JEE Main 2021 · July session

JEE Main 25 July 2021, Shift 2

90 questions: 30 numerical-answer and 60 multiple choice. 3 were rated hard and 43 easy. Its hard-question count was the joint 23rd highest of the 24 2021 shifts Lumi analysed. The biggest chapter was Binomial Theorem, with 5 questions.

Questions: 90; none dropped
Numerical answer: 30; 33% of the shift
Hard: 3; 3% of the shift
Chapters asked: 63; 14 questions with a figure

Key findings · 25 Jul 2021, Shift 2

1
3 hard questions (3%): the joint 23rd highest count of the 24 2021 shifts Lumi analysed. Across those shifts the hard share ran from 3% to 17%.
2
Maths was the hardest section, with 3 of 30 rated hard; Chemistry had the most easy questions (20 of 30).
3
46 questions tested a new idea, one no earlier year had asked: 63% of the 73 questions tagged to an idea.
4
26 came back from 2020: the same idea had been asked in the 2020 shifts Lumi analysed.
5
45 shared an idea with another 2021 shift (62% of tagged questions). Shifts in the same year can test the same idea in a new setting.
6
Biggest chapters: Current Electricity in Physics (3), Aldehydes, Ketones and Carboxylic Acids in Chemistry (3) and Binomial Theorem in Maths (5).

The shift at a glance

Subjects, difficulty and chapters

Physics

30 Q · 10 numerical

Easy 15Medium 15Hard 0

Top chapters

Chemistry

30 Q · 10 numerical

Easy 20Medium 10Hard 0

Top chapters

Maths

30 Q · 10 numerical

Easy 8Medium 19Hard 3

Top chapters

Question format

Single correct46 Q
Numerical value30 Q
Diagram based10 Q
Match the list3 Q
Two statements1 Q

What each question asks you to do

Calculation42 Q
Multi-step24 Q
Conceptual10 Q
Application7 Q
Recall7 Q

Most common: Calculation.

Against the other 2021 shifts

This shift next to the 23 other 2021 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
24 Feb, Shift 1	90	36%	7%	33%	68%	24%
24 Feb, Shift 2	88	43%	5%	32%	63%	21%
25 Feb, Shift 1	90	23%	11%	33%	65%	30%
25 Feb, Shift 2	89	35%	10%	34%	67%	24%
26 Feb, Shift 1	88	50%	3%	33%	78%	17%
26 Feb, Shift 2	90	27%	12%	33%	80%	29%
16 Mar, Shift 2	90	31%	7%	33%	69%	33%
17 Mar, Shift 1	89	39%	8%	34%	61%	26%
17 Mar, Shift 2	89	36%	17%	34%	66%	32%
18 Mar, Shift 1	90	40%	9%	33%	74%	24%
18 Mar, Shift 2	90	34%	9%	33%	63%	28%
22 Jul, Shift 2	90	40%	12%	33%	67%	28%
25 Jul, Shift 1	90	31%	13%	33%	56%	34%
25 Jul, Shift 2	90	48%	3%	33%	63%	27%
3 Aug, Shift 1	90	32%	11%	33%	68%	21%
3 Aug, Shift 2	90	32%	7%	33%	61%	24%
4 Aug, Shift 1	90	40%	8%	33%	62%	29%
4 Aug, Shift 2	90	32%	7%	33%	65%	41%
26 Aug, Shift 1	89	27%	15%	34%	66%	30%
26 Aug, Shift 2	89	21%	9%	34%	60%	28%
27 Aug, Shift 1	90	27%	13%	33%	63%	34%
27 Aug, Shift 2	90	29%	9%	33%	72%	29%
31 Aug, Shift 1	90	27%	11%	33%	70%	24%
31 Aug, Shift 2	90	30%	13%	33%	72%	31%

New ideas: share of a shift’s questions testing an idea no earlier year had asked. Lumi analyses 108 of the 174 JEE Main shifts held from 2017 to 2026.

Drill down

What this shift asked, and its history

Every chapter this shift asked, with all its topics and ideas. Counts and squares cover every analysed shift from 2017 to 2026; rows this shift did not ask are faded, and each row lists this shift’s questions with the most recent from other shifts.

subjectAskedShareYears’17’20’21’22’23’24’25’26

Open30 this shift3092 · 33%33%8/830299720720358540200225
Open30 this shift3084 · 33%33%8/830298716720359538199224
Open30 this shift3065 · 33%33%8/829299715710354533200225

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

Every question

All 90 questions of this shift

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

Subject

Difficulty

Idea

Find

90 of 90 questions

Physics30 questions

15 easy15 medium0 hard

Q1
The force is given in terms of time $t$ and displacement $x$ by the equation $F = A\ \cos Bx + C\ \sin Dt$ The dimensional formula of $\frac{AD}{B}$ is :
Units and Measurements
MediumSingle correct
Q2
A balloon was moving upwards with a uniform velocity of 10 m/s. An object of finite mass is dropped from the balloon when it was at a height of 75 m from the ground level. The height of the balloon from the ground when object strikes the ground was around :…
Motion in a Straight Line Kinematics in 1D Dropping Objects from Moving Frames
MediumSingle correctIdea asked in 3 yearsAsked the year before
Q3
The instantaneous velocity of a particle moving in a straight line is given as $v = \alpha t + \beta t^2$ , where $\alpha$ and $\beta$ are constants. The distance travelled by the particle between 1 s and 2 s is :
Motion in a Straight Line Kinematics in 1D Calculus in Kinematics
EasySingle correctNew ideaIdea asked in 5 years
Q4
The relation between time t and distance $x$ for a moving body is given as $t = mx^2 + nx$ , where $m$ and $n$ are constants. The retardation of the motion is : (Where $v$ stands for velocity)
Motion in a Straight Line Kinematics in 1D Implicit Time-Position Equations
MediumSingle correctIdea asked in 3 yearsAsked the year before
Q5
Two vectors $\vec{X}$ and $\vec{Y}$ have equal magnitude. The magnitude of $(\vec{X} - \vec{Y})$ is n times the magnitude of $(\vec{X} + \vec{Y})$ . The angle between $\vec{X}$ and $\vec{Y}$ is :
Motion in a Plane
MediumSingle correct
Q6
A force $\vec{F} = (40\hat{i} + 10\hat{j})$ N acts on a body of mass 5 kg. If the body starts from rest, its position vector $\vec{r}$ at time $t$ =10 s, will be :
Laws of Motion Newton's Laws of Motion F=ma for Single Body
EasySingle correctNew ideaIdea asked in 6 years
Q7
Consider a planet in some solar system which has a mass double the mass of earth and density equal to the average density of earth. If the weight of an object on earth is W, the weight of the same object on that planet will be :
Gravitation
EasySingle correct
Q8
Two spherical soap bubbles of radii $r_1$ and $r_2$ in vacuum combine under isothermal conditions. The resulting bubble has a radius equal to :
Mechanical Properties of Fluids Surface Tension Isothermal Coalescence of Bubbles
MediumSingle correctNew idea
Q9
A heat engine has an efficiency of $\frac{1}{6}$ . When the temperature of sink is reduced by 62°C, its efficiency get doubled. The temperature of the source is :
Thermodynamics Heat Engines and Refrigerators Carnot Efficiency Limit
MediumSingle correctNew ideaIdea asked in 3 years
Q10
In a simple harmonic oscillation, what fraction of total mechanical energy is in the form of kinetic energy, when the particle is midway between mean and extreme position.
Oscillations Energy in SHM Fractional Energy Displacements
EasySingle correctNew ideaIdea asked in 2 years
Q11
If $q_f$ is the free charge on the capacitor plates and $q_b$ is the bound charge on the dielectric slab of dielectric constant $k$ placed between the capacitor plates, then bound charge $q_b$ can be expressed as :
Electrostatic Potential and Capacitance Dielectrics Induced Surface Charge Density
EasySingle correctNew ideaIdea asked in 2 years
Q12
Two ideal electric dipoles A and B, having their dipole moment $p_1$ and $p_2$ respectively are placed on a plane with their centres at O as shown in the figure. At point C on the axis of dipole A, the resultant electric field is making an angle of 37° with…
Electric Charges and Fields Electric Dipoles Electric Field on Axial Line
MediumDiagram based · figureNew ideaIdea asked in 2 years
Q13
The given potentiometer has its wire of resistance 10 $\Omega$ . When the sliding contact is in the middle of the potentiometer wire, the potential drop across 2 $\Omega$ resistor is :
Current Electricity Kirchhoff's Laws Voltage Divider Rule
MediumDiagram based · figureNew ideaIdea asked in 2 years
Q14
Two ions having same mass have charges in the ratio 1 : 2. They are projected normally in a uniform magnetic field with their speeds in the ratio 2 : 3. The ratio of the radii of their circular trajectories is :
Moving Charges and Magnetism Lorentz Force Circular Motion in Uniform B-Field
EasySingle correctIdea asked in 4 yearsAsked the year before
Q15
A 10 $\Omega$ resistance is connected across 220 V - 50 Hz AC supply. The time taken by the current to change from its maximum value to the rms value is :
Alternating Current RMS and Peak Values Time from Peak to RMS
EasySingle correctNew ideaIdea asked in 2 years
Q16
A ray of light entering from air into a denser medium of refractive index $\frac{4}{3}$ , as shown in figure. The light ray suffers total internal reflection at the adjacent surface as shown. The maximum value of angle $\theta$ should be equal to :
Ray Optics and Optical Instruments Refraction and Snell's Law Total Internal Reflection
MediumDiagram based · figureIdea asked in 6 yearsAsked the year before
Q17
A prism of refractive index $\mu$ and angle of prism A is placed in the position of minimum angle of deviation. If minimum angle of deviation is also A, then in terms of refractive index value of A is :
Ray Optics and Optical Instruments Prism Deviation Minimum Deviation Condition
MediumSingle correctNew ideaIdea asked in 5 years
Q18
When radiation of wavelength $\lambda$ is incident on a metallic surface, the stopping potential of ejected photoelectrons is 4.8 V. If the same surface is illuminated by radiation of double the previous wavelength, then the stopping potential becomes 1.6 V.…
Dual Nature of Radiation and Matter Photoelectric Effect Two-Wavelength System of Equations
MediumSingle correctIdea asked in 5 yearsAsked the year before
Q19
An electron moving with speed $v$ and a photon moving with speed $c$ , have same D-Broglie wavelength. The ratio of kinetic energy of electron to that of photon is :
Dual Nature of Radiation and Matter Matter Waves Electron vs Photon (Same Wavelength)
EasySingle correctNew ideaIdea asked in 3 years
Q20
In the given potentiometer circuit arrangement, the balancing length AC is measured to be 250 cm. When the galvanometer connection is shifted from point (1) to point (2) in the given diagram, the balancing length becomes 400 cm. The ratio of the emf of two…
Current Electricity Potentiometer Comparing EMF of Two Cells
MediumDiagram based · figureNew ideaIdea asked in 2 years
Q21
A force of $F=(5y+20)\hat{j}$ N acts on a particle. The workdone by this force when the particle is moved from $y$ =0 m to $y$ =10 m is __________ J.
Work, Energy and Power Work and Kinetic Energy Work by Variable Force
EasyNumerical valueIdea asked in 6 yearsAsked the year before
Q22
A solid disc of radius 20 cm and mass 10 kg is rotating with an angular velocity of 600 rpm, about an axis normal to its circular plane and passing through its centre of mass. The retarding torque required to bring the disc at rest in 10 s is __________…
System of Particles and Rotational Motion Moment of Inertia MOI of Standard Shapes
EasyNumerical valueIdea asked in 5 yearsAsked the year before
Q23
A system consists of two types of gas molecules A and B having same number density $2\times10^{25}$ /m $^3$ . The diameter of A and B are 10 Å and 5 Å respectively. They suffer collision at room temperature. The ratio of average distance covered by the molecule…
Kinetic Theory Kinetic Theory of Gases Mean Free Path
EasyNumerical valueIdea asked in 4 yearsAsked the year before
Q24
A 16 $\Omega$ wire is bend to form a square loop. A 9 V supply having internal resistance of 1 $\Omega$ is connected across one of its sides. The potential drop across the diagonals of the square loop is __________ $\times10^{-1}$ V.
Current Electricity Ohm's Law and Resistance Resistance of Bent Wires and Loops
MediumNumerical valueNew idea
Q25
Two circuits are shown in the figure (a) & (b). At a frequency of __________ rad/s the average power dissipated in one cycle will be same in both the circuits.
Alternating Current LCR Resonance Resonant Frequency Calculation
MediumNumerical value · figureNew ideaIdea asked in 4 years
Q26
A light beam of wavelength 500 nm is incident on a metal having work function of 1.25 eV, placed in a magnetic field of intensity B. The electrons emitted perpendicular to the magnetic field B, with maximum kinetic energy are bent into circular arc of radius…
Dual Nature of Radiation and Matter Photoelectric Effect Photoelectrons in a Magnetic Field
MediumNumerical valueIdea asked in 2 yearsAsked the year before
Q27
The nuclear activity of a radioactive element becomes $\left(\frac{1}{8}\right)^{\text{th}}$ of its initial value in 30 years. The half-life of radioactive element is __________ years.
Nuclei Radioactive Decay Integer Half-Life Shortcuts
EasyNumerical valueNew ideaIdea asked in 2 years
Q28
From the given data, the amount of energy required to break the nucleus of aluminium $^{27}_{13}\mathrm{Al}$ is __________ $x\times10^{-3}$ J. Mass of neutron=1.00866 u Mass of proton=1.00726 u Mass of Aluminium nucleus=27.18846 u (Assume 1 u corresponds to…
Nuclei Nuclear Binding Energy Total Binding Energy
EasyNumerical valueNew ideaIdea asked in 2 years
Q29
In a semiconductor, the number density of intrinsic charge carriers at 27°C is $1.5\times10^{16}$ /m $^3$ . If the semiconductor is doped with impurity atom, the hole density increases to $4.5\times10^{22}$ /m $^3$ . The electron density in the doped semiconductor…
Semiconductor Electronics: Materials, Devices and Simple Circuits
EasyNumerical value
Q30
A message signal of frequency 20 kHz and peak voltage of 20 volt is used to modulate a carrier wave of frequency 1 MHz and peak voltage of 20 volt. The modulation index will be __________.
Communication Systems
EasyNumerical value

Chemistry30 questions

20 easy10 medium0 hard

Q31
In the following the correct bond order sequence is :
Chemical Bonding and Molecular Structure Molecular Orbital Theory Oxygen Ion Series Comparison
EasySingle correctNew ideaIdea asked in 3 years
Q32
Identify the process in which change in the oxidation state is five :
Redox Reactions Oxidation Number Assignment
EasySingle correctNew ideaIdea asked in 4 years
Q33
Match List I with List II : List - I Example of Colloids (a) Cheese (b) Pumice stone (c) Hair cream (d) Cloud List - II Classification (i) dispersion of liquid in liquid (ii) dispersion of liquid in gas (iii) dispersion of gas in solid (iv) dispersion of…
Surface Chemistry Colloids Colloid Classification by State
EasyMatch the listIdea asked in 4 yearsAsked the year before
Q34
The ionic radii of $\mathrm{F^{-}}$ and $\mathrm{O^{2-}}$ respectively are 1.33 Å and 1.4 Å, while the covalent radius of N is 0.74 Å. The correct statement for the ionic radius of $\mathrm{N^{3-}}$ from the following is :
Classification of Elements and Periodicity in Properties Periodic Trends Isoelectronic Species Radii Comparison
EasySingle correctIdea asked in 5 yearsAsked the year before
Q35
Match List I with List II : (Both having metallurgical terms) List - I (a) Concentration of Ag ore (b) Blast furnace (c) Blister copper (d) Froth floatation method List - II (i) Reverberatory furnace (ii) Pig iron (iii) Leaching with dilute NaCN solution…
General Principles and Processes of Isolation of Elements Metallurgical Extraction MacArthur-Forrest Cyanide Process
EasyMatch the listNew ideaIdea asked in 2 years
Q36
Which one of the following metals forms interstitial hydride easily ?
Hydrogen Hydrogen and Hydrides Metallic or Interstitial
EasySingle correctNew idea
Q37
Match List I with List II : List - I Elements (a) Li (b) Na (c) K (d) Cs List - II Properties (i) Poor water solubility of $\mathrm{I^{-}}$ salt (ii) Most abundant element in cell fluid (iii) Bicarbonate salt used in fire extinguisher (iv) Carbonate salt…
The s-Block Elements s-Block Elements Existence of Solid Bicarbonates
MediumMatch the listNew idea
Q38
Identify the species having one $\pi$ -bond and maximum number of canonical forms from the following :
Chemical Bonding and Molecular Structure Resonance and Hyperconjugation Resonance Structures
MediumSingle correctNew idea
Q39
The spin only magnetic moments (in BM) for free $\mathrm{Ti^{3+}}$ , $\mathrm{V^{2+}}$ and $\mathrm{Sc^{3+}}$ ions respectively are (At. No. Sc : 21 ; Ti : 22 ; V : 23)
The d- and f-Block Elements Crystal Field Theory Spin-only Magnetic Moment
EasySingle correctIdea asked in 7 yearsAsked the year before
Q40
Which one of the following metal complexes is most stable ?
Coordination Compounds
EasySingle correct
Q41
Given below are two statements : Statement I : Chlorofluoro carbons breakdown by radiation in the visible energy region and release chlorine gas in the atmosphere which then reacts with stratospheric ozone. Statement II : Atmospheric ozone reacts with nitric…
Environmental Chemistry Ozone Layer Depletion
MediumTwo statementsIdea asked in 4 yearsAsked the year before
Q42
Which among the following is the strongest acid ?
Organic Chemistry: Some Basic Principles and Techniques Organic Acidity and Basicity Aromaticity-Driven Carbon Acidity
MediumDiagram based · figureNew idea
Q43
The correct decreasing order of densities of the following compounds is : [figure: four drawn structures labelled (A), (B), (C), (D)]
Haloalkanes and Haloarenes
EasyDiagram based · figure
Q44
[figure: structure labelled "Maleic anhydride"] Maleic anhydride can be prepared by :
Aldehydes, Ketones and Carboxylic Acids Carboxylic Acids and Derivatives Heating Effects on Dicarboxylic Acids
EasySingle correct · figureNew idea
Q45
[reaction scheme] drawn structure: label Br joined to a bend, then label CHO; with reagent EtOH excess / dry HCl gas gives "A" (major product); "A" with reagent $\mathrm{{}^tBuO^{-}K^{+}}$ gives "B" (major product) [where Et $\Rightarrow -\mathrm{C_2H_5}$ …
Aldehydes, Ketones and Carboxylic Acids Nucleophilic Addition Acetal and Ketal Formation
MediumDiagram based · figureNew idea
Q46
What is the major product "P" of the following reaction ? [reaction scheme: structure] (i) $\mathrm{NaNO_2}$ , HCl, 278 K (ii) $\mathrm{H_2O}$ $\rightarrow$ P (major product)
Amines Diazonium Salts Replacement by H or OH
EasyDiagram based · figureNew ideaIdea asked in 3 years
Q47
[reaction scheme] $\mathrm{C_6H_5NO_2}$ with reagent Sn + HCl gives "A"; "A" with reagent $\mathrm{C_6H_5\overset{\oplus}{N_2}\overset{\ominus}{Cl}}$ / $\mathrm{H^{\oplus}}$ gives P (Yellow coloured Compound) Consider the above reaction, the Product "P" is :
Amines Diazonium Salts Azo Coupling Reactions
MediumDiagram based · figureNew ideaIdea asked in 5 years
Q48
A biodegradable polyamide can be made from :
Polymers Nylon-2-nylon-6 Monomers
EasySingle correctNew idea
Q49
Which one of the following is correct structure for cytosine ?
Biomolecules
EasyDiagram based · figure
Q50
A reaction of benzonitrile with one equivalent $\mathrm{CH_3MgBr}$ followed by hydrolysis produces a yellow liquid "P". The compound "P" will give positive ________.
Aldehydes, Ketones and Carboxylic Acids Nucleophilic Addition Grignard Reagent Addition
MediumSingle correctIdea asked in 4 yearsAsked the year before
Q51
The number of significant figures in 0.00340 is ___________.
Some Basic Concepts of Chemistry
EasyNumerical value
Q52
An LPG cylinder contains gas at a pressure of 300 kPa at 27°C. The cylinder can withstand the pressure of $1.2 \times 10^{6}$ Pa. The room in which the cylinder is kept catches fire. The minimum temperature at which the bursting of cylinder will take place…
States of Matter Ideal Gas Laws Gay-Lussac's Law
EasyNumerical valueNew ideaIdea asked in 2 years
Q53
An accelerated electron has a speed of $5 \times 10^{6}$ $\mathrm{ms^{-1}}$ with an uncertainty of 0.02%. The uncertainty in finding its location while in motion is $x \times 10^{-9}$ m. The value of $x$ is ________. (Nearest integer) [Use mass of electron…
Structure of Atom
MediumNumerical value
Q54
A system does 200 J of work and at the same time absorbs 150 J of heat. The magnitude of the change in internal energy is ________ J. (Nearest integer)
Thermodynamics (Chemistry)Enthalpy and Hess's Law Sign Conventions for Heat Flow
EasyNumerical valueNew ideaIdea asked in 2 years
Q55
When 3.00 g of a substance 'X' is dissolved in 100 g of $\mathrm{CCl_4}$ , it raises the boiling point by 0.60 K. The molar mass of the substance 'X' is ________ $\mathrm{g\ mol^{-1}}$ . (Nearest integer) [Given $\mathrm{K_b}$ for $\mathrm{CCl_4}$ is 5.0 K kg…
Solutions Colligative Properties Boiling Point Elevation
EasyNumerical valueNew ideaIdea asked in 6 years
Q56
Assuming that $\mathrm{Ba(OH)_2}$ is completely ionised in aqueous solution under the given conditions the concentration of $\mathrm{H_3O^{+}}$ ions in 0.005 M aqueous solution of $\mathrm{Ba(OH)_2}$ at 298 K is ________ $\times 10^{-12}$ …
Equilibrium pH and Buffers pH of Strong Acids and Bases
EasyNumerical valueNew ideaIdea asked in 4 years
Q57
For a chemical reaction $\mathrm{A \rightarrow B}$ , it was found that concentration of B is increased by 0.2 $\mathrm{mol\ L^{-1}}$ in 30 min. The average rate of the reaction is ________ $\times 10^{-1}$ $\mathrm{mol\ L^{-1}\ h^{-1}}$ . (in nearest integer)
Chemical Kinetics Rate Laws Reaction Rate Definition
EasyNumerical valueNew idea
Q58
Number of electrons present in 4f orbital of $\mathrm{Ho^{3+}}$ ion is ________. (Given Atomic No. of Ho = 67)
The d- and f-Block Elements f-Block Elements Lanthanide Electronic Configuration
EasyNumerical valueIdea asked in 5 yearsAsked the year before
Q59
0.8 g of an organic compound was analysed by Kjeldahl's method for the estimation of nitrogen. If the percentage of nitrogen in the compound was found to be 42%, then ________ mL of 1 M $\mathrm{H_2SO_4}$ would have been neutralized by the ammonia evolved…
Organic Chemistry: Some Basic Principles and Techniques Stoichiometry Stoichiometry of Kjeldahl Method
MediumNumerical valueIdea asked in 4 yearsAsked the year before
Q60
[reaction scheme: drawn structure] + $\mathrm{Br_2}$ $\xrightarrow{\mathrm{CCl_4}}$ Product "P" Consider the above chemical reaction. The total number of stereoisomers possible for Product 'P' is ________.
Hydrocarbons Alkenes and Alkynes Halogenation Anti-Addition Stereochemistry
MediumNumerical value · figureNew idea

Maths30 questions

8 easy19 medium3 hard

Q61
The number of distinct real roots of $\begin{vmatrix}\sin x & \cos x & \cos x\\ \cos x & \sin x & \cos x\\ \cos x & \cos x & \sin x\end{vmatrix}=0$ in the interval $-\frac{\pi}{4}\le x\le\frac{\pi}{4}$ is :
Determinants
MediumSingle correct
Q62
The lowest integer which is greater than $\left(1+\frac{1}{10^{100}}\right)^{10^{100}}$ is __________.
Binomial Theorem Binomial Coefficients Bounding Binomial Coefficients
HardSingle correctNew idea
Q63
If ${}^nP_r={}^nP_{r+1}$ and ${}^nC_r={}^nC_{r-1}$ , then the value of r is equal to :
Permutations and Combinations Permutations of Distinct Objects
EasySingle correctIdea asked in 4 yearsAsked the year before
Q64
If $[x]$ be the greatest integer less than or equal to $x$ , then $\sum_{n=8}^{100}\left[\frac{(-1)^n n}{2}\right]$ is equal to :
Relations and Functions
MediumSingle correct
Q65
Consider the statement “The match will be played only if the weather is good and ground is not wet”. Select the correct negation from the following :
Mathematical Reasoning
MediumSingle correct
Q66
The sum of all those terms which are rational numbers in the expansion of $\left(2^{1/3}+3^{1/4}\right)^{12}$ is :
Binomial Theorem
MediumSingle correct
Q67
Let $y=y(x)$ be the solution of the differential equation $x\,dy=(y+x^3\cos x)\,dx$ with $y(\pi)=0$ , then $y\left(\frac{\pi}{2}\right)$ is equal to :
Differential Equations Linear Differential Equations Particular Solution Solving
MediumSingle correctNew ideaIdea asked in 4 years
Q68
The first of the two samples in a group has 100 items with mean 15 and standard deviation 3. If the whole group has 250 items with mean 15.6 and standard deviation $\sqrt{13.44}$ , then the standard deviation of the second sample is :
Statistics Variance and SD Combined Variance of Two Sets
MediumSingle correctNew ideaIdea asked in 3 years
Q69
The value of the integral $\int_{-1}^{1}\log\left(x+\sqrt{x^2+1}\right)dx$ is :
Integrals Definite Integral Properties Even and Odd Functions
EasySingle correctNew ideaIdea asked in 3 years
Q70
Let X be a random variable such that the probability function of a distribution is given by $P(X=0)=\frac{1}{2}$ , $P(X=j)=\frac{1}{3^j}$ $(j=1,2,3,.....,\infty)$ Then the mean of the distribution and P(X is positive and even) respectively are :
Probability Arithmetico-Geometric Series AGP in Expected Value Scenarios
MediumSingle correctNew ideaIdea asked in 2 years
Q71
If the greatest value of the term independent of ‘x’ in the expansion of $\left(x\sin\alpha+a\frac{\cos\alpha}{x}\right)^{10}$ is $\frac{10!}{(5!)^2}$ , then the value of ‘a’ is equal to :
Binomial Theorem
HardSingle correct
Q72
If $f(x)=\begin{cases}\int_0^x(5+|1-t|)\,dt, & x>2\\ 5x+1, & x\le 2\end{cases}$ , then
Continuity and Differentiability Piecewise Differentiability
MediumSingle correctIdea asked in 4 yearsAsked the year before
Q73
The value of $\cot\frac{\pi}{24}$ is :
Trigonometric Functions Trigonometric Identities Values of Non-Standard Specific Angles
MediumSingle correctNew ideaIdea asked in 3 years
Q74
The number of real solutions of the equation, $x^2-|x|-12=0$ is :
Complex Numbers and Quadratic Equations Quadratic Roots Roots of Modulus Quadratics
EasySingle correctIdea asked in 6 yearsAsked the year before
Q75
If $P=\begin{bmatrix}1&0\\ \frac{1}{2}&1\end{bmatrix}$ , then $P^{50}$ is :
Matrices Matrix Multiplication Powers of Square Matrices
EasySingle correctIdea asked in 6 yearsAsked the year before
Q76
If a tangent to the ellipse $x^2+4y^2=4$ meets the tangents at the extremities of its major axis at B and C, then the circle with BC as diameter passes through the point :
Conic Sections Ellipse Eccentricity Coordinates of foci
HardSingle correctNew ideaIdea asked in 4 years
Q77
Let the equation of the pair of lines, $y=px$ and $y=qx$ , can be written as $(y-px)(y-qx)=0$ . Then the equation of the pair of the angle bisectors of the lines $x^2-4xy-5y^2=0$ is :
Straight Lines Pair of Straight Lines Equation of angle bisectors
EasySingle correctNew idea
Q78
If $|\vec{a}|=2$ , $|\vec{b}|=5$ and $|\vec{a}\times\vec{b}|=8$ , then $|\vec{a}\cdot\vec{b}|$ is equal to :
Vector Algebra Cross Product Lagrange's Identity
EasySingle correctIdea asked in 5 yearsAsked the year before
Q79
Consider functions $f:A\to B$ and $g:B\to C$ ( $A,B,C\subseteq\mathbf{R}$ ) such that $(g\circ f)^{-1}$ exists, then :
Relations and Functions
MediumSingle correct
Q80
Let a, b and c be distinct positive numbers. If the vectors $a\hat{i}+a\hat{j}+c\hat{k}$ , $\hat{i}+\hat{k}$ and $c\hat{i}+c\hat{j}+b\hat{k}$ are co-planar, then c is equal to :
Vector Algebra Triple Products Solving for Unknown in Coplanarity
MediumSingle correctIdea asked in 4 yearsAsked the year before
Q81
The equation of a circle is $\mathrm{Re}(z^2)+2(\mathrm{Im}(z))^2+2\mathrm{Re}(z)=0$ , where $z=x+iy$ . A line which passes through the center of the given circle and the vertex of the parabola, $x^2-6x-y+13=0$ , has y-intercept equal to __________.
Conic Sections General Conic Section Vertex of a General Parabola
MediumNumerical valueNew idea
Q82
If the lines $\frac{x-k}{1}=\frac{y-2}{2}=\frac{z-3}{3}$ and $\frac{x+1}{3}=\frac{y+2}{2}=\frac{z+3}{1}$ are co-planar, then the value of k is __________.
Three Dimensional Geometry Skew Lines Coplanarity Determinant Test
EasyNumerical valueNew ideaIdea asked in 3 years
Q83
Let $n\,\epsilon\,\mathbf{N}$ and $[x]$ denote the greatest integer less than or equal to $x$ . If the sum of $(n+1)$ terms ${}^nC_0,\ 3\cdot{}^nC_1,\ 5\cdot{}^nC_2,\ 7\cdot{}^nC_3,\ .....$ is equal to $2^{100}\cdot101$ , then $2\left[\frac{n-1}{2}\right]$ is…
Binomial Theorem Binomial Coefficients Binomial Coefficients with AP Multiplier
MediumNumerical valueNew ideaIdea asked in 2 years
Q84
If $a+b+c=1$ , $ab+bc+ca=2$ and $abc=3$ , then the value of $a^4+b^4+c^4$ is equal to __________.
Complex Numbers and Quadratic Equations Quadratic Roots Newton's Sums
MediumNumerical valueIdea asked in 7 yearsAsked the year before
Q85
A fair coin is tossed n-times such that the probability of getting at least one head is at least 0.9. Then the minimum value of n is __________.
Probability
EasyNumerical value
Q86
Let a curve $y=f(x)$ pass through the point $(2,(\log_e 2)^2)$ and have slope $\frac{2y}{x\log_e x}$ for all positive real value of $x$ . Then the value of $f(e)$ is equal to __________.
Differential Equations Variable Separable Form Initial Value Problem
MediumNumerical valueIdea asked in 7 yearsAsked the year before
Q87
If the co-efficients of $x^7$ and $x^8$ in the expansion of $\left(2+\frac{x}{3}\right)^n$ are equal, then the value of n is equal to __________.
Binomial Theorem Binomial Coefficients Ratio of Consecutive Coefficients
MediumNumerical valueIdea asked in 5 yearsAsked the year before
Q88
If $\left(\vec{a}+3\vec{b}\right)$ is perpendicular to $\left(7\vec{a}-5\vec{b}\right)$ and $\left(\vec{a}-4\vec{b}\right)$ is perpendicular to $\left(7\vec{a}-2\vec{b}\right)$ , then the angle between $\vec{a}$ and $\vec{b}$ (in degrees) is __________.
Vector Algebra Dot Product Angle Between Vectors
MediumNumerical valueIdea asked in 4 yearsAsked the year before
Q89
If a rectangle is inscribed in an equilateral triangle of side length $2\sqrt{2}$ as shown in the figure, then the square of the largest area of such a rectangle is __________.
Application of Derivatives Maxima and Minima Optimization Problems
MediumNumerical value · figureIdea asked in 5 years
Q90
Consider the function $f(x)=\begin{cases}\dfrac{P(x)}{\sin(x-2)}, & x\ne2\\ 7, & x=2\end{cases}$ where $P(x)$ is a polynomial such that $P''(x)$ is always a constant and $P(3)=9$ . If $f(x)$ is continuous at $x=2$ , then $P(5)$ is equal to __________.
Continuity and Differentiability Finding Unknown Constants via Continuity
MediumNumerical valueNew idea

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 2021 shifts.