JEE Main 2024 · January session

JEE Main 31 January 2024, Shift 2

89 questions (1 more dropped by NTA): 30 numerical-answer and 59 multiple choice. 12 were rated hard and 25 easy. Its hard-question count was the 5th highest of the 18 2024 shifts Lumi analysed. The biggest chapter was Amines, with 3 questions.

Questions: 89; 1 dropped by NTA
Numerical answer: 30; 34% of the shift
Hard: 12; 13% of the shift
Chapters asked: 61; 14 questions with a figure

Key findings · 31 Jan 2024, Shift 2

1
12 hard questions (13%): the 5th highest count of the 18 2024 shifts Lumi analysed. Across those shifts the hard share ran from 8% to 20%.
2
Maths was the hardest section, with 10 of 30 rated hard; Physics had the most easy questions (13 of 29).
3
17 questions tested a new idea, one no earlier year had asked: 23% of the 74 questions tagged to an idea.
4
30 came back from 2023: the same idea had been asked in the 2023 shifts Lumi analysed.
5
47 shared an idea with another 2024 shift (64% of tagged questions). Shifts in the same year can test the same idea in a new setting.
6
Biggest chapters: Current Electricity in Physics (3), Amines in Chemistry (3) and Three Dimensional Geometry in Maths (3).

The shift at a glance

Subjects, difficulty and chapters

Physics

29 Q · 10 numerical

Easy 13Medium 16Hard 0

Top chapters

Chemistry

30 Q · 10 numerical

Easy 9Medium 19Hard 2

Top chapters

Maths

30 Q · 10 numerical

Easy 3Medium 17Hard 10

Top chapters

Question format

Single correct44 Q
Numerical value30 Q
Diagram based7 Q
Two statements4 Q
Other3 Q
Match the list1 Q

What each question asks you to do

Multi-step27 Q
Calculation27 Q
Application15 Q
Conceptual11 Q
Recall7 Q
Diagram reading1 Q
Data or graph1 Q

Most common: Multi-step.

Against the other 2024 shifts

This shift next to the 17 other 2024 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
27 Jan, Shift 1	90	50%	9%	33%	29%	29%
29 Jan, Shift 1	89	38%	10%	34%	28%	34%
30 Jan, Shift 1	90	37%	11%	33%	22%	17%
30 Jan, Shift 2	89	42%	10%	34%	32%	29%
31 Jan, Shift 1	90	39%	12%	33%	31%	34%
31 Jan, Shift 2	89	28%	13%	34%	23%	30%
1 Feb, Shift 1	90	36%	11%	33%	21%	19%
1 Feb, Shift 2	88	40%	8%	34%	19%	21%
4 Apr, Shift 1	89	26%	20%	33%	26%	36%
4 Apr, Shift 2	90	33%	18%	33%	13%	26%
5 Apr, Shift 1	90	40%	8%	33%	22%	27%
5 Apr, Shift 2	90	37%	10%	33%	23%	32%
6 Apr, Shift 1	88	43%	11%	34%	18%	22%
6 Apr, Shift 2	89	35%	15%	33%	28%	33%
8 Apr, Shift 1	90	31%	9%	33%	19%	37%
8 Apr, Shift 2	90	33%	12%	33%	19%	28%
9 Apr, Shift 1	90	31%	11%	33%	20%	50%
9 Apr, Shift 2	90	31%	16%	33%	27%	39%

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

Drill down

What this shift asked, and its history

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

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

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

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

Every question

All 89 questions of this shift

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

Subject

Difficulty

Idea

Find

89 of 89 questions

Physics29 questions

13 easy16 medium0 hard

Q31
Consider two physical quantities $A$ and $B$ related to each other as $E=\frac{B-x^2}{At}$ where $E$ , $x$ and $t$ have dimensions of energy, length and time respectively. The dimension of $AB$ is
Units and Measurements
MediumSingle correct
Q32
If two vectors $\vec{A}$ and $\vec{B}$ having equal magnitude $R$ are inclined at an angle $\theta$ , then
Motion in a Plane
EasySingle correct
Q33
A light string passing over a smooth light fixed pulley connects two blocks of masses $m_1$ and $m_2$ . If the acceleration of the system is $g/8$ , then the ratio of masses is:
Laws of Motion Newton's Laws of Motion Standard Atwood Machine
EasySingle correct · figureIdea asked in 3 years
Q35
A body of mass 2 kg begins to move under the action of a time dependent force given by $\vec{F}=\left(6t\,\hat{i}+6t^2\,\hat{j}\right)$ N. The power developed by the force at the time $t$ is given by:
Work, Energy and Power Power 3D Vector Calculation of Power
MediumSingle correctNew ideaIdea asked in 3 years
Q36
The mass of the moon is $\frac{1}{144}$ times the mass of a planet and its diameter is $\frac{1}{16}$ times the diameter of a planet. If the escape velocity on the planet is v, the escape velocity on the moon will be :
Gravitation Orbital Mechanics Escape Velocity
EasySingle correctIdea asked in 5 yearsAsked the year before
Q37
A small spherical ball of radius $r$ , falling through a viscous medium of negligible density has terminal velocity 'v'. Another ball of the same mass but of radius $2r$ , falling through the same viscous medium will have terminal velocity:
Mechanical Properties of Fluids Viscosity Radius Dependence of Terminal Velocity
MediumSingle correctIdea asked in 2 years
Q38
The speed of sound in oxygen at S.T.P. will be approximately: (given, $R=8.3\ \mathrm{JK^{-1}}$ , $\gamma=1.4$ )
Waves Sound Velocity Laplace Correction for Sound in Gases
MediumSingle correctNew idea
Q39
A gas mixture consists of 8 moles of argon and 6 moles of oxygen at temperature T. Neglecting all vibrational modes, the total internal energy of the system is:
Thermodynamics Kinetic Theory of Gases Internal Energy of Gas Mixtures
EasySingle correctIdea asked in 5 yearsAsked the year before
Q40
Force between two point charges $q_1$ and $q_2$ placed in vacuum at 'r' cm apart is $F$ . Force between them when placed in a medium having dielectric constant $K=5$ at 'r/5' cm apart will be:
Electric Charges and Fields Coulomb's Law Medium Dielectric Effect
EasySingle correctNew idea
Q41
By what percentage will the illumination of the lamp decrease if the current drops by 20%?
Current Electricity Ohm's Law and Resistance Joule Heating and Power
EasySingle correctIdea asked in 6 yearsAsked the year before
Q42
A uniform magnetic field of $2\times10^{-3}$ T acts along positive $Y$ -direction. A rectangular loop of sides 20 cm and 10 cm with current of 5 A is in $Y$ - $Z$ plane. The current is in anticlockwise sense with reference to negative $X$ axis. Magnitude and…
Moving Charges and Magnetism Magnetic Dipole Moment Torque on a dipole in uniform field
MediumSingle correctIdea asked in 4 years
Q43
An AC voltage $V=20\sin 200\pi t$ is applied to a series LCR circuit which drives a current $I=10\sin\left(200\pi t+\frac{\pi}{3}\right)$ . The average power dissipated is:
Alternating Current Phasor Diagrams Power Factor from Phasors
EasySingle correctIdea asked in 5 years
Q44
Given below are two statements: Statement I: Electromagnetic waves carry energy as they travel through space and this energy is equally shared by the electric and magnetic fields. Statement II: When electromagnetic waves strike a surface, a pressure is…
Electromagnetic Waves
EasyTwo statements
Q45
When unpolarized light is incident at an angle of $60^\circ$ on a transparent medium from air, the reflected ray is completely polarized. The angle of refraction in the medium is:
Wave Optics Polarization Brewster's Angle
EasySingle correctIdea asked in 3 years
Q46
In a photoelectric effect experiment a light of frequency 1.5 times the threshold frequency is made to fall on the surface of photosensitive material. Now if the frequency is halved and intensity is doubled, the number of photo electrons emitted will be:
Dual Nature of Radiation and Matter Photoelectric Effect Work Function and Threshold
EasySingle correctIdea asked in 4 yearsAsked the year before
Q47
The mass number of nucleus having radius equal to half of the radius of nucleus with mass number 192 is:
Nuclei Nuclear Binding Energy Nuclear Radius and Density
EasySingle correctIdea asked in 5 yearsAsked the year before
Q48
The output of the given circuit diagram is -
Semiconductor Electronics: Materials, Devices and Simple Circuits Logic Gates Combinational Logic Circuits
MediumDiagram based · figureIdea asked in 7 yearsAsked the year before
Q49
The resistance per centimeter of a meter bridge wire is $r$ , with $X\ \Omega$ resistance in left gap. Balancing length from left end is at 40 cm with $25\ \Omega$ resistance in right gap. Now the wire is replaced by another wire of $2r$ resistance per…
Current Electricity Wheatstone Bridge Meter Bridge Principles
MediumSingle correctIdea asked in 4 years
Q50
The measured value of the length of a simple pendulum is 20 cm with 2 mm accuracy. The time for 50 oscillations was measured to be 40 seconds with 1 second resolution. From these measurements, the accuracy in the measurement of acceleration due to gravity is…
Units and Measurements
MediumSingle correct
Q51
A body of mass ' $m$ ' is projected with a speed ' $u$ ' making an angle of $45^\circ$ with the ground. The angular momentum of the body about the point of projection, at the highest point is expressed as $\frac{\sqrt{2}mu^3}{Xg}$ . The value of ' $X$ ' is ________.
Motion in a Plane Angular Momentum Angular Momentum of Projectile at Maximum Height
MediumNumerical valueNew idea
Q52
Two identical spheres each of mass 2 kg and radius 50 cm are fixed at the ends of a light rod so that the separation between the centers is 150 cm. Then, moment of inertia of the system about an axis perpendicular to the rod and passing through its middle…
System of Particles and Rotational Motion Moment of Inertia Parallel Axis Theorem
MediumNumerical valueIdea asked in 4 years
Q53
Two blocks of mass 2 kg and 4 kg are connected by a metal wire going over a smooth pulley as shown in figure. The radius of wire is $4.0\times10^{-5}$ m and Young's modulus of the metal is $2.0\times10^{11}$ N/m $^2$ . The longitudinal strain developed in the…
Mechanical Properties of Solids Elasticity Stress and Strain Types
MediumNumerical value · figureIdea asked in 4 yearsAsked the year before
Q54
The time period of simple harmonic motion of mass $M$ in the given figure is $\pi\sqrt{\frac{\alpha M}{5k}}$ , where the value of $\alpha$ is __________.
Oscillations SHM Equations Springs in Series
MediumNumerical value · figureIdea asked in 3 years
Q55
The distance between charges $+q$ and $-q$ is $2l$ and between $+2q$ and $-2q$ is $4l$ . The electrostatic potential at point $P$ at a distance $r$ from center $O$ is $-\alpha\left[\frac{ql}{r^2}\right]\times10^9\,V$ , where the value of $\alpha$ is…
Electrostatic Potential and Capacitance Electric Potential Electric Potential of a Dipole
MediumNumerical value · figureNew idea
Q56
In the following circuit, the battery has an emf of 2V and an internal resistance of $\frac{2}{3}\Omega$ . The power consumption in the entire circuit is __________ W.
Current Electricity Kirchhoff's Laws Cell EMF and Internal Resistance
MediumNumerical value · figureIdea asked in 6 yearsAsked the year before
Q57
Two circular coils $P$ and $Q$ of 100 turns each have same radius of $\pi$ cm. The currents in $P$ and $R$ are $1A$ and $2A$ respectively. $P$ and $Q$ are placed with their planes mutually perpendicular with their centers coincide. The resultant magnetic…
Moving Charges and Magnetism Biot-Savart Law Field at center of circular arc
MediumNumerical valueIdea asked in 6 yearsAsked the year before
Q58
The magnetic flux $\phi$ (in weber) linked with a closed circuit of resistance $8\ \Omega$ varies with time (in seconds) as $\phi=5t^2-36t+1$ . The induced current in the circuit at $t=2$ s is ____________ $A$ .
Electromagnetic Induction Faraday's Law Faraday's Law Magnitude
EasyNumerical valueIdea asked in 6 yearsAsked the year before
Q59
Light from a point source in air falls on a convex curved surface of radius 20 cm and refractive index 1.5. If the source is located at 100 cm from the convex surface, the image will be formed at ____________ cm from the object.
Ray Optics and Optical Instruments
MediumNumerical value
Q60
A nucleus has mass number $A_1$ and volume $V_1$ . Another nucleus has mass number $A_2$ and Volume $V_2$ . If relation between mass number is $A_2=4A_1$ , then $\frac{V_2}{V_1}=$ ____________.
Nuclei Nuclear Binding Energy Nuclear Radius and Density
EasyNumerical valueIdea asked in 5 yearsAsked the year before

Chemistry30 questions

9 easy19 medium2 hard

Q61
A sample of $\mathrm{CaCO_3}$ and $\mathrm{MgCO_3}$ weighed 2.21 g is ignited to constant weight of 1.152 g. The composition of mixture is : (Given molar mass in g mol $^{-1}$ $\mathrm{CaCO_3}$ : 100, $\mathrm{MgCO_3}$ : 84)
Some Basic Concepts of Chemistry Stoichiometry Thermal Decomposition of Carbonates
MediumSingle correctNew idea
Q62
The four quantum numbers for the electron in the outer most orbital of potassium (atomic no. 19) are
Structure of Atom Quantum Numbers Quantum Numbers of the Differentiating Electron
EasySingle correctNew idea
Q63
Which of the following is least ionic?
Chemical Bonding and Molecular Structure s-Block Elements Covalent Character in Halides
MediumSingle correctIdea asked in 2 years
Q64
$\mathrm{A_{(g)}\rightleftharpoons B_{(g)}+\frac{C}{2}{}_{(g)}}$ The correct relationship between $\mathrm{K_p}$ , $\alpha$ and equilibrium pressure P is
Equilibrium Chemical Equilibrium Kp for PCl5 Dissociation Type Reactions
MediumSingle correctIdea asked in 4 years
Q65
Given below are two statements : Statement I : $\mathrm{S_8}$ solid undergoes disproportionation reaction under alkaline conditions to form $\mathrm{S^{2-}}$ and $\mathrm{S_2O_3^{2-}}$ . Statement II : $\mathrm{ClO_4^-}$ can undergo disproportionation…
Redox Reactions Disproportionation Balancing
MediumTwo statementsIdea asked in 3 years
Q66
Consider the following elements. Group $\downarrow$ A' B' $\rightarrow$ Period C' D' Which of the following is/are true about A', B', C' and D'? A. Order of atomic radii: B' < A' < D' < C' B. Order of metallic character: B' < A' < D' < C' C. Size of the…
Classification of Elements and Periodicity in Properties Periodic Trends Atomic and Ionic Radii Trends
MediumOther · figureIdea asked in 3 years
Q67
Given below are two statements : Statement I : Group 13 trivalent halides get easily hydrolyzed by water due to their covalent nature. Statement II : $\mathrm{AlCl_3}$ upon hydrolysis in acidified aqueous solution forms octahedral…
The p-Block Elements
MediumTwo statements
Q68
Choose the correct statements from the following A. All group 16 elements form oxides of general formula $\mathrm{EO_2}$ and $\mathrm{EO_3}$ , where E = S, Se, Te and Po. Both the types of oxides are acidic in nature. B. $\mathrm{TeO_2}$ is an oxidising agent…
The p-Block Elements Group 15 & 16 Group 16 Hydrides Trends
MediumOtherIdea asked in 4 years
Q69
Choose the correct statements from the following A. $\mathrm{Mn_2O_7}$ is an oil at room temperature B. $\mathrm{V_2O_4}$ reacts with acid to give $\mathrm{VO_2^{2+}}$ C. $\mathrm{CrO}$ is a basic oxide D. $\mathrm{V_2O_5}$ does not react with acid Choose…
The d- and f-Block Elements d-Block Elements Acid-Base Character of Oxides
MediumOtherIdea asked in 4 years
Q70
Select the option with correct property -
Coordination Compounds Crystal Field Theory Spin-only Magnetic Moment
MediumSingle correctIdea asked in 7 yearsAsked the year before
Q71
Match List I with List II LIST I (Complex ion) | LIST II (Electronic Configuration) A. $\mathrm{[Cr(H_2O)_6]^{3+}}$ | I. $t_{2g}^{2}e_g^{0}$ B. $\mathrm{[Fe(H_2O)_6]^{3+}}$ | II. $t_{2g}^{3}e_g^{0}$ C. $\mathrm{[Ni(H_2O)_6]^{2+}}$ | III. $t_{2g}^{3}e_g^{2}$ …
Coordination Compounds Crystal Field Theory High and Low Spin States
MediumMatch the listIdea asked in 7 yearsAsked the year before
Q72
The fragrance of flowers is due to the presence of some steam volatile organic compounds called essential oils. These are generally insoluble in water at room temperature but are miscible with water vapour in vapour phase. A suitable method for the…
Organic Chemistry: Some Basic Principles and Techniques
EasySingle correct
Q73
The correct order of reactivity in electrophilic substitution reaction of the following compounds is :
Hydrocarbons Aromatic Hydrocarbons Ranking Activating Groups
EasyDiagram based · figureIdea asked in 3 yearsAsked the year before
Q74
Identify structure of 2,3-dibromo-1-phenylpentane.
Haloalkanes and Haloarenes IUPAC Nomenclature Lowest Locant Rule
EasyDiagram based · figureNew idea
Q75
Major product of the following reaction is -
Hydrocarbons Alkenes and Alkynes Markovnikov's Rule Application
HardDiagram based · figureIdea asked in 6 years
Q76
Identify A and B in the following reaction sequence. [reaction: bromobenzene --(Conc HNO $_3$ )--> A --((i) NaOH, (ii) HCl)--> B]
Haloalkanes and Haloarenes Aromatic Hydrocarbons SNAr and Meisenheimer Complex
HardDiagram based · figureIdea asked in 3 yearsAsked the year before
Q77
Identify major product 'P' formed in the following reaction. [reaction: benzene + benzoyl chloride (C $_6$ H $_5$ COCl) --(Anhydrous AlCl $_3$ )--> 'P' (Major Product)]
Aldehydes, Ketones and Carboxylic Acids Aromatic Hydrocarbons Friedel-Crafts Acylation and Acylium Ion
EasyDiagram based · figureIdea asked in 2 years
Q78
Identify the name reaction. [reaction: benzene + CO, HCl --(anhydrous AlCl $_3$ /CuCl)--> benzaldehyde (C $_6$ H $_5$ CHO)]
Aldehydes, Ketones and Carboxylic Acids Aromatic Hydrocarbons Gattermann-Koch Formylation
EasySingle correct · figureIdea asked in 3 years
Q79
Given below are two statements : Statement I : Aniline reacts with con. $\mathrm{H_2SO_4}$ , followed by heating at 453 - 473 K gives p-aminobenzene sulphonic acid, which gives blood red colour in the 'Lassaigne's test'. Statement II : In Friedel - Craft's…
Amines Sulphonation of Aniline and Zwitterion Formation
MediumTwo statementsNew idea
Q80
The azo-dye (Y) formed in the following reactions is Sulphanilic acid + $\mathrm{NaNO_2}$ + $\mathrm{CH_3COOH}$ $\rightarrow$ X. X + 1-naphthylamine $\rightarrow$ Y
Amines Diazonium Salts Azo Coupling Reactions
MediumDiagram based · figureIdea asked in 5 yearsAsked the year before
Q81
A diatomic molecule has a dipole moment of 1.2 D. If the bond distance is 1A°, then fractional charge on each atom is ________ $\times 10^{-1}$ esu. (Given 1 D = $10^{-18}$ esu cm)
Chemical Bonding and Molecular Structure
MediumNumerical value
Q82
If 5 moles of an ideal gas expands from 10 L to a volume of 100 L at 300 K under isothermal and reversible condition then work, w, is $-x$ J. The value of $x$ is ___________. (Given R = 8.314 J K $^{-1}$ mol $^{-1}$ )
Thermodynamics (Chemistry)
EasyNumerical value
Q83
The molarity of 1 L orthophosphoric acid ( $\mathrm{H_3PO_4}$ ) having 70% purity by weight (specific gravity 1.54 g cm $^{-3}$ ) is _________ M. (Molar mass of $\mathrm{H_3PO_4}$ = 98 g mol $^{-1}$ )
Some Basic Concepts of Chemistry Mole Concept Specific Gravity in Concentration
EasyNumerical valueNew idea
Q84
The values of conductivity of some materials at 298.15 K in Sm $^{-1}$ are $2.1 \times 10^{3}$ , $1.0 \times 10^{-16}$ , $1.2 \times 10$ , $3.91$ , $1.5 \times 10^{-2}$ , $1 \times 10^{-7}$ , $1.0 \times 10^{3}$ . The number of conductors among the materials is…
The Solid State Solid State Defects Band Theory of Solids
MediumNumerical valueNew idea
Q85
r = k [A] for a reaction, 50% of A is decomposed in 120 minutes. The time taken for 90% decomposition of A is _________ minutes.
Chemical Kinetics Integrated Rate Equations First Order Fractional Life Relationships
MediumNumerical valueNew idea
Q86
Number of moles of $\mathrm{H^+}$ ions required by 1 mole of $\mathrm{MnO_4^-}$ to oxidise oxalate ion to $\mathrm{CO_2}$ is ________.
Redox Reactions Ion-Electron Acidic
MediumNumerical valueIdea asked in 2 yearsAsked the year before
Q87
Number of isomeric products formed by monochlorination of 2-methylbutane in presence of sunlight is __________.
Hydrocarbons
MediumNumerical value
Q88
A compound ( $x$ ) with molar mass 108 g mol $^{-1}$ undergoes acetylation to give product with molar mass 192 g mol $^{-1}$ . The number of amino groups in the compund ( $x$ ) is ________.
Amines
MediumNumerical value
Q89
From the vitamins A, $\mathrm{B_1}$ , $\mathrm{B_6}$ , $\mathrm{B_{12}}$ , C, D, E and K, the number of vitamins that can be stored in our body is_________.
Biomolecules
MediumNumerical value
Q90
In the reaction of potassium dichromate, potassium chloride and sulfuric acid (conc.), the oxidation state of the chromium in the product is (+)_________.
The d- and f-Block Elements d-Block Elements Chromyl Chloride Test Reactions
EasyNumerical valueNew idea

Maths30 questions

3 easy17 medium10 hard

Q1
If the function $f:(-\infty,-1]\to(a,b]$ defined by $f(x)=e^{x^3-3x+1}$ is one - one and onto, then the distance of the point $P(2b+4,\,a+2)$ from the line $x+e^{-3}y=4$ is :
Application of Derivatives Functions and Domain Range via Monotonicity and Derivatives
MediumSingle correctNew idea
Q2
The number of solutions, of the equation $e^{\sin x}-2e^{-\sin x}=2$ , is :
Trigonometric Functions General Solutions Quadratic Substitution in Trigonometry
MediumSingle correctIdea asked in 5 yearsAsked the year before
Q3
Let $z_1$ and $z_2$ be two complex numbers such that $z_1+z_2=5$ and $z_1^3+z_2^3=20+15i$ . Then, $|z_1^4+z_2^4|$ equals -
Complex Numbers and Quadratic Equations Quadratic Roots Symmetric Functions
HardSingle correctIdea asked in 5 years
Q4
Let $A$ be a $3\times3$ real matrix such that $A\begin{pmatrix}1\\0\\1\end{pmatrix}=2\begin{pmatrix}1\\0\\1\end{pmatrix}$ , $A\begin{pmatrix}-1\\0\\1\end{pmatrix}=4\begin{pmatrix}-1\\0\\1\end{pmatrix}$ ,…
Determinants Cramer's Rule Condition for Unique Solution
MediumSingle correctIdea asked in 3 years
Q5
The number of ways in which 21 identical apples can be distributed among three children such that each child gets at least 2 apples, is
Permutations and Combinations Distribution of Identical Objects
EasySingle correctIdea asked in 6 yearsAsked the year before
Q6
If for some $m,n$ ; ${}^6C_m+2\left({}^6C_{m+1}\right)+{}^6C_{m+2}>{}^8C_3$ and ${}^{n-1}P_3:{}^nP_4=1:8$ , then ${}^nP_{m+1}+{}^{n+1}C_m$ is equal to
Permutations and Combinations Pascal's Rule and nCr Properties
MediumSingle correctNew idea
Q7
Let $2^{nd}$ , $8^{th}$ and $44^{th}$ terms of a non-constant A. P. be respectively the $1^{st}$ , $2^{nd}$ and $3^{rd}$ terms of a G. P. If the first term of the A. P. is 1, then the sum of its first 20 terms is equal to -
Sequences and Series AP and GP GP General Term
MediumSingle correctIdea asked in 5 yearsAsked the year before
Q8
Consider the function $f:(0,\infty)\to\mathbb{R}$ defined by $f(x)=e^{-|\log_e x|}$ . If $m$ and $n$ be respectively the number of points at which $f$ is **not** continuous and $f$ is **not** differentiable, then $m+n$ is
Continuity and Differentiability Differentiability of Modulus Functions
MediumSingle correctIdea asked in 5 yearsAsked the year before
Q9
Let $f:\mathbb{R}\to(0,\infty)$ be strictly increasing function such that $\lim_{x\to\infty}\frac{f(7x)}{f(x)}=1$ . Then, the value of $\lim_{x\to\infty}\left[\frac{f(5x)}{f(x)}-1\right]$ is equal to
Continuity and Differentiability
HardSingle correct
Q10
Let a variable line passing through the centre of the circle $x^2+y^2-16x-4y=0$ , meet the positive co-ordinate axes at the points $A$ and $B$ . Then the minimum value of $OA+OB$ , where $O$ is the origin, is equal to
Conic Sections Maxima and Minima Optimization via AM-GM Inequality
MediumSingle correctIdea asked in 4 years
Q11
The area of the region enclosed by the parabolas $y=4x-x^2$ and $3y=(x-4)^2$ is equal to
Application of Integrals Area Under Curves Area Between Curves
MediumSingle correctIdea asked in 7 yearsAsked the year before
Q12
Let $f,g:(0,\infty)\to\mathbb{R}$ be two functions defined by $f(x)=\int_{-x}^{x}\left(|t|-t^2\right)e^{-t^2}dt$ and $g(x)=\int_0^{x^2}t^{1/2}e^{-t}dt$ . Then, the value of $9\left(f\left(\sqrt{\log_e 9}\right)+g\left(\sqrt{\log_e 9}\right)\right)$ is equal to
Integrals Definite Integral Properties Even and Odd Functions
HardSingle correctIdea asked in 3 years
Q13
The temperature $T(t)$ of a body at time $t=0$ is $160^\circ$ F and it decreases continuously as per the differential equation $\frac{dT}{dt}=-K(T-80)$ , where $K$ is a positive constant. If $T(15)=120^\circ$ F, then $T(45)$ is equal to
Differential Equations Variable Separable Form Physical Modeling Applications
EasySingle correctIdea asked in 2 years
Q14
Let $P$ be a parabola with vertex $(2,3)$ and directrix $2x+y=6$ . Let an ellipse $E:\frac{x^2}{a^2}+\frac{y^2}{b^2}=1,\ a>b$ , of eccentricity $\frac{1}{\sqrt{2}}$ pass through the focus of the parabola $P$ . Then, the square of the length of the latus rectum…
Conic Sections Ellipse Eccentricity Length of Latus Rectum
HardSingle correctIdea asked in 6 yearsAsked the year before
Q15
Let $A(a,b)$ , $B(3,4)$ and $C(-6,-8)$ respectively denote the centroid, circumcentre and orthocentre of a triangle. Then, the distance of the point $P(2a+3,\,7b+5)$ from the line $2x+3y-4=0$ measured parallel to the line $x-2y-1=0$ is
Straight Lines Straight Line Distance Distance Measured Parallel to a Line
MediumSingle correctIdea asked in 2 years
Q16
Let $(\alpha,\beta,\gamma)$ be the mirror image of the point $(2,3,5)$ in the line $\frac{x-1}{2}=\frac{y-2}{3}=\frac{z-3}{4}$ . Then, $2\alpha+3\beta+4\gamma$ is equal to
Three Dimensional Geometry Lines in 3D Image of a Point in a Line
MediumSingle correctIdea asked in 6 years
Q17
The shortest distance, between lines $L_1$ and $L_2$ , where $L_1:\frac{x-1}{2}=\frac{y+1}{-3}=\frac{z+4}{2}$ and $L_2$ is the line, passing through the points $A(-4,4,3)$ , $B(-1,6,3)$ and perpendicular to the line $\frac{x-3}{-2}=\frac{y}{3}=\frac{z-1}{1}$ , is
Three Dimensional Geometry Skew Lines Cartesian Shortest Distance
MediumSingle correctIdea asked in 6 yearsAsked the year before
Q18
Let the mean and the variance of 6 observations $a, b, 68, 44, 48, 60$ be 55 and 194, respectively. If $a>b$ , then $a+3b$ is
Statistics Variance and SD Finding Unknown Observations from Mean and Variance
MediumSingle correctIdea asked in 7 yearsAsked the year before
Q19
A coin is biased so that a head is twice as likely to occur as a tail. If the coin is tossed 3 times, then the probability of getting two tails and one head is
Probability
EasySingle correct
Q20
If $a=\sin^{-1}(\sin(5))$ and $b=\cos^{-1}(\cos(5))$ , then $a^2+b^2$ is equal to
Inverse Trigonometric Functions Inverse Trig Identities Periodic Reduction of sin-1(sin x) and cos-1(cos x)
MediumSingle correctIdea asked in 2 years
Q21
Let $A=\{1,2,3,\ldots,100\}$ . Let $R$ be a relation on $A$ defined by $(x,y)\in R$ if and only if $2x=3y$ . Let $R_1$ be a symmetric relation on $A$ such that $R\subset R_1$ and the number of elements in $R_1$ is n. Then, the minimum value of n is ____________.
Relations and Functions Sets and Relations Relation Types
MediumNumerical valueIdea asked in 5 yearsAsked the year before
Q22
Let A be a $3\times3$ matrix and $\det(A)=2$ . If $n=\det\left(\underbrace{\operatorname{adj}(\operatorname{adj}(\ldots(\operatorname{adj}A)))}_{2024\text{-times}}\right)$ , then the remainder when $n$ is divided by 9 is equal to ____________.
Determinants Matrix Inverse Inverse Relation to Double Adjoint
MediumNumerical valueIdea asked in 6 yearsAsked the year before
Q23
Let the coefficient of $x^r$ in the expansion of $(x+3)^{n-1}+(x+3)^{n-2}(x+2)+(x+3)^{n-3}(x+2)^2+\ldots+(x+2)^{n-1}$ be $\alpha_r$ . If $\sum_{r=0}^{n}\alpha_r=\beta^n-\gamma^n$ , $\beta,\gamma\in\mathbb{N}$ , then the value of $\beta^2+\gamma^2$ equals…
Binomial Theorem Binomial Coefficients Sum of all Coefficients
HardNumerical valueIdea asked in 4 yearsAsked the year before
Q24
Let $a, b, c$ be the lengths of three sides of a triangle satisfying the condition $(a^2+b^2)x^2-2b(a+c)x+(b^2+c^2)=0$ . If the set of all possible values of $x$ is the interval $(\alpha,\beta)$ , then $12(\alpha^2+\beta^2)$ is equal to ________.
Complex Numbers and Quadratic Equations Quadratic Roots Nature of Roots
HardNumerical valueIdea asked in 3 years
Q25
If $\lim_{x\to0}\frac{ax^2e^x-b\log_e(1+x)+cxe^{-x}}{x^2\sin x}=1$ , then $16(a^2+b^2+c^2)$ is equal to ________.
Limits and Derivatives L'Hopital's Rule Choosing Series Expansion over L'Hopital
HardNumerical valueIdea asked in 7 yearsAsked the year before
Q26
$\left|\frac{120}{\pi^3}\int_0^{\pi}\frac{x^2\sin x\cos x}{\sin^4x+\cos^4x}dx\right|$ is equal to ____________.
Integrals Definite Integral Properties Removing 'x' using King's Property
HardNumerical valueNew ideaIdea asked in 2 years
Q27
Let $y=y(x)$ be the solution of the differential equation $\sec^2x\,dx+\left(e^{2y}\tan^2x+\tan x\right)dy=0$ , $0<x<\frac{\pi}{2}$ , $y\left(\frac{\pi}{4}\right)=0$ . If $y\left(\frac{\pi}{6}\right)=\alpha$ , then $e^{8\alpha}$ is equal to ________.
Differential Equations Linear Differential Equations Bernoulli Equation Reduction
HardNumerical valueIdea asked in 2 years
Q28
Let $A(-2,-1)$ , $B(1,0)$ , $C(\alpha,\beta)$ and $D(\gamma,\delta)$ be the vertices of a parallelogram $ABCD$ . If the point $C$ lies on $2x-y=5$ and the point $D$ lies on $3x-2y=6$ , then the value of $|\alpha+\beta+\gamma+\delta|$ is equal to ________.
Straight Lines
MediumNumerical value
Q29
A line passes through $A(4,-6,-2)$ and $B(16,-2,4)$ . The point $P(a,b,c)$ , where $a, b, c$ are non-negative integers, on the line $AB$ lies at a distance of 21 units, from the point $A$ . The distance between the points $P(a,b,c)$ and $Q(4,-12,3)$ is equal to…
Three Dimensional Geometry Lines in 3D Parametric Points at Distance r
MediumNumerical valueNew idea
Q30
Let $\vec{a}=3\hat{i}+2\hat{j}+\hat{k}$ , $\vec{b}=2\hat{i}-\hat{j}+3\hat{k}$ and $\vec{c}$ be a vector such that $(\vec{a}+\vec{b})\times\vec{c}=2(\vec{a}\times\vec{b})+24\hat{j}-6\hat{k}$ and $(\vec{a}-\vec{b}+\hat{i})\cdot\vec{c}=-3$ . Then $|\vec{c}|^2$ is…
Vector Algebra Cross Product Solving Vector Equations with Cross Products
HardNumerical valueIdea asked in 6 yearsAsked the year before

Dropped from the final key and left out of every count: Q34.

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 2024 shifts.