JEE Main 2021 · August session

JEE Main 27 August 2021, Shift 2

90 questions: 30 numerical-answer and 60 multiple choice. 8 were rated hard and 26 easy. Its hard-question count was the joint 12th highest of the 24 2021 shifts Lumi analysed. The biggest chapter was Aldehydes, Ketones and Carboxylic Acids, with 3 questions.

Questions: 90; none dropped
Numerical answer: 30; 33% of the shift
Hard: 8; 9% of the shift
Chapters asked: 67; 19 questions with a figure

Key findings · 27 Aug 2021, Shift 2

1
8 hard questions (9%): the joint 12th 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 5 of 30 rated hard; Physics had the most easy questions (13 of 30).
3
52 questions tested a new idea, one no earlier year had asked: 72% of the 72 questions tagged to an idea.
4
17 came back from 2020: the same idea had been asked in the 2020 shifts Lumi analysed.
5
36 shared an idea with another 2021 shift (50% 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), Aldehydes, Ketones and Carboxylic Acids in Chemistry (3) and Conic Sections in Maths (3).

The shift at a glance

Subjects, difficulty and chapters

Physics

30 Q · 10 numerical

Easy 13Medium 16Hard 1

Top chapters

Chemistry

30 Q · 10 numerical

Easy 12Medium 16Hard 2

Top chapters

Maths

30 Q · 10 numerical

Easy 1Medium 24Hard 5

Top chapters

Question format

Single correct48 Q
Numerical value30 Q
Diagram based9 Q
Match the list2 Q
Two statements1 Q

What each question asks you to do

Calculation28 Q
Multi-step26 Q
Application12 Q
Recall10 Q
Conceptual8 Q
Diagram reading6 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

13 easy16 medium1 hard

Q1
Water drops are falling from a nozzle of a shower onto the floor, from a height of 9.8 m. The drops fall at a regular interval of time. When the first drop strikes the floor, at that instant, the third drop begins to fall. Locate the position of second drop…
Motion in a Straight Line Kinematics in 1D Regular Interval Free Fall (Leaky Tap)
MediumSingle correctNew idea
Q2
Match List - I with List - II. List - I List - II (a) $\mathrm{R_H}$ (Rydberg constant) (i) $\mathrm{kg\ m^{-1}s^{-1}}$ (b) h(Planck's constant) (ii) $\mathrm{kg\ m^2\ s^{-1}}$ (c) $\mu_B$ (Magnetic field energy density) (iii) $\mathrm{m^{-1}}$ (d)…
Units and Measurements Viscosity Units and Dimensions of Viscosity
MediumMatch the listNew ideaIdea asked in 2 years
Q3
Two discs have moments of intertia $I_1$ and $I_2$ about their respective axes perpendicular to the plane and passing through the centre. They are rotating with angular speeds, $\omega_1$ and $\omega_2$ respectively and are brought into contact face to face…
System of Particles and Rotational Motion Angular Momentum Kinetic Energy Loss in Rotational Coupling
EasySingle correctIdea asked in 3 yearsAsked the year before
Q4
A player kicks a football with an initial speed of 25 $\mathrm{ms^{-1}}$ at an angle of $45^\circ$ from the ground. What are the maximum height and the time taken by the football to reach at the highest point during motion ? (Take $g=10\ \mathrm{ms^{-2}}$ )
Motion in a Plane Projectile Motion Ground-to-Ground Parameters
EasySingle correctNew ideaIdea asked in 6 years
Q5
The boxes of masses 2 kg and 8 kg are connected by a massless string passing over smooth pulleys. Calculate the time taken by box of mass 8 kg to strike the ground starting from rest. (use $g=10\ \mathrm{m/s^2}$ ) :
Laws of Motion Newton's Laws of Motion Constrained Motion
HardDiagram based · figureNew ideaIdea asked in 2 years
Q6
A mass of 50 kg is placed at the centre of a uniform spherical shell of mass 100 kg and radius 50 m. If the gravitational potential at a point, 25 m from the centre is V kg/m. The value of V is :
Gravitation Gravitational Potential Potential of Spherical Shell
MediumSingle correctNew idea
Q7
The height of victoria falls is 63 m. What is the difference in temperature of water at the top and at the bottom of fall ? [Given 1 cal $=4.2$ J and specific heat of water $=1$ cal $\mathrm{g^{-1}\ ^\circ C^{-1}}$ ]
Thermodynamics Calorimetry Specific Heat Capacity
EasySingle correctNew ideaIdea asked in 2 years
Q8
If the rms speed of oxygen molecules at $0^\circ\mathrm{C}$ is 160 m/s, find the rms speed of hydrogen molecules at $0^\circ\mathrm{C}$ .
Kinetic Theory Kinetic Theory of Gases Molecular Speed Ratios
EasySingle correctNew ideaIdea asked in 4 years
Q9
Three capacitors $C_1=2\ \mu\mathrm{F}$ , $C_2=6\ \mu\mathrm{F}$ and $C_3=12\ \mu\mathrm{F}$ are connected as shown in figure. Find the ratio of the charges on capacitors $C_1$ , $C_2$ and $C_3$ respectively :
Electrostatic Potential and Capacitance Parallel Plate Capacitors Series and Parallel Combinations
MediumSingle correct · figureIdea asked in 6 yearsAsked the year before
Q10
Figure shows a rod AB, which is bent in a $120^\circ$ circular arc of radius R. A charge $(-Q)$ is uniformly distributed over rod AB. What is the electric field $\vec{E}$ at the centre of curvature O ?
Electric Charges and Fields Electric Field Charged Arc Field
MediumSingle correct · figureNew ideaIdea asked in 4 years
Q11
A coaxial cable consists of an inner wire of radius ‘a’ surrounded by an outer shell of inner and outer radii ‘b’ and ‘c’ respectively. The inner wire carries an electric current $i_0$ , which is distributed uniformly across cross-sectional area. The outer…
Moving Charges and Magnetism Ampere's Circuital Law Field Inside Thick Wire
MediumSingle correctIdea asked in 4 yearsAsked the year before
Q12
The colour coding on a carbon resistor is shown in the given figure. The resistance value of the given resistor is :
Current Electricity Ohm's Law and Resistance Carbon Resistor Color Coding
EasyDiagram based · figureNew idea
Q13
For full scale deflection of total 50 divisions, 50 mV voltage is required in galvanometer. The resistance of galvanometer if its current sensitivity is 2 div/mA will be :
Moving Charges and Magnetism
EasySingle correct
Q14
Curved surfaces of a plano-convex lens of refractive index $\mu_1$ and a plano-concave lens of refractive index $\mu_2$ have equal radius of curvature as shown in figure. Find the ratio of radius of curvature to the focal length of the combined lenses.
Ray Optics and Optical Instruments Lens Maker's Formula Lenses in Contact
MediumSingle correct · figureNew ideaIdea asked in 5 years
Q15
If force (F), length (L) and time (T) are taken as the fundamental quantities. Then what will be the dimension of density :
Units and Measurements
MediumSingle correct
Q16
A constant magnetic field of 1 T is applied in the $x>0$ region. A metallic circular ring of radius 1 m is moving with a constant velocity of 1 m/s along the $x$ -axis. At $t=0$ s, the centre O of the ring is at $x=-1$ m. What will be the value of the induced…
Electromagnetic Induction Faraday's Law Motional EMF in Arbitrary Shapes
MediumDiagram based · figureNew idea
Q17
For a transistor $\alpha$ and $\beta$ are given as $\alpha=\frac{I_C}{I_E}$ and $\beta=\frac{I_C}{I_B}$ . Then the correct relation between $\alpha$ and $\beta$ will be :
Semiconductor Electronics: Materials, Devices and Simple Circuits Transistors Current Gains Alpha and Beta
EasySingle correctNew ideaIdea asked in 2 years
Q18
The light waves from two coherent sources have same intensity $I_1=I_2=I_0$ . In interference pattern the intensity of light at minima is zero. What will be the intensity of light at maxima ?
Wave Optics Young's Double Slit Resultant Intensity Formula
EasySingle correctNew ideaIdea asked in 5 years
Q19
A monochromatic neon lamp with wavelength of 670.5 nm illuminates a photo-sensitive material which has a stopping voltage of 0.48 V. What will be the stopping voltage if the source light is changed with another source of wavelength of 474.6 nm ?
Dual Nature of Radiation and Matter Photoelectric Effect Stopping Potential
MediumSingle correctNew ideaIdea asked in 2 years
Q20
An antenna is mounted on a 400 m tall building. What will be the wavelength of signal that can be radiated effectively by the transmission tower upto a range of 44 km ?
Communication Systems
MediumSingle correct
Q21
The ratio of the equivalent resistance of the network (shown in figure) between the points a and b when switch is open and switch is closed is $x:8$ . The value of $x$ is __________.
Current Electricity Kirchhoff's Laws Circuit Symmetry Analysis
MediumNumerical value · figureNew ideaIdea asked in 4 years
Q22
An ac circuit has an inductor and a resistor of resistance R in series, such that $X_L=3\ R$ . Now, a capacitor is added in series such that $X_C=2\ R$ . The ratio of new power factor with the old power factor of the circuit is $\sqrt{5}:x$ . The value of $x$ …
Alternating Current Phasor Diagrams Power Factor from Phasors
MediumNumerical valueIdea asked in 5 yearsAsked the year before
Q23
A bullet of 10 g, moving with velocity $v$ , collides head-on with the stationary bob of a pendulum and recoils with velocity 100 m/s. The length of the pendulum is 0.5 m and mass of the bob is 1 kg. The minimum value of $v=$ __________ m/s so that the…
Work, Energy and Power Circular Motion Vertical Circular Motion
MediumNumerical value · figureNew ideaIdea asked in 2 years
Q24
Wires $W_1$ and $W_2$ are made of same material having the breaking stress of $1.25\times10^9\ \mathrm{N/m^2}$ . $W_1$ and $W_2$ have cross-sectional area of $8\times10^{-7}\ \mathrm{m^2}$ and $4\times10^{-7}\ \mathrm{m^2}$ , respectively. Masses of 20 kg and…
Mechanical Properties of Solids Elasticity Breaking Stress and Breaking Force
MediumNumerical value · figureIdea asked in 4 yearsAsked the year before
Q25
A tuning fork is vibrating at 250 Hz. The length of the shortest closed organ pipe that will resonate with the tuning fork will be __________ cm. (Take speed of sound in air as 340 $\mathrm{ms^{-1}}$ )
Waves Resonance in Air Columns Closed Organ Pipe Harmonics
EasyNumerical valueNew ideaIdea asked in 4 years
Q26
A heat engine operates between a cold reservoir at temperature $T_2=400$ K and a hot reservoir at temperature $T_1$ . It takes 300 J of heat from the hot reservoir and delivers 240 J of heat to the cold reservoir in a cycle. The minimum temperature of the hot…
Thermodynamics Heat Engines and Refrigerators Carnot Efficiency Limit
EasyNumerical valueNew ideaIdea asked in 3 years
Q27
Two simple harmonic motion, are represented by the equations $y_1=10\sin\left(3\pi t+\frac{\pi}{3}\right)$ $y_2=5\,(\sin 3\pi t+\sqrt{3}\cos 3\pi t)$ Ratio of amplitude of $y_1$ to $y_2=x:1$ . The value of $x$ is __________.
Oscillations SHM Equations Collinear Superposition of SHMs
EasyNumerical valueNew ideaIdea asked in 4 years
Q28
A plane electromagnetic wave with frequency of 30 MHz travels in free space. At particular point in space and time, electric field is 6 V/m. The magnetic field at this point will be $x\times10^{-8}$ T. The value of $x$ is __________.
Electromagnetic Waves
EasyNumerical value
Q29
X different wavelengths may be observed in the spectrum from a hydrogen sample if the atoms are exited to states with principal quantum number $n=6$ ? The value of X is __________.
Atoms Bohr's Atomic Model Number of Spectral Lines
EasyNumerical valueNew ideaIdea asked in 3 years
Q30
A zener diode of power rating 2 W is to be used as a voltage regulator. If the zener diode has a breakdown of 10 V and it has to regulate voltage fluctuated between 6 V and 14 V, the value of $R_S$ for safe operation should be __________ $\Omega$ .
Semiconductor Electronics: Materials, Devices and Simple Circuits Semiconductor Diodes Zener Diode Power Dissipation Limits
MediumNumerical value · figureNew ideaIdea asked in 5 years

Chemistry30 questions

12 easy16 medium2 hard

Q31
Lyophilic sols are more stable than lyophobic sols because,
Surface Chemistry Colloids Lyophilic and Lyophobic Sols
EasySingle correctNew ideaIdea asked in 3 years
Q32
The correct order of ionic radii for the ions, $\mathrm{P^{3-}}$ , $\mathrm{S^{2-}}$ , $\mathrm{Ca^{2+}}$ , $\mathrm{K^{+}}$ , $\mathrm{Cl^{-}}$ is :
Classification of Elements and Periodicity in Properties Periodic Trends Isoelectronic Species Radii Comparison
EasySingle correctIdea asked in 5 yearsAsked the year before
Q33
Match List - I with List - II : List - I (Name of ore/mineral) : (a) Calamine (b) Malachite (c) Siderite (d) Sphalerite List - II (Chemical formula) : (i) $\mathrm{ZnS}$ (ii) $\mathrm{FeCO_3}$ (iii) $\mathrm{ZnCO_3}$ (iv) $\mathrm{CuCO_3 \cdot Cu(OH)_2}$ …
General Principles and Processes of Isolation of Elements
EasyMatch the list
Q34
The oxide that gives $\mathrm{H_2O_2}$ most readily on treatment with $\mathrm{H_2O}$ is :
Hydrogen Hydrogen and Hydrides H2O2 Preparation from Barium Peroxide
MediumSingle correctNew ideaIdea asked in 2 years
Q35
Choose the correct statement from the following :
The s-Block Elements s-Block Elements Anomalous Solubility of Specific Alkali Halides
MediumSingle correctNew ideaIdea asked in 2 years
Q36
Which one of the following is formed (mainly) when red phosphorus is heated in a sealed tube at 803 K ?
The p-Block Elements Group 15 & 16 Phosphorus Allotropes
MediumSingle correctNew ideaIdea asked in 2 years
Q37
Potassium permanganate on heating at 513 K gives a product which is :
The d- and f-Block Elements d-Block Elements Permanganate Reactions
MediumSingle correctNew ideaIdea asked in 3 years
Q38
Which one of the following is used to remove most of plutonium from spent nuclear fuel ?
The p-Block Elements
HardSingle correct
Q39
In stratosphere most of the ozone formation is assisted by :
Environmental Chemistry Stratospheric Ozone Formation
EasySingle correctNew idea
Q40
Which one of the following tests used for the identification of functional groups in organic compounds does not use copper reagent ?
Biomolecules Carbohydrates Seliwanoff's Test
MediumSingle correctIdea asked in 2 yearsAsked the year before
Q41
The major product of the following reaction, if it occurs by $\mathrm{S_N2}$ mechanism is : [reaction drawn: a benzene ring bearing OH, plus a skeletal bromo-substituted chain drawn with a C=C double bond, with $\mathrm{K_2CO_3}$ / acetone over the arrow]
Alcohols, Phenols and Ethers Ethers Williamson Synthesis Limits
MediumDiagram based · figureNew ideaIdea asked in 3 years
Q42
Which one of the following reactions will not yield propionic acid ?
Aldehydes, Ketones and Carboxylic Acids Carboxylic Acids and Derivatives Carboxylic Acid Synthesis via Grignard and CO2
MediumSingle correctNew ideaIdea asked in 3 years
Q43
The major product (A) formed in the reaction given below is : [reaction drawn: $\mathrm{CH_3-CH_2-CH-CH_2-Br}$ with the CH carbon bonded to a benzene ring below it, + $\mathrm{CH_3O}$ with a circled minus charge, arrow labelled $\mathrm{CH_3OH}$ , giving A…
Haloalkanes and Haloarenes
MediumDiagram based · figure
Q44
Which one of the following is the major product of the given reaction ? [reaction drawn: a fused bicyclic compound - a benzene ring (drawn with an inner circle) bearing NC, fused to a six-membered ring that has $\mathrm{CH_3}$ on the top ring carbon, then a…
Aldehydes, Ketones and Carboxylic Acids Nucleophilic Addition Grignard Reagent Addition
HardDiagram based · figureIdea asked in 4 yearsAsked the year before
Q45
Given below are two statements : Statement I : Ethyl pent-4-yn-oate on reaction with $\mathrm{CH_3MgBr}$ gives a $3^\circ$ -alcohol. Statement II : In this reaction one mole of ethyl pent-4-yn-oate utilizes two moles of $\mathrm{CH_3MgBr}$ . In the light of…
Aldehydes, Ketones and Carboxylic Acids Carboxylic Acids and Derivatives Reaction of Esters with Grignard Reagents
MediumTwo statementsIdea asked in 3 years
Q46
The compound/s which will show significant intermolecular H-bonding is/are : [three structures drawn: (a) a benzene ring with $\mathrm{NO_2}$ at the top and OH on the adjacent ring carbon (upper right); (b) a benzene ring with HO at the lower left and, at…
Chemical Bonding and Molecular Structure Organic Acidity and Basicity Intramolecular H-Bonding in Phenols
MediumDiagram based · figureIdea asked in 2 yearsAsked the year before
Q47
The correct structures of A and B formed in the following reactions are : [reaction drawn: a benzene ring with OH at the top and $\mathrm{NO_2}$ at the bottom, arrow labelled $\mathrm{H_2/Pd}$ , $\mathrm{C_2H_5OH}$ giving A; then arrow with a drawn structure…
Amines
MediumDiagram based · figure
Q48
Which one of the following chemicals is responsible for the production of HCl in the stomach leading to irritation and pain ?
Chemistry in Everyday Life Evolution of Antacids: Neutralizers vs Receptor Blockers
EasyDiagram based · figureNew ideaIdea asked in 2 years
Q49
Hydrolysis of sucrose gives :
Biomolecules Carbohydrates Inversion of Cane Sugar
EasySingle correctNew ideaIdea asked in 2 years
Q50
The addition of dilute NaOH to $\mathrm{Cr^{3+}}$ salt solution will give :
The d- and f-Block Elements Cation Analysis Group III Hydroxides
EasySingle correctNew ideaIdea asked in 2 years
Q51
100 g of propane is completely reacted with 1000 g of oxygen. The mole fraction of carbon dioxide in the resulting mixture is $x \times 10^{-2}$ . The value of $x$ is __________. (Nearest integer) [Atomic weight : H = 1.008; C = 12.00; O = 16.00]
Some Basic Concepts of Chemistry Stoichiometry Limiting Reagent Identification
MediumNumerical valueNew ideaIdea asked in 4 years
Q52
Two flasks I and II shown below are connected by a valve of negligible volume. [figure: two circular flasks joined through a valve; flask I contains 2.8 g $\mathrm{N_2}$ , 300 K, 1 L; flask II contains 0.2 g $\mathrm{N_2}$ , 60 K, 2 L] When the valve is…
States of Matter Ideal Gas Laws Connected Gas Bulbs Equilibrium
MediumNumerical value · figureNew ideaIdea asked in 4 years
Q53
The number of photons emitted by a monochromatic (single frequency) infrared range finder of power 1 mW and wavelength of 1000 nm, in 0.1 second is $x \times 10^{13}$ . The value of $x$ is __________. (Nearest integer) (h = $6.63 \times 10^{-34}$ Js, c =…
Structure of Atom Photoelectric Effect Photon Energy and Momentum
EasyNumerical valueNew ideaIdea asked in 4 years
Q54
The number of species having non-pyramidal shape among the following is __________. (A) $\mathrm{SO_3}$ (B) $\mathrm{NO_3^{-}}$ (C) $\mathrm{PCl_3}$ (D) $\mathrm{CO_3^{2-}}$
Chemical Bonding and Molecular Structure VSEPR Theory Steric Number 3 Shapes
EasyNumerical valueNew idea
Q55
Data given for the following reaction is as follows : $\mathrm{FeO_{(s)} + C_{(graphite)} \longrightarrow Fe_{(s)} + CO_{(g)}}$ Table (Substance : $\Delta_f H^\circ$ ( $\mathrm{kJ\ mol^{-1}}$ ) ; $\Delta S^\circ$ ( $\mathrm{J\ mol^{-1}\ K^{-1}}$ )) :…
Thermodynamics (Chemistry)Gibbs Free Energy Temperature Dependence
MediumNumerical value · figureNew ideaIdea asked in 2 years
Q56
40 g of glucose (Molar mass = 180) is mixed with 200 mL of water. The freezing point of solution is __________ K. (Nearest integer) [Given : $\mathrm{K_f}$ = 1.86 K kg $\mathrm{mol^{-1}}$ ; Density of water = 1.00 g $\mathrm{cm^{-3}}$ ; Freezing point of water…
Solutions Colligative Properties Freezing Point Depression
EasyNumerical valueIdea asked in 4 yearsAsked the year before
Q57
When 5.1 g of solid $\mathrm{NH_4HS}$ is introduced into a two litre evacuated flask at $27^\circ$ C, 20% of the solid decomposes into gaseous ammonia and hydrogen sulphide. The $\mathrm{K_p}$ for the reaction at $27^\circ$ C is $x \times 10^{-2}$ . The value…
Equilibrium Chemical Equilibrium Heterogeneous Equilibrium
MediumNumerical valueNew ideaIdea asked in 2 years
Q58
The resistance of a conductivity cell with cell constant 1.14 $\mathrm{cm^{-1}}$ , containing 0.001 M KCl at 298 K is 1500 $\Omega$ . The molar conductivity of 0.001 M KCl solution at 298 K in S $\mathrm{cm^2}$ $\mathrm{mol^{-1}}$ is __________. (Integer answer)
Electrochemistry
EasyNumerical value
Q59
The first order rate constant for the decomposition of $\mathrm{CaCO_3}$ at 700 K is $6.36 \times 10^{-3}$ $\mathrm{s^{-1}}$ and activation energy is 209 kJ $\mathrm{mol^{-1}}$ . Its rate constant (in $\mathrm{s^{-1}}$ ) at 600 K is $x \times 10^{-6}$ . The…
Chemical Kinetics Arrhenius Equation Two-Point Arrhenius Equation
MediumNumerical valueIdea asked in 4 yearsAsked the year before
Q60
The number of optical isomers possible for $\mathrm{[Cr(C_2O_4)_3]^{3-}}$ is __________.
Coordination Compounds Coordination Isomerism Enantiomers of Tris-Chelate M(AA)3 Complexes
EasyNumerical valueNew idea

Maths30 questions

1 easy24 medium5 hard

Q61
Let $\mathbb{Z}$ be the set of all integers, $A = \{(x, y) \in \mathbb{Z} \times \mathbb{Z} : (x-2)^2 + y^2 \le 4\}$ , $B = \{(x, y) \in \mathbb{Z} \times \mathbb{Z} : x^2 + y^2 \le 4\}$ and…
Relations and Functions Sets and Relations Counting Relations
MediumSingle correctNew ideaIdea asked in 6 years
Q62
The set of all values of $k > -1$ , for which the equation $(3x^2 + 4x + 3)^2 - (k+1)(3x^2 + 4x + 3)(3x^2 + 4x + 2) + k(3x^2 + 4x + 2)^2 = 0$ has real roots, is :
Complex Numbers and Quadratic Equations Quadratic Roots Nature of Roots
HardSingle correctIdea asked in 3 yearsAsked the year before
Q63
Let $[\lambda]$ be the greatest integer less than or equal to $\lambda$ . The set of all values of $\lambda$ for which the system of linear equations $x + y + z = 4$ , $3x + 2y + 5z = 3$ , $9x + 4y + (28 + [\lambda])z = [\lambda]$ has a solution is :
Determinants Cramer's Rule Evaluating Parameters for Consistency
MediumSingle correctIdea asked in 6 yearsAsked the year before
Q64
Let $A = \begin{pmatrix} [x+1] & [x+2] & [x+3] \\ [x] & [x+3] & [x+3] \\ [x] & [x+2] & [x+4] \end{pmatrix}$ , where $[t]$ denotes the greatest integer less than or equal to t. If $\det(A) = 192$ , then the set of values of $x$ is the interval :
Determinants
MediumSingle correct
Q65
If $0 < x < 1$ and $y = \frac{1}{2}x^2 + \frac{2}{3}x^3 + \frac{3}{4}x^4 + \dots$ , then the value of $e^{1+y}$ at $x = \frac{1}{2}$ is :
Sequences and Series
MediumSingle correct
Q66
If $\lim_{x \to \infty} \left(\sqrt{x^2 - x + 1} - ax\right) = b$ , then the ordered pair (a, b) is :
Limits and Derivatives L'Hopital's Rule Conjugation vs L'Hopital for Radicals
MediumSingle correctNew ideaIdea asked in 3 years
Q67
If $y(x) = \cot^{-1}\left(\frac{\sqrt{1+\sin x} + \sqrt{1-\sin x}}{\sqrt{1+\sin x} - \sqrt{1-\sin x}}\right), x \in \left(\frac{\pi}{2}, \pi\right)$ , then $\frac{dy}{dx}$ at $x = \frac{5\pi}{6}$ is :
Continuity and Differentiability Inverse Trig Identities Periodic Reduction of tan-1(tan x) and cot-1(cot x)
MediumSingle correctIdea asked in 2 yearsAsked the year before
Q68
Let M and m respectively be the maximum and minimum values of the function $f(x) = \tan^{-1}(\sin x + \cos x)$ in $\left[0, \frac{\pi}{2}\right]$ . Then the value of $\tan(M - m)$ is equal to :
Application of Derivatives Maxima and Minima Extrema of Trigonometric Compositions
MediumSingle correctNew ideaIdea asked in 2 years
Q69
A box open from top is made from a rectangular sheet of dimension $a \times b$ by cutting squares each of side $x$ from each of the four corners and folding up the flaps. If the volume of the box is maximum, then $x$ is equal to :
Application of Derivatives Maxima and Minima Optimization Problems
MediumSingle correctIdea asked in 5 years
Q70
The value of the integral $\int_0^1 \frac{\sqrt{x}\, dx}{(1+x)(1+3x)(3+x)}$ is :
Integrals Integration by Substitution Fractional Powers LCM Substitution
HardSingle correctIdea asked in 4 yearsAsked the year before
Q71
The area of the region bounded by the parabola $(y-2)^2 = (x-1)$ , the tangent to it at the point whose ordinate is 3 and the $x$ -axis is :
Application of Integrals Area Under Curves Area Involving Tangents and Normals
MediumSingle correctNew ideaIdea asked in 3 years
Q72
A differential equation representing the family of parabolas with axis parallel to $y$ -axis and whose length of latus rectum is the distance of the point $(2, -3)$ form the line $3x + 4y = 5$ , is given by :
Differential Equations
MediumSingle correct
Q73
If the solution curve of the differential equation $(2x - 10y^3)dy + y\,dx = 0$ , passes through the points $(0, 1)$ and $(2, \beta)$ , then $\beta$ is a root of the equation :
Differential Equations Linear Differential Equations Linear in x Form
MediumSingle correctNew ideaIdea asked in 4 years
Q74
Let $A(a, 0)$ , $B(b, 2b+1)$ and $C(0, b)$ , $b \ne 0$ , $|b| \ne 1$ , be points such that the area of triangle ABC is 1 sq. unit, then the sum of all possible values of a is :
Straight Lines
MediumSingle correct
Q75
If two tangents drawn from a point P to the parabola $y^2 = 16(x-3)$ are at right angles, then the locus of point P is :
Conic Sections General Conic Section Equation of Director Circle
MediumSingle correctNew idea
Q76
The angle between the straight lines, whose direction cosines are given by the equations $2l + 2m - n = 0$ and $mn + nl + lm = 0$ , is :
Three Dimensional Geometry Angle Between Lines Angle via Direction Cosines
MediumSingle correctNew idea
Q77
The equation of the plane passing through the line of intersection of the planes $\vec{r} \cdot (\hat{i} + \hat{j} + \hat{k}) = 1$ and $\vec{r} \cdot (2\hat{i} + 3\hat{j} - \hat{k}) + 4 = 0$ and parallel to the $x$ -axis is :
Three Dimensional Geometry Planes in 3D Family of Planes
MediumSingle correctIdea asked in 4 yearsAsked the year before
Q78
Each of the persons A and B independently tosses three fair coins. The probability that both of them get the same number of heads is :
Probability
EasySingle correct
Q79
The Boolean expression $(p \wedge q) \Rightarrow ((r \wedge q) \wedge p)$ is equivalent to :
Mathematical Reasoning
MediumSingle correct
Q80
Two poles, AB of length a metres and CD of length $a + b$ ( $b \neq a$ ) metres are erected at the same horizontal level with bases at B and D. If $BD = x$ and $\tan\underline{\lfloor ACB} = \frac{1}{2}$ , then :
Trigonometric Functions
MediumSingle correct
Q81
Let $z_1$ and $z_2$ be two complex numbers such that $\arg(z_1 - z_2) = \frac{\pi}{4}$ and $z_1, z_2$ satisfy the equation $|z - 3| = \mathrm{Re}(z)$ . Then the imaginary part of $z_1 + z_2$ is equal to __________.
Complex Numbers and Quadratic Equations
HardNumerical value
Q82
Let $S = \{1, 2, 3, 4, 5, 6, 9\}$ . Then the number of elements in the set $T = \{A \subseteq S : A \neq \phi$ and the sum of all the elements of A is not a multiple of 3 $\}$ is __________.
Relations and Functions Permutations and Combinations Selection of One or More Objects
HardNumerical valueNew ideaIdea asked in 3 years
Q83
$3 \times 7^{22} + 2 \times 10^{22} - 44$ when divided by 18 leaves the remainder __________.
Binomial Theorem
MediumNumerical value
Q84
If $\int \frac{2e^x + 3e^{-x}}{4e^x + 7e^{-x}}\, dx = \frac{1}{14}\left(ux + v \log_e(4e^x + 7e^{-x})\right) + C$ , where C is a constant of integration, then $u + v$ is equal to __________.
Integrals Integration by Substitution Numerator-Denominator Derivative Matching
MediumNumerical valueNew ideaIdea asked in 2 years
Q85
Two circles each of radius 5 units touch each other at the point $(1, 2)$ . If the equation of their common tangent is $4x + 3y = 10$ , and $C_1(\alpha, \beta)$ and $C_2(\gamma, \delta)$ , $C_1 \neq C_2$ are their centres, then…
Conic Sections Circle Tangents Common Tangent of Touching Circles
MediumNumerical valueNew ideaIdea asked in 4 years
Q86
Let $A(\sec\theta, 2\tan\theta)$ and $B(\sec\phi, 2\tan\phi)$ , where $\theta + \phi = \pi/2$ , be two points on the hyperbola $2x^2 - y^2 = 2$ . If $(\alpha, \beta)$ is the point of the intersection of the normals to the hyperbola at A and B, then $(2\beta)^2$ …
Conic Sections Normals to Conics Hyperbola Normal Parametric Form
HardNumerical valueNew idea
Q87
Let S be the mirror image of the point $Q(1, 3, 4)$ with respect to the plane $2x - y + z + 3 = 0$ and let $R(3, 5, \gamma)$ be a point of this plane. Then the square of the length of the line segment SR is __________.
Three Dimensional Geometry Point-Plane Distance Image of a Point
MediumNumerical valueIdea asked in 4 years
Q88
The probability distribution of random variable X is given by : | X | 1 | 2 | 3 | 4 | 5 | | P(X) | K | 2K | 2K | 3K | K | Let $p = P(1 < X < 4 \mid X < 3)$ . If $5p = \lambda K$ , then $\lambda$ is equal to __________.
Probability Conditional Probability Reduced Sample Space Method
MediumNumerical value · figureNew idea
Q89
Let S be the sum of all solutions (in radians) of the equation $\sin^4\theta + \cos^4\theta - \sin\theta\cos\theta = 0$ in $[0, 4\pi]$ . Then $\frac{8S}{\pi}$ is equal to __________.
Trigonometric Functions General Solutions Fourth Power Identity Application
MediumNumerical valueIdea asked in 2 yearsAsked the year before
Q90
An online exam is attempted by 50 candidates out of which 20 are boys. The average marks obtained by boys is 12 with a variance 2. The variance of marks obtained by 30 girls is also 2. The average marks of all 50 candidates is 15. If $\mu$ is the average…
Statistics Variance and SD Combined Variance of Two Sets
MediumNumerical valueNew ideaIdea asked in 3 years

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.