JEE Main 2023 · April session

JEE Main 10 April 2023, Shift 1

88 questions (2 more dropped by NTA): 29 numerical-answer and 59 multiple choice. 8 were rated hard and 34 easy. Its hard-question count was the joint 11th highest of the 12 2023 shifts Lumi analysed. The biggest chapter was Conic Sections, with 3 questions.

Questions: 88; 2 dropped by NTA
Numerical answer: 29; 33% of the shift
Hard: 8; 9% of the shift
Chapters asked: 67; 10 questions with a figure

Key findings · 10 Apr 2023, Shift 1

1
8 hard questions (9%): the joint 11th highest count of the 12 2023 shifts Lumi analysed. Across those shifts the hard share ran from 9% to 24%.
2
Maths was the hardest section, with 6 of 29 rated hard; Physics had the most easy questions (17 of 30).
3
8 questions tested a new idea, one no earlier year had asked: 12% of the 67 questions tagged to an idea.
4
49 came back from 2022: the same idea had been asked in the 2022 shifts Lumi analysed.
5
29 shared an idea with another 2023 shift (43% 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), Coordination Compounds in Chemistry (3) and Conic Sections in Maths (3).

The shift at a glance

Subjects, difficulty and chapters

Physics

30 Q · 10 numerical

Easy 17Medium 12Hard 1

Top chapters

Current Electricity2 Q
Electrostatic Potential and Capacitance2 Q
Gravitation2 Q
Work, Energy and Power2 Q
Alternating Current1 Q
Atoms1 Q

Chemistry

29 Q · 10 numerical

Easy 12Medium 16Hard 1

Top chapters

Maths

29 Q · 9 numerical

Easy 5Medium 18Hard 6

Top chapters

Conic Sections3 Q
Three Dimensional Geometry3 Q
Binomial Theorem2 Q
Complex Numbers and Quadratic Equations2 Q
Determinants2 Q
Differential Equations2 Q

Question format

Single correct46 Q
Numerical value29 Q
Two statements4 Q
Diagram based4 Q
Match the list3 Q
Other2 Q

What each question asks you to do

Calculation25 Q
Multi-step22 Q
Application20 Q
Conceptual12 Q
Recall8 Q
Data or graph1 Q

Most common: Calculation.

Against the other 2023 shifts

This shift next to the 11 other 2023 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
6 Apr, Shift 1	90	31%	11%	33%	30%	27%
6 Apr, Shift 2	89	33%	17%	34%	25%	20%
8 Apr, Shift 1	89	38%	9%	34%	29%	21%
8 Apr, Shift 2	89	42%	16%	34%	23%	30%
10 Apr, Shift 1	88	39%	9%	33%	12%	25%
10 Apr, Shift 2	90	38%	12%	33%	20%	21%
11 Apr, Shift 1	90	34%	16%	33%	25%	33%
11 Apr, Shift 2	88	24%	18%	34%	28%	39%
12 Apr, Shift 1	89	30%	24%	34%	29%	35%
13 Apr, Shift 1	89	24%	19%	34%	21%	37%
13 Apr, Shift 2	90	33%	13%	33%	35%	32%
15 Apr, Shift 1	90	43%	14%	33%	18%	30%

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 shift3084 · 33%33%8/830298716720359538199224
Open29 this shift3092 · 33%33%8/830299720720358540200225
Open29 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

17 easy12 medium1 hard

Q31
A particle executes S.H.M. of amplitude A along $x$ -axis. At $t = 0$ , the position of the particle is $x = \frac{A}{2}$ and it moves along positive $x$ -axis. The displacement of particle in time $t$ is $x = A\sin(wt + \delta)$ , then the value $\delta$ will be
Oscillations SHM Equations Phase and Initial Conditions
EasySingle correctIdea asked in 3 yearsAsked the year before
Q32
Match List I with List II : List I List II (A) 3 Translational degrees of freedom (I) Monoatomic gases (B) 3 Translational, 2 rotational degrees of freedoms (II) Polyatomic gases (C) 3 Translational, 2 rotational and 1 vibrational degrees of freedom (III)…
Kinetic Theory Kinetic Theory of Gases Degrees of Freedom
EasyMatch the listIdea asked in 3 yearsAsked the year before
Q33
Consider two containers A and B containing monoatomic gases at the same Pressure (P), Volume (V) and Temperature (T). The gas in A is compressed isothermally to $\frac{1}{8}$ of its original volume while the gas in B is compressed adiabatically to…
Thermodynamics Thermodynamic Processes Adiabatic Process Analysis
MediumSingle correctIdea asked in 7 yearsAsked the year before
Q34
Two satellites of masses $m$ and $3m$ revolve around the earth in circular orbits of radii $r$ & $3\,r$ respectively. The ratio of orbital speeds of the satellites respectively is
Gravitation Orbital Mechanics Orbital Velocity
EasySingle correctIdea asked in 5 yearsAsked the year before
Q35
A physical quantity P is given as $P = \frac{a^2 b^3}{c\sqrt{d}}$ The percentage error in the measurement of $a$ , $b$ , $c$ and $d$ are 1%, 2%, 3% and 4% respectively. The percentage error in the measurement of quantity P will be
Units and Measurements
EasySingle correct
Q36
Given below are two statements : Statement I : Pressure in a reservoir of water is same at all points at the same level of water. Statement II : The pressure applied to enclosed water is transmitted in all directions equally. In the light of the above…
Mechanical Properties of Fluids Fluid Statics Pascal's Law
EasyTwo statementsIdea asked in 3 years
Q37
Assuming the earth to be a sphere of uniform mass density, the weight of a body at a depth $d = \frac{R}{2}$ from the surface of earth, if its weight on the surface of earth is 200 N, will be : (Given R = radius of earth)
Gravitation
EasySingle correct
Q38
The position-time graphs for two students A and B returning from the school to their homes are shown in figure. (A) A lives closer to the school (B) B lives closer to the school (C) A takes lesser time to reach home (D) A travels faster than B (E) B travels…
Motion in a Straight Line Kinematics in 1D Kinematic Graph Slopes
MediumOther · figureNew idea
Q39
The range of the projectile projected at an angle of $15^\circ$ with horizontal is 50 m. If the projectile is projected with same velocity at an angle of $45^\circ$ with horizontal, then its range will be
Motion in a Plane Projectile Motion Ground-to-Ground Parameters
EasySingle correctIdea asked in 6 yearsAsked the year before
Q40
A particle of mass $m$ moving with velocity $v$ collides with a stationary particle of mass $2m$ . After collision, they stick together and continue to move together with velocity
Work, Energy and Power Collisions Inelastic Collisions
EasySingle correctIdea asked in 3 yearsAsked the year before
Q41
The equivalent capacitance of the combination shown is
Electrostatic Potential and Capacitance Parallel Plate Capacitors Series and Parallel Combinations
MediumDiagram based · figureIdea asked in 6 yearsAsked the year before
Q42
A carrier wave of amplitude 15 V is modulated by a sinusoidal base band signal of amplitude 3 V. The ratio of maximum amplitude to minimum amplitude in an amplitude modulated wave is
Communication Systems
EasySingle correct
Q43
A zener diode of power rating 1.6 W is to be used as voltage regulator. If the zener diode has a breakdown of 8 V and it has to regulate voltage fluctuating between 3 V and 10 V. The value of resistance $R_s$ for safe operation of diode will be
Semiconductor Electronics: Materials, Devices and Simple Circuits Semiconductor Diodes Zener Diode Power Dissipation Limits
MediumDiagram based · figureIdea asked in 5 yearsAsked the year before
Q44
The angular momentum for the electron in Bohr's orbit is $L$ . If the electron is assumed to revolve in second orbit of hydrogen atom, then the change in angular momentum will be
Atoms Bohr's Atomic Model Angular Momentum Quantization
EasySingle correctIdea asked in 5 yearsAsked the year before
Q45
Given below are two statements : Statement I : Maximum power is dissipated in a circuit containing an inductor, a capacitor and a resistor connected in series with an AC source, when resonance occurs Statement II : Maximum power is dissipated in a circuit…
Alternating Current LCR Resonance Power Factor at Resonance
EasyTwo statementsIdea asked in 3 yearsAsked the year before
Q46
The de Broglie wavelength of a molecule in a gas at room temperature (300 K) is $\lambda_1$ . If the temperature of the gas is increased to 600 K, then the de Broglie wavelength of the same gas molecule becomes
Dual Nature of Radiation and Matter Matter Waves Thermal De Broglie Wavelength
EasySingle correctNew idea
Q47
An object is placed at a distance of 12 cm in front of a plane mirror. The virtual and erect image is formed by the mirror. Now the mirror is moved by 4 cm towards the stationary object. The distance by which the position of image would be shifted, will be
Ray Optics and Optical Instruments Spherical Mirrors Image Velocity (Axial Motion)
EasySingle correctIdea asked in 3 years
Q48
The energy of an electromagnetic wave contained in a small volume oscillates with
Electromagnetic Waves
MediumSingle correct
Q49
Given below are two statements : Statement I : If the number of turns in the coil of a moving coil galvanometer is doubled then the current sensitivity becomes double. Statement II : Increasing current sensitivity of a moving coil galvanometer by only…
Moving Charges and Magnetism
MediumTwo statements
Q50
The equivalent resistance of the circuit shown below between points $a$ and $b$ is :
Current Electricity Kirchhoff's Laws Circuit Symmetry Analysis
MediumDiagram based · figureIdea asked in 4 yearsAsked the year before
Q51
Two wires each of radius 0.2 cm and negligible mass, one made of steel and the other made of brass are loaded as shown in the figure. The elongation of the steel wire is ____________ $\times 10^{-6}\,m$ . [Young's modulus for steel…
Mechanical Properties of Solids Elasticity Series and Parallel Composite Wires
MediumNumerical value · figureIdea asked in 4 yearsAsked the year before
Q52
A transverse harmonic wave on a string is given by $y(x,t) = 5\sin(6t + 0.003\,x)$ where $x$ and $y$ are in cm and $t$ in sec. The wave velocity is ____________ $ms^{-1}$ .
Waves Waves on Strings Wave Speed on String
EasyNumerical valueIdea asked in 5 yearsAsked the year before
Q53
If the earth suddenly shrinks to $\frac{1}{64}$ th of its original volume with its mass remaining the same, the period of rotation of earth becomes $\frac{24}{x}\,h$ . The value of $x$ is ____________.
System of Particles and Rotational Motion Angular Momentum Conservation of Angular Momentum
MediumNumerical valueIdea asked in 5 yearsAsked the year before
Q54
A closed circular tube of average radius 15 cm, whose inner walls are rough, is kept in vertical plane. A block of mass 1 kg just fit inside the tube. The speed of block is 22 m/s, when it is introduced at the top of tube. After completing five oscillations,…
Work, Energy and Power Work and Kinetic Energy Work-Energy Theorem
HardNumerical value · figureIdea asked in 6 yearsAsked the year before
Q55
Three concentric spherical metallic shells $X$ , $Y$ and $Z$ of radius $a$ , $b$ and $c$ respectively $[a < b < c]$ have surface charge densities $\sigma$ , $-\sigma$ and $\sigma$ , respectively. The shells $X$ and $Z$ are at same potential. If the radii of $X$ …
Electrostatic Potential and Capacitance Conductors in Fields Potential of Concentric Conducting Shells
MediumNumerical valueNew idea
Q56
The decay constant for a radioactive nuclide is $1.5\times10^{-5}\,s^{-1}$ . Atomic weight of the substance is 60 g mole $^{-1}$ , ( $N_A = 6\times10^{23}$ ). The activity of 1.0 $\mu$ g of the substance is ____________ $\times10^{10}$ Bq.
Nuclei Radioactive Decay Activity and Rate of Decay
EasyNumerical valueIdea asked in 3 yearsAsked the year before
Q57
Unpolarised light of intensity $32\,Wm^{-2}$ passes through the combination of three polaroids such that the pass axis of the last polaroid is perpendicular to that of the pass axis of first polaroid. If intensity of emerging light is $3\,Wm^{-2}$ , then the…
Wave Optics Polarization Malus's Law Calculation
MediumNumerical valueIdea asked in 5 yearsAsked the year before
Q58
A 1 m long metal rod $XY$ completes the circuit as shown in figure. The plane of the circuit is perpendicular to the magnetic field of flux density 0.15 T. If the resistance of the circuit is $5\,\Omega$ , the force needed to move the rod in direction, as…
Electromagnetic Induction Faraday's Law Linear Motional EMF
MediumNumerical value · figureIdea asked in 3 yearsAsked the year before
Q59
The current required to be passed through a solenoid of 15 cm length and 60 turns in order to demagnetise a bar magnet of magnetic intensity $2.4\times10^{3}\,Am^{-1}$ is ____________ A.
Magnetism and Matter
EasyNumerical value
Q60
10 resistors each of resistance $10\,\Omega$ can be connected in such as to get maximum and minimum equivalent resistance. The ratio of maximum and minimum equivalent resistance will be ____________.
Current Electricity
EasyNumerical value

Chemistry29 questions

12 easy16 medium1 hard

Q61
The number of molecules and moles in 2.8375 litres of $\mathrm{O_2}$ at STP are respectively
Some Basic Concepts of Chemistry Mole Concept Mole and Avogadro Calculations
EasySingle correctIdea asked in 5 yearsAsked the year before
Q62
Given (A) $2\mathrm{CO(g)} + \mathrm{O_2(g)} \rightarrow 2\mathrm{CO_2(g)}$ $\Delta \mathrm{H}_1^{\theta} = -x\ \mathrm{kJ\,mol^{-1}}$ (B) $\mathrm{C(graphite)} + \mathrm{O_2(g)} \rightarrow \mathrm{CO_2(g)}$ …
Thermodynamics (Chemistry)Enthalpy and Hess's Law Manipulating Thermochemical Equations
EasySingle correctNew idea
Q63
The enthalpy change for the adsorption process and micelle formation respectively are
Surface Chemistry Adsorption Thermodynamics of Adsorption
MediumSingle correctIdea asked in 2 years
Q64
Which of the following is used as a stabilizer during the concentration of sulphide ores?
General Principles and Processes of Isolation of Elements Adsorption Adsorption in Froth Flotation
EasySingle correctIdea asked in 3 yearsAsked the year before
Q65
Given below are two reactions, involved in the commercial production of dihydrogen ( $\mathrm{H_2}$ ). The two reactions are carried out at temperature “ $T_1$ ” and “ $T_2$ ”, respectively…
Hydrogen Hydrogen and Hydrides Water-Gas Shift Reaction
EasySingle correctIdea asked in 2 years
Q66
Lime reacts exothermally with water to give ‘A’ which has low solubility in water. Aqueous solution of ‘A’ is often used for the test of $\mathrm{CO_2}$ , a test in which insoluble B is formed. If B is further reacted with $\mathrm{CO_2}$ then soluble…
The s-Block Elements
EasySingle correct
Q67
The compound which does not exist is
The s-Block Elements s-Block Elements Formation of Oxides, Peroxides, and Superoxides
MediumSingle correctIdea asked in 4 yearsAsked the year before
Q68
Which of the following statements are correct? (A) The $\mathrm{M^{3+}/M^{2+}}$ reduction potential for iron is greater than manganese. (B) The higher oxidation states of first row d-block elements get stabilized by oxide ion. (C) Aqueous solution of…
The d- and f-Block Elements d-Block Elements Standard Electrode Potential Trends
HardSingle correctIdea asked in 3 yearsAsked the year before
Q69
The octahedral diamagnetic low spin complex among the following is
Coordination Compounds Crystal Field Theory High and Low Spin States
MediumSingle correctIdea asked in 7 yearsAsked the year before
Q70
Match List I with List II List I (Industry) List II (Waste Generated) (A) Steel plants (I) Gypsum (B) Thermal power plants (II) Fly ash (C) Fertilizer industries (III) Slag (D) Paper mills (IV) Bio-degradable wastes Choose the correct answer from the options…
Environmental Chemistry Non-Biodegradable Industrial Solid Wastes
EasyMatch the listIdea asked in 2 yearsAsked the year before
Q71
Using column chromatography, mixture of two compounds ‘A’ and ‘B’ was separated. ‘A’ eluted first, this indicates ‘B’ has
Organic Chemistry: Some Basic Principles and Techniques
MediumSingle correct
Q72
The pair from the following pairs having both compounds with net non-zero dipole moment is
Chemical Bonding and Molecular Structure VSEPR Theory Net Dipole Moment from VSEPR Symmetry
MediumSingle correctIdea asked in 4 yearsAsked the year before
Q73
Identify the correct order of reactivity for the following pairs towards the respective mechanism (A) $\mathrm{S_N2}$ [structure: $(\mathrm{CH_3})_2\mathrm{CHCH_2Br}$ (isobutyl bromide, skeletal)] > [structure: $(\mathrm{CH_3})_3\mathrm{CBr}$ (tert-butyl…
Haloalkanes and Haloarenes SN1 and SN2 Mechanisms Substrate Steric Effects
MediumOther · figureIdea asked in 3 years
Q74
The major product ‘P’ formed in the given reaction is [structure: naphthalene ring bearing $\mathrm{CH_2CH_3}$ at one position of one ring, $\mathrm{CH{=}CH_2}$ at the para (opposite) position of the same ring, and $\mathrm{CH_2COOCH_3}$ on the other ring]…
Hydrocarbons Aromatic Hydrocarbons Side-chain Oxidation
MediumDiagram based · figureIdea asked in 5 yearsAsked the year before
Q75
Suitable reaction condition for preparation of Methyl phenyl ether is
Alcohols, Phenols and Ethers Ethers Williamson Synthesis Limits
EasySingle correctIdea asked in 3 years
Q77
Match List I with List II List I (Polymer) List II (Type / Class) (A) Nylon-2-Nylon-6 (I) Thermosetting polymer (B) Buna-N (II) Biodegradable polymer (C) Urea-formaldehyde resin (III) Synthetic rubber (D) Dacron (IV) Polyester Choose the correct answer from…
Polymers Polymer Classification
EasyMatch the listIdea asked in 3 yearsAsked the year before
Q78
The one that does not stabilize $2^{\circ}$ and $3^{\circ}$ structures of proteins is
Biomolecules
EasySingle correct
Q79
Given below are two statements: Statement I : Aqueous solution of $\mathrm{K_2Cr_2O_7}$ is preferred as a primary standard in volumetric analysis over $\mathrm{Na_2Cr_2O_7}$ aqueous solution. Statement II : $\mathrm{K_2Cr_2O_7}$ has a higher solubility in…
The d- and f-Block Elements d-Block Elements Preparation Steps of Potassium Dichromate
MediumTwo statementsIdea asked in 3 yearsAsked the year before
Q80
Prolonged heating is avoided during the preparation of ferrous ammonium sulphate to
Coordination Compounds
EasySingle correct
Q81
At constant temperature, a gas is at a pressure of 940.3 mm Hg. The pressure at which its volume decreases by 40% is __________ mm Hg. (Nearest integer)
States of Matter Ideal Gas Laws Individual Gas Laws
EasyNumerical valueIdea asked in 2 yearsAsked the year before
Q82
The number of incorrect statement/s about the black body from the following is __________ (A) Emit or absorb energy in the form of electromagnetic radiation. (B) Frequency distribution of the emitted radiation depends on temperature. (C) At a given…
Structure of Atom
MediumNumerical value
Q83
The number of bent-shaped molecule/s from the following is ______________ $\mathrm{N_3^-}$ , $\mathrm{NO_2^-}$ , $\mathrm{I_3^-}$ , $\mathrm{O_3}$ , $\mathrm{SO_2}$
Chemical Bonding and Molecular Structure VSEPR Theory Geometry vs Shape Distinction
MediumNumerical valueIdea asked in 2 yearsAsked the year before
Q84
If the degree of dissociation of aqueous solution of weak monobasic acid is determined to be 0.3, then the observed freezing point will be __________% higher than the expected/theoretical freezing point. (Nearest integer)
Solutions Colligative Properties Degree of Dissociation from Van't Hoff Factor
MediumNumerical valueIdea asked in 5 yearsAsked the year before
Q85
The number of correct statement/s involving equilibria in physical processes from the following is __________ (A) Equilibrium is possible only in a closed system at a given temperature. (B) Both the opposing processes occur at the same rate. (C) When…
Equilibrium
MediumNumerical value
Q86
$\mathrm{FeO_4^{2-}} \xrightarrow{+2.2\,\mathrm{V}} \mathrm{Fe^{3+}} \xrightarrow{+0.70\,\mathrm{V}} \mathrm{Fe^{2+}} \xrightarrow{-0.45\,\mathrm{V}} \mathrm{Fe^{0}}$ $E^{0}_{\mathrm{FeO_4^{2-}/Fe^{2+}}}$ is $x \times 10^{-3}$ V. The value of $x$ is…
Electrochemistry Nernst Equation Additivity of Delta G0 for E0 Calculation
MediumNumerical valueIdea asked in 2 yearsAsked the year before
Q87
A molecule undergoes two independent first order reactions whose respective half lives are 12 min and 3 min. If both the reactions are occurring then the time taken for the 50% consumption of the reactant is ____________ min. (Nearest integer)
Chemical Kinetics Integrated Rate Equations Parallel First-Order Effective Rate Constant
MediumNumerical valueNew idea
Q88
In the following reactions, the total number of oxygen atoms in X and Y is ____________ $\mathrm{Na_2O + H_2O \rightarrow 2X}$ $\mathrm{Cl_2O_7 + H_2O \rightarrow 2Y}$
The p-Block Elements Group 17 & 18 Halogen Oxoacids
EasyNumerical valueNew idea
Q89
The sum of lone pairs present on the central atom of the interhalogen $\mathrm{IF_5}$ and $\mathrm{IF_7}$ is __________
The p-Block Elements VSEPR Theory Geometry vs Shape Distinction
MediumNumerical valueIdea asked in 2 yearsAsked the year before
Q90
In potassium ferrocyanide, there are ____________ pairs of electrons in the $t_{2g}$ set of orbitals.
Coordination Compounds Crystal Field Theory High and Low Spin States
MediumNumerical valueIdea asked in 7 yearsAsked the year before

Maths29 questions

5 easy18 medium6 hard

Q1
If $f(x)=\frac{(\tan 1^\circ)x+\log_e(123)}{x\log_e(1234)-(\tan 1^\circ)}$ , $x>0$ , then the least value of $f(f(x))+f\left(f\left(\frac{4}{x}\right)\right)$ is
Relations and Functions Functions and Domain Range of Composite Functions
HardSingle correctIdea asked in 3 yearsAsked the year before
Q2
Let the complex number $z=x+iy$ be such that $\frac{2z-3i}{2z+i}$ is purely imaginary. If $x+y^2=0$ , then $y^4+y^2-y$ is equal to:
Complex Numbers and Quadratic Equations
MediumSingle correct
Q3
For the system of linear equations $2x-y+3z=5$ $3x+2y-z=7$ $4x+5y+\alpha z=\beta$ , which of the following is NOT correct?
Determinants Cramer's Rule Condition for Infinite Solutions
MediumSingle correctIdea asked in 7 yearsAsked the year before
Q4
If $A$ is a $3\times3$ matrix and $|A|=2$ , then $\left|3\,adj\left(|3A|A^2\right)\right|$ is equal to
Determinants Matrix Inverse Trap involving |adj(A)| vs |A^-1|
MediumSingle correctIdea asked in 6 yearsAsked the year before
Q5
If the coefficient of $x^7$ in $\left(ax-\frac{1}{bx^2}\right)^{13}$ and the coefficient of $x^{-5}$ in $\left(ax+\frac{1}{bx^2}\right)^{13}$ are equal, then $a^4b^4$ is equal to:
Binomial Theorem
MediumSingle correct
Q6
Let the first term $a$ and the common ratio $r$ of a geometric progression be positive integers. If the sum of squares of its first three terms is 33033, then the sum of these three terms is equal to
Sequences and Series AP and GP GP Sum of Squares
MediumSingle correctIdea asked in 2 yearsAsked the year before
Q7
A square piece of tin of side 30 cm is to be made into a box without top by cutting a square from each corner and folding up the flaps to form a box. If the volume of the box is maximum, then its surface area (in cm $^2$ ) is equal to
Application of Derivatives Maxima and Minima Optimization Problems
MediumSingle correctIdea asked in 5 yearsAsked the year before
Q8
If $I(x)=\int e^{\sin^2x}(\cos x\sin 2x-\sin x)\,dx$ and $I(0)=1$ , then $I\left(\frac{\pi}{3}\right)$ is equal to
Integrals Integration by Substitution Trigonometric Substitution
HardSingle correctIdea asked in 3 years
Q9
Let $f$ be a differentiable function such that $x^2f(x)-x=4\int_0^x tf(t)\,dt$ , $f(1)=\frac{2}{3}$ . Then $18f(3)$ is equal to
Differential Equations Leibniz Rule Solving Integral Equations
MediumSingle correctIdea asked in 6 yearsAsked the year before
Q10
The slope of tangent at any point $(x,y)$ on a curve $y=y(x)$ is $\frac{x^2+y^2}{2xy}$ , $x>0$ . If $y(2)=0$ , then a value of $y(8)$ is
Differential Equations Homogeneous Equations Substitution y = vx
MediumSingle correctIdea asked in 6 yearsAsked the year before
Q11
Let the ellipse $E: x^2+9y^2=9$ intersect the positive $x$ - and $y$ -axes at the points A and B respectively. Let the major axis of E be a diameter of the circle C. Let the line passing through A and B meet the circle C at the point P. If the area of the…
Conic Sections
MediumSingle correct
Q12
A line segment AB of length $\lambda$ moves such that the points A and B remain on the periphery of a circle of radius $\lambda$ . Then the locus of the point, that divides the line segment AB in the ratio $2:3$ , is a circle of radius
Conic Sections
HardSingle correct
Q13
Let two vertices of a triangle ABC be $(2,4,6)$ and $(0,-2,-5)$ , and its centroid be $(2,1,-1)$ . If the image of the third vertex in the plane $x+2y+4z=11$ is $(\alpha,\beta,\gamma)$ , then $\alpha\beta+\beta\gamma+\gamma\alpha$ is equal to
Three Dimensional Geometry Point-Plane Distance Image of a Point
MediumSingle correctIdea asked in 4 yearsAsked the year before
Q14
Let P be the point of intersection of the line $\frac{x+3}{3}=\frac{y+2}{1}=\frac{1-z}{2}$ and the plane $x+y+z=2$ . If the distance of the point P from the plane $3x-4y+12z=32$ is $q$ , then $q$ and $2q$ are the roots of the equation
Three Dimensional Geometry Point-Plane Distance Distance from Point
MediumSingle correctIdea asked in 2 years
Q15
The shortest distance between the lines $\frac{x+2}{1}=\frac{y}{-2}=\frac{z-5}{2}$ and $\frac{x-4}{1}=\frac{y-1}{2}=\frac{z+3}{0}$ is
Three Dimensional Geometry Skew Lines Cartesian Shortest Distance
EasySingle correctIdea asked in 6 yearsAsked the year before
Q16
Let O be the origin and the position vector of the point P be $-\hat{i}-2\hat{j}+3\hat{k}$ . If the position vectors of the points A, B and C are $-2\hat{i}+\hat{j}-3\hat{k}$ , $2\hat{i}+4\hat{j}-2\hat{k}$ and $-4\hat{i}+2\hat{j}-\hat{k}$ respectively, then…
Vector Algebra Cross Product Determinant Calculation Method
MediumSingle correctIdea asked in 2 years
Q17
An arc PQ of a circle subtends a right angle at its centre O. The mid point of the arc PQ is R. If $\overrightarrow{OP}=\vec{u}$ , $\overrightarrow{OR}=\vec{v}$ and $\overrightarrow{OQ}=\alpha\vec{u}+\beta\vec{v}$ , then $\alpha,\beta^2$ are the roots of the…
Vector Algebra Linear Independence Unique Linear Combinations
MediumSingle correctNew idea
Q18
Let N denote the sum of the numbers obtained when two dice are rolled. If the probability that $2^N<N!$ is $\frac{m}{n}$ , where $m$ and $n$ are coprime, then $4m-3n$ is equal to
Probability
EasySingle correct
Q19
$96\cos\frac{\pi}{33}\cos\frac{2\pi}{33}\cos\frac{4\pi}{33}\cos\frac{8\pi}{33}\cos\frac{16\pi}{33}$ is equal to
Trigonometric Functions Trigonometric Identities Product of Cosines with Doubling Angles
MediumSingle correctIdea asked in 2 yearsAsked the year before
Q20
The negation of the statement $(p\vee q)\wedge(q\vee(\sim r))$ is
Mathematical Reasoning
EasySingle correct
Q22
The number of elements in the set $\{n\in\mathbb{Z}:|n^2-10n+19|<6\}$ is ____________.
Complex Numbers and Quadratic Equations
EasyNumerical value
Q23
The number of permutations, of the digits 1, 2, 3, ..., 7 without repetition, which neither contain the string 153 nor the string 2467, is ____________.
Permutations and Combinations The Block/Tie Method
HardNumerical valueIdea asked in 4 years
Q24
Some couples participated in a mixed doubles badminton tournament. If the number of matches played, so that no couple played in a match, is 840, then the total numbers of persons, who participated in the tournament, is ____________.
Permutations and Combinations Combinations of Distinct Objects
MediumNumerical valueIdea asked in 8 yearsAsked the year before
Q25
The coefficient of $x^7$ in $(1-x+2x^3)^{10}$ is ____________.
Binomial Theorem
MediumNumerical value
Q26
The sum of all those terms, of the arithmetic progression 3, 8, 13, ..., 373, which are not divisible by 3, is equal to ____________.
Sequences and Series AP and GP AP Sum of n terms
EasyNumerical valueIdea asked in 7 yearsAsked the year before
Q27
Let $f:(-2,2)\to\mathbb{R}$ be defined by $f(x)=\begin{cases}x[x], & -2<x<0\\ (x-1)[x], & 0\le x<2\end{cases}$ where $[x]$ denotes the greatest integer function. If $m$ and $n$ respectively are the number of points in $(-2,2)$ at which $y=|f(x)|$ is not…
Continuity and Differentiability Differentiability of Modulus Functions
HardNumerical valueIdea asked in 5 yearsAsked the year before
Q28
Let $y=p(x)$ be the parabola passing through the points $(-1,0)$ , $(0,1)$ and $(1,0)$ . If the area of the region $\{(x,y):(x+1)^2+(y-1)^2\le1,\ y\le p(x)\}$ is A, then $12(\pi-4A)$ is equal to ____________.
Application of Integrals Area Under Curves Area Between Curves
HardNumerical valueIdea asked in 7 yearsAsked the year before
Q29
Let a common tangent to the curves $y^2=4x$ and $(x-4)^2+y^2=16$ touch the curves at the points P and Q. Then $(PQ)^2$ is equal to ____________.
Conic Sections Circle Tangents Common Tangent to Circle and Conic
MediumNumerical valueIdea asked in 3 yearsAsked the year before
Q30
If the mean of the frequency distribution | Class : | 0-10 | 10-20 | 20-30 | 30-40 | 40-50 | | Frequency : | 2 | 3 | $x$ | 5 | 4 | is 28, then its variance is ____________.
Statistics Variance and SD Variance of Continuous Grouped Data
MediumNumerical value · figureNew idea

Dropped from the final key and left out of every count: Q21, Q76.

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 2023 shifts.