JEE Main 2023 · April session

JEE Main 8 April 2023, Shift 2

89 questions (1 more dropped by NTA): 30 numerical-answer and 59 multiple choice. 14 were rated hard and 37 easy. Its hard-question count was the joint 5th 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: 14; 16% of the shift
Chapters asked: 69; 10 questions with a figure

Key findings · 8 Apr 2023, Shift 2

1
14 hard questions (16%): the joint 5th 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 13 of 29 rated hard; Physics had the most easy questions (20 of 30).
3
17 questions tested a new idea, one no earlier year had asked: 23% of the 75 questions tagged to an idea.
4
36 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), Coordination Compounds in Chemistry (2) and Conic Sections in Maths (3).

The shift at a glance

Subjects, difficulty and chapters

Physics

30 Q · 10 numerical

Easy 20Medium 9Hard 1

Top chapters

Chemistry

30 Q · 10 numerical

Easy 15Medium 15Hard 0

Top chapters

Maths

29 Q · 10 numerical

Easy 2Medium 14Hard 13

Top chapters

Question format

Single correct42 Q
Numerical value30 Q
Other5 Q
Diagram based4 Q
Match the list3 Q
Two statements3 Q
Assertion–reason2 Q

What each question asks you to do

Multi-step27 Q
Calculation26 Q
Application13 Q
Recall12 Q
Conceptual9 Q
Diagram reading1 Q
Data or graph1 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

20 easy9 medium1 hard

Q31
Match List I with List II LIST I: A. Torque; B. Stress; C. Pressure gradient; D. Coefficient of viscosity LIST II: I. $\mathrm{ML^{-2}T^{-2}}$ ; II. $\mathrm{ML^{2}T^{-2}}$ ; III. $\mathrm{ML^{-1}T^{-1}}$ ; IV. $\mathrm{ML^{-1}T^{-2}}$ Choose the correct answer…
Units and Measurements
EasyMatch the list
Q32
For particle $P$ revolving round the centre $O$ with radius of circular path $r$ and angular velocity $\omega$ , as shown in below figure, the projection of $OP$ on the $x$ -axis at time $t$ is
Oscillations SHM Equations Phasor Diagram Technique
MediumSingle correct · figureIdea asked in 3 years
Q33
The temperature at which the kinetic energy of oxygen molecules becomes double than its value at $27^\circ$ C is
Kinetic Theory Kinetic Theory of Gases Kinetic Energy and Temp
EasySingle correctIdea asked in 7 yearsAsked the year before
Q34
Work done by a Carnot engine operating between temperatures $127^\circ$ C and $27^\circ$ C is 2 kJ. The amount of heat transferred to the engine by the reservoir is :
Thermodynamics Heat Engines and Refrigerators Carnot Efficiency Limit
EasySingle correctIdea asked in 3 yearsAsked the year before
Q35
A hydraulic automobile lift is designed to lift vehicles of mass 5000 kg. The area of cross section of the cylinder carrying the load is 250 cm $^2$ . The maximum pressure the smaller piston would have to bear is [Assume $g = 10$ m/s $^2$ ]:
Mechanical Properties of Fluids Fluid Statics Pascal's Law
EasySingle correctIdea asked in 3 years
Q36
The acceleration due to gravity at height $h$ above the earth if $h \ll R$ (Radius of earth) is given by
Gravitation
EasySingle correct
Q37
The orbital angular momentum of a satellite is L, when it is revolving in a circular orbit at height h from earth surface. If the distance of satellite from the earth centre is increased by eight times to its initial value, then the new angular momentum will…
Gravitation Orbital Mechanics Angular Momentum Dependence on Radius
MediumSingle correctNew idea
Q38
Given below are two statements: Statement I : Area under velocity- time graph gives the distance travelled by the body in a given time. Statement II : Area under acceleration- time graph is equal to the change in velocity- in the given time. In the light of…
Motion in a Straight Line Kinematics in 1D Kinematic Graph Areas
EasyTwo statementsIdea asked in 3 years
Q39
The trajectory of projectile, projected from the ground is given by $y = x - \frac{x^2}{20}$ . Where $x$ and $y$ are measured in meter. The maximum height attained by the projectile will be.
Motion in a Plane Projectile Motion Trajectory Equation Application
MediumSingle correctIdea asked in 3 yearsAsked the year before
Q40
A bullet of mass 0.1 kg moving horizontally with speed 400 ms $^{-1}$ hits a wooden block of mass 3.9 kg kept on a horizontal rough surface. The bullet gets embedded into the block and moves 20 m before coming to rest. The coefficient of friction between the…
Work, Energy and Power Collisions Inelastic Collisions
MediumSingle correctIdea asked in 3 yearsAsked the year before
Q41
Electric potential at a point 'P' due to a point charge of $5\times10^{-9}$ C is 50 V. The distance of 'P' from the point charge is: (Assume, $\frac{1}{4\pi\varepsilon_0} = 9 \times 10^{+9}\ \mathrm{Nm^2C^{-2}}$ )
Electrostatic Potential and Capacitance Electric Potential Point Charge Potential
EasySingle correctIdea asked in 3 years
Q42
The power radiated from a linear antenna of length $l$ is proportional to (Given, $\lambda$ = Wavelength of wave):
Electromagnetic Waves
HardSingle correct
Q43
For a given transistor amplifier circuit in CE configuration $V_{CC} = 1$ V, $R_C = 1$ k $\Omega$ , $R_b = 100$ k $\Omega$ and $\beta = 100$ . Value of base current $I_b$ is
Semiconductor Electronics: Materials, Devices and Simple Circuits Transistors CE Amplifier Voltage Gain
EasySingle correct · figureIdea asked in 3 yearsAsked the year before
Q44
A radio active material is reduced to 1/8 of its original amount in 3 days. If $8\times10^{-3}$ kg of the material is left after 5 days the initial amount of the material is
Nuclei Radioactive Decay Half-Life Calculations
MediumSingle correctIdea asked in 4 yearsAsked the year before
Q45
In photo electric effect A. The photocurrent is proportional to the intensity of the incident radiation. B. Maximum Kinetic energy with which photoelectrons are emitted depends on the intensity of incident light. C. Max. K.E with which photoelectrons are…
Dual Nature of Radiation and Matter Photoelectric Effect Intensity vs Frequency Effects
EasyOtherIdea asked in 3 years
Q46
The width of fringe is 2 mm on the screen in a double slits experiment for the light of wavelength of 400 nm. The width of the fringe for the light of wavelength 600 nm will be:
Wave Optics Young's Double Slit YDSE Fringe Width
EasySingle correctIdea asked in 5 yearsAsked the year before
Q47
The waves emitted when a metal target is bombarded with high energy electrons are
Electromagnetic Waves X-Ray Production Continuous Spectrum Origin
EasySingle correctNew idea
Q48
An emf of 0.08 V is induced in a metal rod of length 10 cm held normal to a uniform magnetic field of 0.4 T, when moves with a velocity of:
Electromagnetic Induction Faraday's Law Linear Motional EMF
EasySingle correctIdea asked in 3 yearsAsked the year before
Q49
Given below are two statements: one is labelled as Assertion A and the other is labelled as Reason R Assertion A : Electromagnets are made of soft iron. Reason R : Soft iron has high permeability and low retentivity. In the light of above, statements, choose…
Magnetism and Matter
EasyAssertion–reason
Q50
The equivalent resistance between A and B as shown in figure is:
Current Electricity Kirchhoff's Laws Short Circuits and Equipotential Nodes
EasySingle correct · figureIdea asked in 3 yearsAsked the year before
Q51
A guitar string of length 90 cm vibrates with a fundamental frequency of 120 Hz. The length of the string producing a fundamental frequency of 180 Hz will be ____________ cm.
Waves Waves on Strings Sonometer Laws
EasyNumerical valueIdea asked in 4 years
Q52
A steel rod of length 1 m and cross sectional area $10^{-4}$ m $^2$ is heated from $0^\circ$ C to $200^\circ$ C without being allowed to extend or bend. The compressive tension produced in the rod is ____________ $\times 10^4$ N. (Given Young's modulus of steel…
Thermal Properties of Matter Thermal Expansion Thermal Stress
EasyNumerical valueIdea asked in 3 years
Q53
A hollow spherical ball of uniform density rolls up a curved surface with an initial velocity 3 m/s (as shown in figure). Maximum height with respect to the initial position covered by it will be __________ cm (take, $g = 10$ m/s $^2$ )
System of Particles and Rotational Motion
MediumNumerical value · figure
Q54
A body of mass 5 kg is moving with a momentum of 10 kg ms $^{-1}$ . Now a force of 2 N acts on the body in the direction of its motion for 5 s. The increase in the Kinetic energy of the body is __________ J.
Work, Energy and Power Work and Kinetic Energy Momentum and Kinetic Energy Relation
EasyNumerical valueIdea asked in 4 yearsAsked the year before
Q55
A 600 pF capacitor is charged by 200 V supply. It is then disconnected from the supply and is connected to another uncharged 600 pF capacitor. Electrostatic energy lost in the process is ____________ $\mu$ J.
Electrostatic Potential and Capacitance Parallel Plate Capacitors Charge Sharing and Heat Loss
MediumNumerical valueIdea asked in 6 yearsAsked the year before
Q56
The ratio of wavelength of spectral lines $\mathrm{H}_\alpha$ and $\mathrm{H}_\beta$ in the Balmer series is $\frac{x}{20}$ . The value of $x$ is __________
Atoms Bohr's Atomic Model Balmer Series Specific Lines
EasyNumerical valueNew idea
Q57
Two transparent media having refractive indices 1.0 and 1.5 are separated by a spherical refracting surface of radius of curvature 30 cm. The centre of curvature of surface is towards denser medium and a point object is placed on the principle axis in rarer…
Ray Optics and Optical Instruments
MediumNumerical value
Q58
A series combination of resistor of resistance 100 $\Omega$ , inductor of inductance 1 H and capacitor of capacitance 6.25 $\mu$ F is connected to an ac source. The quality factor of the circuit will be ____________
Alternating Current LCR Resonance Q-Factor Formula in L, C, R
EasyNumerical valueNew idea
Q59
The ratio of magnetic field at the centre of a current carrying coil of radius $r$ to the magnetic field at distance $r$ from the centre of coil on its axis is $\sqrt{x} : 1$ . The value of $x$ is __________
Moving Charges and Magnetism Biot-Savart Law Ratio of Axial to Central Magnetic Field
MediumNumerical valueIdea asked in 4 yearsAsked the year before
Q60
The number density of free electrons in copper is nearly $8 \times 10^{28}$ m $^{-3}$ . A copper wire has its area of cross section = $2 \times 10^{-6}$ m $^2$ and is carrying a current of 3.2 A. The drift speed of the electrons is __________ $\times 10^{-6}$ …
Current Electricity Ohm's Law and Resistance Drift Velocity and Current
EasyNumerical valueIdea asked in 3 yearsAsked the year before

Chemistry30 questions

15 easy15 medium0 hard

Q61
Which of the following have same number of significant figures? A. 0.00253 B. 1.0003 C. 15.0 D. 163 Choose the correct answer from the options given below
Some Basic Concepts of Chemistry
EasyOther
Q62
Arrange the following gases in increasing order of van der Waals constant 'a' A. Ar B. $\mathrm{CH_4}$ C. $\mathrm{H_2O}$ D. $\mathrm{C_6H_6}$ Choose the correct option from the following.
States of Matter Real Gases Predicting 'a' from Molecular Structure
EasyOtherNew idea
Q63
The correct reaction profile diagram for a positive catalyst reaction.
Chemical Kinetics Arrhenius Equation Reaction Energy Profiles
EasyDiagram based · figureIdea asked in 3 years
Q64
Henry Moseley studied characteristic X-ray spectra of elements. The graph which represents his observation correctly is Given $\nu$ = frequency of X-ray emitted $Z$ = atomic number
Structure of Atom X-Ray Production Moseley's Law Graph Analysis
EasyDiagram based · figureNew idea
Q65
In Hall - Heroult process, the following is used for reducing $\mathrm{Al_2O_3}$ :-
General Principles and Processes of Isolation of Elements Metallurgical Extraction Hall-Heroult Process
EasySingle correctIdea asked in 3 yearsAsked the year before
Q66
Which of the following can reduce decomposition of $\mathrm{H_2O_2}$ on exposure to light
Hydrogen Hydrogen and Hydrides Storage and Stabilization of H2O2
MediumSingle correctIdea asked in 2 yearsAsked the year before
Q67
Given below are two statements: One is labelled as Assertion A and the other is labelled as Reason R Assertion A : Sodium is about 30 times as abundant as potassium in the oceans. Reason R : Potassium is bigger in size than sodium. In the light of the above…
The s-Block Elements s-Block Elements Biological Role of Na and K
MediumAssertion–reasonNew idea
Q68
For a good quality cement, the ratio of lime to the total of the oxides of Si, Al and Fe should be as close as to
The p-Block Elements s-Block Elements Portland Cement Composition and Setting
MediumSingle correctIdea asked in 2 years
Q69
Match List I with List II LIST I (Coordination Complex): A. $[\mathrm{Cr(CN)_6}]^{3-}$ ; B. $[\mathrm{Fe(H_2O)_6}]^{2+}$ ; C. $[\mathrm{Co(NH_3)_6}]^{3+}$ ; D. $[\mathrm{Ni(NH_3)_6}]^{2+}$ LIST II (Number of unpaired electrons): I. 0; II. 3; III. 2; IV. 4…
Coordination Compounds Crystal Field Theory Spin-only Magnetic Moment
MediumMatch the listIdea asked in 7 yearsAsked the year before
Q70
Which of these reactions is not a part of breakdown of ozone in stratosphere?
Environmental Chemistry Ozone Layer Depletion
MediumSingle correctIdea asked in 4 yearsAsked the year before
Q71
Major product 'P' formed in the following reaction is: [reaction: $\mathrm{CH_2{=}CH{-}CH_2{-}CH_2{-}COOH}$ (pent-4-enoic acid, drawn as skeletal structure) $\xrightarrow{Br_2,\ NaHCO_3}$ P (Major product)]
Haloalkanes and Haloarenes Alkenes and Alkynes Halohydrin Formation Regiochemistry
MediumDiagram based · figureNew idea
Q72
The correct IUPAC nomenclature for the following compound is: [structure: $\mathrm{CH_3{-}CO{-}CH_2{-}CH_2{-}CH(COOH){-}CH_3}$ ]
Organic Chemistry: Some Basic Principles and Techniques IUPAC Nomenclature Functional Group Priority
EasySingle correct · figureIdea asked in 6 yearsAsked the year before
Q73
The correct order of reactivity of following haloarenes towards nucleophilic substitution with aqueous NaOH is : A. [chlorobenzene] B. [1-chloro-2-nitrobenzene (NO $_2$ ortho to Cl)] C. [1-chloro-2-methoxybenzene (OMe ortho to Cl)] D.…
Haloalkanes and Haloarenes Aromatic Hydrocarbons SNAr and Meisenheimer Complex
MediumSingle correct · figureIdea asked in 3 years
Q74
The descending order of acidity for the following carboxylic acid is- A. $\mathrm{CH_3COOH}$ B. $\mathrm{F_3C{-}COOH}$ C. $\mathrm{ClCH_2{-}COOH}$ D. $\mathrm{FCH_2{-}COOH}$ E. $\mathrm{BrCH_2{-}COOH}$ Choose the correct answer from the options given below:
Aldehydes, Ketones and Carboxylic Acids Carboxylic Acids and Derivatives Carboxylic Acid Acidity
EasyOtherNew idea
Q75
A compound 'X' when treated with phthalic anhydride in presence of concentrated $\mathrm{H_2SO_4}$ yields 'Y'. 'Y' is used as an acid/base indicator. 'X' and 'Y' are respectively
Alcohols, Phenols and Ethers Alcohols and Phenols Phthalein Dye Test
EasySingle correctIdea asked in 3 years
Q76
The product (P) formed from the following multistep reaction is: [reaction: 4-nitrotoluene (benzene ring with $\mathrm{NO_2}$ and $\mathrm{CH_3}$ para to each other)…
Amines Diazonium Salts Replacement by H or OH
MediumDiagram based · figureIdea asked in 3 years
Q77
The statement/s which are true about antagonists from the following is/are: A. They bind to the receptor site. B. Get transferred inside the cell for their action. C. Inhibit the natural communication of the body. D. Mimic the natural messenger. Choose the…
Chemistry in Everyday Life Receptor Agonists and Antagonists
EasyOtherIdea asked in 2 yearsAsked the year before
Q78
Match List I with List II LIST I (Natural amino acid): A. Glutamic acid; B. Glutamine; C. Tyrosine; D. Tryptophan LIST II (One letter code): I. Q; II. W; III. E; IV. Y Choose the correct answer from the options given below:
Biomolecules
EasyMatch the list
Q79
Given below are two statements: Statement I : Methyl orange is a weak acid. Statement II : The benzenoid form of methyl orange is more intense/deeply coloured than the quinonoid form. In the light of the above statement, choose the most appropriate answer…
Equilibrium
MediumTwo statements
Q80
Given below are two statements: Statement I : In redox titration, the indicators used are sensitive to change in pH of the solution. Statement II : In acid-base titration, the indicators used are sensitive to change in oxidation potential. In the light of…
Redox Reactions
EasyTwo statements
Q81
The number of atomic orbitals from the following having 5 radial nodes is ____________. 7s, 7p, 6s, 8p, 8d
Structure of Atom Quantum Numbers Calculation of Radial and Angular Nodes
EasyNumerical valueIdea asked in 5 yearsAsked the year before
Q82
The number of species from the following carrying a single lone pair on central atom Xenon is ____________. $\mathrm{XeF_5^+}$ , $\mathrm{XeO_3}$ , $\mathrm{XeO_2F_2}$ , $\mathrm{XeF_5^-}$ , $\mathrm{XeO_3F_2}$ , $\mathrm{XeOF_4}$ , $\mathrm{XeF_4}$
Chemical Bonding and Molecular Structure VSEPR Theory Steric Number Calculation
MediumNumerical valueIdea asked in 6 yearsAsked the year before
Q83
For complete combustion of ethene, $\mathrm{C_2H_4(g) + 3O_2(g) \rightarrow 2CO_2(g) + 2H_2O(l)}$ the amount of heat produced as measured in bomb calorimeter is 1406 kJ mol $^{-1}$ at 300K. The minimum value of $T\Delta S$ needed to reach equilibrium is ( $-$ )…
Thermodynamics (Chemistry)Enthalpy and Hess's Law Bomb Calorimetry Calculations
MediumNumerical valueIdea asked in 6 yearsAsked the year before
Q84
If the boiling points of two solvents X and Y (having same molecular weights) are in the ratio 2:1 and their enthalpy of vaporizations are in the ratio 1:2, then the boiling point elevation constant of X is $m$ times the boiling point elevation constant of…
Solutions Colligative Properties Thermodynamic Derivation of Ebullioscopic Constant (Kb)
MediumNumerical valueNew idea
Q85
The solubility product of $\mathrm{BaSO_4}$ is $1\times10^{-10}$ at 298K. The solubility of $\mathrm{BaSO_4}$ in 0.1 M $\mathrm{K_2SO_4}$ (aq) solution is ____________ $\times 10^{-9}$ g L $^{-1}$ (nearest integer). Given: Molar mass of $\mathrm{BaSO_4}$ is…
Equilibrium Solubility Product Common Ion Effect
EasyNumerical valueIdea asked in 4 yearsAsked the year before
Q86
The number of incorrect statements from the following is ____________ A. The electrical work that a reaction can perform at constant pressure and temperature is equal to the reaction Gibbs energy. B. $E^{\circ}_{cell}$ is dependent on the pressure. C.…
Electrochemistry Nernst Equation Gibbs Energy and EMF
MediumNumerical valueIdea asked in 3 yearsAsked the year before
Q87
Coagulating value of the electrolytes $\mathrm{AlCl_3}$ and NaCl for $\mathrm{As_2S_3}$ are 0.09 and 50.04 respectively. The coagulating power of $\mathrm{AlCl_3}$ is $x$ times the coagulating power of NaCl. The value of $x$ is ____________.
Surface Chemistry Colloids Flocculation Value Calculation
EasyNumerical valueIdea asked in 2 yearsAsked the year before
Q88
The ratio of sigma and $\pi$ bonds present in pyrophosphoric acid is ____________.
The p-Block Elements Group 15 & 16 Phosphorus Oxoacids
MediumNumerical valueIdea asked in 4 yearsAsked the year before
Q89
The sum of oxidation state of the metals in $\mathrm{Fe(CO)_5}$ , $\mathrm{VO^{2+}}$ and $\mathrm{WO_3}$ is ____________.
The d- and f-Block Elements d-Block Elements Oxidation State Trends
EasyNumerical valueIdea asked in 4 years
Q90
The observed magnetic moment of the complex $[\mathrm{Mn(NCS)_6}]^{x-}$ is 6.06 BM. The numerical value of $x$ is ____________.
Coordination Compounds Crystal Field Theory Spin-only Magnetic Moment
MediumNumerical valueIdea asked in 7 yearsAsked the year before

Maths29 questions

2 easy14 medium13 hard

Q1
Let $A = \{1, 2, 3, 4, 5, 6, 7\}$ . Then the relation $R = \{(x, y) \in A \times A : x + y = 7\}$ is
Relations and Functions Sets and Relations Relation Types
EasySingle correctIdea asked in 5 years
Q2
Let $A = \left\{\theta \in (0, 2\pi) : \frac{1 + 2i\sin\theta}{1 - i\sin\theta} \text{ is purely imaginary}\right\}$ . Then the sum of the elements in $A$ is
Complex Numbers and Quadratic Equations Complex Numbers - Polar Form Purely Real and Imaginary Euler Conditions
MediumSingle correctNew ideaIdea asked in 3 years
Q3
Let S be the set of all values of $\theta \in [-\pi, \pi]$ for which the system of linear equations $x + y + \sqrt{3}z = 0$ $-x + (\tan\theta)y + \sqrt{7}z = 0$ $x + y + (\tan\theta)z = 0$ has non-trivial solution. Then…
Determinants Cramer's Rule Parameter Evaluation in Homogeneous Systems
MediumSingle correctIdea asked in 5 years
Q4
If $A = \begin{bmatrix} 1 & 5 \\ \lambda & 10 \end{bmatrix}$ , $A^{-1} = \alpha A + \beta I$ and $\alpha + \beta = -2$ , then $4\alpha^2 + \beta^2 + \lambda^2$ is equal to :
Matrices Matrix Inverse Inverse via Characteristic Equation
MediumSingle correctIdea asked in 2 years
Q5
If the number of words, with or without meaning, which can be made using all the letters of the word MATHEMATICS in which C and S do not come together, is $(6!)k$ , then $k$ is equal to
Permutations and Combinations The Block/Tie Method
MediumSingle correctIdea asked in 4 years
Q6
$25^{190} - 19^{190} - 8^{190} + 2^{190}$ is divisible by
Binomial Theorem
MediumSingle correct
Q7
The absolute difference of the coefficients of $x^{10}$ and $x^7$ in the expansion of $\left(2x^2 + \frac{1}{2x}\right)^{11}$ is equal to
Binomial Theorem
MediumSingle correct
Q8
Let $a_n$ be the $n^{th}$ term of the series $5 + 8 + 14 + 23 + 35 + 50 + \ldots$ and $S_n = \sum_{k=1}^{n} a_k$ . Then $S_{30} - a_{40}$ is equal to
Sequences and Series AP and GP Method of Differences for Telescoping Series
MediumSingle correctIdea asked in 6 yearsAsked the year before
Q9
If $\alpha > \beta > 0$ are the roots of the equation $ax^2 + bx + 1 = 0$ , and $\lim_{x \to \frac{1}{\alpha}} \left(\frac{1 - \cos(x^2 + bx + a)}{2(1 - \alpha x)^2}\right)^{\frac{1}{2}} = \frac{1}{k}\left(\frac{1}{\beta} - \frac{1}{\alpha}\right)$ , then $k$ …
Limits and Derivatives L'Hopital's Rule Simplifying with Standard Limits Before L'Hopital
HardSingle correctIdea asked in 6 yearsAsked the year before
Q11
Let $O$ be the origin and $OP$ and $OQ$ be the tangents to the circle $x^2 + y^2 - 6x + 4y + 8 = 0$ at the points $P$ and $Q$ on it. If the circumcircle of the triangle $OPQ$ passes through the point $\left(\alpha, \frac{1}{2}\right)$ , then a value of…
Conic Sections Circle Tangents Circumcircle of Tangent Triangle
MediumSingle correctNew idea
Q12
Let $A(0, 1)$ , $B(1, 1)$ and $C(1, 0)$ be the mid-points of the sides of a triangle with incentre at the point $D$ . If the focus of the parabola $y^2 = 4ax$ passing through $D$ is $(\alpha + \beta\sqrt{2}, 0)$ , where $\alpha$ and $\beta$ are rational…
Conic Sections Straight Line Distance Incenter via Equidistance
HardSingle correctIdea asked in 3 years
Q13
Let $P$ be the plane passing through the line $\frac{x-1}{1} = \frac{y-2}{-3} = \frac{z+5}{7}$ and the point $(2, 4, -3)$ . If the image of the point $(-1, 3, 4)$ in the plane $P$ is $(\alpha, \beta, \gamma)$ then $\alpha + \beta + \gamma$ is equal to
Three Dimensional Geometry Planes in 3D Plane Containing a Line and a Point
MediumSingle correctIdea asked in 3 yearsAsked the year before
Q14
For $a, b \in \mathbb{Z}$ and $|a - b| \leq 10$ , let the angle between the plane $P: ax + y - z = b$ and the line $l: x - 1 = a - y = z + 1$ be $\cos^{-1}\left(\frac{1}{3}\right)$ . If the distance of the point $(6, -6, 4)$ from the plane $P$ is $3\sqrt{6}$ ,…
Three Dimensional Geometry Planes in 3D Angle Between a Line and a Plane
HardSingle correctNew idea
Q15
The area of the quadrilateral $ABCD$ with vertices $A(2, 1, 1)$ , $B(1, 2, 5)$ , $C(-2, -3, 5)$ and $D(1, -6, -7)$ is equal to
Vector Algebra Cross Product Area of Quadrilateral via Diagonals
MediumSingle correctIdea asked in 3 yearsAsked the year before
Q16
Let the vectors $\vec{u}_1 = \hat{i} + \hat{j} + a\hat{k}$ , $\vec{u}_2 = \hat{i} + b\hat{j} + \hat{k}$ and $\vec{u}_3 = c\hat{i} + \hat{j} + \hat{k}$ be coplanar. If the vectors $\vec{v}_1 = (a + b)\hat{i} + c\hat{j} + c\hat{k}$ ,…
Vector Algebra Triple Products Parametric Coplanarity via STP
HardSingle correctNew idea
Q17
If the probability that the random variable $X$ takes values $x$ is given by $P(X = x) = k(x + 1)3^{-x}$ , $x = 0, 1, 2, 3, \ldots$ , where $k$ is a constant, then $P(X \geq 2)$ is equal to
Probability Arithmetico-Geometric Series AGP in Expected Value Scenarios
MediumSingle correctIdea asked in 2 years
Q18
Let the mean and variance of 12 observations be $\frac{9}{2}$ and $4$ respectively. Later on, it was observed that two observations were considered as 9 and 10 instead of 7 and 14 respectively. If the correct variance is $\frac{m}{n}$ , where $m$ and $n$ are…
Statistics Variance and SD Correcting Incorrect Variance
MediumSingle correctIdea asked in 6 yearsAsked the year before
Q19
The value of $36(4\cos^2 9^\circ - 1)(4\cos^2 27^\circ - 1)(4\cos^2 81^\circ - 1)(4\cos^2 243^\circ - 1)$ is
Trigonometric Functions Trigonometric Identities Triple Angle Formulas
HardSingle correctIdea asked in 2 years
Q20
The negation of $(p \wedge (\sim q)) \vee (\sim p)$ is equivalent to
Mathematical Reasoning
EasySingle correct
Q21
If domain of the function $\log_e\left(\frac{6x^2 + 5x + 1}{2x - 1}\right) + \cos^{-1}\left(\frac{2x^2 - 3x + 4}{3x - 5}\right)$ is $(\alpha, \beta) \cup (\gamma, \delta]$ , then $18(\alpha^2 + \beta^2 + \gamma^2 + \delta^2)$ is equal to ____________.
Relations and Functions Functions and Domain Domain via Logarithmic Inequalities
HardNumerical valueNew ideaIdea asked in 2 years
Q22
Let $m$ and $n$ be the numbers of real roots of the quadratic equations $x^2 - 12x + [x] + 31 = 0$ and $x^2 - 5|x + 2| - 4 = 0$ respectively, where $[x]$ denotes the greatest integer $\leq x$ . Then $m^2 + mn + n^2$ is equal to __________.
Complex Numbers and Quadratic Equations Quadratic Roots Roots of Modulus Quadratics
HardNumerical valueIdea asked in 6 yearsAsked the year before
Q23
Let $R = \{a, b, c, d, e\}$ and $S = \{1, 2, 3, 4\}$ . Total number of onto functions $f: R \to S$ such that $f(a) \neq 1$ , is equal to ____________.
Relations and Functions Permutations and Combinations Number of Onto Functions Using PIE
HardNumerical valueNew idea
Q24
Let $0 < z < y < x$ be three real numbers such that $\frac{1}{x}, \frac{1}{y}, \frac{1}{z}$ are in an arithmetic progression and $x, \sqrt{2}y, z$ are in a geometric progression. If $xy + yz + zx = \frac{3}{\sqrt{2}}xyz$ , then $3(x + y + z)^2$ is equal to…
Sequences and Series AP and GP Simultaneous AP, GP, HP Condition
HardNumerical valueNew idea
Q25
Let $k$ and $m$ be positive real numbers such that the function $f(x) = \begin{cases} 3x^2 + k\sqrt{x + 1}, & 0 < x < 1 \\ mx^2 + k^2, & x \geq 1 \end{cases}$ is differentiable for all $x > 0$ . Then $\frac{8f'(8)}{f'\left(\frac{1}{8}\right)}$ is equal to…
Continuity and Differentiability Finding Unknowns using Differentiability
MediumNumerical valueIdea asked in 3 years
Q26
Let $[t]$ denote the greatest integer function. If $\int_0^{2.4}[x^2]\,dx = \alpha + \beta\sqrt{2} + \gamma\sqrt{3} + \delta\sqrt{5}$ , then $\alpha + \beta + \gamma + \delta$ is equal to ________.
Integrals Definite Integral Properties Greatest Integer Function Integrals
MediumNumerical valueIdea asked in 7 yearsAsked the year before
Q27
Let the area enclosed by the lines $x + y = 2$ , $y = 0$ , $x = 0$ and the curve $f(x) = \min\left\{x^2 + \frac{3}{4}, 1 + [x]\right\}$ where $[x]$ denotes the greatest integer $\leq x$ , be $A$ . Then the value of $12A$ is ________
Application of Integrals Area Under Curves Area Involving Max/Min Functions
HardNumerical valueIdea asked in 6 yearsAsked the year before
Q28
Let the solution curve $x = x(y)$ , $0 < y < \frac{\pi}{2}$ , of the differential equation $(\log_e(\cos y))^2 \cos y\,dx - (1 + 3x\log_e(\cos y))\sin y\,dy = 0$ satisfy $x\left(\frac{\pi}{3}\right) = \frac{1}{2\log_e 2}$ . If…
Differential Equations Linear Differential Equations Linear in x Form
HardNumerical valueIdea asked in 4 yearsAsked the year before
Q29
The ordinates of the points $P$ and $Q$ on the parabola with focus $(3, 0)$ and directrix $x = -3$ are in the ratio $3:1$ . If $R(\alpha, \beta)$ is the point of intersection of the tangents to the parabola at $P$ and $Q$ , then $\frac{\beta^2}{\alpha}$ is…
Conic Sections
HardNumerical value
Q30
Let $P_1$ be the plane $3x - y - 7z = 11$ and $P_2$ be the plane passing through the points $(2, -1, 0)$ , $(2, 0, -1)$ , and $(5, 1, 1)$ . If the foot of the perpendicular drawn from the point $(7, 4, -1)$ on the line of intersection of the planes $P_1$ and…
Three Dimensional Geometry Planes in 3D Plane Through Three Points
HardNumerical valueIdea asked in 2 years

Dropped from the final key and left out of every count: Q10.

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.