JEE Main 2023 · April session

JEE Main 13 April 2023, Shift 1

89 questions (1 more dropped by NTA): 30 numerical-answer and 59 multiple choice. 17 were rated hard and 21 easy. Its hard-question count was the 2nd highest of the 12 2023 shifts Lumi analysed. The biggest chapter was Conic Sections, with 3 questions.

Questions: 89; 1 dropped by NTA
Numerical answer: 30; 34% of the shift
Hard: 17; 19% of the shift
Chapters asked: 68; 13 questions with a figure

Key findings · 13 Apr 2023, Shift 1

1
17 hard questions (19%): the 2nd 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 12 of 29 rated hard; Physics had the most easy questions (11 of 30).
3
16 questions tested a new idea, one no earlier year had asked: 21% of the 75 questions tagged to an idea.
4
46 came back from 2022: the same idea had been asked in the 2022 shifts Lumi analysed.
5
28 shared an idea with another 2023 shift (37% 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), Alcohols, Phenols and Ethers in Chemistry (2) and Conic Sections in Maths (3).

The shift at a glance

Subjects, difficulty and chapters

Physics

30 Q · 10 numerical

Easy 11Medium 16Hard 3

Top chapters

Chemistry

30 Q · 10 numerical

Easy 8Medium 20Hard 2

Top chapters

Maths

29 Q · 10 numerical

Easy 2Medium 15Hard 12

Top chapters

Question format

Single correct45 Q
Numerical value30 Q
Diagram based7 Q
Other3 Q
Two statements2 Q
Match the list2 Q

What each question asks you to do

Multi-step33 Q
Calculation25 Q
Conceptual11 Q
Application10 Q
Recall7 Q
Data or graph3 Q

Most common: Multi-step.

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 shift3092 · 33%33%8/830299720720358540200225
Open30 this shift3084 · 33%33%8/830298716720359538199224
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 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

Physics30 questions

11 easy16 medium3 hard

Q31
A body of mass $(5 \pm 0.5)$ kg is moving with a velocity of $(20 \pm 0.4)$ m/s. Its kinetic energy will be
Units and Measurements
EasySingle correct
Q32
Two trains 'A' and 'B' of length ' $l$ ' and ' $4l$ ' are travelling into a tunnel of length 'L' in parallel tracks from opposite directions with velocities 108 km/h and 72 km/h, respectively. If train 'A' takes 35s less time than train 'B' to cross the tunnel…
Motion in a Straight Line Relative Velocity Extended Bodies Overtaking and Crossing
MediumSingle correctNew idea
Q33
A disc is rolling without slipping on a surface. The radius of the disc is R. At t = 0, the top most point on the disc is A as shown in figure. When the disc completes half of its rotation, the displacement of point A from its initial position is
System of Particles and Rotational Motion
MediumSingle correct · figure
Q34
Two bodies are having kinetic energies in the ratio 16 : 9. If they have same linear momentum, the ratio of their masses respectively is :
Work, Energy and Power Work and Kinetic Energy Momentum and Kinetic Energy Relation
EasySingle correctIdea asked in 4 yearsAsked the year before
Q35
A bullet of 10 g leaves the barrel of gun with a velocity of 600 m/s. If the barrel of gun is 50 cm long and mass of gun is 3 kg, then value of impulse supplied to the gun will be :
Laws of Motion Newton's Laws of Motion Momentum Formulation (F = dp/dt)
EasySingle correctIdea asked in 4 yearsAsked the year before
Q36
A planet having mass 9 Me and radius $4R_e$ , where Me and Re are mass and radius of earth respectively, has escape velocity in km/s given by: (Given escape velocity on earth $V_e = 11.2 \times 10^3$ m/s)
Gravitation Gravitational Potential Escape Velocity and Potential Relation
EasySingle correctIdea asked in 6 years
Q37
The figure shows a liquid of given density flowing steadily in horizontal tube of varying cross - section. Cross sectional areas at A is $1.5\ \mathrm{cm^2}$ , and B is $25\ \mathrm{mm^2}$ , if the speed of liquid at B is 60 cm/s then $(P_A - P_B)$ is : (Given…
Mechanical Properties of Fluids Fluid Dynamics Bernoulli's Equation
MediumSingle correct · figureIdea asked in 5 yearsAsked the year before
Q38
Under isothermal condition, the pressure of a gas is given by $P = aV^{-3}$ , where $a$ is a constant and V is the volume of the gas. The bulk modulus at constant temperature is equal to
Mechanical Properties of Solids Elasticity Isothermal and Adiabatic Bulk Moduli
EasySingle correctNew idea
Q39
The rms speed of oxygen molecule in a vessel at particular temperature is $\left(1+\frac{5}{x}\right)^{\frac{1}{2}} v$ , where $v$ is the average speed of the molecule. The value of $x$ will be : (Take $\pi = \frac{22}{7}$ )
Kinetic Theory Kinetic Theory of Gases Molecular Speed Ratios
MediumSingle correctIdea asked in 4 yearsAsked the year before
Q40
Which graph represents the difference between total energy and potential energy of a particle executing SHM vs it's distance from mean position ?
Oscillations Energy in SHM Energy-Displacement Graphs
MediumDiagram based · figureNew idea
Q41
The ratio of powers of two motors is $\frac{3\sqrt{x}}{\sqrt{x}+1}$ , that are capable of raising 300 kg water in 5 minutes and 50 kg water in 2 minutes respectively from a well of 100 m deep. The value of $x$ will be
Work, Energy and Power Power Power in Fluid Flow
MediumSingle correctIdea asked in 2 yearsAsked the year before
Q42
Which of the following Maxwell's equation is valid for time varying conditions but not valid for static conditions :
Electromagnetic Waves
MediumSingle correct
Q43
The source of time varying magnetic field may be (A) a permanent magnet (B) an electric field changing linearly with time (C) direct current (D) a decelerating charge particle (E) an antenna fed with a digital signal Choose the correct answer from the…
Electromagnetic Waves
MediumOther
Q44
A vessel of depth 'd' is half filled with oil of refractive index $n_1$ and the other half is filled with water of refractive index $n_2$ . The apparent depth of this vessel when viewed from above will be-
Ray Optics and Optical Instruments Refraction and Snell's Law Apparent Depth of Immiscible Liquids
EasySingle correctIdea asked in 4 years
Q45
The difference between threshold wavelengths for two metal surfaces A and B having work function $\phi_A = 9\ eV$ and $\phi_B = 4{\cdot}5\ eV$ in nm is : {Given, hc = 1242 eV nm}
Dual Nature of Radiation and Matter Photoelectric Effect Work Function and Threshold
EasySingle correctIdea asked in 4 yearsAsked the year before
Q46
$^{238}_{92}A \rightarrow\, ^{234}_{90}B + \,^{4}_{2}D + Q$ In the given nuclear reaction, the approximate amount of energy released will be: [Given, mass of $^{238}_{92}A = 238{\cdot}05079 \times 931.5\ \mathrm{MeV/c^2}$ , mass of…
Nuclei Nuclear Binding Energy Alpha Decay Q-Value with Atomic Masses
EasySingle correctNew idea
Q47
For the following circuit and given inputs A and B, choose the correct option for output 'Y'
Semiconductor Electronics: Materials, Devices and Simple Circuits Logic Gates Timing Diagrams and Waveform Analysis
HardDiagram based · figureIdea asked in 4 yearsAsked the year before
Q48
Match List - I with List - II List - I (Layer of atmosphere) | List - II (Approximate height over earth's surface) (A) $F_1$ - Layer | (I) 10 km (B) D - Layer | (II) 170 - 190 km (C) Troposphere | (III) 100 km (D) E - layer | (IV) 65 - 75 km Choose the…
Communication Systems
EasyMatch the list
Q49
Two charges each of magnitude 0.01 C and separated by a distance of $0{\cdot}4$ mm constitute an electric dipole. If the dipole is placed in an uniform electric field ' $\vec{E}$ ' of 10 dyne/C making $30^\circ$ angle with $\vec{E}$ , the magnitude of torque…
Electrostatic Potential and Capacitance Electric Dipoles Torque in Uniform Field
MediumSingle correctIdea asked in 4 yearsAsked the year before
Q50
Different combination of 3 resistors of equal resistance R are shown in the figures. The increasing order for power dissipation is: (A) [figure A] (B) [figure B] (C) [figure C] (D) [figure D]
Current Electricity Ohm's Law and Resistance Joule Heating and Power
MediumDiagram based · figureIdea asked in 6 yearsAsked the year before
Q51
The radius of $2^{\text{nd}}$ orbit of $\mathrm{He^+}$ of Bohr's model is $r_1$ and that of fourth orbit of $\mathrm{Be^{3+}}$ is represented as $r_2$ . Now the ratio $\frac{r_2}{r_1}$ is $x : 1$ . The value of $x$ is ____________
Atoms Bohr's Atomic Model Bohr Orbit Radius
EasyNumerical valueIdea asked in 4 yearsAsked the year before
Q52
A fish rising vertically upward with a uniform velocity of $8\ \mathrm{ms^{-1}}$ , observes that a bird is diving vertically downward towards the fish with the velocity of $12\ \mathrm{ms^{-1}}$ . If the refractive index of water is $\frac{4}{3}$ , then the…
Ray Optics and Optical Instruments Refraction and Snell's Law Apparent Distance in Two Media (Bird-Fish Problem)
HardNumerical valueNew idea
Q53
In the given figure, an inductor and a resistor are connected in series with a battery of emf E volt. $\frac{E^a}{2b}$ J/s represents the maximum rate at which the energy is stored in the magnetic field (inductor). The numerical value of $\frac{b}{a}$ will…
Electromagnetic Induction Inductance Power Distribution in a Non-Ideal Inductor
HardNumerical value · figureNew idea
Q54
A thin infinite sheet charge and an infinite line charge of respective charge densities $+\sigma$ and $+\lambda$ are placed parallel at 5 m distance from each other. Points 'P' and 'Q' are at $\frac{3}{\pi}$ m and $\frac{4}{\pi}$ m perpendicular distances…
Electric Charges and Fields Electric Field Superposition Principle for Electric Fields
MediumNumerical valueIdea asked in 4 yearsAsked the year before
Q55
The elastic potential energy stored in a steel wire of length 20 m stretched through 2 cm is 80 J. The cross sectional area of the wire is ____________ $\mathrm{mm^2}$ . (Given, $y = 2.0 \times 10^{11}\ \mathrm{Nm^{-2}}$ )
Mechanical Properties of Solids Elasticity Elastic Potential Energy
MediumNumerical valueIdea asked in 3 years
Q56
A solid sphere is rolling on a horizontal plane without slipping. If the ratio of angular momentum about axis of rotation of the sphere to the total energy of moving sphere is $\pi :\ 22$ then, the value of its angular speed will be ________________ rad/s.
System of Particles and Rotational Motion Angular Momentum Angular Momentum of Purely Rolling Body
MediumNumerical valueIdea asked in 2 yearsAsked the year before
Q57
At a given point of time the value of displacement of a simple harmonic oscillator is given as $y = A\cos(30^\circ)$ . If amplitude is 40 cm and kinetic energy at that time is 200 J, the value of force constant is $1.0 \times 10^{x}\ \mathrm{Nm^{-1}}$ . The…
Oscillations Energy in SHM Kinetic Energy in SHM
MediumNumerical valueNew ideaIdea asked in 2 years
Q58
When a resistance of 5 $\Omega$ is shunted with a moving coil galvanometer, it shows a full scale deflection for a current of 250 mA, however when 1050 $\Omega$ resistance is connected with it in series, it gives full scale deflection for 25 volt. The…
Moving Charges and Magnetism
MediumNumerical value
Q59
From the given transfer characteristic of a transistor in CE configuration, the value of power gain of this configuration is $10^x$ , for $R_B = 10\ \mathrm{k}\Omega$ , and $R_C = 1\ \mathrm{k}\Omega$ . The value of $x$ is ______________
Semiconductor Electronics: Materials, Devices and Simple Circuits Transistors CE Amplifier Voltage Gain
MediumNumerical value · figureIdea asked in 3 yearsAsked the year before
Q60
A potential $V_o$ is applied across a uniform wire of resistance R. The power dissipation is $P_1$ . The wire is then cut into two equal halves and a potential of $V_o$ is applied across the length of each half. The total power dissipation across two wires is…
Current Electricity Ohm's Law and Resistance Joule Heating and Power
EasyNumerical valueIdea asked in 6 yearsAsked the year before

Chemistry30 questions

8 easy20 medium2 hard

Q61
The energy of an electron in the first Bohr orbit of hydrogen atom is $-2.18\times10^{-18}$ J. Its energy in the third Bohr orbit is_________.
Structure of Atom Bohr's Atomic Theory Total, Kinetic, and Potential Energy
EasySingle correctIdea asked in 4 years
Q62
In which of the following processes, the bond order increases and paramagnetic character changes to diamagnetic one ?
Chemical Bonding and Molecular Structure Molecular Orbital Theory Magnetic Property Prediction
MediumSingle correctIdea asked in 6 yearsAsked the year before
Q63
What happens when a lyophilic sol is added to a lyophobic sol?
Surface Chemistry Colloids Lyophilic and Lyophobic Sols
EasySingle correctIdea asked in 3 yearsAsked the year before
Q64
Which of the following statements are not correct ? A. The electron gain enthalpy of F is more negative than that of Cl. B. Ionization enthalpy decreases in a group of periodic table. C. The electronegativity of an atom depends upon the atoms bonded to it.…
Classification of Elements and Periodicity in Properties Periodic Trends Electron Gain Enthalpy
MediumOtherIdea asked in 5 yearsAsked the year before
Q65
Which one of the following is most likely a mismatch ?
General Principles and Processes of Isolation of Elements Metallurgical Extraction Metal Refining Techniques
MediumSingle correctIdea asked in 4 yearsAsked the year before
Q66
Given below are two statements : Statement I: Permutit process is more efficient compared to the synthetic resin method for the softening of water. Statement II: Synthetic resin method results in the formation of soluble sodium salts. In the light of the…
Hydrogen Hydrogen and Hydrides Ion-Exchange Resins for Water Softening
MediumTwo statementsIdea asked in 2 years
Q67
$\mathrm{Be(OH)_2}$ reacts with $\mathrm{Sr(OH)_2}$ to yield an ionic salt. Choose the incorrect option related to this reaction from the following :
The s-Block Elements s-Block Elements Amphoterism of Beryllium Compounds
MediumSingle correctIdea asked in 4 yearsAsked the year before
Q68
$\mathrm{ClF_5}$ at room temperature is a:
The p-Block Elements Group 17 & 18 Interhalogen Compounds
MediumSingle correctIdea asked in 3 yearsAsked the year before
Q69
The incorrect statement from the following for borazine is:
The p-Block Elements Group 13 & 14 Borazine Structure and Reactivity
MediumSingle correctIdea asked in 3 yearsAsked the year before
Q70
The pair of lanthanides in which both elements have high third - ionization energy is:
The d- and f-Block Elements f-Block Elements Lanthanide Oxidation States
MediumSingle correctIdea asked in 3 yearsAsked the year before
Q71
The mismatched combinations are A. Chlorophyll - Co B. Water hardness - EDTA C. Photography - $[\mathrm{Ag(CN)_2}]^-$ D. Wilkinson catalyst - $[\mathrm{(Ph_3P)_3RhCl}]$ E. Chelating ligand - D-Penicillamine Choose the correct answer from the options given…
Coordination Compounds
MediumOther
Q72
The radical which mainly causes ozone depletion in the presence of UV radiations is :
Environmental Chemistry Ozone Layer Depletion
EasySingle correctIdea asked in 4 yearsAsked the year before
Q73
Among the following compounds, the one which shows highest dipole moment is
Organic Chemistry: Some Basic Principles and Techniques Inductive and Electromeric Effects Inductive Effect on Dipole Moment
MediumDiagram based · figureNew idea
Q74
In the following reaction 'X' is $\mathrm{CH_3(CH_2)_4CH_3}\xrightarrow[\Delta]{\text{Anhy.AlCl}_3/\text{HCl}}$ 'X' (major product)
Hydrocarbons Alkanes and Free Radicals Catalytic Isomerization of Alkanes
MediumDiagram based · figureNew ideaIdea asked in 2 years
Q75
2-Methyl propyl bromide reacts with $\mathrm{C_2H_5O^-}$ and gives 'A' whereas on reaction with $\mathrm{C_2H_5OH}$ it gives 'B'. The mechanism followed in these reactions and the products 'A' and 'B' respectively are :
Haloalkanes and Haloarenes SN1 and SN2 Mechanisms Solvent Effects on Substitution
MediumSingle correctNew ideaIdea asked in 3 years
Q76
[figure: cyclobutyl ring attached to a carbon bearing OH, which is attached to a cyclopentyl ring] $\xrightarrow[\Delta]{H^+}$ major product In the above reaction, left hand side and right hand side rings are named as 'A' and 'B' respectively. They undergo…
Alcohols, Phenols and Ethers Alcohols and Phenols Acid-Catalyzed Hydration Rearrangements
HardSingle correct · figureNew idea
Q77
In the reaction given below [figure: N-methyl five-membered lactam (N-methylpyrrolidin-2-one)] $\xrightarrow[\text{(ii) }H^+]{\text{(i) NaOH, }\Delta}$ 'A' Major Product 'A' is
Amines
MediumDiagram based · figure
Q78
In the reaction given below [figure: tetralin ring system spiro-fused at C1 to a cyclopentane ring, with OH on the spiro carbon and a $\mathrm{H_2N}$ -CH2CH2- chain on the same carbon] $\xrightarrow[\text{ii) KOH}]{\text{i) HCl}}$ 'B' Major Product 'B' is :
Amines
HardDiagram based · figure
Q79
Match the following Column-A | Column-B a) Nylon 6 | I. Natural Rubber b) Vulcanized Rubber | II. Cross Linked c) cis-1, 4-polyisoprene | III. Caprolactam d) Polychloroprene | IV. Neoprene Choose the correct answer from options given below :
Polymers Common Condensation Polymers
EasyMatch the listIdea asked in 2 yearsAsked the year before
Q80
D -(+)- Glyceraldehyde $\xrightarrow[\text{iii) HNO}_3]{\text{i) HCN; ii) H}_2\text{O / H}^+}$ The products formed in the above reaction are
Biomolecules Carbohydrates Kiliani-Fischer Synthesis
MediumSingle correctNew idea
Q81
$\mathrm{KMnO_4}$ is titrated with ferrous ammonium sulphate hexahydrate in presence of dilute $\mathrm{H_2SO_4}$ . Number of water molecules produced for 2 molecules of $\mathrm{KMnO_4}$ is_________
Redox Reactions Permanganate n-Factor Across Media
EasyNumerical valueIdea asked in 3 yearsAsked the year before
Q82
A certain quantity of real gas occupies a volume of $0.15\ \mathrm{dm^3}$ at 100 atm and 500 K when its compressibility factor is 1.07. Its volume at 300 atm and 300 K (When its compressibility factor is 1.4) is __________ $\times\ 10^{-4}\ \mathrm{dm^3}$ …
States of Matter Real Gases Compressibility Factor Z
MediumNumerical valueIdea asked in 2 yearsAsked the year before
Q83
$\mathrm{A_2}+\mathrm{B_2}\rightarrow 2\mathrm{AB}$ . $\Delta H_f^0 = -200\ \mathrm{kJ\ mol^{-1}}$ AB, $\mathrm{A_2}$ and $\mathrm{B_2}$ are diatomic molecules. If the bond enthalpies of $\mathrm{A_2}$ , $\mathrm{B_2}$ and AB are in the ratio 1:0.5:1, then the…
Thermodynamics (Chemistry)Enthalpy and Hess's Law Bond Enthalpy Calculations
MediumNumerical valueIdea asked in 4 years
Q84
Solution of 12 g of non-electrolyte (A) prepared by dissolving it in 1000 mL of water exerts the same osmotic pressure as that of 0.05 M glucose solution at the same temperature. The empirical formula of A is $\mathrm{CH_2O}$ . The molecular mass of A is…
Solutions Colligative Properties Osmotic Pressure
EasyNumerical valueIdea asked in 4 yearsAsked the year before
Q85
25.0 mL of 0.050 M $\mathrm{Ba(NO_3)_2}$ is mixed with 25.0 mL of 0.020 M NaF. $\mathrm{K_{sp}}$ of $\mathrm{BaF_2}$ is $0.5\times10^{-6}$ at 298 K. The ratio of $[\mathrm{Ba^{2+}}]\ [\mathrm{F^-}]^2$ and $\mathrm{K_{sp}}$ is ___________. (Nearest integer)
Equilibrium Solubility Product Precipitation Condition
MediumNumerical valueIdea asked in 4 years
Q86
A metal surface of 100 $\mathrm{cm^2}$ area has to be coated with nickel layer of thickness 0.001 mm. A current of 2A was passed through a solution of $\mathrm{Ni(NO_3)_2}$ for 'x' seconds to coat the desired layer. The value of x is __________. (Nearest…
Electrochemistry Faraday's Laws of Electrolysis Electroplating Thickness and Area
MediumNumerical valueNew idea
Q87
$t_{87.5}$ is the time required for the reaction to undergo 87.5% completion and $t_{50}$ is the time required for the reaction to undergo 50% completion.The relation between $t_{87.5}$ and $t_{50}$ for a first order reaction is $t_{87.5}$ …
Chemical Kinetics Integrated Rate Equations First Order Half-Life
EasyNumerical valueIdea asked in 3 yearsAsked the year before
Q88
An organic compound gives 0.220 g of $\mathrm{CO_2}$ and 0.126 g of $\mathrm{H_2O}$ on complete combustion. If the % of carbon is 24 then the % of hydrogen is __________ $\times10^{-1}$ . ( Nearest integer )
Organic Chemistry: Some Basic Principles and Techniques Stoichiometry Empirical Formula from Combustion Data
EasyNumerical valueIdea asked in 4 yearsAsked the year before
Q89
For the given reaction $\mathrm{CH_3-C(CH_3)(CH_3)-CH(OH)-C(H)(CH_3)-CH_3}\xrightarrow[\Delta]{H^+}$ (the carbon bearing OH is labelled 'A') The total number of possible products formed by tertiary carbocation of A is ___________.
Alcohols, Phenols and Ethers Alcohols and Phenols Alcohol Dehydration Ease
MediumNumerical value · figureIdea asked in 6 yearsAsked the year before
Q90
20 mL of calcium hydroxide was consumed when it was reacted with 10 mL of unknown solution of $\mathrm{H_2SO_4}$ . Also 20 mL standard solution of 0.5 M HCl containing 2 drops of phenolphthalein was titrated with calcium hydroxide, the mixture showed pink…
Some Basic Concepts of Chemistry Stoichiometry Mole-to-Mole Relationships
MediumNumerical valueIdea asked in 3 yearsAsked the year before

Maths29 questions

2 easy15 medium12 hard

Q1
For the differentiable function $f : \mathbb{R}-\{0\} \rightarrow \mathbb{R}$ , let $3f(x)+2f\left(\frac{1}{x}\right)=\frac{1}{x}-10$ , then $\left|f(3)+f'\left(\frac{1}{4}\right)\right|$ is equal to
Continuity and Differentiability Chain Rule Differentiating Functional Equations
MediumSingle correctIdea asked in 2 years
Q2
The set of all $a \in \mathbb{R}$ for which the equation $x|x-1|+|x+2|+a=0$ has exactly one real root, is
Relations and Functions Quadratic Roots Roots of Modulus Quadratics
HardSingle correctIdea asked in 6 yearsAsked the year before
Q3
Let $B=\begin{bmatrix}1&3&\alpha\\1&2&3\\\alpha&\alpha&4\end{bmatrix}$ , $\alpha>2$ be the adjoint of a matrix $A$ and $|A|=2$ . Then $\begin{bmatrix}\alpha&-2\alpha&\alpha\end{bmatrix}B\begin{bmatrix}\alpha\\-2\alpha\\\alpha\end{bmatrix}$ is equal to
Determinants Matrix Inverse Inverse Relation to Double Adjoint
HardSingle correctIdea asked in 6 yearsAsked the year before
Q4
For the system of linear equations $2x+4y+2az=b$ $x+2y+3z=4$ $2x-5y+2z=8$ which of the following is NOT correct?
Determinants Cramer's Rule Evaluating Parameters for Consistency
MediumSingle correctIdea asked in 6 yearsAsked the year before
Q5
The number of symmetric matrices of order 3, with all the entries from the set $\{0, 1, 2, 3, 4, 5, 6, 7, 8, 9\}$ , is
Matrices Permutations and Combinations Permutations with Repetition
EasySingle correctIdea asked in 4 yearsAsked the year before
Q6
Let $s_1, s_2, s_3, \ldots, s_{10}$ respectively be the sum to 12 terms of 10 A.P. s whose first terms are $1, 2, 3, \ldots, 10$ and the common differences are $1, 3, 5, \ldots, 19$ respectively. Then $\sum_{i=1}^{10} s_i$ is equal to
Sequences and Series AP and GP AP Sum of n terms
MediumSingle correctIdea asked in 7 yearsAsked the year before
Q7
Among (S1) : $\lim_{n\to\infty}\frac{1}{n^2}(2+4+6+\ldots\ldots+2n)=1$ (S2) : $\lim_{n\to\infty}\frac{1}{n^{16}}\left(1^{15}+2^{15}+3^{15}+\ldots\ldots+n^{15}\right)=\frac{1}{16}$
Limits and Derivatives Integral as Limit of Sum Summation to Riemann Form
MediumTwo statementsIdea asked in 4 yearsAsked the year before
Q8
Let the equation of plane passing through the line of intersection of the planes $x+2y+az=2$ and $x-y+z=3$ be $5x-11y+bz=6a-1$ . For $c \in \mathbb{Z}$ , if the distance of this plane from the point $(a,-c,c)$ is $\frac{2}{\sqrt{a}}$ , then $\frac{a+b}{c}$ is…
Three Dimensional Geometry Planes in 3D Family of Planes
MediumSingle correctIdea asked in 4 yearsAsked the year before
Q9
Let $\vec{a}=\hat{i}+4\hat{j}+2\hat{k}$ , $\vec{b}=3\hat{i}-2\hat{j}+7\hat{k}$ and $\vec{c}=2\hat{i}-\hat{j}+4\hat{k}$ . If a vector $\vec{d}$ satisfies $\vec{d}\times\vec{b}=\vec{c}\times\vec{b}$ and $\vec{d}\cdot\vec{a}=24$ , then $|\vec{d}|^2$ is equal to
Vector Algebra Cross Product Simultaneous Cross and Dot Equations
MediumSingle correctIdea asked in 5 yearsAsked the year before
Q10
$\max_{0\leq x\leq\pi}\left\{x-2\sin x\cos x+\frac{1}{3}\sin 3x\right\}=$
Application of Derivatives Maxima and Minima Closed Interval Extrema
HardSingle correctIdea asked in 6 yearsAsked the year before
Q11
$\int_0^{\infty}\frac{6}{e^{3x}+6e^{2x}+11e^x+6}\,dx=$
Integrals Integration by Substitution Algebraic Manipulation
MediumSingle correctNew idea
Q12
The area of the region enclosed by the curve $f(x)=\max\{\sin x,\cos x\}$ , $-\pi\leq x\leq\pi$ and the $x$ -axis is
Application of Integrals Area Under Curves Area Involving Max/Min Functions
MediumSingle correctIdea asked in 6 yearsAsked the year before
Q13
Fractional part of the number $\frac{4^{2022}}{15}$ is equal to
Binomial Theorem
MediumSingle correct
Q14
Let $y=y_1(x)$ and $y=y_2(x)$ be the solution curves of the differential equation $\frac{dy}{dx}=y+7$ with initial conditions $y_1(0)=0$ and $y_2(0)=1$ respectively. Then the curves $y=y_1(x)$ and $y=y_2(x)$ intersect at
Differential Equations Variable Separable Form Initial Value Problem
EasySingle correctIdea asked in 7 yearsAsked the year before
Q15
Let the tangent and normal at the point $(3\sqrt{3},1)$ on the ellipse $\frac{x^2}{36}+\frac{y^2}{4}=1$ meet the $y$ -axis at the points $A$ and $B$ respectively. Let the circle $C$ be drawn taking $AB$ as a diameter and the line $x=2\sqrt{5}$ intersect $C$ …
Conic Sections Circle Tangents Equation of Polar
HardSingle correctNew idea
Q16
Let PQ be a focal chord of the parabola $y^2=36x$ of length 100, making an acute angle with the positive $x$ -axis. Let the ordinate of P be positive and M be the point on the line segment PQ such that PM:MQ = 3:1. Then which of the following points does NOT…
Conic Sections Parabola Focal Chords Focal Chord Length via Slope
HardSingle correctNew ideaIdea asked in 2 years
Q17
The distance of the point $(-1,2,3)$ from the plane $\vec{r}\cdot\left(\hat{i}-2\hat{j}+3\hat{k}\right)=10$ parallel to the line of the shortest distance between the lines $\vec{r}=\left(\hat{i}-\hat{j}\right)+\lambda\left(2\hat{i}+\hat{k}\right)$ and…
Three Dimensional Geometry Point-Plane Distance Distance Along a Line
HardSingle correctIdea asked in 5 yearsAsked the year before
Q18
A coin is biased so that the head is 3 times as likely to occur as tail. This coin is tossed until a head or three tails occur. If X denotes the number of tosses of the coin, then the mean of X is
Probability
MediumSingle correct
Q20
The negation of the statement $((A\wedge(B\vee C))\Rightarrow(A\vee B))\Rightarrow A$ is
Mathematical Reasoning
MediumSingle correct
Q21
Let $w=z\bar{z}+k_1z+k_2iz+\lambda(1+i)$ , $k_1,k_2\in\mathbb{R}$ . Let $\mathrm{Re}(w)=0$ be the circle C of radius 1 in the first quadrant touching the line $y=1$ and the $y$ -axis. If the curve $\mathrm{Im}(w)=0$ intersects C at A and B, then $30(AB)^2$ is…
Complex Numbers and Quadratic Equations Power of a Point Radical axis of two circles
HardNumerical valueIdea asked in 3 years
Q22
The number of seven digit positive integers formed using the digits 1, 2, 3 and 4 only and sum of the digits equal to 12 is ________.
Permutations and Combinations Multinomial Theorem in Counting
HardNumerical valueIdea asked in 2 years
Q23
Let $\alpha$ be the constant term in the binomial expansion of $\left(\sqrt{x}-\frac{6}{x^{3/2}}\right)^n$ , $n\leq15$ . If the sum of the coefficients of the remaining terms in the expansion is 649 and the coefficient of $x^{-n}$ is $\lambda\alpha$ , then…
Binomial Theorem Binomial Coefficients Sum of all Coefficients
HardNumerical valueIdea asked in 4 yearsAsked the year before
Q24
The sum to 20 terms of the series $2\cdot2^2-3^2+2\cdot4^2-5^2+2\cdot6^2-\ldots\ldots$ is equal to ________.
Sequences and Series AP and GP AP Sum of n terms
MediumNumerical valueIdea asked in 7 yearsAsked the year before
Q25
Let for $x\in\mathbb{R}$ , $S_0(x)=x$ , $S_k(x)=C_kx+k\int_0^xS_{k-1}(t)\,dt$ , where $C_0=1$ , $C_k=1-\int_0^1S_{k-1}(x)\,dx$ , $k=1,2,3,\ldots$ . Then $S_2(3)+6C_3$ is equal to ________.
Integrals
HardNumerical value
Q26
Let $m_1$ and $m_2$ be the slopes of the tangents drawn from the point P(4, 1) to the hyperbola $H:\frac{y^2}{25}-\frac{x^2}{16}=1$ . If Q is the point from which the tangents drawn to H have slopes $|m_1|$ and $|m_2|$ and they make positive intercepts…
Conic Sections
HardNumerical value
Q27
Let the image of the point $\left(\frac{5}{3},\frac{5}{3},\frac{8}{3}\right)$ in the plane $x-2y+z-2=0$ be P. If the distance of the point $Q(6,-2,\alpha)$ , $\alpha>0$ , from P is 13, then $\alpha$ is equal to ________.
Three Dimensional Geometry Planes in 3D Image of a Point in a Plane
MediumNumerical valueIdea asked in 2 years
Q28
Let $\vec{a}=3\hat{i}+\hat{j}-\hat{k}$ and $\vec{c}=2\hat{i}-3\hat{j}+3\hat{k}$ . If $\vec{b}$ is a vector such that $\vec{a}=\vec{b}\times\vec{c}$ and $|\vec{b}|^2=50$ , then $\left|72-|\vec{b}+\vec{c}|^2\right|$ is equal to ________.
Vector Algebra Cross Product Solving Vector Equations with Cross Products
MediumNumerical valueIdea asked in 6 yearsAsked the year before
Q29
Let the mean of the data | $x$ | 1 | 3 | 5 | 7 | 9 | | Frequency ( $f$ ) | 4 | 24 | 28 | $\alpha$ | 8 | be 5. If $m$ and $\sigma^2$ are respectively the mean deviation about the mean and the variance of the data, then $\frac{3\alpha}{m+\sigma^2}$ is equal to…
Statistics Variance and SD Finding Unknown Frequency via Variance
MediumNumerical valueIdea asked in 2 years
Q30
If $S=\left\{x\in\mathbb{R}:\sin^{-1}\left(\frac{x+1}{\sqrt{x^2+2x+2}}\right)-\sin^{-1}\left(\frac{x}{\sqrt{x^2+1}}\right)=\frac{\pi}{4}\right\}$ , then $\sum_{x\in s}\left(\sin\left((x^2+x+5)\frac{\pi}{2}\right)-\cos\left((x^2+x+5)\pi\right)\right)$ is equal…
Inverse Trigonometric Functions Inverse Trig Identities Sine Inverse Sum and Difference
HardNumerical valueIdea asked in 3 years

Dropped from the final key and left out of every count: Q19.

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.