JEE Main 2025 · April session

JEE Main 3 April 2025, Shift 1

74 questions (1 more dropped by NTA): 15 numerical-answer and 59 multiple choice. 10 were rated hard and 21 easy. Its hard-question count was the 3rd highest of the 8 2025 shifts Lumi analysed. The biggest chapter was Organic Chemistry: Some Basic Principles and Techniques, with 4 questions.

Questions: 74; 1 dropped by NTA
Numerical answer: 15; 20% of the shift
Hard: 10; 14% of the shift
Chapters asked: 52; 17 questions with a figure

Key findings · 3 Apr 2025, Shift 1

1
10 hard questions (14%): the 3rd highest count of the 8 2025 shifts Lumi analysed. Across those shifts the hard share ran from 8% to 20%.
2
Maths was the hardest section, with 6 of 25 rated hard; Physics had the most easy questions (10 of 24).
3
16 questions tested a new idea, one no earlier year had asked: 25% of the 64 questions tagged to an idea.
4
37 came back from 2024: the same idea had been asked in the 2024 shifts Lumi analysed.
5
22 shared an idea with another 2025 shift (34% of tagged questions). Shifts in the same year can test the same idea in a new setting.
6
Biggest chapters: Thermodynamics 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

24 Q · 5 numerical

Easy 10Medium 10Hard 4

Top chapters

Chemistry

25 Q · 5 numerical

Easy 10Medium 15Hard 0

Top chapters

Maths

25 Q · 5 numerical

Easy 1Medium 18Hard 6

Top chapters

Conic Sections3 Q
Binomial Theorem2 Q
Complex Numbers and Quadratic Equations2 Q
Determinants2 Q
Integrals2 Q
Sequences and Series2 Q

Question format

Single correct37 Q
Numerical value15 Q
Diagram based12 Q
Other5 Q
Match the list3 Q
Two statements2 Q

What each question asks you to do

Multi-step28 Q
Calculation20 Q
Conceptual11 Q
Application8 Q
Recall3 Q
Data or graph2 Q
Diagram reading2 Q

Most common: Multi-step.

Against the other 2025 shifts

This shift next to the 7 other 2025 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
2 Apr, Shift 1	75	21%	12%	20%	21%	44%
2 Apr, Shift 2	75	31%	8%	20%	22%	33%
3 Apr, Shift 1	74	28%	14%	20%	25%	38%
3 Apr, Shift 2	75	29%	8%	20%	31%	36%
4 Apr, Shift 2	75	24%	20%	20%	21%	45%
7 Apr, Shift 1	75	32%	12%	20%	18%	43%
7 Apr, Shift 2	75	24%	12%	20%	25%	45%
8 Apr, Shift 2	75	27%	17%	20%	20%	40%

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

Open25 this shift3092 · 33%33%8/830299720720358540200225
Open25 this shift3065 · 33%33%8/829299715710354533200225
Open24 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 74 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

74 of 74 questions

Physics24 questions

10 easy10 medium4 hard

Q26
A person measures mass of 3 different particles as 435.42 g, 226.3 g and 0.125 g. According to the rules for arithmetic operations with significant figures, the addition of the masses of 3 particles will be.
Units and Measurements
EasySingle correct
Q27
Match the LIST-I with LIST-II LIST-I A. Gravitational constant B. Gravitational potential energy C. Gravitational potential D. Acceleration due to gravity LIST-II I. $[\mathrm{LT^{-2}}]$ II. $[\mathrm{L^2T^{-2}}]$ III. $[\mathrm{ML^2T^{-2}}]$ IV.…
Units and Measurements
EasyMatch the list
Q28
A particle is released from height S above the surface of the earth. At certain height its kinetic energy is three times its potential energy. The height from the surface of the earth and the speed of the particle at that instant are respectively.
Work, Energy and Power Work and Kinetic Energy Conservation of Mechanical Energy
EasySingle correctIdea asked in 4 yearsAsked the year before
Q29
The angle of projection of a particle is measured from the vertical axis as $\phi$ and the maximum height reached by the particle is $\mathrm{h_m}$ . Here $\mathrm{h_m}$ as function of $\phi$ can be presented as
Motion in a Plane Projectile Motion Ground-to-Ground Parameters
MediumDiagram based · figureIdea asked in 6 yearsAsked the year before
Q30
Which of the following curves possibly represent one-dimensional motion of a particle? A. [figure: Phase vs time, an inclined straight line crossing the time axis at a point near origin] B. [figure: Velocity vs displacement, a circle centred at O] C.…
Motion in a Straight Line Kinematics in 1D Impossible Kinematic Graphs
MediumDiagram based · figureNew idea
Q31
A force of 49 N acts tangentially at the highest point of a sphere (solid) of mass 20 kg, kept on a rough horizontal plane. If the sphere rolls without slipping, then the acceleration of the center of the sphere is
System of Particles and Rotational Motion
HardDiagram based · figure
Q32
Consider a completely full cylindrical water tank of height 1.6 m and of cross-sectional area $0.5\ \mathrm{m^2}$ . It has a small hole in its side at a height 90 cm from the bottom. Assume, the cross-sectional area of the hole to be negligibly small as…
Mechanical Properties of Fluids Fluid Dynamics Efflux from a Pressurized Tank
MediumSingle correctNew idea
Q33
During the melting of a slab of ice at 273 K at atmospheric pressure:
Thermodynamics Calorimetry Volume Change on Melting
MediumSingle correctNew idea
Q34
A gas is kept in a container having walls which are thermally non-conducting. Initially the gas has a volume of $800\ \mathrm{cm^3}$ and temperature $27^\circ\mathrm{C}$ . The change in temperature when the gas is adiabatically compressed to…
Thermodynamics First Law of Thermodynamics Adiabatic T-V Relation
EasySingle correctNew idea
Q35
A piston of mass M is hung from a massless spring whose restoring force law goes as $F = -kx^3$ , where k is the spring constant of appropriate dimension. The piston separates the vertical chamber into two parts, where the bottom part is filled with 'n' moles…
Thermodynamics First Law of Thermodynamics Work and Heat in Isothermal Processes
HardDiagram based · figureIdea asked in 3 yearsAsked the year before
Q36
Two blocks of masses m and M, $(M > m)$ , are placed on a frictionless table as shown in figure. A massless spring with spring constant k is attached with the lower block. If the system is slightly displaced and released, then ( $\mu$ = coefficient of friction…
Oscillations SHM Equations SHM Definition and Force Law
HardOther · figureIdea asked in 4 yearsAsked the year before
Q38
A parallel plate capacitor is filled equally(half) with two dielectrics of dielectric constants $\epsilon_1$ and $\epsilon_2$ , as shown in figures. The distance between the plates is d and area of each plate is A. If capacitance in first configuration and…
Electrostatic Potential and Capacitance Dielectrics Identifying Series vs Parallel Slab Configurations
MediumDiagram based · figureNew idea
Q39
The electrostatic potential on the surface of uniformly charged spherical shell of radius R = 10 cm is 120 V. The potential at the centre of shell, at a distance r = 5 cm from centre, and at a distance r = 15 cm from the centre of the shell respectively, are:
Electrostatic Potential and Capacitance Electric Potential Potential of a Spherical Shell
EasySingle correctIdea asked in 3 years
Q40
The radiation pressure exerted by a 450 W light source on a perfectly reflecting surface placed at 2m away from it, is
Electromagnetic Waves Thermal Radiation Thermal Radiation Pressure
MediumSingle correctIdea asked in 3 yearsAsked the year before
Q41
Consider following statements for refraction of light through prism, when angle of deviation is minimum. A. The refracted ray inside prism becomes parallel to the base. B. Larger angle prisms provide smaller angle of minimum deviation. C. Angle of incidence…
Ray Optics and Optical Instruments Prism Deviation Minimum Deviation Condition
MediumOtherIdea asked in 5 yearsAsked the year before
Q42
The radii of curvature for a thin convex lens are 10 cm and 15 cm respectively. The focal length of the lens is 12 cm. The refractive index of the lens material is
Ray Optics and Optical Instruments Lens Maker's Formula Lens Maker's Equation
EasySingle correctIdea asked in 3 yearsAsked the year before
Q43
The work function of a metal is 3 eV. The color of the visible light that is required to cause emission of photoelectrons is
Dual Nature of Radiation and Matter Photoelectric Effect Work Function and Threshold
EasySingle correctIdea asked in 4 yearsAsked the year before
Q44
Match the LIST-I with LIST-II LIST-I A. $^{1}_{0}\mathrm{n} + {}^{235}_{92}\mathrm{U} \rightarrow {}^{140}_{54}\mathrm{Xe} + {}^{94}_{38}\mathrm{Sr} + 2\,{}^{1}_{0}\mathrm{n}$ B. $2\mathrm{H_2} + \mathrm{O_2} \rightarrow 2\mathrm{H_2O}$ C.…
Nuclei Fission and Fusion Q-Value Sign Convention
MediumMatch the listNew idea
Q45
Choose the correct logic circuit for the given truth table having inputs A and B. Inputs A, B / Output Y: A=0,B=0: Y=0 A=0,B=1: Y=0 A=1,B=0: Y=1 A=1,B=1: Y=1
Semiconductor Electronics: Materials, Devices and Simple Circuits Logic Gates Combinational Logic Circuits
MediumDiagram based · figureIdea asked in 7 yearsAsked the year before
Q46
Three identical spheres of mass m, are placed at the vertices of an equilateral triangle of length $a$ . When released, they interact only through gravitational force and collide after a time $\mathrm{T} = 4$ seconds. If the sides of the triangle are…
Gravitation
HardNumerical value
Q47
A 4.0 cm long straight wire carrying a current of 8A is placed perpendicular to a uniform magnetic field of strength 0.15 T. The magnetic force on the wire is ______ mN.
Moving Charges and Magnetism
EasyNumerical value
Q48
A loop ABCDA, carrying current $\mathrm{I} = 12$ A, is placed in a plane, consists of two semi-circular segments of radius $\mathrm{R_1} = 6\pi$ m and $\mathrm{R_2} = 4\pi$ m. The magnitude of the resultant magnetic field at center O is $k\times10^{-7}$ T.…
Moving Charges and Magnetism Biot-Savart Law Field at center of circular arc
MediumNumerical value · figureIdea asked in 6 yearsAsked the year before
Q49
In the figure shown below, a resistance of $150.4\ \Omega$ is connected in series to an ammeter A of resistance $240\ \Omega$ . A shunt resistance of $10\ \Omega$ is connected in parallel with the ammeter. The reading of the ammeter is ______ mA.
Current Electricity Kirchhoff's Laws Current Division Rule
EasyNumerical value · figureIdea asked in 3 years
Q50
Two coherent monochromatic light beams of intensities 4I and 9I are superimposed. The difference between the maximum and minimum intensities in the resulting interference pattern is $x$ I. The value of $x$ is ______.
Wave Optics Young's Double Slit Resultant Intensity Formula
EasyNumerical valueIdea asked in 5 yearsAsked the year before

Chemistry25 questions

10 easy15 medium0 hard

Q51
Among $10^{-9}$ g (each) of the following elements, which one will have the highest number of atoms? Element: Pb, Po, Pr and Pt
Some Basic Concepts of Chemistry Mole Concept Mole and Avogadro Calculations
EasySingle correctIdea asked in 5 years
Q52
Which of the following postulate of Bohr's model of hydrogen atom is not in agreement with quantum mechanical model of an atom?
Structure of Atom Bohr's Atomic Theory Limitations of Bohr Model
EasySingle correctNew idea
Q53
Which of the following properties will change when system containing solution 1 will become solution 2? Solution 1: 10 mol of solute x + 10 L of water Solution 2: 1 L of solution 1 + 1 mol of solute x + 1 L of water
Thermodynamics (Chemistry)Gibbs Free Energy Extensive Nature of Gibbs Energy
MediumDiagram based · figureNew idea
Q54
2 moles each of ethylene glycol and glucose are dissolved in 500 g of water. The boiling point of the resulting solution is: (Given : Ebullioscopic constant of water = $0.52\ \mathrm{K\,kg\,mol^{-1}}$ )
Solutions Colligative Properties Boiling Point Elevation
EasySingle correctIdea asked in 6 yearsAsked the year before
Q55
In the following system, $\mathrm{PCl_5(g)} \rightleftharpoons \mathrm{PCl_3(g)} + \mathrm{Cl_2(g)}$ at equilibrium, upon addition of xenon gas at constant T & p, the concentration of
Equilibrium Chemical Equilibrium Inert Gas Effect at Constant Pressure
EasySingle correctNew idea
Q56
Correct order of limiting molar conductivity for cations in water at 298 K is :
Electrochemistry
MediumSingle correct
Q57
Given below are two statements: Statement I : A catalyst cannot alter the equilibrium constant ( $\mathrm{K_c}$ ) of the reaction, temperature remaining constant. Statement II : A homogenous catalyst can change the equilibrium composition of a system,…
Equilibrium Chemical Equilibrium Effect of Catalyst on Equilibrium
EasyTwo statementsNew idea
Q58
In a reaction $\mathrm{A} + \mathrm{B} \rightarrow \mathrm{C}$ , initial concentrations of A and B are related as $[\mathrm{A}]_0 = 8[\mathrm{B}]_0$ . The half lives of A and B are 10 min and 40 min, respectively. If they start to disappear at the same time,…
Chemical Kinetics Integrated Rate Equations Concentration After n Half-Lives
MediumSingle correctNew ideaIdea asked in 2 years
Q59
Match the LIST-I with LIST-II LIST-I (Molecules/ ion) A. $\mathrm{PF_5}$ B. $\mathrm{SF_6}$ C. $\mathrm{Ni(CO)_4}$ D. $\mathrm{[PtCl_4]^{2-}}$ LIST-II (Hybridisation of central atom) I. $dsp^2$ II. $sp^3d$ III. $sp^3d^2$ IV. $sp^3$ Choose the correct answer…
Chemical Bonding and Molecular Structure
MediumMatch the list
Q60
Which of the following statements are correct ? A. The process of adding an electron to a neutral gaseous atom is always exothermic. B. The process of removing an electron from an isolated gaseous atom is always endothermic. C. The $1^{\mathrm{st}}$ …
Classification of Elements and Periodicity in Properties Periodic Trends Ionization Enthalpy Trends
MediumOtherIdea asked in 6 yearsAsked the year before
Q61
Given below are two statements: Statement I : The N - N single bond is weaker and longer than that of P - P single bond. Statement II : Compounds of group 15 elements in + 3 oxidation states readily undergo disproportionation reactions. In the light of the…
The p-Block Elements Group 15 & 16 Anomalous Properties of Nitrogen
MediumTwo statementsIdea asked in 3 years
Q62
The metal ions that have the calculated spin-only magnetic moment value of 4.9 B.M. are : A. $\mathrm{Cr^{2+}}$ B. $\mathrm{Fe^{2+}}$ C. $\mathrm{Fe^{3+}}$ D. $\mathrm{Co^{2+}}$ E. $\mathrm{Mn^{3+}}$ Choose the correct answer from the options given below:
The d- and f-Block Elements d-Block Elements Spin-only Magnetism
EasyOtherIdea asked in 6 yearsAsked the year before
Q63
The correct order of the complexes $\mathrm{[Co(NH_3)_5(H_2O)]^{3+}}$ (A), $\mathrm{[Co(NH_3)_6]^{3+}}$ (B), $\mathrm{[Co(CN)_6]^{3-}}$ (C) and $\mathrm{[CoCl(NH_3)_5]^{2+}}$ (D) in terms of wavelength of light absorbed is
Coordination Compounds Crystal Field Theory Color and d-d Transitions
MediumSingle correctIdea asked in 5 yearsAsked the year before
Q64
Identify the correct statements from the following. A. [structure: pentan-3-one (skeletal, C=O at the middle carbon)] and [structure: pentan-2-one (skeletal, C=O at the second carbon)] are metamers B. [structure: skeletal chain ending in -CN] and [structure:…
Organic Chemistry: Some Basic Principles and Techniques Structural Isomerism Metamerism
MediumOther · figureIdea asked in 3 yearsAsked the year before
Q65
Which compound would give 3-methyl-6-oxoheptanal upon ozonolysis?
Hydrocarbons Alkenes and Alkynes Reductive Ozonolysis Products
MediumDiagram based · figureIdea asked in 7 yearsAsked the year before
Q66
Identify [A], [B] and [C], respectively in the following reaction sequence: [A] --(NaNO $_2$ , HCl, 273 - 278 K)--> benzenediazonium chloride ( $\mathrm{C_6H_5N_2^{+}Cl^{-}}$ ) --(KI)--> [B] --(2Na, Dry ether)--> [C]
Amines Diazonium Salts Synthesis of Iodobenzene
EasyDiagram based · figureIdea asked in 2 years
Q67
The least acidic compound, among the following is: (A) [structure: 3-ethoxyphenol, EtO and OH meta on a benzene ring] (B) [structure: benzenesulfonic acid, $\mathrm{C_6H_5SO_3H}$ ] (C) [structure: skeletal CH3CH2CH2-COOH (butanoic acid)] (D) [structure:…
Organic Chemistry: Some Basic Principles and Techniques Organic Acidity and Basicity Acidity of Terminal Alkynes
MediumDiagram based · figureNew idea
Q68
Number of molecules from below which cannot give iodoform reaction is: Ethanol, Isopropyl alcohol, Bromoacetone, 2 $-$ Butanol, 2 $-$ Butanone, Butanal, 2 $-$ Pentanone, 3 $-$ Pentanone, Pentanal and 3 $-$ Pentanol.
Aldehydes, Ketones and Carboxylic Acids Alcohols and Phenols Iodoform Test for Specific Alcohols
MediumSingle correctIdea asked in 4 yearsAsked the year before
Q69
In the following reactions, which one is NOT correct?
Amines Diazonium Salts Reduction to Benzene via Ethanol
EasyDiagram based · figureIdea asked in 4 years
Q70
Which of the following is the correct structure of L $-$ Fructose?
Biomolecules Carbohydrates D and L Configuration of Sugars
MediumDiagram based · figureIdea asked in 3 yearsAsked the year before
Q71
Given : $\Delta H^{\ominus}_{sub}$ [C (graphite)] = 710 kJ mol $^{-1}$ $\Delta_{C-H}H^{\ominus}$ = 414 kJ mol $^{-1}$ $\Delta_{H-H}H^{\ominus}$ = 436 kJ mol $^{-1}$ $\Delta_{C=C}H^{\ominus}$ = 611 kJ mol $^{-1}$ The $\Delta H_f^{\ominus}$ for…
Thermodynamics (Chemistry)Enthalpy and Hess's Law Bond Enthalpy Calculations
MediumNumerical valueIdea asked in 4 yearsAsked the year before
Q72
The number of optical isomers exhibited by the iron complex (A) obtained from the following reaction is ______. $\mathrm{FeCl_3 + KOH + H_2C_2O_4 \rightarrow A}$
Coordination Compounds Coordination Isomerism Optical Isomerism in Chelates
MediumNumerical valueNew idea
Q73
Consider the following reactions $\mathrm{A + NaCl + H_2SO_4 \rightarrow CrO_2Cl_2 + Side\ Products}$ (A: little amount) $\mathrm{CrO_2Cl_2}$ (Vapour) $+ \mathrm{NaOH} \rightarrow \mathrm{B + NaCl + H_2O}$ $\mathrm{B + H^+ \rightarrow C + H_2O}$ The number…
The d- and f-Block Elements d-Block Elements Chromyl Chloride Test
MediumNumerical valueNew idea
Q74
0.5 g of an organic compound on combustion gave 1.46 g of $\mathrm{CO_2}$ and 0.9 g of $\mathrm{H_2O}$ . The percentage of carbon in the compound is ______. (Nearest integer) [Given : Molar mass (in $\mathrm{g\,mol^{-1}}$ ) C : 12, H : 1, O : 16]
Organic Chemistry: Some Basic Principles and Techniques Stoichiometry Empirical Formula from Combustion Data
EasyNumerical valueIdea asked in 4 years
Q75
During estimation of nitrogen by Dumas' method of compound X (0.42 g) [structure X: piperazine, six-membered ring with two NH groups at opposite (1,4) positions] ______ mL of $\mathrm{N_2}$ gas will be liberated at STP. (nearest integer) (Given molar mass in…
Organic Chemistry: Some Basic Principles and Techniques Stoichiometry Stoichiometry of Dumas Method
EasyNumerical value · figureIdea asked in 4 years

Maths25 questions

1 easy18 medium6 hard

Q1
Let $A = \{-3, -2, -1, 0, 1, 2, 3\}$ . Let R be a relation on A defined by $x\mathrm{R}y$ if and only if $0 \le x^2 + 2y \le 4$ . Let $l$ be the number of elements in R and $m$ be the minimum number of elements required to be added in R to make it a reflexive…
Relations and Functions Sets and Relations Counting Relations
MediumSingle correctIdea asked in 6 yearsAsked the year before
Q2
If the domain of the function $f(x) = \log_e\left(\frac{2x-3}{5+4x}\right) + \sin^{-1}\left(\frac{4+3x}{2-x}\right)$ is $[\alpha, \beta)$ , then $\alpha^2 + 4\beta$ is equal to
Inverse Trigonometric Functions Functions and Domain Inverse Trig Function Domain
MediumSingle correctIdea asked in 4 yearsAsked the year before
Q3
Let $\alpha$ and $\beta$ be the roots of $x^2 + \sqrt{3}x - 16 = 0$ , and $\gamma$ and $\delta$ be the roots of $x^2 + 3x - 1 = 0$ . If $P_n = \alpha^n + \beta^n$ and $Q_n = \gamma^n + \delta^n$ , then…
Complex Numbers and Quadratic Equations Quadratic Roots Newton's Sums
MediumSingle correctIdea asked in 7 yearsAsked the year before
Q4
Let $z \in \mathbb{C}$ be such that $\frac{z^2 + 3i}{z - 2 + i} = 2 + 3i$ . Then the sum of all possible values of $z^2$ is
Complex Numbers and Quadratic Equations Quadratic Roots Symmetric Functions
MediumSingle correctIdea asked in 5 yearsAsked the year before
Q5
Let $A$ be a matrix of order $3 \times 3$ and $|A| = 5$ . If $|2\,adj(3A\,adj(2A))| = 2^{\alpha}\cdot 3^{\beta}\cdot 5^{\gamma}$ , $\alpha, \beta, \gamma \in \mathbb{N}$ , then $\alpha + \beta + \gamma$ is equal to
Determinants Matrix Inverse Trap involving |adj(A)| vs |A^-1|
MediumSingle correctIdea asked in 6 yearsAsked the year before
Q6
Let $a_1, a_2, a_3, \ldots$ be a G.P. of increasing positive numbers. If $a_3 a_5 = 729$ and $a_2 + a_4 = \frac{111}{4}$ , then $24\,(a_1 + a_2 + a_3)$ is equal to
Sequences and Series AP and GP GP General Term
EasySingle correctIdea asked in 5 yearsAsked the year before
Q7
The sum $1 + 3 + 11 + 25 + 45 + 71 + \ldots$ upto 20 terms, is equal to
Sequences and Series AP and GP Method of Differences for Telescoping Series
MediumSingle correctIdea asked in 6 yearsAsked the year before
Q8
If $\sum_{r=1}^{9}\left(\frac{r+3}{2^r}\right)\cdot {}^9C_r = \alpha\left(\frac{3}{2}\right)^9 - \beta$ , $\alpha, \beta \in \mathbb{N}$ , then $(\alpha + \beta)^2$ is equal to
Binomial Theorem Binomial Coefficients Binomial Coefficients with AP Multiplier
MediumSingle correctIdea asked in 2 years
Q9
The sum of all rational terms in the expansion of $\left(2+\sqrt{3}\right)^8$ is
Binomial Theorem
MediumSingle correct
Q10
The number of solutions of the equation $2x + 3\tan x = \pi$ , $x \in [-2\pi, 2\pi] - \left\{\pm\frac{\pi}{2}, \pm\frac{3\pi}{2}\right\}$ is:
Trigonometric Functions General Solutions Graphical Root Counting for Mixed Equations
MediumSingle correctIdea asked in 4 yearsAsked the year before
Q11
The radius of the smallest circle which touches the parabolas $y = x^2 + 2$ and $x = y^2 + 2$ is
Conic Sections Normals to Conics Shortest Distance via Common Normal
HardSingle correctNew idea
Q12
A line passing through the point $P(\sqrt{5}, \sqrt{5})$ intersects the ellipse $\frac{x^2}{36} + \frac{y^2}{25} = 1$ at $A$ and $B$ such that $(PA)\cdot(PB)$ is maximum. Then $5(PA^2 + PB^2)$ is equal to :
Conic Sections
HardSingle correct
Q13
A line passes through the origin and makes equal angles with the positive coordinate axes. It intersects the lines $\mathrm{L}_1 : 2x + y + 6 = 0$ and $\mathrm{L}_2 : 4x + 2y - p = 0,\ p > 0$ , at the points A and B, respectively. If…
Straight Lines Straight Line Distance Distance between parallel lines
MediumSingle correctIdea asked in 4 yearsAsked the year before
Q14
Line $\mathrm{L}_1$ passes through the point $(1, 2, 3)$ and is parallel to $z$ -axis. Line $\mathrm{L}_2$ passes through the point $(\lambda, 5, 6)$ and is parallel to $y$ -axis. Let for $\lambda = \lambda_1, \lambda_2$ , $\lambda_2 < \lambda_1$ , the shortest…
Three Dimensional Geometry Skew Lines Solve for Parameter Given Shortest Distance
MediumSingle correctNew idea
Q15
Let a line passing through the point $(4, 1, 0)$ intersect the line $\mathrm{L}_1 : \frac{x-1}{2} = \frac{y-2}{3} = \frac{z-3}{4}$ at the point $A(\alpha, \beta, \gamma)$ and the line $\mathrm{L}_2 : x - 6 = y = -z + 4$ at the point $B(a, b, c)$ . Then…
Three Dimensional Geometry Lines in 3D Line Through Point Intersecting Two Skew Lines
HardSingle correctIdea asked in 2 yearsAsked the year before
Q16
If $y(x) = \begin{vmatrix} \sin x & \cos x & \sin x + \cos x + 1 \\ 27 & 28 & 27 \\ 1 & 1 & 1 \end{vmatrix}$ , $x \in \mathbb{R}$ , then $\frac{d^2y}{dx^2} + y$ is equal to
Determinants Chain Rule Differentiating Determinants via Chain Rule
MediumSingle correctIdea asked in 2 yearsAsked the year before
Q17
Let $f(x) = \begin{cases} (1+ax)^{1/x}, & x < 0 \\ 1+b, & x = 0 \\ \dfrac{(x+4)^{1/2} - 2}{(x+c)^{1/3} - 2}, & x > 0 \end{cases}$ be continuous at $x = 0$ . Then $e^a bc$ is equal to:
Continuity and Differentiability Finding Unknowns using Continuity
MediumSingle correctIdea asked in 3 years
Q18
Let $f(x) = \int x^3\sqrt{3 - x^2}\,dx$ . If $5f(\sqrt{2}) = -4$ , then $f(1)$ is equal to
Integrals Integration by Substitution Composite Function Substitution
MediumSingle correctIdea asked in 3 yearsAsked the year before
Q19
Let the domain of the function $f(x) = \log_2 \log_4 \log_6 (3 + 4x - x^2)$ be $(a, b)$ . If $\int_0^{b-a} [x^2]\,dx = p - \sqrt{q} - \sqrt{r}$ , $p, q, r \in \mathbb{N}$ , $\gcd(p, q, r) = 1$ , where $[\cdot]$ is the greatest integer function, then $p + q + r$ …
Integrals Definite Integral Properties Greatest Integer Function Integrals
HardSingle correctIdea asked in 7 yearsAsked the year before
Q20
Let $g$ be a differentiable function such that $\int_0^x g(t)\,dt = x - \int_0^x t\,g(t)\,dt$ , $x \ge 0$ and let $y = y(x)$ satisfy the differential equation $\frac{dy}{dx} - y\tan x = 2(x+1)\sec x\; g(x)$ , $x \in \left[0, \frac{\pi}{2}\right)$ . If…
Differential Equations Linear Differential Equations Integrating Factor Calculation
HardSingle correctIdea asked in 7 yearsAsked the year before
Q21
If the number of seven-digit numbers, such that the sum of their digits is even, is $m \cdot n \cdot 10^n$ ; $m, n \in \{1, 2, 3, \ldots, 9\}$ , then $m + n$ is equal to ______
Permutations and Combinations Fundamental Counting Principle
MediumNumerical valueIdea asked in 5 yearsAsked the year before
Q22
All five letter words are made using all the letters A, B, C, D, E and arranged as in an English dictionary with serial numbers. Let the word at serial number $n$ be denoted by $\mathrm{W}_n$ . Let the probability $\mathrm{P}(\mathrm{W}_n)$ of choosing the…
Probability AP and GP GP Sum of n terms
HardNumerical valueIdea asked in 5 yearsAsked the year before
Q23
Let the product of the focal distances of the point $\mathrm{P}\left(4, 2\sqrt{3}\right)$ on the hyperbola $\mathrm{H}: \frac{x^2}{a^2} - \frac{y^2}{b^2} = 1$ be 32. Let the length of the conjugate axis of H be $p$ and the length of its latus rectum be $q$ .…
Conic Sections
MediumNumerical value
Q24
Let $\vec{a} = \hat{i} + \hat{j} + \hat{k}, \vec{b} = 3\hat{i} + 2\hat{j} - \hat{k}, \vec{c} = \lambda\hat{j} + \mu\hat{k}$ and $\hat{d}$ be a unit vector such that $\vec{a} \times \hat{d} = \vec{b} \times \hat{d}$ and $\vec{c}\cdot\hat{d} = 1$ . If $\vec{c}$ …
Vector Algebra Cross Product Condition for Parallel Vectors
MediumNumerical valueIdea asked in 2 years
Q25
The area of the region bounded by the curve $y = \max\{|x|, x|x-2|\}$ , the $x$ -axis and the lines $x = -2$ and $x = 4$ is equal to ______
Application of Integrals Area Under Curves Area Involving Max/Min Functions
MediumNumerical valueIdea asked in 6 yearsAsked the year before

Dropped from the final key and left out of every count: Q37.

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 2025 shifts.