JEE Main 2024 · January session

JEE Main 29 January 2024, Shift 1

89 questions (1 more dropped by NTA): 30 numerical-answer and 59 multiple choice. 9 were rated hard and 34 easy. Its hard-question count was the joint 12th highest of the 18 2024 shifts Lumi analysed. The biggest chapter was Organic Chemistry: Some Basic Principles and Techniques, with 4 questions.

Questions: 89; 1 dropped by NTA
Numerical answer: 30; 34% of the shift
Hard: 9; 10% of the shift
Chapters asked: 63; 7 questions with a figure

Key findings · 29 Jan 2024, Shift 1

1
9 hard questions (10%): the joint 12th 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 8 of 30 rated hard; Physics had the most easy questions (16 of 29).
3
21 questions tested a new idea, one no earlier year had asked: 28% of the 76 questions tagged to an idea.
4
25 came back from 2023: the same idea had been asked in the 2023 shifts Lumi analysed.
5
35 shared an idea with another 2024 shift (46% of tagged questions). Shifts in the same year can test the same idea in a new setting.
6
Biggest chapters: Current Electricity in Physics (2), Organic Chemistry: Some Basic Principles and Techniques in Chemistry (4) and Application of Derivatives in Maths (2).

The shift at a glance

Subjects, difficulty and chapters

Physics

29 Q · 10 numerical

Easy 16Medium 12Hard 1

Top chapters

Chemistry

30 Q · 10 numerical

Easy 16Medium 14Hard 0

Top chapters

Maths

30 Q · 10 numerical

Easy 2Medium 20Hard 8

Top chapters

Question format

Single correct47 Q
Numerical value30 Q
Diagram based5 Q
Two statements3 Q
Match the list2 Q
Assertion–reason2 Q

What each question asks you to do

Multi-step30 Q
Calculation29 Q
Application11 Q
Recall10 Q
Conceptual7 Q
Data or graph1 Q
Diagram reading1 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

16 easy12 medium1 hard

Q31
The resistance $R=\frac{V}{I}$ where $V=(200\pm5)$ V and $I=(20\pm0.2)$ A, the percentage error in the measurement of R is :
Units and Measurements Ohm's Law and Resistance Error Analysis in Ohm's Law
EasySingle correctIdea asked in 5 yearsAsked the year before
Q32
A body starts moving from rest with constant acceleration covers displacement $S_1$ in first $(p-1)$ seconds and $S_2$ in first $p$ seconds. The displacement $S_1+S_2$ will be made in time :
Motion in a Straight Line Kinematics in 1D Uniform Acceleration Equations
MediumSingle correctIdea asked in 4 yearsAsked the year before
Q33
If the radius of curvature of the path of two particles of same mass are in the ratio 3:4, then in order to have constant centripetal force, their velocities will be in the ratio of :
Motion in a Plane Circular Motion Centripetal Acceleration
EasySingle correctIdea asked in 4 yearsAsked the year before
Q34
A block of mass 100 kg slides over a distance of 10 m on a horizontal surface. If the co-efficient of friction between the surfaces is 0.4, then the work done against friction (in J) is :
Work, Energy and Power Work and Kinetic Energy Work Done by Friction
EasySingle correctIdea asked in 3 yearsAsked the year before
Q35
The potential energy function (in J) of a particle in a region of space is given as $U=(2x^2+3y^3+2z)$ . Here $x$ , $y$ and $z$ are in meter. The magnitude of $x$ - component of force (in N) acting on the particle at point $P(1,2,3)$ m is :
Work, Energy and Power Work and Kinetic Energy Potential Energy from Conservative Force
EasySingle correctNew idea
Q36
At what distance above and below the surface of the earth a body will have same weight. (take radius of earth as $R$ .)
Gravitation
MediumSingle correct
Q37
Given below are two statements: Statement I : If a capillary tube is immersed first in cold water and then in hot water, the height of capillary rise will be smaller in hot water. Statement II : If a capillary tube is immersed first in cold water and then in…
Mechanical Properties of Fluids Surface Tension Capillary Rise (Ascent Formula)
EasyTwo statementsIdea asked in 2 years
Q39
Two vessels A and B are of the same size and are at same temperature. A contains 1g of hydrogen and B contains 1g of oxygen. $P_A$ and $P_B$ are the pressures of the gases in A and B respectively, then $\frac{P_A}{P_B}$ is :
Kinetic Theory Kinetic Theory of Gases Avogadro's Hypothesis Application
EasySingle correctIdea asked in 2 years
Q40
Two charges of $5Q$ and $-2Q$ are situated at the points $(3a, 0)$ and $(-5a, 0)$ respectively. The electric flux through a sphere of radius ' $4a$ ' having center at origin is :
Electric Charges and Fields Gauss's Law Gauss Law Application
EasySingle correctIdea asked in 4 yearsAsked the year before
Q41
The electric current through a wire varies with time as $I=I_0+\beta t$ , where $I_0=20$ A and $\beta=3$ A/s. The amount of electric charge crossed through a section of the wire in 20 s is :
Current Electricity
EasySingle correct
Q42
A galvanometer having coil resistance $10\,\Omega$ shows a full scale deflection for a current of 3mA. For it to measure a current of 8A, the value of the shunt should be:
Moving Charges and Magnetism Wheatstone Bridge Shunting Galvanometer for Protection
EasySingle correctNew ideaIdea asked in 2 years
Q43
A capacitor of capacitance $100\ \mu$ F is charged to a potential of 12 V and connected to a 6.4 mH inductor to produce oscillations. The maximum current in the circuit would be :
Alternating Current LCR Resonance LC Energy Oscillation at Resonance
EasySingle correctIdea asked in 3 yearsAsked the year before
Q44
Match List I with List II LIST I A. $\oint\vec{B}\cdot d\vec{l}=\mu_o i_c+\mu_o\varepsilon_o\frac{d\phi_E}{dt}$ B. $\oint\vec{E}\cdot d\vec{l}=\frac{d\phi_B}{dt}$ C. $\oint\vec{E}\cdot d\vec{A}=\frac{Q}{\varepsilon_o}$ D. $\oint\vec{B}\cdot d\vec{A}=0$ LIST…
Electromagnetic Waves
EasyMatch the list
Q45
A biconvex lens of refractive index 1.5 has a focal length of 20 cm in air. Its focal length when immersed in a liquid of refractive index 1.6 will be:
Ray Optics and Optical Instruments Lens Maker's Formula Lens Immersed in a Liquid
MediumSingle correctIdea asked in 5 yearsAsked the year before
Q46
The de-Broglie wavelength of an electron is the same as that of a photon. If velocity of electron is 25% of the velocity of light, then the ratio of K.E. of electron and K.E. of photon will be:
Dual Nature of Radiation and Matter Matter Waves Electron vs Photon (Same Wavelength)
MediumSingle correctIdea asked in 3 years
Q47
The explosive in a Hydrogen bomb is a mixture of ${}_1\mathrm{H}^2$ , ${}_1\mathrm{H}^3$ and ${}_3\mathrm{Li}^6$ in some condensed form. The chain reaction is given by ${}_3\mathrm{Li}^6+{}_0\mathrm{n}^1\rightarrow{}_2\mathrm{He}^4+{}_1\mathrm{H}^3$ …
Nuclei Fission and Fusion Mass Defect to MeV Conversion
MediumSingle correctNew idea
Q48
In the given circuit, the breakdown voltage of the Zener diode is 3.0 V. What is the value of $I_z$ ?
Semiconductor Electronics: Materials, Devices and Simple Circuits Semiconductor Diodes Zener Diode Regulation
MediumDiagram based · figureIdea asked in 5 years
Q49
A convex mirror of radius of curvature 30 cm forms an image that is half the size of the object. The object distance is :
Ray Optics and Optical Instruments Spherical Mirrors Linear Magnification
EasySingle correctIdea asked in 3 years
Q50
The deflection in moving coil galvanometer falls from 25 divisions to 5 division when a shunt of $24\,\Omega$ is applied. The resistance of galvanometer coil will be :
Moving Charges and Magnetism Wheatstone Bridge Effect of Shunting the Galvanometer
MediumSingle correctNew idea
Q51
A ball rolls off the top of a stairway with horizontal velocity $u$ . The steps are 0.1 m high and 0.1 m wide. The minimum velocity $u$ with which that ball just hits the step 5 of the stairway will be $\sqrt{x}\ \mathrm{ms^{-1}}$ where $x$ = ______ [use g =…
Motion in a Plane Projectile Motion Horizontal Projection onto a Staircase
MediumNumerical valueNew idea
Q52
A cylinder is rolling down on an inclined plane of inclination $60^\circ$ . It's acceleration during rolling down will be $\frac{x}{\sqrt{3}}\ \mathrm{m/s^2}$ , where $x$ = ______ (use g = 10 m/s $^2$ ).
System of Particles and Rotational Motion
MediumNumerical value
Q53
In a test experiment on a model aeroplane in wind tunnel, the flow speeds on the upper and lower surfaces of the wings are $70\ \mathrm{ms^{-1}}$ and $65\ \mathrm{ms^{-1}}$ respectively. If the wing area is $2\ \mathrm{m^2}$ , the lift of the wing is ______…
Mechanical Properties of Fluids Fluid Dynamics Dynamic Lift and Magnus Effect
EasyNumerical valueNew idea
Q54
When the displacement of a simple harmonic oscillator is one third of its amplitude, the ratio of total energy to the kinetic energy is $\frac{x}{8}$ , where $x$ = ______.
Oscillations Energy in SHM Kinetic to Potential Energy Ratio
EasyNumerical valueIdea asked in 2 yearsAsked the year before
Q55
An electron is moving under the influence of the electric field of a uniformly charged infinite plane sheet S having surface charge density $+\sigma$ . The electron at $t$ =0 is at a distance of 1 m from S and has a speed of 1 m/s. The maximum value of…
Electric Charges and Fields Electric Field Field of an Infinite Non-Conducting Sheet
HardNumerical valueIdea asked in 4 years
Q56
A $16\,\Omega$ wire is bend to form a square loop. A 9V battery with internal resistance $1\,\Omega$ is connected across one of its sides. If a $4\,\mu F$ capacitor is connected across one of its diagonals, the energy stored by the capacitor will be…
Current Electricity Kirchhoff's Laws Uniform Wire Ring Resistance
MediumNumerical valueNew idea
Q57
The magnetic potential due to a magnetic dipole at a point on its axis situated at a distance of 20 cm from its center is $1.5\times10^{-5}\ T\,m$ . The magnetic moment of the dipole is ______ $A\,m^2$ . (Given : $\frac{\mu_o}{4\pi}=10^{-7}\ T\,m\,A^{-1}$ )
Magnetism and Matter Magnetic Dipole Moment Magnetic Field of a Short Dipole
EasyNumerical valueIdea asked in 3 years
Q58
A square loop of side 10 cm and resistance $0.7\,\Omega$ is placed vertically in east-west plane. A uniform magnetic field of $0.20\ T$ is set up across the plane in north east direction. The magnetic field is decreased to zero in 1 s at a steady rate. Then,…
Electromagnetic Induction Faraday's Law Faraday's Law Magnitude
EasyNumerical valueIdea asked in 6 yearsAsked the year before
Q59
In a double slit experiment shown in figure, when light of wavelength 400 nm is used, dark fringe is observed at $P$ . If D=0.2 m, the minimum distance between the slits $S_1$ and $S_2$ is ______ mm.
Wave Optics Young's Double Slit Exact Path Difference without Approximation
MediumNumerical value · figureNew idea
Q60
When a hydrogen atom going from $n$ =2 to $n$ =1 emits a photon, its recoil speed is $\frac{x}{5}\ m/s$ . Where $x$ = ______. (Use, mass of hydrogen atom = $1.6\times10^{-27}$ kg)
Atoms Bohr's Atomic Model Atom Recoil During Emission
MediumNumerical valueIdea asked in 2 years

Chemistry30 questions

16 easy14 medium0 hard

Q61
The correct set of four quantum numbers for the valence electron of rubidium atom (Z = 37) is :
Structure of Atom Quantum Numbers Quantum Numbers of the Differentiating Electron
EasySingle correctNew idea
Q62
Which of the following is not correct?
Thermodynamics (Chemistry)Gibbs Free Energy Spontaneity Criteria
EasySingle correctIdea asked in 2 years
Q63
Chlorine undergoes disproportionation in alkaline medium as shown below : $\mathrm{a\ Cl_{2(g)}+b\ OH^-_{(aq)}\rightarrow c\ ClO^-_{(aq)}+d\ Cl^-_{(aq)}+e\ H_2O_{(l)}}$ The values of a, b, c and d in a balanced redox reaction are respectively :
Redox Reactions Disproportionation Balancing
EasySingle correctIdea asked in 3 years
Q64
Given below are two statements : one is labelled as Assertion A and the other is labelled as Reason R : Assertion A : The first ionisation enthalpy decreases across a period. Reason R : The increasing nuclear charge outweighs the shielding across the period.…
Classification of Elements and Periodicity in Properties Periodic Trends Ionization Enthalpy Trends
EasyAssertion–reasonIdea asked in 6 years
Q65
Identify the incorrect pair from the following :
General Principles and Processes of Isolation of Elements
EasySingle correct
Q66
Given below are two statements : Statement I : The electronegativity of group 14 elements from Si to Pb, gradually decreases. Statement II : Group 14 contains non-metallic, metallic, as well as metalloid elements. In the light of the above statements, choose…
The p-Block Elements Periodic Trends Electronegativity Scales
MediumTwo statementsIdea asked in 4 yearsAsked the year before
Q67
$\mathrm{KMnO_4}$ decomposes on heating at 513K to form $\mathrm{O_2}$ along with
The d- and f-Block Elements d-Block Elements Thermal Decomposition of KMnO4
EasySingle correctNew idea
Q68
In alkaline medium, $\mathrm{MnO_4^-}$ oxidises $\mathrm{I^-}$ to
The d- and f-Block Elements d-Block Elements Specific KMnO4 Oxidations in Neutral/Alkaline Medium
MediumSingle correctIdea asked in 2 years
Q69
In which one of the following metal carbonyls, CO forms a bridge between metal atoms?
Coordination Compounds
MediumSingle correct
Q70
Match List I with List II LIST I (Substances) A. Ziegler catalyst B. Blood Pigment C. Wilkinson catalyst D. Vitamin $\mathrm{B_{12}}$ LIST II (Element Present) I. Rhodium II. Cobalt III. Iron IV. Titanium Choose the correct answer from the options given below:
Coordination Compounds d-Block Elements Specific d-Block Industrial Catalysts
EasyMatch the listIdea asked in 3 years
Q71
Appearance of blood red colour, on treatment of the sodium fusion extract of an organic compound with $\mathrm{FeSO_4}$ in presence of concentrated $\mathrm{H_2SO_4}$ indicates the presence of element/s
Organic Chemistry: Some Basic Principles and Techniques
MediumSingle correct
Q72
The interaction between $\pi$ bond and lone pair of electrons present on an adjacent atom is responsible for
Organic Chemistry: Some Basic Principles and Techniques Resonance and Hyperconjugation Mesomeric Effect
EasySingle correctIdea asked in 3 years
Q73
The difference in energy between the actual structure and the lowest energy resonance structure for the given compound is
Organic Chemistry: Some Basic Principles and Techniques Resonance and Hyperconjugation Relative Stability
EasySingle correctIdea asked in 2 years
Q74
Identify product A and product B :
Hydrocarbons Alkenes and Alkynes Allylic Halogenation with NBS
MediumDiagram based · figureIdea asked in 3 years
Q75
Given below are two statements : one is labelled as Assertion A and the other is labelled as Reason R : Assertion A : Aryl halides cannot be prepared by replacement of hydroxyl group of phenol by halogen atom. Reason R : Phenols react with halogen acids…
Haloalkanes and Haloarenes
MediumAssertion–reason
Q76
The final product A formed in the following multistep reaction sequence is
Aldehydes, Ketones and Carboxylic Acids Alkanes and Free Radicals Wolff-Kishner Reduction to Alkanes
MediumDiagram based · figureIdea asked in 3 yearsAsked the year before
Q77
The major product (P) in the following reaction is
Alcohols, Phenols and Ethers Ethers Cleavage of Alkyl-Aryl Ethers
MediumDiagram based · figureIdea asked in 3 years
Q78
Type of amino acids obtained by hydrolysis of proteins is :
Biomolecules
EasySingle correct
Q79
In chromyl chloride test for confirmation of $\mathrm{Cl^-}$ ion, a yellow solution is obtained. Acidification of the solution and addition of amyl alcohol and 10% $\mathrm{H_2O_2}$ turns organic layer blue indicating formation of chromium pentoxide. The…
Redox Reactions Anion Analysis Chromate Hydrogen Peroxide Test
MediumSingle correctNew idea
Q80
The arenium ion which is not involved in the bromination of Aniline is ________.
Amines Aromatic Hydrocarbons EAS Sigma Complex Mechanism
MediumDiagram based · figureNew idea
Q81
A solution of $\mathrm{H_2SO_4}$ is 31.4% $\mathrm{H_2SO_4}$ by mass and has a density of 1.25g/mL. The molarity of the $\mathrm{H_2SO_4}$ solution is ________ M (nearest integer) [Given molar mass of $\mathrm{H_2SO_4}$ = 98g mol $^{-1}$ ]
Some Basic Concepts of Chemistry Mole Concept Molarity Calculation
EasyNumerical valueIdea asked in 5 yearsAsked the year before
Q82
The number of species from the following which are paramagnetic and with bond order equal to one is ________. $\mathrm{H_2,He_2^+,O_2^+,N_2^{2-},O_2^{2-},F_2,Ne_2^+,B_2}$
Chemical Bonding and Molecular Structure Molecular Orbital Theory Bond Order Calculation
MediumNumerical valueIdea asked in 3 years
Q83
Number of compounds with one lone pair of electrons on central atom amongst following is ________ $\mathrm{O_3, H_2O, SF_4, ClF_3, NH_3, BrF_5, XeF_4}$
Chemical Bonding and Molecular Structure VSEPR Theory Steric Number 5 Derivative Shapes
MediumNumerical valueIdea asked in 3 years
Q84
The osmotic pressure of a dilute solution is $7\times10^{5}$ Pa at 273K. Osmotic pressure of the same solution at 283K is ________ $\times10^{4}$ Nm $^{-2}$ .
Solutions Colligative Properties Osmotic Pressure
EasyNumerical valueIdea asked in 4 yearsAsked the year before
Q85
For the reaction $\mathrm{N_2O_{4(g)} \rightleftharpoons 2NO_{2(g)}}$ , $\mathrm{K_p}$ = 0.492 atm at 300K. $\mathrm{K_c}$ for the reaction at same temperature is ________ $\times10^{-2}$ . (Given : R = 0.082 L atm mol $^{-1}$ K $^{-1}$ )
Equilibrium Chemical Equilibrium Kc and Kp Relationship
EasyNumerical valueIdea asked in 2 years
Q86
The mass of zinc produced by the electrolysis of zinc sulphate solution with a steady current of 0.015 A for 15 minutes is ________ $\times10^{-4}$ g. (Atomic mass of zinc = 65.4 amu)
Electrochemistry Faraday's Laws of Electrolysis Faraday's First Law
EasyNumerical valueIdea asked in 2 years
Q87
For a reaction taking place in three steps at same temperature, overall rate constant $K=\frac{K_1K_2}{K_3}$ . If $Ea_1$ , $Ea_2$ and $Ea_3$ are 40, 50 and 60 kJ/mol respectively, the overall Ea is ________ kJ/mol.
Chemical Kinetics Arrhenius Equation Effective Activation Energy for Complex Reactions
MediumNumerical valueNew idea
Q88
Number of compounds among the following which contain sulphur as heteroatom is ________. Furan, Thiophene, Pyridine, Pyrrole, Cysteine, Tyrosine
Organic Chemistry: Some Basic Principles and Techniques
EasyNumerical value
Q89
[figure: trans-2-butene drawn as CH $_3$ and H on the left carbon, H and CH $_3$ on the right carbon (CH $_3$ top-left, H bottom-left, H top-right, CH $_3$ bottom-right), reacting with (i) O $_3$ , (ii) Zn+H $_2$ O to give product (P)] Consider the given reaction.…
Hydrocarbons Alkenes and Alkynes Reductive Ozonolysis Products
EasyNumerical value · figureIdea asked in 7 yearsAsked the year before
Q90
From the compounds given below, number of compounds which give positive Fehling's test is ________. Benzaldehyde, Acetaldehyde, Acetone, Acetophenone, Methanal, 4-nitrobenzaldehyde, cyclohexane carbaldehyde.
Aldehydes, Ketones and Carboxylic Acids
MediumNumerical value

Maths30 questions

2 easy20 medium8 hard

Q1
If $f(x)=\begin{cases}2+2x, & -1\leq x<0\\ 1-\frac{x}{3}, & 0\leq x\leq 3\end{cases}$ ; $g(x)=\begin{cases}-x, & -3\leq x\leq 0\\ x, & 0<x\leq 1\end{cases}$ , then range of $(f\circ g)(x)$ is
Relations and Functions Functions and Domain Range of Composite Functions
MediumSingle correctIdea asked in 3 yearsAsked the year before
Q2
Let R be a relation on $\mathbb{Z}\times\mathbb{Z}$ defined by $(a, b)\,R\,(c, d)$ if and only if $ad-bc$ is divisible by 5. Then R is
Relations and Functions Sets and Relations Relation Types
MediumSingle correctIdea asked in 5 yearsAsked the year before
Q3
If $z=\frac{1}{2}-2i$ is such that $|z+1|=\alpha z+\beta(1+i)$ , $i=\sqrt{-1}$ and $\alpha,\beta\in\mathbb{R}$ , then $\alpha+\beta$ is equal to
Complex Numbers and Quadratic Equations Complex Numbers - Polar Form Modulus and Argument
EasySingle correctIdea asked in 5 yearsAsked the year before
Q4
Let A be a square matrix such that $AA^T=I$ . Then $\frac{1}{2}A\left[(A+A^T)^2+(A-A^T)^2\right]$ is equal to
Matrices Matrix Multiplication Orthogonal Matrix Multiplication Definition
MediumSingle correctIdea asked in 2 years
Q5
Let $A=\begin{bmatrix}1&0&0\\0&\alpha&\beta\\0&\beta&\alpha\end{bmatrix}$ and $|2A|^3=2^{21}$ where $\alpha,\beta\in\mathbb{Z}$ , Then a value of $\alpha$ is
Determinants
EasySingle correct
Q6
In an A.P., the sixth term $a_6=2$ . If the product $a_1a_4a_5$ is the greatest, then the common difference of the A.P. is equal to
Sequences and Series
HardSingle correct
Q7
If in a G.P. of 64 terms, the sum of all the terms is 7 times the sum of the odd terms of the G.P, then the common ratio of the G.P. is equal to
Sequences and Series AP and GP GP Sum of n terms
MediumSingle correctIdea asked in 5 years
Q8
$\lim_{x\to\frac{\pi}{2}}\left(\frac{1}{\left(x-\frac{\pi}{2}\right)^2}\int_{x^3}^{\left(\frac{\pi}{2}\right)^3}\cos\left(t^{\frac{1}{3}}\right)dt\right)$ is equal to
Limits and Derivatives L'Hopital's Rule L'Hopital with Leibniz Rule
MediumSingle correctIdea asked in 2 years
Q9
Suppose $f(x)=\frac{(2^x+2^{-x})\tan x\sqrt{\tan^{-1}(x^2-x+1)}}{(7x^2+3x+1)^3}$ . Then the value of $f'(0)$ is equal to
Continuity and Differentiability Chain Rule Product Rule with Chain Rule
HardSingle correctNew idea
Q10
Consider the function $f:\left[\frac{1}{2},1\right]\to\mathbb{R}$ defined by $f(x)=4\sqrt{2}x^3-3\sqrt{2}x-1$ . Consider the statements (I) The curve $y=f(x)$ intersects the x-axis exactly at one point. (II) The curve $y=f(x)$ intersects the x-axis at…
Application of Derivatives Curve Sketching Root Counting via Intersection Sketching
MediumTwo statementsIdea asked in 2 yearsAsked the year before
Q11
If the value of the integral $\int_{-\frac{\pi}{2}}^{\frac{\pi}{2}}\left(\frac{x^2\cos x}{1+\pi^x}+\frac{1+\sin^2x}{1+e^{\sin x^{2023}}}\right)dx=\frac{\pi}{4}(\pi+a)-2$ , then the value of $a$ is
Integrals Definite Integral Properties Symmetric Limits with Exponential Denominator Trick
HardSingle correctIdea asked in 5 years
Q12
For $x\in\left(-\frac{\pi}{2},\frac{\pi}{2}\right)$ , if $y(x)=\int\frac{\mathrm{cosec}\,x+\sin x}{\mathrm{cosec}\,x\sec x+\tan x\sin^2x}dx$ , and $\lim_{x\to\left(\frac{\pi}{2}\right)^-}y(x)=0$ then $y\left(\frac{\pi}{4}\right)$ is equal to
Integrals Integration by Substitution Odd Power of Cosine Substitution
HardSingle correctNew idea
Q13
A function $y=f(x)$ satisfies $f(x)\sin 2x+\sin x-(1+\cos^2x)f'(x)=0$ with condition $f(0)=0$ . Then, $f\left(\frac{\pi}{2}\right)$ is equal to
Differential Equations Linear Differential Equations Integrating Factors with Trigonometric Functions
MediumSingle correctIdea asked in 4 years
Q14
Let $\left(5,\frac{a}{4}\right)$ be the circumcenter of a triangle with vertices $A(a,-2)$ , $B(a,6)$ and $C\left(\frac{a}{4},-2\right)$ . Let $\alpha$ denote the circumradius, $\beta$ denote the area and $\gamma$ denote the perimeter of the triangle. Then…
Straight Lines Properties of Triangles Inradius and Circumradius
MediumSingle correctIdea asked in 3 years
Q15
In a $\Delta ABC$ , suppose $y=x$ is the equation of the bisector of the angle $B$ and the equation of the side $AC$ is $2x-y=2$ . If $2AB=BC$ and the points $A$ and $B$ are respectively $(4,6)$ and $(\alpha,\beta)$ , then $\alpha+2\beta$ is equal to
Straight Lines Straight Line Distance Image of a point in a line
HardSingle correctIdea asked in 4 years
Q16
Let PQR be a triangle with $R(-1,4,2)$ . Suppose $M(2,1,2)$ is the mid point of PQ. The distance of the centroid of $\Delta PQR$ from the point of intersection of the lines $\frac{x-2}{0}=\frac{y}{2}=\frac{z+3}{-1}$ and…
Three Dimensional Geometry Lines in 3D Point of Intersection Calculation
MediumSingle correctIdea asked in 6 yearsAsked the year before
Q17
Let $\vec{a},\vec{b}$ and $\vec{c}$ be three non-zero vectors such that $\vec{b}$ and $\vec{c}$ are non-collinear. If $\vec{a}+5\vec{b}$ is collinear with $\vec{c}$ , $\vec{b}+6\vec{c}$ is collinear with $\vec{a}$ and…
Vector Algebra Linear Independence Linear Combination Concept
MediumSingle correctNew idea
Q18
Let O be the origin and the position vectors of A and B be $2\hat{i}+2\hat{j}+\hat{k}$ and $2\hat{i}+4\hat{j}+4\hat{k}$ respectively. If the internal bisector of $\angle AOB$ meets the line AB at C, then the length of OC is
Vector Algebra Dot Product Angle Bisector Construction
MediumSingle correctNew idea
Q19
A fair die is thrown until 2 appears. Then the probability, that 2 appears in even number of throws, is
Probability Conditional Probability Conditional Probability in Infinite Games
MediumSingle correctIdea asked in 2 yearsAsked the year before
Q20
If $\alpha,-\frac{\pi}{2}<\alpha<\frac{\pi}{2}$ is the solution of $4\cos\theta+5\sin\theta=1$ , then the value of $\tan\alpha$ is
Trigonometric Functions General Solutions Linear Combination Solution
MediumSingle correctNew idea
Q21
Let $\alpha,\beta$ be the roots of the equation $x^2-x+2=0$ with $Im(\alpha)>Im(\beta)$ . Then $\alpha^6+\alpha^4+\beta^4-5\alpha^2$ is equal to ______.
Complex Numbers and Quadratic Equations Quadratic Roots Higher Degree Evaluation via Root Substitution
MediumNumerical valueIdea asked in 2 years
Q22
All the letters of the word "GTWENTY" are written in all possible ways with or without meaning and these words are written as in a dictionary. The serial number of the word "GTWENTY" is ______.
Permutations and Combinations Lexicographic Rank of a Word
MediumNumerical valueIdea asked in 4 yearsAsked the year before
Q23
If $\frac{{}^{11}C_1}{2}+\frac{{}^{11}C_2}{3}+\ldots+\frac{{}^{11}C_9}{10}=\frac{n}{m}$ with $\gcd(n,m)=1$ , then $n+m$ is equal to ______.
Binomial Theorem Binomial Coefficients Integration Method for Sums
MediumNumerical valueIdea asked in 3 yearsAsked the year before
Q24
Let $f(x)=2^x-x^2$ , $x\in\mathbb{R}$ . If $m$ and $n$ are respectively the number of points at which the curves $y=f(x)$ and $y=f'(x)$ intersect the x-axis, then the value of $m+n$ is ______.
Application of Derivatives Curve Sketching Graphical Root Counting
MediumNumerical valueIdea asked in 2 yearsAsked the year before
Q25
The area (in sq. units) of the part of the circle $x^2+y^2=169$ which is below the line $5x-y=13$ is $\frac{\pi\alpha}{2\beta}-\frac{65}{2}+\frac{\alpha}{\beta}\sin^{-1}\left(\frac{12}{13}\right)$ , where $\alpha,\beta$ are coprime numbers. Then…
Application of Integrals Area Under Curves Area of Ellipse and its Segments
HardNumerical valueNew idea
Q26
If the solution curve $y=y(x)$ of the differential equation $(1+y^2)(1+\log_e x)dx+x\,dy=0$ , $x>0$ passes through the point $(1,1)$ and $y(e)=\frac{\alpha-\tan\left(\frac{3}{2}\right)}{\beta+\tan\left(\frac{3}{2}\right)}$ , then $\alpha+2\beta$ is ______.
Differential Equations Variable Separable Form Initial Value Problem
MediumNumerical valueIdea asked in 7 yearsAsked the year before
Q27
Equations of two diameters of a circle are $2x-3y=5$ and $3x-4y=7$ . The line joining the points $\left(-\frac{22}{7},-4\right)$ and $\left(-\frac{1}{7},3\right)$ intersects the circle at only one point $P(\alpha,\beta)$ . Then, $17\beta-\alpha$ is equal to…
Conic Sections Circle Tangents Condition of tangency
MediumNumerical valueIdea asked in 4 years
Q28
If the points of intersection of two distinct conics $x^2+y^2=4b$ and $\frac{x^2}{16}+\frac{y^2}{b^2}=1$ lie on the curve $y^2=3x^2$ , then $3\sqrt{3}$ times the area of the rectangle formed by the intersection points is ______.
Conic Sections General Conic Section Family of Conics: Intersection of Two Conics
HardNumerical valueNew idea
Q29
A line with direction ratios $2,1,2$ meets the lines $x=y+2=z$ and $x+2=2y=2z$ respectively at the points P and Q. If the length of the perpendicular from the point $(1,2,12)$ to the line PQ is $l$ , then $l^2$ is ______.
Three Dimensional Geometry Lines in 3D Line Parallel to Vector Intersecting Two Lines
HardNumerical valueNew ideaIdea asked in 2 years
Q30
If the mean and variance of the data $65, 68, 58, 44, 48, 45, 60, \alpha, \beta, 60$ where $\alpha>\beta$ , are 56 and 66.2 respectively, then $\alpha^2+\beta^2$ is equal to ______.
Statistics Variance and SD Finding Unknown Observations from Mean and Variance
MediumNumerical valueIdea asked in 7 yearsAsked the year before

Dropped from the final key and left out of every count: 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 2024 shifts.