JEE Main 2024 · January session

JEE Main 1 February 2024, Shift 2

88 questions (2 more dropped by NTA): 30 numerical-answer and 58 multiple choice. 7 were rated hard and 35 easy. Its hard-question count was the joint 17th highest of the 18 2024 shifts Lumi analysed. The biggest chapter was Current Electricity, with 3 questions.

Questions: 88; 2 dropped by NTA
Numerical answer: 30; 34% of the shift
Hard: 7; 8% of the shift
Chapters asked: 58; 9 questions with a figure

Key findings · 1 Feb 2024, Shift 2

1
7 hard questions (8%): the joint 17th 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 6 of 28 rated hard; Chemistry had the most easy questions (19 of 30).
3
14 questions tested a new idea, one no earlier year had asked: 19% 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
36 shared an idea with another 2024 shift (49% 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), Hydrocarbons in Chemistry (3) and Integrals in Maths (3).

The shift at a glance

Subjects, difficulty and chapters

Physics

30 Q · 10 numerical

Easy 15Medium 14Hard 1

Top chapters

Chemistry

30 Q · 10 numerical

Easy 19Medium 11Hard 0

Top chapters

Maths

28 Q · 10 numerical

Easy 1Medium 21Hard 6

Top chapters

Question format

Single correct43 Q
Numerical value30 Q
Diagram based4 Q
Two statements4 Q
Match the list3 Q
Other3 Q
Assertion–reason1 Q

What each question asks you to do

Calculation42 Q
Multi-step18 Q
Conceptual14 Q
Recall8 Q
Application6 Q

Most common: Calculation.

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 shift3084 · 33%33%8/830298716720359538199224
Open28 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

Physics30 questions

15 easy14 medium1 hard

Q31
Match List - I with List - II. List - I (Number) List - II (Significant figure) (A) 1001 (I) 3 (B) 010.1 (II) 4 (C) 100.100 (III) 5 (D) 0.0010010 (IV) 6 Choose the correct answer from the options given below :
Electric Charges and Fields
EasyMatch the list
Q32
Train A is moving along two parallel rail tracks towards north with speed $72$ km/h and train B is moving towards south with speed $108$ km/h. Velocity of train B with respect to A and velocity of ground with respect to B are (in $\mathrm{ms^{-1}}$ ) :
Motion in a Straight Line Relative Velocity Relative Motion in 1D
EasySingle correctIdea asked in 3 years
Q33
A cricket player catches a ball of mass $120$ g moving with $25$ m/s speed. If the catching process is completed in $0.1$ s then the magnitude of force exerted by the ball on the hand of player will be (in SI unit) :
Laws of Motion Collisions Average Impact Force Calculation
EasySingle correctIdea asked in 2 years
Q34
A body of mass $4$ kg experiences two forces $\vec{F}_1 = 5\hat{i} + 8\hat{j} + 7\hat{k}$ and $\vec{F}_2 = 3\hat{i} - 4\hat{j} - 3\hat{k}$ . The acceleration acting on the body is :
Laws of Motion Newton's Laws of Motion F=ma for Single Body
EasySingle correctIdea asked in 6 yearsAsked the year before
Q35
A disc of radius R and mass M is rolling horizontally without slipping with speed $v$ . It then moves up an inclined smooth surface as shown in figure. The maximum height that the disc can go up the incline is :
System of Particles and Rotational Motion
MediumDiagram based · figure
Q36
A light planet is revolving around a massive star in a circular orbit of radius R with a period of revolution T. If the force of attraction between planet and star is proportional to $R^{-3/2}$ then choose the correct option :
Gravitation Orbital Mechanics Kepler's Laws
MediumSingle correctIdea asked in 4 yearsAsked the year before
Q37
A big drop is formed by coalescing 1000 small droplets of water. The surface energy will become :
Mechanical Properties of Fluids Surface Tension Splitting and Coalescence
EasySingle correctIdea asked in 5 years
Q38
A diatomic gas ( $\gamma = 1.4$ ) does 200 J of work when it is expanded isobarically. The heat given to the gas in the process is :
Thermodynamics First Law of Thermodynamics Heat in Isochoric and Isobaric Processes
EasySingle correctIdea asked in 4 years
Q39
If the root mean square velocity of hydrogen molecule at a given temperature and pressure is 2 km/s, the root mean square velocity of oxygen at the same condition in km/s is :
Kinetic Theory Kinetic Theory of Gases Molecular Speed Ratios
EasySingle correctIdea asked in 4 yearsAsked the year before
Q40
$C_1$ and $C_2$ are two hollow concentric cubes enclosing charges 2Q and 3Q respectively as shown in figure. The ratio of electric flux passing through $C_1$ and $C_2$ is :
Electric Charges and Fields Gauss's Law Gauss Law Application
EasyDiagram based · figureIdea asked in 4 yearsAsked the year before
Q41
A galvanometer (G) of $2\,\Omega$ resistance is connected in the given circuit. The ratio of charge stored in $C_1$ and $C_2$ is :
Electrostatic Potential and Capacitance RC Circuits Initial and Steady State Behavior
MediumDiagram based · figureIdea asked in 5 yearsAsked the year before
Q42
In an ammeter, 5% of the main current passes through the galvanometer. If resistance of the galvanometer is G, the resistance of ammeter will be :
Moving Charges and Magnetism
EasySingle correct
Q43
A transformer has an efficiency of 80% and works at 10 V and 4 kW. If the secondary voltage is 240 V, then the current in the secondary coil is :
Alternating Current
EasySingle correct
Q44
If frequency of electromagnetic wave is 60 MHz and it travels in air along $z$ direction then the corresponding electric and magnetic field vectors will be mutually perpendicular to each other and the wavelength of the wave (in m) is :
Electromagnetic Waves
EasySingle correct
Q45
A microwave of wavelength 2.0 cm falls normally on a slit of width 4.0 cm. The angular spread of the central maxima of the diffraction pattern obtained on a screen 1.5 m away from the slit, will be :
Wave Optics Single Slit Diffraction Angular Width of Central Max
EasySingle correctIdea asked in 3 years
Q46
Monochromatic light of frequency $6\times10^{14}$ Hz is produced by a laser. The power emitted is $2\times10^{-3}$ W. How many photons per second on an average, are emitted by the source ? (Given $h = 6.63\times10^{-34}$ Js)
Dual Nature of Radiation and Matter Photoelectric Effect Photon Flux and Beam Power
EasySingle correctIdea asked in 4 years
Q47
From the statements given below : (A) The angular momentum of an electron in $n^{th}$ orbit is an integral multiple of $\hbar$ . (B) Nuclear forces do not obey inverse square law. (C) Nuclear forces are spin dependent. (D) Nuclear forces are central and…
Nuclei Nuclear Binding Energy Stability Curve Interpretation
MediumOtherIdea asked in 2 yearsAsked the year before
Q48
Conductivity of a photodiode starts changing only if the wavelength of incident light is less than 660 nm. The band gap of photodiode is found to be $\left(\frac{X}{8}\right)$ eV. The value of X is : (Given, $h = 6.6\times10^{-34}$ Js,…
Semiconductor Electronics: Materials, Devices and Simple Circuits Semiconductor Diodes Optoelectronic Diodes
MediumSingle correctIdea asked in 3 years
Q49
To measure the temperature coefficient of resistivity $\alpha$ of a semiconductor, an electrical arrangement shown in the figure is prepared. The arm BC is made up of the semiconductor. The experiment is being conducted at $25^\circ$ C and resistance of the…
Current Electricity Ohm's Law and Resistance Temperature Dependence in Semiconductors
MediumDiagram based · figureIdea asked in 2 yearsAsked the year before
Q50
In a metre-bridge when a resistance in the left gap is $2\,\Omega$ and unknown resistance in the right gap, the balance length is found to be 40 cm. On shunting the unknown resistance with $2\,\Omega$ , the balance length changes by :
Current Electricity Wheatstone Bridge Meter Bridge Principles
MediumSingle correctIdea asked in 4 years
Q51
A particle initially at rest starts moving from reference point $x = 0$ along $x$ -axis, with velocity $v$ that varies as $v = 4\sqrt{x}$ m/s. The acceleration of the particle is ________ $\mathrm{ms^{-2}}$ .
Motion in a Straight Line Kinematics in 1D Position-Dependent Acceleration (v dv/dx)
MediumNumerical valueIdea asked in 4 years
Q52
A uniform rod AB of mass 2 kg and length 30 cm at rest on a smooth horizontal surface. An impulse of force 0.2 Ns is applied to end B. The time taken by the rod to turn through at right angles will be $\frac{\pi}{x}$ s, where $x =$ ________.
System of Particles and Rotational Motion Angular Momentum Angular Momentum from Linear Impulse
HardNumerical valueNew idea
Q53
One end of a metal wire is fixed to a ceiling and a load of 2 kg hangs from the other end. A similar wire is attached to the bottom of the load and another load of 1 kg hangs from this lower wire. Then the ratio of longitudinal strain of upper wire to that…
Mechanical Properties of Solids Elasticity Stress and Strain Types
MediumNumerical valueIdea asked in 4 yearsAsked the year before
Q54
A mass $m$ is suspended from a spring of negligible mass and the system oscillates with a frequency $f_1$ . The frequency of oscillations if a mass $9m$ is suspended from the same spring is $f_2$ . The value of $\frac{f_1}{f_2}$ is ________.
Oscillations SHM Equations Standard Differential Equation Form
EasyNumerical valueNew idea
Q55
Suppose a uniformly charged wall provides a uniform electric field of $2\times10^4$ N/C normally. A charged particle of mass 2 g being suspended through a silk thread of length 20 cm and remain stayed at a distance of 10 cm from the wall. Then the charge on…
Electric Charges and Fields Electric Field Equilibrium of Suspended Charges
MediumNumerical valueNew idea
Q56
In an electrical circuit drawn below the amount of charge stored in the capacitor is ________ $\mu$ C.
Current Electricity RC Circuits Initial and Steady State Behavior
MediumNumerical value · figureIdea asked in 5 yearsAsked the year before
Q57
A moving coil galvanometer has 100 turns and each turn has an area of $2.0\ \mathrm{cm^2}$ . The magnetic field produced by the magnet is 0.01 T and the deflection in the coil is 0.05 radian when a current of 10 mA is passed through it. The torsional constant…
Moving Charges and Magnetism
MediumNumerical value
Q58
A coil of 200 turns and area $0.20\ \mathrm{m^2}$ is rotated at half a revolution per second and is placed in uniform magnetic field of 0.01 T perpendicular to axis of rotation of the coil. The maximum voltage generated in the coil is $\frac{2\pi}{\beta}$ …
Electromagnetic Induction Faraday's Law AC Generator Principle
EasyNumerical valueIdea asked in 4 years
Q59
In Young's double slit experiment, monochromatic light of wavelength 5000 Å is used. The slits are 1.0 mm apart and screen is placed at 1.0 m away from slits. The distance from the centre of the screen where intensity becomes half of the maximum intensity…
Wave Optics Young's Double Slit Resultant Intensity Formula
MediumNumerical valueIdea asked in 5 yearsAsked the year before
Q60
A particular hydrogen - like ion emits the radiation of frequency $3\times10^{15}$ Hz when it makes transition from n = 2 to n = 1. The frequency of radiation emitted in transition from n = 3 to n = 1 is $\frac{x}{9}\times10^{15}$ Hz, when $x =$ ________.
Atoms Bohr's Atomic Model Identifying Atomic Number from Spectra
MediumNumerical valueIdea asked in 2 years

Chemistry30 questions

19 easy11 medium0 hard

Q61
The number of radial node/s for 3p orbital is :
Structure of Atom Quantum Numbers Calculation of Radial and Angular Nodes
EasySingle correctIdea asked in 5 yearsAsked the year before
Q62
Given below are two statements : Statement (I) : A $\pi$ bonding MO has lower electron density above and below the inter-nuclear axis. Statement (II) : The $\pi^*$ antibonding MO has a node between the nuclei. In the light of the above statements, choose the…
Chemical Bonding and Molecular Structure Molecular Orbital Theory Symmetry of Pi Molecular Orbitals
MediumTwo statementsNew idea
Q63
Select the compound from the following that will show intramolecular hydrogen bonding.
Chemical Bonding and Molecular Structure Alcohols and Phenols Hydrogen Bonding Trends in Alcohols
EasySingle correct · figureIdea asked in 3 yearsAsked the year before
Q64
Solubility of calcium phosphate (molecular mass, M) in water is $\mathrm{W_g}$ per 100 mL at 25 $^\circ$ C. Its solubility product at 25 $^\circ$ C will be approximately.
Equilibrium Solubility Product Solubility-Ksp Relationship
MediumSingle correctIdea asked in 5 years
Q65
Given below are two statements : Statement (I) : Both metals and non-metals exist in p and d-block elements. Statement (II) : Non-metals have higher ionisation enthalpy and higher electronegativity than the metals. In the light of the above statements,…
Classification of Elements and Periodicity in Properties Periodic Trends Ionization Enthalpy Trends
EasyTwo statementsIdea asked in 6 years
Q66
The strongest reducing agent among the following is :
The p-Block Elements Group 15 & 16 Group 15 Hydrides Trends
EasySingle correctIdea asked in 3 years
Q67
Given below are two statements : Statement (I) : $\mathrm{SiO_2}$ and $\mathrm{GeO_2}$ are acidic while SnO and PbO are amphoteric in nature. Statement (II) : Allotropic forms of carbon are due to property of catenation and $p\pi$ - $d\pi$ bond formation. In…
The p-Block Elements Group 13 & 14 Catenation Trend in Group 14
MediumTwo statementsNew idea
Q68
The transition metal having highest $3^{rd}$ ionisation enthalpy is :
The d- and f-Block Elements d-Block Elements Ionization Enthalpy Irregularities
MediumSingle correctNew ideaIdea asked in 3 years
Q69
Given below are two statements : one is labelled as Assertion (A) and the other is labelled as Reason (R). Assertion (A) : In aqueous solutions $\mathrm{Cr^{2+}}$ is reducing while $\mathrm{Mn^{3+}}$ is oxidising in nature. Reason (R) : Extra stability to…
The d- and f-Block Elements d-Block Elements Oxidation State Trends
MediumAssertion–reasonIdea asked in 4 yearsAsked the year before
Q70
$\mathrm{[Co(NH_3)_6]^{3+}}$ and $\mathrm{[CoF_6]^{3-}}$ are respectively known as :
Coordination Compounds Crystal Field Theory High and Low Spin States
EasySingle correctIdea asked in 7 yearsAsked the year before
Q71
Which of the following compounds show colour due to d-d transition ?
The d- and f-Block Elements d-Block Elements Color and Catalysis
EasySingle correctIdea asked in 4 years
Q72
Lassaigne's test is used for detection of :
Organic Chemistry: Some Basic Principles and Techniques
EasySingle correct
Q73
The functional group that shows negative resonance effect is :
Organic Chemistry: Some Basic Principles and Techniques Resonance and Hyperconjugation Mesomeric Effect
EasySingle correctIdea asked in 3 years
Q74
The set of meta directing functional groups from the following sets is :
Hydrocarbons Aromatic Hydrocarbons Ortho-Para vs Meta Directing
EasySingle correctIdea asked in 5 years
Q75
In the given reactions identify A and B $\mathrm{H_2 + A \xrightarrow{Pd/C}}$ [figure: alkene with $\mathrm{CH_3}$ (upper left) and $\mathrm{C_2H_5}$ (upper right) on the two carbons of C=C, and H (lower left), H (lower right)]…
Hydrocarbons Alkenes and Alkynes Stereoselective Alkyne Reduction
EasyOther · figureIdea asked in 3 years
Q76
Match List - I with List - II. List - I (Compound) : (A) Carbon tetrachloride, (B) Methylene chloride, (C) DDT, (D) Freons List - II (Use) : (I) Paint remover, (II) Refrigerators and air conditioners, (III) Fire extinguisher, (IV) Non Biodegradable…
Haloalkanes and Haloarenes
EasyMatch the list
Q77
Match List - I with List - II. List - I (Reactants) : (A) Phenol, Zn/ $\Delta$ , (B) Phenol, $\mathrm{CHCl_3}$ , NaOH, HCl, (C) Phenol, $\mathrm{CO_2}$ , NaOH, HCl, (D) Phenol, Conc. $\mathrm{HNO_3}$ List - II (Product) : (I) Salicylaldehyde, (II) Salicylic…
Alcohols, Phenols and Ethers Alcohols and Phenols Kolbe's Electrolytic Reaction
EasyMatch the listNew idea
Q78
$\mathrm{C_2H_5Br \xrightarrow{alc.\ KOH} A \xrightarrow[CCl_4]{Br_2} B \xrightarrow[Excess]{KCN} C \xrightarrow[Excess]{H_3O^+} D}$ Acid D formed in above reaction is :
Haloalkanes and Haloarenes
MediumOther · figure
Q79
Which among the followng has highest boiling point ?
Alcohols, Phenols and Ethers Alcohols and Phenols Hydrogen Bonding Trends in Alcohols
EasySingle correctIdea asked in 3 yearsAsked the year before
Q80
Given below are two statements : Statement (I) : Dimethyl glyoxime forms a six-membered covalent chelate when treated with $\mathrm{NiCl_2}$ solution in presence of $\mathrm{NH_4OH}$ . Statement (II) : Prussian blue precipitate contains iron both in $(+2)$ …
Coordination Compounds Cation Analysis Confirmatory Tests
MediumTwo statementsIdea asked in 4 yearsAsked the year before
Q81
10 mL of gaseous hydrocarbon on combustion gives 40 mL of $\mathrm{CO_2(g)}$ and 50 mL of water vapour. Total number of carbon and hydrogen atoms in the hydrocarbon is ________.
Some Basic Concepts of Chemistry Stoichiometry Eudiometry and Gas Mixture Analysis
EasyNumerical valueIdea asked in 2 years
Q82
For a certain reaction at 300K, K = 10, then $\Delta G^\circ$ for the same reaction is $-$ ________ $\times 10^{-1}$ kJ mol $^{-1}$ . (Given R = 8.314 J K $^{-1}$ mol $^{-1}$ )
Thermodynamics (Chemistry)Gibbs Free Energy Gibbs Energy and Equil
EasyNumerical valueIdea asked in 5 yearsAsked the year before
Q83
Mass of ethylene glycol (antifreeze) to be added to 18.6 kg of water to protect the freezing point at $-24^\circ\mathrm{C}$ is ________ kg (Molar mass in g mol $^{-1}$ for ethylene glycol 62, $\mathrm{K_f}$ of water = 1.86 K kg mol $^{-1}$ )
Solutions Colligative Properties Antifreeze Mass Calculations
EasyNumerical valueIdea asked in 2 years
Q84
The amount of electricity in Coulomb required for the oxidation of 1 mol of $\mathrm{H_2O}$ to $\mathrm{O_2}$ is ________ $\times 10^5$ C.
Electrochemistry Faraday's Laws of Electrolysis Water Consumption Stoichiometry
EasyNumerical valueNew idea
Q85
Consider the following redox reaction : $\mathrm{MnO_4^- + H^+ + H_2C_2O_4 \rightleftharpoons Mn^{2+} + H_2O + CO_2}$ The standard reduction potentials are given as below ( $\mathrm{E^\circ_{red}}$ ) : $\mathrm{E^\circ_{MnO_4^-/Mn^{2+}}} = +1.51$ V…
Electrochemistry Nernst Equation E°cell and Equilibrium
MediumNumerical valueIdea asked in 3 years
Q86
The following data were obtained during the first order thermal decomposition of a gas A at constant volume : $\mathrm{A(g) \rightarrow 2B(g) + C(g)}$ S.No. | Time/s | Total pressure/(atm) 1. | 0 | 0.1 2. | 115 | 0.28 The rate constant of the reaction is…
Chemical Kinetics Integrated Rate Equations Gas Kinetics with Complex Stoichiometry
MediumNumerical valueIdea asked in 5 yearsAsked the year before
Q87
Following Kjeldahl's method, 1g of organic compound released ammonia, that neutralised 10 mL of 2M $\mathrm{H_2SO_4}$ . The percentage of nitrogen in the compound is ________%.
Organic Chemistry: Some Basic Principles and Techniques Stoichiometry Stoichiometry of Kjeldahl Method
EasyNumerical valueIdea asked in 4 years
Q88
Total number of isomeric compounds (including stereoisomers) formed by monochlorination of 2-methylbutane is ________.
Hydrocarbons Alkanes and Free Radicals Stereochemistry of Free Radical Halogenation
MediumNumerical valueIdea asked in 2 years
Q89
Number of compounds which give reaction with Hinsberg's reagent is ________. [figure: eleven structures in three rows]
Amines Hinsberg Test
MediumNumerical value · figureIdea asked in 5 yearsAsked the year before
Q90
The number of tripeptides formed by three different amino acids using each amino acid once is ________.
Biomolecules
EasyNumerical value

Maths28 questions

1 easy21 medium6 hard

Q1
If the domain of the function $f(x)=\frac{\sqrt{x^2-25}}{(4-x^2)}+\log_{10}(x^2+2x-15)$ is $(-\infty,\alpha)\cup[\beta,\infty)$ , then $\alpha^2+\beta^3$ is equal to :
Relations and Functions Functions and Domain Square Root Function Domain
MediumSingle correctNew idea
Q2
Consider the relations $R_1$ and $R_2$ defined as $aR_1b \Leftrightarrow a^2+b^2=1$ for all $a, b\in\mathbf{R}$ and $(a,b)R_2(c,d)\Leftrightarrow a+d=b+c$ for all $(a,b),(c,d)\in\mathbf{N}\times\mathbf{N}$ . Then
Relations and Functions Sets and Relations Equivalence Relations
MediumSingle correctIdea asked in 4 years
Q4
Let $\alpha$ and $\beta$ be the roots of the equation $px^2+qx-r=0$ , where $p\neq 0$ . If $p$ , $q$ and $r$ be the consecutive terms of a non constant G.P. and $\frac{1}{\alpha}+\frac{1}{\beta}=\frac{3}{4}$ , then the value of $(\alpha-\beta)^2$ is :
Complex Numbers and Quadratic Equations Quadratic Roots Root-Coefficient Relation
MediumSingle correctIdea asked in 6 yearsAsked the year before
Q5
Let the system of equations $x+2y+3z=5$ , $2x+3y+z=9$ , $4x+3y+\lambda z=\mu$ have infinite number of solutions. Then $\lambda+2\mu$ is equal to :
Determinants Cramer's Rule Evaluating Parameters for Consistency
MediumSingle correctIdea asked in 6 yearsAsked the year before
Q6
Let $m$ and $n$ be the coefficients of seventh and thirteenth terms respectively in the expansion of $\left(\frac{1}{3}x^{\frac{1}{3}}+\frac{1}{2x^{\frac{2}{3}}}\right)^{18}$ . Then $\left(\frac{n}{m}\right)^{\frac{1}{3}}$ is :
Binomial Theorem
MediumSingle correct
Q7
Let $S_n$ denote the sum of the first $n$ terms of an arithmetic progression. If $S_{10}=390$ and the ratio of the tenth and the fifth terms is $15:7$ , then $S_{15}-S_5$ is equal to :
Sequences and Series AP and GP AP Sum of n terms
MediumSingle correctIdea asked in 7 yearsAsked the year before
Q8
Let $f(x)=|2x^2+5|x|-3|$ , $x\in\mathbf{R}$ . If $m$ and $n$ denote the number of points where $f$ is not continuous and not differentiable respectively, then $m+n$ is equal to :
Continuity and Differentiability Differentiability of Modulus Functions
MediumSingle correctIdea asked in 5 yearsAsked the year before
Q9
Let $f(x)=\begin{cases}x-1, & x \text{ is even,}\\ 2x, & x \text{ is odd,}\end{cases}$ $x\in\mathbf{N}$ . If for some $a\in\mathbf{N}$ , $f(f(f(a)))=21$ , then $\lim_{x\to a^-}\left\{\frac{|x|^3}{a}-\left[\frac{x}{a}\right]\right\}$ , where $[t]$ denotes the…
Limits and Derivatives L'Hopital's Rule Handling Greatest Integer Functions
MediumSingle correctIdea asked in 3 years
Q10
If $\int_0^{\frac{\pi}{3}}\cos^4x\,dx=a\pi+b\sqrt{3}$ , where $a$ and $b$ are rational numbers, then $9a+8b$ is equal to :
Integrals Definite Integral Properties Reduction Formulae in Definite Integrals
MediumSingle correctIdea asked in 4 yearsAsked the year before
Q11
The value of $\int_{-0}^{1}(2x^3-3x^2-x+1)^{\frac{1}{3}}\,dx$ is equal to :
Integrals Definite Integral Properties King's Property
MediumSingle correctIdea asked in 6 yearsAsked the year before
Q13
Let $P$ be a point on the ellipse $\frac{x^2}{9}+\frac{y^2}{4}=1$ . Let the line passing through $P$ and parallel to $y$ -axis meet the circle $x^2+y^2=9$ at point $Q$ such that $P$ and $Q$ are on the same side of the $x$ -axis. Then, the eccentricity of the…
Conic Sections Ellipse Eccentricity Eccentricity calculation
HardSingle correctIdea asked in 6 yearsAsked the year before
Q14
Let the locus of the midpoints of the chords of the circle $x^2+(y-1)^2=1$ drawn from the origin intersect the line $x+y=1$ at $P$ and $Q$ . Then, the length of $PQ$ is :
Conic Sections Circle Tangents Locus of Midpoint of Chords of Contact
MediumSingle correctNew idea
Q15
If the mirror image of the point $P(3,4,9)$ in the line $\frac{x-1}{3}=\frac{y+1}{2}=\frac{z-2}{1}$ is $(\alpha,\beta,\gamma)$ , then $14(\alpha+\beta+\gamma)$ is :
Three Dimensional Geometry Lines in 3D Image of a Point in a Line
MediumSingle correctIdea asked in 6 years
Q16
Let $P$ and $Q$ be the points on the line $\frac{x+3}{8}=\frac{y-4}{2}=\frac{z+1}{2}$ which are at a distance of $6$ units from the point $R(1,2,3)$ . If the centroid of the triangle $PQR$ is $(\alpha,\beta,\gamma)$ , then $\alpha^2+\beta^2+\gamma^2$ is :
Three Dimensional Geometry Lines in 3D Parametric Points at Distance r
MediumSingle correctNew idea
Q17
Consider a $\Delta ABC$ where $A(1,3,2)$ , $B(-2,8,0)$ and $C(3,6,7)$ . If the angle bisector of $\angle BAC$ meets the line $BC$ at $D$ , then the length of the projection of the vector $\overrightarrow{AD}$ on the vector $\overrightarrow{AC}$ is :
Vector Algebra Dot Product Scalar and Vector Projection
MediumSingle correctIdea asked in 5 years
Q18
Let Ajay will not appear in JEE exam with probability $p=\frac{2}{7}$ , while both Ajay and Vijay will appear in the exam with probability $q=\frac{1}{5}$ . Then the probability, that Ajay will appear in the exam and Vijay will not appear is :
Probability
EasySingle correct
Q19
Consider $10$ observations $x_1, x_2, \ldots, x_{10}$ such that $\sum_{i=1}^{10}(x_i-\alpha)=2$ and $\sum_{i=1}^{10}(x_i-\beta)^2=40$ , where $\alpha$ , $\beta$ are positive integers. Let the mean and the variance of the observations be $\frac{6}{5}$ and…
Statistics Variance and SD Variance via Arbitrary Point Deviations
MediumSingle correctIdea asked in 3 years
Q20
The number of solutions of the equation $4\sin^2x-4\cos^3x+9-4\cos x=0$ ; $x\in[-2\pi,2\pi]$ is :
Trigonometric Functions General Solutions Quadratic Substitution in Trigonometry
HardSingle correctIdea asked in 5 yearsAsked the year before
Q21
Let $A=I_2-2MM^T$ , where $M$ is a real matrix of order $2\times1$ such that the relation $M^TM=I_1$ holds. If $\lambda$ is a real number such that the relation $AX=\lambda X$ holds for some non-zero real matrix $X$ of order $2\times1$ , then the sum of…
Matrices
HardNumerical value
Q22
The lines $L_1, L_2, \ldots, L_{20}$ are distinct. For $n=1,2,3,\ldots,10$ all the lines $L_{2n-1}$ are parallel to each other and all the lines $L_{2n}$ pass through a given point $P$ . The maximum number of points of intersection of pairs of lines from the…
Permutations and Combinations Fundamental Counting Principle
MediumNumerical valueIdea asked in 5 yearsAsked the year before
Q23
If three successive terms of a G.P. with common ratio $r$ ( $r>1$ ) are the lengths of the sides of a triangle and $[r]$ denotes the greatest integer less than or equal to $r$ , then $3[r]+[-r]$ is equal to ________.
Sequences and Series AP and GP Conditions for Three Terms in AP or GP
MediumNumerical valueIdea asked in 5 years
Q24
Let $f:(0,\infty)\to\mathbf{R}$ and $F(x)=\int_0^x t f(t)\,dt$ . If $F(x^2)=x^4+x^5$ , then $\sum_{r=1}^{12}f(r^2)$ is equal to ________.
Integrals Leibniz Rule Solving Integral Equations
MediumNumerical valueIdea asked in 6 yearsAsked the year before
Q25
If $y=\frac{(\sqrt{x}+1)(x^2-\sqrt{x})}{x\sqrt{x}+x+\sqrt{x}}+\frac{1}{15}(3\cos^2x-5)\cos^3x$ , then $96\,y'\left(\frac{\pi}{6}\right)$ is equal to ________.
Continuity and Differentiability Chain Rule Chain Rule for Trigonometric Powers and Arguments
HardNumerical valueNew idea
Q26
The sum of squares of all possible values of $k$ , for which area of the region bounded by the parabolas $2y^2=kx$ and $ky^2=2(y-x)$ is maximum, is equal to ________.
Application of Integrals Area Under Curves Area Between Two Standard Parabolas
HardNumerical valueIdea asked in 4 years
Q27
If $\frac{dx}{dy}=\frac{1+x-y^2}{y}$ , $x(1)=1$ , then $5x(2)$ is equal to ________.
Differential Equations Linear Differential Equations Linear in x Form
MediumNumerical valueIdea asked in 4 yearsAsked the year before
Q28
Three points $O(0,0)$ , $P(a,a^2)$ , $Q(-b,b^2)$ , $a>0$ , $b>0$ , are on the parabola $y=x^2$ . Let $S_1$ be the area of the region bounded by the line $PQ$ and the parabola, and $S_2$ be the area of the triangle $OPQ$ . If the minimum value of $\frac{S_1}{S_2}$ …
Application of Integrals Area Under Curves Area of Parabolic Segment
HardNumerical valueNew idea
Q29
Let $ABC$ be an isosceles triangle in which $A$ is at $(-1,0)$ , $\angle A=\frac{2\pi}{3}$ , $AB=AC$ and $B$ is on the positive $x$ -axis. If $BC=4\sqrt{3}$ and the line $BC$ intersects the line $y=x+3$ at $(\alpha,\beta)$ , then $\frac{\beta^4}{\alpha^2}$ is…
Straight Lines
MediumNumerical value
Q30
Let $\vec{a}=\hat{i}+\hat{j}+\hat{k}$ , $\vec{b}=-\hat{i}-8\hat{j}+2\hat{k}$ and $\vec{c}=4\hat{i}+c_2\hat{j}+c_3\hat{k}$ be three vectors such that $\vec{b}\times\vec{a}=\vec{c}\times\vec{a}$ . If the angle between the vector $\vec{c}$ and the vector…
Vector Algebra Cross Product General Solution of a x x = b
MediumNumerical valueNew idea

Dropped from the final key and left out of every count: Q3, Q12.

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.