNew ideaIdea asked in 2 years
Q53
A circular plate is rotating in horizontal plane, about an axis passing through its center and perpendicular to the plate, with an angular velocity $\omega$ . A person sits at the center having two dumbbells in his hands. When he stretches out his hands, the…
System of Particles and Rotational Motion Angular Momentum Conservation of Angular Momentum
MediumNumerical valueIdea asked in 5 yearsAsked the year before
Q54
A block of mass $5\ \mathrm{kg}$ starting from rest pulled up on a smooth incline plane making an angle of $30^\circ$ with horizontal with an affective acceleration of $1\ \mathrm{ms^{-2}}$ . The power delivered by the pulling force at t = 10 s from the start…
Work, Energy and Power Power Instantaneous Power
MediumNumerical valueIdea asked in 2 years
Q55
A nucleus disintegrates into two nuclear parts, in such a way that ratio of their nuclear sizes is $1 : 2^{1/3}$ . Their respective speed have a ratio of n : 1. The value of n is ____________
Nuclei Nuclear Binding Energy Nuclear Radius and Density
MediumNumerical valueIdea asked in 5 yearsAsked the year before
Q56
As shown in the figure, a plane mirror is fixed at a height of $50\ \mathrm{cm}$ from the bottom of tank containing water $\left(\mu = \frac{4}{3}\right)$ . The height of water in the tank is $8\ \mathrm{cm}$ . A small bulb is placed at the bottom of the water…
Ray Optics and Optical Instruments Refraction and Snell's Law Apparent Depth and Shift
HardNumerical value · figureIdea asked in 3 yearsAsked the year before
Q57
A coil has an inductance of $2\mathrm{H}$ and resistance of $4\ \Omega$ . A $10\ \mathrm{V}$ is applied across the coil. The energy stored in the magnetic field after the current has built up to its equilibrium value will be ____________…
Electromagnetic Induction Inductance Energy Stored in Inductors
EasyNumerical valueIdea asked in 5 yearsAsked the year before
Q58
A metallic cube of side $15\ \mathrm{cm}$ moving along y-axis at a uniform velocity of $2\ \mathrm{ms^{-1}}$ . In a region of uniform magnetic field of magnitude $0.5\mathrm{T}$ directed along z-axis. In equilibrium the potential difference between the faces…
Electromagnetic Induction Faraday's Law Linear Motional EMF
MediumNumerical value · figureIdea asked in 3 yearsAsked the year before
Q59
Two identical cells each of emf $1.5\ \mathrm{V}$ are connected in series across a $10\ \Omega$ resistance. An ideal voltmeter connected across $10\ \Omega$ resistance reads $1.5\ \mathrm{V}$ . The internal resistance of each cell is ____________ $\Omega$ .
Current Electricity Kirchhoff's Laws Cell EMF and Internal Resistance
MediumNumerical valueIdea asked in 6 yearsAsked the year before
Q60
In the given circuit, $\mathrm{C_1 = 2\ \mu F, C_2 = 0.2\ \mu F, C_3 = 2\ \mu F, C_4 = 4\ \mu F, C_5 = 2\ \mu F, C_6 = 2\ \mu F}$ , The charge stored on capacitor $\mathrm{C_4}$ is ____________ $\mu\mathrm{C}$ .
Electrostatic Potential and Capacitance Parallel Plate Capacitors Series and Parallel Combinations
HardNumerical value · figureIdea asked in 6 yearsAsked the year before

Chemistry30 questions

10 easy18 medium2 hard

Q61
Which one of the following pairs is an example of polar molecular solids?
The Solid State
EasySingle correct
Q62
What weight of glucose must be dissolved in 100 g of water to lower the vapour pressure by 0.20 mm Hg ? (Assume dilute solution is being formed) Given : Vapour pressure of pure water is 54.2 mm Hg at room temperature. Molar mass of glucose is 180 g mol $^{-1}$
Solutions Colligative Properties Relative Lowering of VP
MediumSingle correctIdea asked in 7 yearsAsked the year before
Q63
For a chemical reaction A + B $\to$ Product, the order is 1 with respect to A and B. | Rate $\mathrm{mol\ L^{-1}\ S^{-1}}$ | [A] $\mathrm{mol\ L^{-1}}$ | [B] $\mathrm{mol\ L^{-1}}$ | | 0.10 | 20 | 0.5 | | 0.40 | $x$ | 0.5 | | 0.80 | 40 | $y$ | What is the…
Chemical Kinetics Rate Laws Initial Rates Method
MediumSingle correct · figureIdea asked in 5 yearsAsked the year before
Q64
A solution is prepared by adding 2 g of “X” to 1 mole of water. Mass percent of “X” in the solution is
Solutions Mole Concept PPM and Mass Percentage
EasySingle correctIdea asked in 5 yearsAsked the year before
Q65
Given below are two statements : Statement I : In the metallurgy process, sulphide ore is converted to oxide before reduction. Statement II : Oxide ores in general are easier to reduce. In the light of the above statements, choose the most appropriate answer…
General Principles and Processes of Isolation of Elements Metallurgical Extraction Reducing Agent Selection
EasyTwo statementsNew idea
Q66
Which hydride among the following is less stable?
Hydrogen Hydrogen and Hydrides Polymeric s-Block Hydrides
EasySingle correctNew idea
Q67
One mole of $\mathrm{P_4}$ reacts with 8 moles of $\mathrm{SOCl_2}$ to give 4 moles of A, $x$ mole of $\mathrm{SO_2}$ and 2 moles of B. A, B and $x$ respectively are
The p-Block Elements
MediumSingle correct
Q68
Alkali metal from the following with least melting point is:
The s-Block Elements s-Block Elements Melting and Boiling Point Trends
EasySingle correctIdea asked in 2 yearsAsked the year before
Q69
The magnetic moment is measured in Bohr Magneton (BM). Spin only magnetic moment of Fe in $\mathrm{[Fe(H_2O)_6]^{3+}}$ and $\mathrm{[Fe(CN)_6]^{3-}}$ complexes respectively is:
Coordination Compounds Crystal Field Theory Spin-only Magnetic Moment
EasySingle correctIdea asked in 7 yearsAsked the year before
Q70
Given below are two statements, one is labelled as Assertion A and the other is labelled as Reason R. Assertion A : $\mathrm{[CoCl(NH_3)_5]^{2+}}$ absorbs at lower wavelength of light with respect to $\mathrm{[Co(NH_3)_5\ (H_2O)]^{3+}}$ Reason R : It is…
Coordination Compounds Crystal Field Theory Color and d-d Transitions
MediumAssertion–reasonIdea asked in 5 yearsAsked the year before
Q71
Which of the following compounds is an example of Freon?
Haloalkanes and Haloarenes Environmental Chemistry Ozone Layer Depletion
EasySingle correctIdea asked in 4 yearsAsked the year before
Q72
If $\mathrm{Ni^{2+}}$ is replaced by $\mathrm{Pt^{2+}}$ in the complex $\mathrm{[NiCl_2Br_2]^{2-}}$ , which of the following properties are expected to get changed ? A. Geometry B. Geometrical isomerism C. Optical isomerism D. Magnetic properties
Coordination Compounds Coordination Isomerism Square Planar Geometry
MediumSingle correctIdea asked in 3 years
Q73
Match List I with List II LIST I (Complex) | LIST II (Colour) A. $\mathrm{Mg(NH_4)PO_4}$ | I. brown B. $\mathrm{K_3[Co(NO_2)_6]}$ | II. white C. $\mathrm{MnO(OH)_2}$ | III. yellow D. $\mathrm{Fe_4[Fe(CN)_6]_3}$ | IV. blue Choose the correct answer from the…
Coordination Compounds Cation Analysis Confirmatory Tests
MediumMatch the listIdea asked in 4 yearsAsked the year before
Q74
Compound from the following that will not produce precipitate on reaction with $\mathrm{AgNO_3}$ is :
Haloalkanes and Haloarenes SN1 and SN2 Mechanisms Inertness of Vinylic and Aryl Halides
MediumDiagram based · figureNew idea
Q75
[reaction scheme: $\mathrm{H_3C-CH_2-CH(OH)-CH_3}$ (butan-2-ol) $\xrightarrow{\text{(i) NaI, } \mathrm{H_3PO_4};\ \text{(ii) Mg, Dry ether};\ \text{(iii) } \mathrm{D_2O}}$ [X] Product] Product [X] formed in the above reaction is :
Haloalkanes and Haloarenes
MediumOther · figure
Q76
The major product formed in the following reaction is $\mathrm{C_6H_5-CH(OH)-CH(CH_3)-CH_2CHO}$ $\xrightarrow[\Delta]{\mathrm{Zn(Hg)/HCl}}$ Major products A. $\mathrm{C_6H_5-CH(OH)-CH(CH_3)-C_2H_5}$ B. $\mathrm{C_6H_5-CH=C(CH_3)-C_2H_5}$ C.…
Aldehydes, Ketones and Carboxylic Acids Alkanes and Free Radicals Clemmensen Reduction to Alkanes
MediumOther · figureIdea asked in 3 yearsAsked the year before
Q77
Given below are two statements, one is labelled as Assertion A and the other is labelled as Reason R. Assertion A : [figure: skeletal ketone, a chain with a methyl branch, a carbon bearing Cl and a C=O group, i.e. (CH3)2CH-CH(Cl)-C(=O)-CH3] can be subjected…
Aldehydes, Ketones and Carboxylic Acids Alkanes and Free Radicals Wolff-Kishner Reduction to Alkanes
MediumAssertion–reason · figureIdea asked in 3 years
Q78
Compound ‘B’ is [reaction scheme: a benzene ring bearing CH3 (top), OH (lower right) and isopropyl (bottom) $\xrightarrow[\mathrm{HCl}]{\mathrm{NaNO_2}}$ A $\xrightarrow[\mathrm{H_2O}]{\mathrm{NH_4SH}}$ B (Major)]
Amines
HardDiagram based · figure
Q79
Given below are two statements, one is labelled as Assertion A and the other is labelled as Reason R. Assertion A : A solution of the product obtained by heating a mole of glycine with a mole of chlorine in presence of red phosphorous generates chiral carbon…
Haloalkanes and Haloarenes Stereoisomerism Chiral Centers
MediumAssertion–reasonIdea asked in 5 yearsAsked the year before
Q80
Given below are two statements : Statement I : Ethene at 333 to 343K and 6-7 atm pressure in the presence of $\mathrm{AlEt_3}$ and $\mathrm{TiCl_4}$ undergoes addition polymerization to give LDP. Statement II : Caprolactam at 533-543K in $\mathrm{H_2O}$ …
Polymers HDPE vs LDPE and Ziegler-Natta Catalyst
MediumTwo statementsNew idea
Q81
The volume of hydrogen liberated at STP by treating 2.4 g of magnesium with excess of hydrochloric acid is __________ $\times 10^{-2}$ L Given : Molar volume of gas is 22.4 L at STP. Molar mass of magnesium is 24 g mol $^{-1}$
Some Basic Concepts of Chemistry Stoichiometry Mass-Volume Stoichiometry at STP
EasyNumerical valueIdea asked in 3 yearsAsked the year before
Q82
The maximum number of lone pairs of electrons on the central atom from the following species is ____________ $\mathrm{ClO_3^-}$ , $\mathrm{XeF_4}$ , $\mathrm{SF_4}$ and $\mathrm{I_3^-}$
Chemical Bonding and Molecular Structure VSEPR Theory Octahedral and Square Planar
MediumNumerical valueIdea asked in 3 years
Q83
The number of correct statements from the following is __________. A. For 1s orbital, the probability density is maximum at the nucleus B. For 2s orbital, the probability density first increases to maximum and then decreases sharply to zero. C. Boundary…
Structure of Atom Quantum Numbers Interpreting Radial Probability Curves
MediumNumerical valueNew idea
Q84
The total number of intensive properties from the following is ____________ Volume, Molar heat capacity, Molarity, $\mathrm{E^{\Theta}}$ cell, Gibbs free energy change, Molar mass, Mole
Thermodynamics (Chemistry)Enthalpy and Hess's Law Extensive Nature of Enthalpy
EasyNumerical valueNew idea
Q85
4.5 moles each of hydrogen and iodine is heated in a sealed ten litre vessel. At equilibrium, 3 moles of HI were found. The equilibrium constant for $\mathrm{H_2(g) + I_2(g) \rightleftharpoons 2HI\ (g)}$ is ____________
Equilibrium Chemical Equilibrium ICE Table Construction
EasyNumerical valueIdea asked in 3 yearsAsked the year before
Q86
The number of correct statements from the following is _______ A. $\mathrm{E_{cell}}$ is an intensive parameter B. A negative $\mathrm{E^{\Theta}}$ means that the redox couple is a stronger reducing agent than the $\mathrm{H^+/H_2}$ couple. C. The amount of…
Electrochemistry Nernst Equation Standard Cell Potential
MediumNumerical valueIdea asked in 7 yearsAsked the year before
Q87
The number of correct statements about modern adsorption theory of heterogeneous catalysis from the following is ________ A. The catalyst is diffused over the surface of reactants. B. Reactants are adsorbed on the surface of the catalyst. C. Occurrence of…
Surface Chemistry Adsorption Adsorption Theory of Heterogeneous Catalysis
MediumNumerical valueIdea asked in 2 yearsAsked the year before
Q88
$\mathrm{Mg(NO_3)_2{\cdot}XH_2O}$ and $\mathrm{Ba(NO_3)_2{\cdot}YH_2O}$ , represent formula of the crystalline forms of nitrate salts. Sum of X and Y is ____________
The s-Block Elements s-Block Elements Hydration Enthalpy Trends
MediumNumerical valueIdea asked in 3 yearsAsked the year before
Q89
Number of compounds from the following which will not produce orange red precipitate with Benedict solution is ____________ Glucose, maltose, sucrose, ribose, 2-deoxyribose, amylose, lactose
Biomolecules Carbohydrates Reducing vs Non-reducing Sugars
MediumNumerical valueIdea asked in 4 yearsAsked the year before
Q90
The number of possible isomeric products formed when 3-chloro-1-butene reacts with HCl through carbocation formation is ____________
Haloalkanes and Haloarenes SN1 and SN2 Mechanisms Carbocation Rearrangements in SN1
HardNumerical valueNew ideaIdea asked in 3 years

Maths29 questions

2 easy15 medium12 hard

Q1
Let $A = \{1, 3, 4, 6, 9\}$ and $B = \{2, 4, 5, 8, 10\}$ . Let $R$ be a relation defined on $A \times B$ such that $R = \{((a_1, b_1), (a_2, b_2)) : a_1 \le b_2$ and $b_1 \le a_2\}$ . Then the number of elements in the set $R$ is
Relations and Functions Sets and Relations Cartesian Products
MediumSingle correctNew idea
Q2
The domain of the function $f(x) = \frac{1}{\sqrt{[x]^2 - 3[x] - 10}}$ is (where $[x]$ denotes the greatest integer less than or equal to $x$ )
Relations and Functions Functions and Domain Domain Constraints of Greatest Integer Function
MediumSingle correctIdea asked in 2 years
Q3
For $a \in \mathbb{C}$ , let $A = \{z \in \mathbb{C} : \mathrm{Re}(a + \overline{z}) > \mathrm{Im}(\overline{a} + z)\}$ and $B = \{z \in \mathbb{C} : \mathrm{Re}(a + \overline{z}) < \mathrm{Im}(\overline{a} + z)\}$ . Then among the two statements: (S1) : If…
Complex Numbers and Quadratic Equations
MediumTwo statements
Q4
If the system of linear equations $7x + 11y + \alpha z = 13$ $5x + 4y + 7z = \beta$ $175x + 194y + 57z = 361$ has infinitely many solutions, then $\alpha + \beta + 2$ is equal to:
Determinants Cramer's Rule Evaluating Parameters for Consistency
MediumSingle correctIdea asked in 6 yearsAsked the year before
Q5
If $\begin{vmatrix} x+1 & x & x \\ x & x+\lambda & x \\ x & x & x+\lambda^2 \end{vmatrix} = \frac{9}{8}(103x + 81)$ , then $\lambda, \frac{\lambda}{3}$ are the roots of the equation
Determinants
MediumSingle correct
Q6
If the letters of the word MATHS are permuted and all possible words so formed are arranged as in a dictionary with serial numbers, then the serial number of the word THAMS is
Permutations and Combinations Lexicographic Rank of a Word
EasySingle correctIdea asked in 4 yearsAsked the year before
Q8
The sum of the coefficients of three consecutive terms in the binomial expansion of $(1 + x)^{n+2}$ , which are in the ratio $1:3:5$ , is equal to
Binomial Theorem Binomial Coefficients Ratio of Consecutive Coefficients
MediumSingle correctIdea asked in 5 yearsAsked the year before
Q9
Let $a, b, c$ and $d$ be positive real numbers such that $a + b + c + d = 11$ . If the maximum value of $a^5 b^3 c^2 d$ is $3750\beta$ , then the value of $\beta$ is
Sequences and Series AP and GP Weighted AM-GM Inequality
MediumSingle correctIdea asked in 2 yearsAsked the year before
Q10
Let $f$ and $g$ be two functions defined by $f(x) = \begin{cases} x+1, & x<0 \\ |x-1|, & x \ge 0 \end{cases}$ and $g(x) = \begin{cases} x+1, & x<0 \\ 1, & x \ge 0 \end{cases}$ Then $(g \circ f)(x)$ is
Continuity and Differentiability Continuity and Differentiability of Composite Functions
MediumSingle correctIdea asked in 5 yearsAsked the year before
Q11
Let the function $f : [0, 2] \to \mathbb{R}$ be defined as $f(x) = \begin{cases} e^{\min\{x^2,\, x - [x]\}}, & x \in [0, 1) \\ e^{[x - \log_e x]}, & x \in [1, 2] \end{cases}$ where $[t]$ denotes the greatest integer less than or equal to $t$ . Then the value…
Integrals Definite Integral Properties Greatest Integer Function Integrals
HardSingle correctIdea asked in 7 yearsAsked the year before
Q12
If $f : \mathbb{R} \to \mathbb{R}$ be a continuous function satisfying $\int_0^{\pi/2} f(\sin 2x)\sin x\,dx + \alpha \int_0^{\pi/4} f(\cos 2x)\cos x\,dx = 0$ , then the value of $\alpha$ is
Integrals
HardSingle correct
Q13
Let $y = y(x)$ be the solution of the differential equation $\frac{dy}{dx} + \frac{5}{x(x^5+1)}y = \frac{(x^5+1)^2}{x^7}$ , $x > 0$ . If $y(1) = 2$ , then $y(2)$ is equal to
Differential Equations Linear Differential Equations Integrating Factor Calculation
HardSingle correctIdea asked in 7 yearsAsked the year before
Q14
If the radius of the largest circle with centre $(2,0)$ inscribed in the ellipse $x^2 + 4y^2 = 36$ is $r$ , then $12r^2$ is equal to
Conic Sections
HardSingle correct
Q15
Let $P$ be the plane passing through the points $(5,3,0)$ , $(13,3,-2)$ and $(1,6,2)$ . For $\alpha \in \mathbb{N}$ , if the distances of the points $A(3, 4, \alpha)$ and $B(2, \alpha, a)$ from the plane $P$ are $2$ and $3$ respectively, then the positive value…
Three Dimensional Geometry Point-Plane Distance Distance to Plane Through Three Points
MediumSingle correctNew idea
Q16
Let the line passing through the points $P(2, -1, 2)$ and $Q(5, 3, 4)$ meet the plane $x - y + z = 4$ at the point $R$ . Then the distance of the point $R$ from the plane $x + 2y + 3z + 2 = 0$ measured parallel to the line…
Three Dimensional Geometry Point-Plane Distance Distance Along a Line
HardSingle correctIdea asked in 5 yearsAsked the year before
Q17
If four distinct points with position vectors $\vec{a}, \vec{b}, \vec{c}$ and $\vec{d}$ are coplanar, then $[\vec{a}\ \vec{b}\ \vec{c}]$ is equal to
Vector Algebra Triple Products Coplanarity of Four Points
MediumSingle correctIdea asked in 3 yearsAsked the year before
Q18
Let the mean of $6$ observations $1, 2, 4, 5, x$ and $y$ be $5$ and their variance be $10$ . Then their mean deviation about the mean is equal to
Statistics Variance and SD Finding Unknown Observations from Mean and Variance
MediumSingle correctIdea asked in 7 yearsAsked the year before
Q19
The angle of elevation of the top $P$ of a tower from the feet of one person standing due South of the tower is $45^\circ$ and from the feet of another person standing due west of the tower is $30^\circ$ . If the height of the tower is $5$ meters, then the…
Trigonometric Functions
MediumSingle correct
Q20
The converse of $((\sim p) \wedge q) \Rightarrow r$ is
Mathematical Reasoning
EasySingle correct
Q21
Let $S = \left\{z \in \mathbb{C} - \{i, 2i\} : \frac{z^2 + 8iz - 15}{z^2 - 3iz - 2} \in \mathbb{R}\right\}$ . If $\alpha - \frac{13}{11}i \in S$ , $\alpha \in \mathbb{R} - \{0\}$ , then $242\alpha^2$ is equal to ________.
Complex Numbers and Quadratic Equations Complex Numbers - Polar Form Purely Real and Imaginary Euler Conditions
HardNumerical valueNew ideaIdea asked in 3 years
Q22
Let $A = \{1, 2, 3, 4, 5\}$ and $B = \{1, 2, 3, 4, 5, 6\}$ . Then the number of functions $f : A \to B$ satisfying $f(1) + f(2) = f(4) - 1$ is equal to ________.
Permutations and Combinations Total Number of Functions
MediumNumerical valueIdea asked in 4 yearsAsked the year before
Q23
For $k \in \mathbb{N}$ , if the sum of the series $1 + \frac{4}{k} + \frac{8}{k^2} + \frac{13}{k^3} + \frac{19}{k^4} + \ldots$ is $10$ , then the value of $k$ is ________
Sequences and Series Arithmetico-Geometric Series Second-Order AGP Summation
HardNumerical valueIdea asked in 3 yearsAsked the year before
Q24
The number of points, where the curve $f(x) = e^{8x} - e^{6x} - 3e^{4x} - e^{2x} + 1$ , $x \in \mathbb{R}$ cuts $x$ -axis, is equal to ________
Application of Derivatives Curve Sketching Root Counting via Intersection Sketching
MediumNumerical valueNew ideaIdea asked in 2 years
Q25
If $A$ is the area in the first quadrant enclosed by the curve $C : 2x^2 - y + 1 = 0$ , the tangent to $C$ at the point $(1, 3)$ and the line $x + y = 1$ , then the value of $60A$ is ________.
Application of Integrals Area Under Curves Area Involving Tangents and Normals
MediumNumerical valueIdea asked in 3 years
Q26
If the line $l_1 : 3y - 2x = 3$ is the angular bisector of the lines $l_2 : x - y + 1 = 0$ and $l_3 : \alpha x + \beta y + 17 = 0$ , then $\alpha^2 + \beta^2 - \alpha - \beta$ is equal to ________.
Straight Lines Straight Line Distance Angle Bisector Locus via Distances
HardNumerical valueIdea asked in 3 years
Q27
Let the tangent to the parabola $y^2 = 12x$ at the point $(3, \alpha)$ be perpendicular to the line $2x + 2y = 3$ . Then the square of distance of the point $(6, -4)$ from the normal to the hyperbola $\alpha^2 x^2 - 9y^2 = 9\alpha^2$ at its point…
Conic Sections Normals to Conics Hyperbola Normal Point Form
HardNumerical valueIdea asked in 3 yearsAsked the year before
Q28
Let the line $l : x = \frac{1-y}{-2} = \frac{z-3}{\lambda}$ , $\lambda \in \mathbb{R}$ meet the plane $P : x + 2y + 3z = 4$ at the point $(\alpha, \beta, \gamma)$ . If the angle between the line $l$ and the plane $P$ is…
Three Dimensional Geometry Angle Between Lines Angle: Line and Plane
HardNumerical valueNew idea
Q29
Let $\vec{a} = \hat{i} + 2\hat{j} + 3\hat{k}$ and $\vec{b} = \hat{i} + \hat{j} - \hat{k}$ . If $\vec{c}$ is a vector such that $\vec{a} \cdot \vec{c} = 11$ , $\vec{b} \cdot (\vec{a} \times \vec{c}) = 27$ and $\vec{b} \cdot \vec{c} = -\sqrt{3}|\vec{b}|$ , then…
Vector Algebra Cross Product Magnitude and Sine Formula
HardNumerical valueIdea asked in 7 yearsAsked the year before
Q30
Let the probability of getting head for a biased coin be $\frac{1}{4}$ . It is tossed repeatedly until a head appears. Let $N$ be the number of tosses required. If the probability that the equation $64x^2 + 5Nx + 1 = 0$ has no real root is $\frac{p}{q}$ ,…
Probability Conditional Probability Conditional Probability in Infinite Games
HardNumerical valueNew ideaIdea asked in 2 years

Dropped from the final key and left out of every count: Q7, Q38.

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 2023 shifts.