JEE Main 2024 · January session

JEE Main 30 January 2024, Shift 1

90 questions: 30 numerical-answer and 60 multiple choice. 10 were rated hard and 33 easy. Its hard-question count was the joint 8th highest of the 18 2024 shifts Lumi analysed. The biggest chapter was Organic Chemistry: Some Basic Principles and Techniques, with 4 questions.

Questions: 90; none dropped
Numerical answer: 30; 33% of the shift
Hard: 10; 11% of the shift
Chapters asked: 56; 20 questions with a figure

Key findings · 30 Jan 2024, Shift 1

1
10 hard questions (11%): the joint 8th 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 7 of 30 rated hard; Chemistry had the most easy questions (19 of 30).
3
17 questions tested a new idea, one no earlier year had asked: 22% of the 76 questions tagged to an idea.
4
29 came back from 2023: the same idea had been asked in the 2023 shifts Lumi analysed.
5
40 shared an idea with another 2024 shift (53% 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), Organic Chemistry: Some Basic Principles and Techniques in Chemistry (4) and Conic Sections in Maths (3).

The shift at a glance

Subjects, difficulty and chapters

Physics

30 Q · 10 numerical

Easy 14Medium 13Hard 3

Top chapters

Chemistry

30 Q · 10 numerical

Easy 19Medium 11Hard 0

Top chapters

Maths

30 Q · 10 numerical

Easy 0Medium 23Hard 7

Top chapters

Conic Sections3 Q
Integrals3 Q
Relations and Functions3 Q
Application of Derivatives2 Q
Complex Numbers and Quadratic Equations2 Q
Determinants2 Q

Question format

Single correct39 Q
Numerical value30 Q
Diagram based11 Q
Match the list3 Q
Other3 Q
Two statements2 Q
Assertion–reason2 Q

What each question asks you to do

Calculation41 Q
Multi-step15 Q
Conceptual12 Q
Recall11 Q
Application9 Q
Diagram reading1 Q
Data or graph1 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
Open30 this shift3065 · 33%33%8/829299715710354533200225

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

Every question

All 90 questions of this shift

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

Subject

Difficulty

Idea

Find

90 of 90 questions

Physics30 questions

14 easy13 medium3 hard

Q31
Match List - I with List - II. List - I List - II (A) Coefficient of viscosity (I) $[\mathrm{M\,L^2\,T^{-2}}]$ (B) Surface tension (II) $[\mathrm{M\,L^2\,T^{-1}}]$ (C) Angular momentum (III) $[\mathrm{M\,L^{-1}\,T^{-1}}]$ (D) Rotational kinetic energy (IV)…
Mechanical Properties of Fluids
EasyMatch the list
Q32
A particle of mass m is projected with a velocity 'u' making an angle of $30^\circ$ with the horizontal. The magnitude of angular momentum of the projectile about the point of projection when the particle is at its maximum height h is :
System of Particles and Rotational Motion Angular Momentum Angular Momentum of Projectile at Maximum Height
MediumSingle correctNew idea
Q33
A spherical body of mass 100 g is dropped from a height of 10 m from the ground. After hitting the ground, the body rebounds to a height of 5 m. The impulse of force imparted by the ground to the body is given by : (given, $g = 9.8\ \mathrm{m/s^2}$ )
Work, Energy and Power Collisions Impulse-Momentum
MediumSingle correctIdea asked in 2 years
Q34
All surfaces shown in figure are assumed to be frictionless and the pulleys and the string are light. The acceleration of the block of mass 2 kg is :
Laws of Motion Newton's Laws of Motion Movable Pulley Acceleration Formula
MediumDiagram based · figureNew idea
Q35
A particle is placed at the point A of a frictionless track ABC as shown in figure. It is gently pushed towards right. The speed of the particle when it reaches the point B is : (Take $g = 10\ \mathrm{m/s^2}$ ).
Work, Energy and Power Work and Kinetic Energy Conservation of Mechanical Energy
EasyDiagram based · figureIdea asked in 4 years
Q36
The gravitational potential at a point above the surface of earth is $-5.12\times10^{7}\ \mathrm{J/kg}$ and the acceleration due to gravity at that point is $6.4\ \mathrm{m/s^2}$ . Assume that the mean radius of earth to be 6400 km. The height of this point…
Gravitation Gravitational Potential Potential-Field Relation
MediumSingle correctIdea asked in 2 years
Q37
Young's modules of material of a wire of length 'L' and cross-sectional area A is Y. If the length of the wire is doubled and cross-sectional area is halved then Young's modules will be :
Mechanical Properties of Solids Elasticity Stress and Strain Types
EasySingle correctIdea asked in 4 yearsAsked the year before
Q38
Two thermodynamical processes are shown in the figure. The molar heat capacity for process A and B are $C_A$ and $C_B$ . The molar heat capacity at constant pressure and constant volume are represented by $C_P$ and $C_V$ , respectively. Choose the correct…
Thermodynamics Thermodynamic Processes Polytropic Process Analysis
HardDiagram based · figureIdea asked in 2 years
Q39
At which temperature the r.m.s. velocity of a hydrogen molecule equal to that of an oxygen molecule at $47\,^\circ\mathrm{C}$ ?
Kinetic Theory Kinetic Theory of Gases Equating Speeds of Different Gases
EasySingle correctNew idea
Q40
The electrostatic potential due to an electric dipole at a distance 'r' varies as :
Electrostatic Potential and Capacitance Electric Dipoles Electric Potential at a General Point
EasySingle correctNew idea
Q41
An electric toaster has resistance of 60 $\Omega$ at room temperature ( $27\,^\circ\mathrm{C}$ ). The toaster is connected to a 220 V supply. If the current flowing through it reaches 2.75 A, the temperature attained by toaster is around : ( if…
Current Electricity Ohm's Law and Resistance Temperature Coefficient of Resistance
MediumSingle correctIdea asked in 3 yearsAsked the year before
Q42
Two insulated circular loop A and B of radius 'a' carrying a current of 'I' in the anti clockwise direction as shown in the figure. The magnitude of the magnetic induction at the centre will be :
Moving Charges and Magnetism Biot-Savart Law Field at center of circular arc
MediumDiagram based · figureIdea asked in 6 yearsAsked the year before
Q43
A series L.R circuit connected with an ac source $E = (25\sin 1000\,t)$ V has a power factor of $\frac{1}{\sqrt{2}}$ . If the source of emf is changed to $E = (20\sin 2000\,t)$ V, the new power factor of the circuit will be :
Alternating Current Phasor Diagrams Power Factor from Phasors
MediumSingle correctIdea asked in 5 years
Q44
The electric field of an electromagnetic wave in free space is represented as $\vec{E} = E_0\cos(\omega t - kz)\,\hat{i}$ . The corresponding magnetic induction vector will be :
Electromagnetic Waves
EasySingle correct
Q45
The diffraction pattern of a light of wavelength 400 nm diffracting from a slit of width 0.2 mm is focused on the focal plane of a convex lens of focal length 100 cm. The width of the $1^{\mathrm{st}}$ secondary maxima will be :
Wave Optics Single Slit Diffraction Width of Secondary Maxima
MediumSingle correctNew idea
Q46
The work function of a substance is 3.0 eV. The longest wavelength of light that can cause the emission of photoelectrons from this substance is approximately;
Dual Nature of Radiation and Matter Photoelectric Effect Work Function and Threshold
EasySingle correctIdea asked in 4 yearsAsked the year before
Q47
The ratio of the magnitude of the kinetic energy to the potential energy of an electron in the $5^{\mathrm{th}}$ excited state of a hydrogen atom is :
Atoms Bohr's Atomic Model Energy Ratios in Bohr Orbits
EasySingle correctIdea asked in 3 years
Q48
A Zener diode of breakdown voltage 10 V is used as a voltage regulator as shown in the figure. The current through the Zener diode is :
Semiconductor Electronics: Materials, Devices and Simple Circuits Semiconductor Diodes Zener Diode Regulation
MediumDiagram based · figureIdea asked in 5 years
Q49
A potential divider circuit is shown in figure. The output voltage $V_0$ is :
Current Electricity Kirchhoff's Laws Voltage Divider Rule
EasyDiagram based · figureIdea asked in 2 years
Q50
Primary coil of a transformer is connected to 220 V ac. Primary and secondary turns of the transforms are 100 and 10 respectively. Secondary coil of transformer is connected to two series resistances shown in figure. The output voltage ( $V_0$ ) is :
Alternating Current Kirchhoff's Laws Voltage Divider Rule
EasyDiagram based · figureIdea asked in 2 years
Q51
The displacement and the increase in the velocity of a moving particle in the time interval of t to $(t+1)$ s are 125 m and 50 m/s, respectively. The distance travelled by the particle in $(t+2)^{\mathrm{th}}$ s is ________ m.
Motion in a Straight Line Kinematics in 1D Displacement in the nth Second
MediumNumerical valueIdea asked in 4 years
Q52
Consider a Disc of mass 5 kg, radius 2 m, rotating with angular velocity of 10 rad/s about an axis perpendicular to the plane of rotation. An identical disc is kept gently over the rotating disc along the same axis. The energy dissipated so that both the…
System of Particles and Rotational Motion Angular Momentum Kinetic Energy Loss in Rotational Coupling
EasyNumerical value · figureIdea asked in 3 years
Q53
Each of three blocks P, Q and R shown in figure has a mass of 3 kg. Each of the wires A and B has cross-sectional area $0.005\ \mathrm{cm^2}$ and Young's modulus $2\times10^{11}\ \mathrm{N\,m^{-2}}$ . Neglecting friction, the longitudinal strain on wire B is…
Mechanical Properties of Solids Elasticity Stress and Strain Types
HardNumerical value · figureIdea asked in 4 yearsAsked the year before
Q54
In a closed organ pipe, the frequency of fundamental note is 30 Hz. A certain amount of water is now poured in the organ pipe so that the fundamental frequency is increased to 110 Hz. If the organ pipe has a cross-sectional area of $2\ \mathrm{cm^2}$ , the…
Waves Resonance in Air Columns Closed Organ Pipe Harmonics
HardNumerical valueIdea asked in 4 yearsAsked the year before
Q55
A capacitor of capacitance C and potential V has energy E. It is connected to another capacitor of capacitance 2 C and potential 2 V. Then the loss of energy is $\frac{x}{3}E$ , where $x$ is ________.
Electrostatic Potential and Capacitance Parallel Plate Capacitors Charge Sharing and Heat Loss
MediumNumerical valueIdea asked in 6 yearsAsked the year before
Q56
Two cells are connected in opposition as shown. Cell $E_1$ is of 8 V emf and $2\ \Omega$ internal resistance; the cell $E_2$ is of 2 V emf and $4\ \Omega$ internal resistance. The terminal potential difference of cell $E_2$ is ________ V.
Current Electricity Kirchhoff's Laws Reversed Cells in Series
EasyNumerical value · figureIdea asked in 2 years
Q57
The horizontal component of earth's magnetic field at a place is $3.5\times10^{-5}$ T. A very long straight conductor carrying current of $\sqrt{2}$ A in the direction from South east to North West is placed. The force per unit length experienced by the…
Moving Charges and Magnetism
MediumNumerical value
Q58
A ceiling fan having 3 blades of length 80 cm each is rotating with an angular velocity of 1200 rpm. The magnetic field of earth in that region is 0.5 G and angle of dip is $30^\circ$ . The emf induced across the blades is N $\pi\times10^{-5}$ V. The value of…
Electromagnetic Induction Faraday's Law Rotational Motional EMF
MediumNumerical valueIdea asked in 4 yearsAsked the year before
Q59
The distance between object and its two times magnified real image as produced by a convex lens is 45 cm. The focal length of the lens used is ________ cm.
Ray Optics and Optical Instruments Lens Maker's Formula Lens Magnification
EasyNumerical valueIdea asked in 2 years
Q60
A electron of hydrogen atom on an excited state is having energy $E_n = -0.85$ eV. The maximum number of allowed transitions to lower energy level is ________.
Atoms Bohr's Atomic Model Number of Spectral Lines
EasyNumerical valueIdea asked in 3 yearsAsked the year before

Chemistry30 questions

19 easy11 medium0 hard

Q61
Given below are two statements : Statement (I) : The orbitals having same energy are called as degenerate orbitals. Statement (II) : In hydrogen atom, 3p and 3d orbitals are not degenerate orbitals. In the light of the above statements, choose the most…
Structure of Atom Quantum Numbers Degeneracy in Hydrogen-like Species
EasyTwo statementsNew idea
Q62
Match List - I with List - II. List - I (Molecule) List - II (Shape) (A) $\mathrm{BrF_5}$ (I) T-shape (B) $\mathrm{H_2O}$ (II) See saw (C) $\mathrm{ClF_3}$ (III) Bent (D) $\mathrm{SF_4}$ (IV) Square pyramidal Choose the correct answer from the options given…
Chemical Bonding and Molecular Structure VSEPR Theory Steric Number 5 Derivative Shapes
EasyMatch the listIdea asked in 3 years
Q63
What happens to freezing point of benzene when small quantity of napthalene is added to benzene ?
Solutions Colligative Properties Freezing Point Depression
EasySingle correctIdea asked in 4 years
Q64
Aluminium chloride in acidified aqueous solution forms an ion having geometry
The d- and f-Block Elements
EasySingle correct
Q65
Given below are two statements : one is labelled as Assertion (A) and the other is labelled as Reason (R). Assertion (A) : There is a considerable increase in covalent radius from N to P. However from As to Bi only a small increase in covalent radius is…
Classification of Elements and Periodicity in Properties Periodic Trends Atomic and Ionic Radii Trends
MediumAssertion–reasonIdea asked in 3 years
Q66
Match List - I with List - II. List - I (Species) List - II (Electronic distribution) (A) $\mathrm{Cr^{+2}}$ (I) $3d^8$ (B) $\mathrm{Mn^{+}}$ (II) $3d^3 4s^1$ (C) $\mathrm{Ni^{+2}}$ (III) $3d^4$ (D) $\mathrm{V^{+}}$ (IV) $3d^5 4s^1$ Choose the correct answer…
Structure of Atom Quantum Numbers Anomalous Electron Configurations
MediumMatch the listIdea asked in 4 years
Q67
Diamagnetic Lanthanoid ions are :
The d- and f-Block Elements f-Block Elements Magnetic and Spectral Properties
MediumSingle correctIdea asked in 2 years
Q68
Choose the correct statements from the following : (A) Ethane-1, 2-diamine is a chelating ligand. (B) Metallic aluminium is produced by electrolysis of aluminium oxide in presence of cryolite. (C) Cyanide ion is used as ligand for leaching of silver. (D)…
Coordination Compounds
MediumOther
Q69
The Lassiagne's extract is boiled with dil $\mathrm{HNO_3}$ before testing for halogens because,
Organic Chemistry: Some Basic Principles and Techniques
EasySingle correct
Q70
Structure of 4-Methylpent-2-enal is :
Organic Chemistry: Some Basic Principles and Techniques IUPAC Nomenclature Oxo vs Formyl Prefixes
EasySingle correct · figureNew idea
Q71
Which of the following molecule/species is most stable ?
Hydrocarbons Aromatic Hydrocarbons Huckel's Rule (4n+2)
MediumDiagram based · figureIdea asked in 6 yearsAsked the year before
Q72
Compound A formed in the following reaction reacts with B gives the product C. Find out A and B. $\mathrm{CH_3-C\equiv CH + Na \longrightarrow A \xrightarrow{B} CH_3-C\equiv C-CH_2-CH_2 + NaBr}$ (C) with a $\mathrm{CH_3}$ group drawn below the last…
Hydrocarbons Alkenes and Alkynes Acetylide Ion Alkylation
EasyOther · figureIdea asked in 2 years
Q73
Example of vinylic halide is :
Haloalkanes and Haloarenes SN1 and SN2 Mechanisms Inertness of Vinylic and Aryl Halides
EasyDiagram based · figureNew idea
Q74
Given below are two statements : one is labelled as Assertion (A) and the other is labelled as Reason (R). Assertion (A) : $\mathrm{CH_2=CH-CH_2-Cl}$ is an example of allyl halide. Reason (R) : Allyl halides are the compounds in which the halogen atom is…
Haloalkanes and Haloarenes SN1 and SN2 Mechanisms Reactivity of Allylic and Benzylic Halides
EasyAssertion–reasonIdea asked in 2 years
Q75
In the given reactions, identify the reagent A and reagent B. [Scheme: toluene reacts via two routes to give benzaldehyde. Route 1: "A" + $\mathrm{(CH_3CO)_2O}$ , 273-283 K, gives [Intermediate], then $\mathrm{H_3O^+}$ , $\Delta$ gives benzaldehyde (CHO).…
Aldehydes, Ketones and Carboxylic Acids Aromatic Hydrocarbons Side-chain Oxidation
MediumOther · figureIdea asked in 5 yearsAsked the year before
Q76
[Reaction: benzoyl chloride (benzene ring bearing C(=O)Cl) $\xrightarrow{\mathrm{H_2,\ Pd-BaSO_4}}$ benzaldehyde (benzene ring bearing CHO)] This reduction reaction is known as :
Aldehydes, Ketones and Carboxylic Acids Carboxylic Acids and Derivatives Rosenmund Reduction
EasySingle correct · figureIdea asked in 4 years
Q77
The final product A, formed in the following multistep reaction sequence is : [Scheme: bromobenzene $\xrightarrow{\text{(i) Mg, ether then }\mathrm{CO_2, H^+};\ \text{(ii) }\mathrm{NH_3, \Delta};\ \text{(iii) }\mathrm{Br_2, NaOH}}$ A]
Amines Hofmann Bromamide Degradation
MediumDiagram based · figureIdea asked in 5 yearsAsked the year before
Q78
Sugar which does not give reddish brown precipitate with Fehling's reagent, is :
Biomolecules Carbohydrates Non-Reducing Nature of Sucrose
EasySingle correctIdea asked in 2 years
Q79
Given below are two statements : Statement (I) : The gas liberated on warming a salt with dil $\mathrm{H_2SO_4}$ , turns a piece of paper dipped in lead acetate into black, it is a confirmatory test for sulphide ion. Statement (II) : In statement-I the colour…
Organic Chemistry: Some Basic Principles and Techniques Anion Analysis Sulfide Lead Acetate Test
EasyTwo statementsNew idea
Q80
Following is a confirmatory test for aromatic primary amines. Identify reagent (A) and (B). [Scheme: aniline ( $\mathrm{C_6H_5NH_2}$ ) $\xrightarrow{(A)}$ benzenediazonium chloride ( $\mathrm{C_6H_5N_2^+Cl^-}$ ) $\xrightarrow[\mathrm{NaOH}]{(B)}$ Scarlet red dye]
Amines Diazonium Salts Azo Coupling Reactions
EasyDiagram based · figureIdea asked in 5 yearsAsked the year before
Q81
The mass of sodium acetate ( $\mathrm{CH_3COONa}$ ) required to prepare 250 mL of 0.35 M aqueous solution is ________ g. (Molar mass of $\mathrm{CH_3COONa}$ is 82.02 g mol $^{-1}$ )
Some Basic Concepts of Chemistry Mole Concept Molarity Calculation
EasyNumerical valueIdea asked in 5 yearsAsked the year before
Q82
The total number of molecular orbitals formed from 2s and 2p atomic orbitals of a diatomic molecule is ________.
Chemical Bonding and Molecular Structure Molecular Orbital Theory LCAO and MO Formation
EasyNumerical valueNew idea
Q83
[Graph: V (dm $^3$ ) on the vertical axis (marks 10, 20, 30) versus P (kPa) on the horizontal axis (marks 10, 20, 30). Point A at P = 10 kPa, V = 10 dm $^3$ ; point B at P = 10 kPa, V = 30 dm $^3$ ; point C at P = 30 kPa, V = 10 dm $^3$ . Path A to B is vertical…
Thermodynamics (Chemistry)
MediumNumerical value · figure
Q84
The pH at which $\mathrm{Mg(OH)_2}$ [ $K_{sp} = 1\times10^{-11}$ ] begins to precipitate from a solution containing 0.10 M $\mathrm{Mg^{2+}}$ ions is ________.
Equilibrium Solubility Product Precipitation Condition
EasyNumerical valueIdea asked in 4 yearsAsked the year before
Q85
$\mathrm{2MnO_4^- + bI^- + cH_2O \rightarrow x\,I_2 + y\,MnO_2 + z\,\overline{O}H}$ If the above equation is balanced with integer coefficients, the value of $z$ is ________.
Redox Reactions Balancing in Neutral/Faintly Alkaline Media
MediumNumerical valueNew idea
Q86
0.05 cm thick coating of silver is deposited on a plate of 0.05 m $^2$ area. The number of silver atoms deposited on plate are ________ $\times 10^{23}$ . (At mass Ag = 108, d = 7.9 g cm $^{-3}$ )
Electrochemistry Faraday's Laws of Electrolysis Number of Deposited Atoms
MediumNumerical valueNew idea
Q87
The rate of First order reaction is 0.04 mol L $^{-1}$ s $^{-1}$ at 10 minutes and 0.03 mol L $^{-1}$ s $^{-1}$ at 20 minutes after initiation. Half life of the reaction is ________ minutes. (Given log2 = 0.3010, log3 = 0.4771)
Chemical Kinetics Integrated Rate Equations First Order Integrated Rate Law
MediumNumerical valueIdea asked in 6 years
Q88
If IUPAC name of an element is "Unununnium" then the element belongs to n $^{th}$ group of Periodic table. The value of n is ________.
Classification of Elements and Periodicity in Properties Periodic Trends IUPAC Nomenclature for Z > 100
EasyNumerical valueIdea asked in 3 years
Q89
On a thin layer chromatographic plate, an organic compound moved by 3.5 cm, while the solvent moved by 5 cm. The retardation factor of the organic compound is ________ $\times 10^{-1}$ .
Organic Chemistry: Some Basic Principles and Techniques
EasyNumerical value
Q90
The compound formed by the reaction of ethanal with semicarbazide contains ________ number of nitrogen atoms.
Aldehydes, Ketones and Carboxylic Acids Nucleophilic Addition Regioselectivity in Semicarbazide
EasyNumerical valueNew idea

Maths30 questions

0 easy23 medium7 hard

Q1
If the domain of the function $f(x)=\cos^{-1}\left(\frac{2-|x|}{4}\right)+\{\log_e(3-x)\}^{-1}$ is $[-\alpha,\beta)-\{\gamma\}$ , then $\alpha+\beta+\gamma$ is equal to :
Relations and Functions Functions and Domain Inverse Trig Function Domain
MediumSingle correctIdea asked in 4 yearsAsked the year before
Q2
If $z=x+iy,\ xy\neq0$ , satisfies the equation $z^2+i\bar{z}=0$ , then $|z^2|$ is equal to :
Complex Numbers and Quadratic Equations Complex Numbers - Polar Form Modulus and Argument
MediumSingle correctIdea asked in 5 yearsAsked the year before
Q3
Consider the system of linear equations $x+y+z=4\mu$ , $x+2y+2\lambda z=10\mu$ , $x+3y+4\lambda^2z=\mu^2+15$ , where $\lambda,\mu\in\mathbf{R}$ . Which one of the following statements is NOT correct ?
Determinants Cramer's Rule Evaluating Parameters for Consistency
MediumSingle correctIdea asked in 6 yearsAsked the year before
Q4
If $f(x)=\begin{vmatrix}2\cos^4x & 2\sin^4x & 3+\sin^22x\\ 3+2\cos^4x & 2\sin^4x & \sin^22x\\ 2\cos^4x & 3+2\sin^4x & \sin^22x\end{vmatrix}$ , then $\frac{1}{5}f'(0)$ is equal to :
Determinants Chain Rule Differentiating Determinants via Chain Rule
MediumSingle correctNew ideaIdea asked in 2 years
Q5
Two integers $x$ and $y$ are chosen with replacement from the set $\{0,1,2,3,\ldots,10\}$ . Then the probability that $|x-y|>5$ , is :
Probability
MediumSingle correct
Q6
Let $S_n$ denote the sum of first $n$ terms of an arithmetic progression. If $S_{20}=790$ and $S_{10}=145$ , then $S_{15}-S_5$ is :
Sequences and Series AP and GP AP Sum of n terms
MediumSingle correctIdea asked in 7 yearsAsked the year before
Q7
The maximum area of a triangle whose one vertex is at $(0,0)$ and the other two vertices lie on the curve $y=-2x^2+54$ at points $(x,y)$ and $(-x,y)$ , where $y>0$ , is :
Application of Derivatives Maxima and Minima Max/Min of Inscribed Geometric Figures
MediumSingle correctNew idea
Q8
Let $f:\left[-\frac{\pi}{2},\frac{\pi}{2}\right]\to\mathbf{R}$ be a differentiable function such that $f(0)=\frac{1}{2}$ . If $\lim_{x\to0}\frac{x\int_0^x f(t)dt}{e^{x^2}-1}=\alpha$ , then $8\alpha^2$ is equal to :
Integrals L'Hopital's Rule L'Hopital with Leibniz Rule
MediumSingle correctIdea asked in 2 years
Q9
Let $g:\mathbf{R}\to\mathbf{R}$ be a non constant twice differentiable function such that $g'\left(\frac{1}{2}\right)=g'\left(\frac{3}{2}\right)$ . If a real valued function $f$ is defined as $f(x)=\frac{1}{2}[g(x)+g(2-x)]$ , then
Application of Derivatives Rolle's Theorem Higher-Order Derivative Roots
HardSingle correctIdea asked in 3 years
Q10
The value of $\lim_{n\to\infty}\sum_{k=1}^{n}\frac{n^3}{(n^2+k^2)(n^2+3k^2)}$ is :
Integrals Integral as Limit of Sum Summation to Riemann Form
MediumSingle correctIdea asked in 4 yearsAsked the year before
Q11
The area (in square units) of the region bounded by the parabola $y^2=4(x-2)$ and the line $y=2x-8$ , is :
Application of Integrals Area Under Curves Area Between Parabola and Line
MediumSingle correctIdea asked in 4 years
Q12
Let $y=y(x)$ be the solution of the differential equation $\sec x\,dy+\{2(1-x)\tan x+x(2-x)\}dx=0$ such that $y(0)=2$ . Then $y(2)$ is equal to :
Differential Equations Linear Differential Equations Exact Differential Recognition in LDEs
MediumSingle correctIdea asked in 3 years
Q13
A line passing through the point $A(9,0)$ makes an angle of $30^\circ$ with the positive direction of $x$ -axis. If this line is rotated about $A$ through an angle of $15^\circ$ in the clockwise direction, then its equation in the new position is :
Straight Lines
MediumSingle correct
Q14
If the circles $(x+1)^2+(y+2)^2=r^2$ and $x^2+y^2-4x-4y+4=0$ intersect at exactly two distinct points, then
Conic Sections
MediumSingle correct
Q15
If the length of the minor axis of an ellipse is equal to half of the distance between the foci, then the eccentricity of the ellipse is :
Conic Sections Ellipse Eccentricity Eccentricity when Foci Distance Equals Minor Axis
MediumSingle correctNew idea
Q16
Let $(\alpha,\beta,\gamma)$ be the foot of perpendicular from the point $(1,2,3)$ on the line $\frac{x+3}{5}=\frac{y-1}{2}=\frac{z+4}{3}$ . Then $19(\alpha+\beta+\gamma)$ is equal to :
Three Dimensional Geometry Lines in 3D Foot of the Perpendicular
MediumSingle correctIdea asked in 6 yearsAsked the year before
Q17
Let $A(2,3,5)$ and $C(-3,4,-2)$ be opposite vertices of a parallelogram $ABCD$ . If the diagonal $\overrightarrow{BD}=\hat{i}+2\hat{j}+3\hat{k}$ , then the area of the parallelogram is equal to :
Vector Algebra Cross Product Area of Quadrilateral via Diagonals
MediumSingle correctIdea asked in 3 yearsAsked the year before
Q18
Let $\vec{a}=a_1\hat{i}+a_2\hat{j}+a_3\hat{k}$ and $\vec{b}=b_1\hat{i}+b_2\hat{j}+b_3\hat{k}$ be two vectors such that $|\vec{a}|=1$ , $\vec{a}\cdot\vec{b}=2$ and $|\vec{b}|=4$ . If $\vec{c}=2(\vec{a}\times\vec{b})-3\vec{b}$ , then the angle between $\vec{b}$ …
Vector Algebra Cross Product Magnitude and Sine Formula
HardSingle correctIdea asked in 7 yearsAsked the year before
Q19
Let $M$ denote the median of the following frequency distribution Class: $0-4$ , $4-8$ , $8-12$ , $12-16$ , $16-20$ Frequency: $3$ , $9$ , $10$ , $8$ , $6$ Then $20M$ is equal to :
Statistics
MediumSingle correct · figure
Q20
If $2\sin^3x+\sin2x\cos x+4\sin x-4=0$ has exactly 3 solutions in the interval $\left[0,\frac{n\pi}{2}\right]$ , $n\in N$ , then the roots of the equation $x^2+nx+(n-3)=0$ belong to :
Trigonometric Functions General Solutions Factorization Method
HardSingle correctNew idea
Q21
Let $A=\{1,2,3,\ldots,7\}$ and let $P(A)$ denote the power set of $A$ . If the number of functions $f:A\to P(A)$ such that $a\in f(a),\ \forall a\in A$ is $m^n$ , $m$ and $n\in N$ and $m$ is least, then $m+n$ is equal to __________.
Relations and Functions Permutations and Combinations Total Number of Functions
HardNumerical valueIdea asked in 4 yearsAsked the year before
Q22
Let $\alpha,\beta\in N$ be roots of the equation $x^2-70x+\lambda=0$ , where $\frac{\lambda}{2},\frac{\lambda}{3}\notin N$ . If $\lambda$ assumes the minimum possible value, then $\frac{(\sqrt{\alpha-1}+\sqrt{\beta-1})(\lambda+35)}{|\alpha-\beta|}$ is equal to :
Complex Numbers and Quadratic Equations Quadratic Roots Integer Roots Condition
MediumNumerical valueIdea asked in 2 years
Q23
Number of integral terms in the expansion of $\left\{7^{\left(\frac{1}{2}\right)}+11^{\left(\frac{1}{6}\right)}\right\}^{824}$ is equal to __________.
Binomial Theorem
HardNumerical value
Q24
Let $\alpha=1^2+4^2+8^2+13^2+19^2+26^2+\ldots$ upto 10 terms and $\beta=\sum_{n=1}^{10}n^4$ . If $4\alpha-\beta=55k+40$ , then $k$ is equal to __________.
Sequences and Series AP and GP Method of Differences for Telescoping Series
MediumNumerical valueIdea asked in 6 yearsAsked the year before
Q25
The value of $9\int_0^9\left[\sqrt{\frac{10x}{x+1}}\right]dx$ , where $[t]$ denotes the greatest integer less than or equal to $t$ , is __________.
Integrals Definite Integral Properties Greatest Integer Function Integrals
HardNumerical valueIdea asked in 7 yearsAsked the year before
Q26
Let $y=y(x)$ be the solution of the differential equation $(1-x^2)dy=\left[xy+(x^3+2)\sqrt{3(1-x^2)}\right]dx$ , $-1<x<1$ , $y(0)=0$ . If $y\left(\frac{1}{2}\right)=\frac{m}{n}$ , $m$ and $n$ are co-prime numbers, then $m+n$ is equal to __________.
Differential Equations Linear Differential Equations Integrating Factor Calculation
MediumNumerical valueIdea asked in 7 yearsAsked the year before
Q27
Let the latus ractum of the hyperbola $\frac{x^2}{9}-\frac{y^2}{b^2}=1$ subtend an angle of $\frac{\pi}{3}$ at the centre of the hyperbola. If $b^2$ is equal to $\frac{l}{m}(1+\sqrt{n})$ , where $l$ and $m$ are co-prime numbers, then $l^2+m^2+n^2$ is equal to…
Conic Sections
MediumNumerical value
Q28
If $d_1$ is the shortest distance between the lines $x+1=2y=-12z$ , $x=y+2=6z-6$ and $d_2$ is the shortest distance between the lines $\frac{x-1}{2}=\frac{y+8}{-7}=\frac{z-4}{5}$ , $\frac{x-1}{2}=\frac{y-2}{1}=\frac{z-6}{-3}$ , then the value of…
Three Dimensional Geometry Skew Lines Cartesian Shortest Distance
MediumNumerical valueIdea asked in 6 yearsAsked the year before
Q29
A group of 40 students appeared in an examination of 3 subjects - Mathematics, Physics and Chemistry. It was found that all students passed in atleast one of the subjects, 20 students passed in Mathematics, 25 students passed in Physics, 16 students passed…
Relations and Functions Sets and Relations Inclusion-Exclusion for Three Sets
HardNumerical valueIdea asked in 4 yearsAsked the year before
Q30
If the function $f(x)=\begin{cases}\frac{1}{|x|}, & |x|\geq2\\ ax^2+2b, & |x|<2\end{cases}$ is differentiable on $\mathbf{R}$ , then $48(a+b)$ is equal to __________.
Continuity and Differentiability Piecewise Differentiability
MediumNumerical valueIdea asked in 4 years

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.