JEE study notes

Units and Measurement

NCERT Class 11 Physics, Chapter 1

1. SI units

   Expressing derived units in base units, plane and solid angle, unit conversions, and the 2018 constant-based definitions.

2. Significant figures

   Trailing-zero ambiguity, the different rules for products and sums, the round-half-to-even rule, and how relative errors add in a product.

3. Errors in measurement

   Accuracy and precision, systematic and random errors, mean absolute and percentage error, and combining errors in sums, products and powers.

4. Least count: vernier callipers and screw gauge

   Least count, reading vernier callipers and a screw gauge, positive and negative zero error and its correction, for Unit 1 and the Unit 20 experiments.

5. Dimensions of physical quantities

   Finding dimensions from defining relations, quantities that share dimensions (work and torque, pressure and energy density), and constants with dimensions such as G and R.

6. Dimensional formulae and equations

   Dimensional formulae of mechanical, thermal and electrical quantities (h, η, Y, k_B, gas constant R, resistance, C, ε_₀, B, μ_₀), dimensional constants, and a length from h, G and c.

7. Dimensional analysis

   Checking and deducing relations (pendulum, Stokes’ drag, string frequency), converting units between systems, and the limits of the method, with 5 solved examples.

Motion in a Straight Line

NCERT Class 11 Physics, Chapter 2

1. Frame of reference and displacement

   Frame of reference, displacement versus path length, average velocity versus average speed (with the 2v_₁v_₂/(v_₁ + v_₂) trap), uniform motion and reading x-t graphs.

2. Velocity and speed

   Velocity as dx/dt, tangent versus chord on x-t graphs, turning points where v = 0, and average speed versus instantaneous speed, with 5 solved examples.

3. Acceleration

   a = dv/dt = d^²x/dt^², the signs of v and a for speeding up or slowing down, and net versus total area under v-t graphs, with 5 solved examples.

4. Equations of motion

   Signed use of v = v_₀ + at, x = v_₀t + ½at^² and v^² = v_₀^² + 2ax, the nth-second formula, free fall, and stopping distance with reaction time.

5. Relative velocity

   Relative velocity in one and two dimensions, closest approach, rain-umbrella and river-crossing problems (shortest time, shortest path, least drift), with 6 solved examples.

Motion in a Plane

NCERT Class 11 Physics, Chapter 3

1. Scalars and vectors

   Why direction alone does not make a vector (current is a scalar), displacement versus path length, equal and free vectors, with 6 solved examples.

2. Multiplying a vector by a number

   |λA| = |λ|A, parallel and antiparallel vectors as B = λA and by component ratios, physical products like p = mv and F = ma, and the unit vector n̂ = A/|A|.

3. Adding vectors graphically

   Triangle and parallelogram laws, the range of a resultant (triangle inequality), A − B as the other diagonal, change in velocity at a corner, with 6 solved examples.

4. Resolution of vectors

   Components and the quadrant trap with tan^⁻¹, unit vectors, direction cosines, and resolving along tilted axes (NCERT Exercise 3.19), with 6 solved examples.

5. Adding vectors by components

   Component addition, R = √(A^² + B^² + 2AB cos θ), direction of the resultant, |A − B|, equal vectors 2A cos(θ/2), and the quadrant trap, with 5 solved examples.

6. Position, velocity and acceleration in a plane

   v = dr/dt and a = dv/dt in components, the angle between v and a (speeding up, slowing down, turning) via v·a, and which relations hold for any motion.

7. Constant acceleration in a plane

   Vector equations of motion, solving x and y separately with a common time, v^² = v_₀^² + 2a·Δr, and when the equations fail (circular motion), with 5 solved examples.

8. Projectile motion

   Path equation, time of flight, maximum height and range for level ground, maximum range at 45° and complementary angles, horizontal launch from a height, a ceiling limit, with 6 solved examples.

9. Uniform circular motion

   a_c = v^²/R = ω^²R = 4π^²ν^²R, why it is not a constant vector, average versus instantaneous acceleration, and the tangential part in non-uniform motion.

Laws of Motion

NCERT Class 11 Physics, Chapter 4

1. Law of inertia

   Why a ball on smooth planes regains its height (a = g sin θ, v = √(2gh)), mass as the measure of inertia, and a ball on a braking train’s floor.

2. Newton’s first law

   Equilibrium (F_net = 0 along each axis), inertial frames, and why a body at rest still has forces acting on it.

3. Newton’s second law

   F = dp/dt and F = ma with free-body diagrams: impulse, apparent weight in an accelerating lift and the momentum form of the law.

4. Newton’s third law

   Action-reaction pairs and the normal force between bodies, why a pair never cancels, rockets, and the recoil of a gun worked out with M V = m v.

5. Conservation of momentum

   Conservation along one axis, momentum in explosions with vector addition, perfectly inelastic collisions and recoil with the bullet’s speed given relative to the gun.

6. Equilibrium of a particle

   Tensions in strings meeting at a knot by components and by Lami’s theorem, the equilibrant of given forces, and why a loaded wire can never be straight.

7. Friction

   Self-adjusting static friction, the angle of repose, blocks on rough inclines, pulling versus pushing at an angle, and a box on an accelerating truck.

8. Dynamics of circular motion

   Highest and lowest safe speeds on a banked road with friction, the optimum speed v_o = √(Rg tan θ), and why N = mg/cos θ at the optimum speed.

9. Solving problems in mechanics

   Connected bodies: contact forces between blocks, tensions in strings, Atwood machines and a block on a rough table pulled over a pulley, solved with one diagram per body.

Work, Energy and Power

NCERT Class 11 Physics, Chapter 5

1. Scalar product

   Angle and projection from components, the perpendicularity test, and identities such as (A + B)·(A − B) = A^² − B^², worked step by step.

2. Work-energy theorem

   Net work as the sum of the works of all the forces, stopping distance d = mv^²/(2F), and the work of resistive forces found from the change in kinetic energy.

3. Work

   Work by each force on a body sliding down a rough incline, work from vectors as F·d, and why equal and opposite forces need not do equal and opposite work.

4. Kinetic energy

   K = p^²/2m and p = √(2mK): comparing bodies with equal kinetic energy or equal momentum, and percentage changes in K from changes in speed.

5. Work done by a variable force

   Integrating F(x) to find work, forces that change sign, and graphs built from rectangles, triangles and trapeziums, with negative areas handled correctly.

6. Work-energy theorem (variable force)

   Speeds and stopping points under position-dependent forces from K_f − K_i = ∫F dx, including NCERT’s rough-patch example with a logarithm.

7. Potential energy

   F = −dV/dx from a potential function, path independence of gravity’s work, zero work on a closed path, and how the reference level affects V but not ΔV.

8. Conservation of mechanical energy

   Vertical circle: speeds and tensions at the bottom, side and top, string versus rod, and when the string goes slack; energy lost to friction on rough slopes.

9. Potential energy of a spring

   Work between two stretches, ½k(x_₂^² − x_₁^²), spring-block speeds, a car hitting a spring buffer with friction, and a block dropped on a spring.

10. Power

    P = F·v for vehicles at steady speed and top speed, instantaneous versus average power under a constant force, and a pump efficiency problem.

11. Collisions

    Elastic formulas with both bodies moving, kinetic energy lost in perfectly inelastic collisions, glancing collisions of equal masses and the coefficient of restitution.

Systems of Particles and Rotational Motion

NCERT Class 11 Physics, Chapter 6

1. Rigid body motion

   Rolling as translation plus rotation: v = ωR without slipping, the top at 2v, the contact point at rest, slipping wheels, and translation on a curved path.

2. Centre of mass

   Composite bodies and cut-outs as negative mass, a non-uniform rod by integration, and standard results for semicircles, hemispheres and cones.

3. Motion of the centre of mass

   Fragments of an exploding shell, the man-on-a-boat problem with relative displacement, skaters on ice, and when the parabola rule stops working.

4. Momentum of a system

   Component-wise conservation, the centre-of-mass frame, energy sharing in a decay from rest, and K = K′ + ½MV^², with an explosion in flight.

5. Vector product

   The determinant form and its sign traps, areas of parallelograms and triangles, unit vectors perpendicular to two vectors, and finding an angle with both products.

6. Angular velocity

   v = ω × r with components, why only the perpendicular distance counts, and tangential and centripetal acceleration of a point on a rotating body.

7. Torque and angular momentum

   Torque and angular momentum from components, a particle in straight-line motion, internal torques for a system, and the torque of gravity on a projectile.

8. Equilibrium of a rigid body

   Choosing the point for moments, couples about any point, the ladder against a smooth wall, a non-uniform bar on strings and a plank on the point of tipping.

9. Experiment: mass with a metre scale

   The Unit 20 experiment step by step: balancing at the centre of gravity, m = Md_₂/d_₁, a scale not supported at its centre, and the error in the result.

10. Moment of inertia

    Derivations of the rod and disc results by integration, the rod about an end, the hollow-sphere value, and how the same body has different I about different axes.

11. Parallel and perpendicular axes

    Proofs of both theorems, shifting axes through the centre of mass, tangents to rings, discs and spheres, a square plate about a diagonal and a triangle of rods.

12. Rotational kinematics

    Derivations by integration, the ω–t graph and average-speed shortcut, the angle in the n-th second, and linking θ, ω, α to a point’s s, v and a.

13. Rotational dynamics

    Blocks hanging from wheels and pulleys with a = Rα, unequal tensions over a massive pulley, and the work-energy theorem for rotation.

14. Angular momentum about an axis

    When Iω is the whole angular momentum, kinetic energy changing while L is conserved, two discs locking together, and a child walking on a turntable.

Gravitation

NCERT Class 11 Physics, Chapter 7

1. Kepler’s laws

   v_P r_P = v_A r_A, the semi-major axis as the average of the nearest and farthest distances, T^² ∝ a^³ from gravity for circular orbits, and the central mass.

2. Universal law of gravitation

   Superposition with symmetry, the point between two bodies where gravity cancels, and why uniform spheres act as point masses but other shapes do not.

3. Gravitational constant

   The torsion-balance equation G = κθd^²/(MmL) worked with numbers, the dimensions of G, and how the reading changes with distance.

4. Acceleration due to gravity

   g in terms of density, comparing planets by mass, radius and density, and the force inside a uniform Earth growing with distance from the centre.

5. Variation of g

   When 1 − 2h/R fails, the height and depth where g is halved, equal falls at height h and depth 2h, and how the real Earth differs from the uniform model.

6. Gravitational potential energy

   Why mgh fails at large heights, U for several particles by pairs, the potential inside a shell, inside a solid sphere and of a ring, and the field as the slope of the potential.

7. Escape speed

   The energy picture of bound and unbound motion, escape from a height, speed left at infinity, the height reached below escape speed, and the two-sphere problem with its striking speed.

8. Earth satellites

   How speed and period scale with radius, weighing a planet from an orbit, density from the surface-skimming period, and the geostationary orbit.

9. Energy of an orbiting satellite

   K = −E and U = 2E, energy to change orbits or launch into orbit, binding energy, the energy of an elliptical orbit and why air drag speeds a satellite up.

Mechanical Properties of Solids

NCERT Class 11 Physics, Chapter 8

1. Elasticity and plasticity

   The restoring force, elastic versus plastic behaviour depending on load, permanent set and what ‘more elastic’ means, with NCERT’s steel and copper data.

2. Stress and strain

   Choosing the right area for each kind of stress, shear strain as tan θ ≈ θ in radians, how stress differs from pressure, and stress ratios for wires of different thickness.

3. Hooke’s law

   The difference between a modulus and a spring constant, a wire’s spring constant kA/L, the proportional limit, and testing Hooke’s law from data.

4. Stress-strain curve

   Proportional limit versus elastic limit, the size of the permanent set from the unloading line, safe and breaking loads, and the area under the curve.

5. Elastic moduli

   Wires in series, Y/G and Poisson’s ratio, the ½ × stress × strain energy result, and the printing slips in NCERT Section 8.5 corrected.

6. Experiment: Young’s modulus of a wire

   The Unit 20 experiment step by step: Searle’s apparatus, Y = MgL/(πr^²l), Y from the graph’s slope, and the largest-error calculation where the diameter counts twice.

Mechanical Properties of Fluids

NCERT Class 11 Physics, Chapter 9

1. Pressure in fluids

   Absolute versus gauge pressure, the hydrostatic paradox, force on a dam wall, and the work balance in a hydraulic lift.

2. Streamline and turbulent flow

   Continuity in rivers, hoses and blood vessels, a falling stream that narrows, and the Reynolds number and critical speed (beyond NCERT).

3. Bernoulli’s principle

   Continuity with Bernoulli in pipes, pressurised tanks, the range of a jet from a tank, and the venturi meter (beyond NCERT).

4. Viscosity

   Rate of strain in the definition, the a^² dependence of terminal velocity, merging drops, rising bubbles and where Stokes’ law holds.

5. Experiment: viscosity from terminal velocity

   The Unit 20 falling-ball experiment step by step: terminal velocity d/t, η from the v_t against a^² graph, why the marks and jar size matter, and the error calculation.

6. Surface tension

   Film and bubble surfaces counted correctly, the contact-angle balance (and NCERT’s slip in it), capillary rise and depression, and the energy to spray a drop.

7. Experiment: surface tension by capillary rise

   The Unit 20 capillary-rise experiment step by step: the pointer and travelling microscope, S = rhρg/2, the meniscus correction, hr constant, the effect of detergents and the error budget.

Thermal Properties of Matter

NCERT Class 11 Physics, Chapter 10

1. Temperature and heat

   Heat versus internal energy, the direction of heat flow, kelvin–Celsius conversions for temperatures and differences, and work turned into heat.

2. Measuring temperature

   Converting temperatures and temperature differences, arbitrary linear scales and faulty thermometers, and the −40° crossing.

3. Ideal-gas equation and absolute temperature

   Using T in kelvin in every gas law, moles and molar volume, finding absolute zero from data, and the combined gas law between two states.

4. Thermal expansion

   Holes that expand, apparent expansion of a liquid in a flask, thermal stress Yα_lΔT, pendulum clocks and bimetallic strips.

5. Specific heat capacity

   Heat capacity versus specific heat, molar specific heats C_p and C_v with C_p − C_v = R, the heating-rate slope P/(ms), and 3R for solids with carbon as the exception.

6. Calorimetry

   Heat lost equals heat gained with the calorimeter included, the final temperature of a mixture, water equivalent and how heat losses bias the result.

7. Experiment: specific heat by the method of mixtures

   The Unit 20 method-of-mixtures experiment for a solid and a liquid: steam heater, calorimeter, the two working formulas, precautions and the error budget.

8. Change of state

   Latent heat in mixing problems, deciding whether all the ice melts, the heating curve to scale, and the phase diagram with triple and critical points.

9. Heat transfer

   Steady conduction through compound bars with thermal resistance, junction temperature and equivalent conductivity, Wien’s law and net radiation eσA(T^⁴ − T_s^⁴).

Thermodynamics

NCERT Class 11 Physics, Chapter 11

1. Thermal equilibrium

   Thermal versus mechanical equilibrium, what changes when two gases in rigid containers reach equilibrium, and kinetic energy turned into internal energy.

2. Zeroth law of thermodynamics

   Transitivity of thermal equilibrium, temperature as the shared variable, the constant-volume gas thermometer, and why a thermometer’s heat capacity matters.

3. Heat, internal energy and work

   Why ΔU is path-independent while heat and work are not, what counts in internal energy, and work turned into internal energy.

4. First law of thermodynamics

   Sign conventions, ΔQ − ΔW as a path-independent quantity, work as the area under a P–V path, and rate form with heaters.

5. Specific heat capacity of solids, water and gases

   C = 3R for solids and its limits, C_p − C_v = R derived from the first law, per-mole versus per-kilogram forms, and heat at constant volume versus constant pressure.

6. State variables and equation of state

   Independent variables from the equation of state, isotherms, the halving test for extensive and intensive variables, and checking equations by kind.

7. Thermodynamic processes

   Work in isothermal and adiabatic processes, PV^γ and TV^γ−1 constant, adiabatic versus isothermal slopes, and heat and ΔU in every process.

8. Second law of thermodynamics

   Efficiency and coefficient of performance, why η < 1 and α is finite, ‘sole result’ in the two statements, and why they are equivalent.

9. Reversible and irreversible processes

   Quasi-static and non-dissipative as the conditions for reversibility, free expansion with W = Q = ΔU = 0 yet irreversible, and the link to the second law.

Kinetic Theory

NCERT Class 11 Physics, Chapter 12

1. Behaviour of gases

   PV = Nk_B T and P = ρRT/M_₀, when real gases deviate from ideal behaviour, and mixtures of gases with partial pressures.

2. Kinetic theory of an ideal gas

   Deriving gas pressure from molecular collisions, ½m⟨v^²⟩ = (3/2)k_B T, v_rms = √(3RT/M_₀), PV = (2/3)E, and Dalton’s law from kinetic theory.

3. Law of equipartition of energy

   Counting squared energy terms, why a vibration adds k_B T, energy per molecule and per mole, and internal energy of gas mixtures.

4. Specific heat capacity from kinetic theory

   C_v = (f/2)R, γ for monatomic, rigid and vibrating diatomic gases, γ of mixtures, linear versus non-linear molecules, and 3R for solids.

5. Mean free path

   Deriving l = 1/(√2 nπd^²), collision time and frequency, l = k_B T/(√2 πd^²P), and how l scales with pressure, temperature and molecular size.

Oscillations

NCERT Class 11 Physics, Chapter 13

1. Periodic and oscillatory motion

   Periods of sums and squares of sines and cosines, A sin ωt + B cos ωt as a single sine, and the difference between ν and ω.

2. Simple harmonic motion

   Reading A, ω and φ from an equation, finding φ from the starting position and direction, phase differences as time delays, and the T/12 and T/6 timing results.

3. SHM and uniform circular motion

   Writing the SHM equation from a reference circle, including clockwise rotation and starting angles, both projections, and why the forces differ.

4. Velocity and acceleration in SHM

   v = ±ω√(A^² − x^²), a = −ω^²x, finding A and ω from speeds at two points or from the greatest speed and acceleration, and phase relations.

5. Force law for SHM

   Recognising SHM from a = −ω^²x, springs in series and parallel, vertical springs, and why the period does not depend on the amplitude.

6. Energy in SHM

   K = ½k(A^² − x^²), where K = U, energy against amplitude and mass, average energies over a period, and the T/2 period of K and U.

7. The simple pendulum

   Restoring torque and T = 2π√(I/mgL), when sin θ ≈ θ fails, measuring g, and how the period changes with length, g and an accelerating lift.

8. Experiment: simple pendulum and loss of energy

   The Unit 20 simple pendulum experiment: timing the period, the effective length and g = 4π^²L/T^² with its error, and the square of the amplitude against time as the energy is dissipated.

Waves

NCERT Class 11 Physics, Chapter 14

1. Wave motion

   Mechanical versus electromagnetic waves, the speed of light in a vacuum, and the spring model of an elastic medium.

2. Transverse and longitudinal waves

   Why transverse waves need a shear-resisting medium, waves in solids and fluids, and using the different speeds of the two kinds to find how far away an earthquake is.

3. Displacement relation for a progressive wave

   Reading a, k, ω and the direction from a wave equation, phase differences in space and time, and combining sine and cosine terms.

4. Speed of a travelling wave

   v = √(T/μ), v = √(B/ρ), Laplace’s √(γP/ρ), the dependence on temperature but not pressure, and the dimensional-analysis route to the formulas.

5. Principle of superposition of waves

   Resultant y = 2a cos(φ/2) sin(kx − ωt + φ/2), finding phase differences from resultant amplitudes, and where the energy goes when pulses cancel.

6. Reflection of waves and standing waves

   Standing wave y = 2a sin kx cos ωt, phases of neighbouring loops, harmonics of strings and open and closed pipes, overtones, and the resonance tube.

7. Experiment: speed of sound with a resonance tube

   The Unit 20 resonance-tube experiment: first and second resonance lengths, v = 2ν(l_₂ − l_₁), the end correction and its cancellation, temperature correction and error estimate.

8. Beats

   Deriving s = [2a cos ω_b t] cos ω_a t, why the beat frequency is ν_₁ − ν_₂ and not half of it, and deciding which frequency is higher by loading or tightening.

Electric Charges and Fields

NCERT Class 12 Physics, Chapter 1

1. Electric charge

   JEE Main Unit 11 starts from electric charges: the sign conventions, charging by electron transfer with no charge created, and a counting example with q = ne.

2. Conductors and insulators

   JEE Main Unit 11 lists conductors and insulators: free electrons, why charge spreads over a conductor’s surface, and why a hand-held metal does not stay charged.

3. Basic properties of electric charge

   JEE Main Unit 11, conservation of charge: algebraic addition, charge sharing between identical spheres, q = ne with integer n, and when quantisation can be ignored.

4. Coulomb’s law

   JEE Main Unit 11, Coulomb’s law: the inverse-square force, k and ε_₀, the vector form with r̂_₂₁, charge sharing between identical spheres, and electric against gravitational force.

5. Forces between multiple charges

   JEE Main Unit 11, forces between multiple charges: superposition, vector sums by the parallelogram law, the zero-force point between two charges, and symmetric arrangements.

6. Electric field

   JEE Main Unit 11, electric field: E = F/q, the field of a point charge, superposition of fields, charged particles in a uniform field, and the field at points near two charges.

7. Electric field lines

   JEE Main Unit 11, electric field lines: tangent gives the field, density gives strength, lines of a single charge, two like charges and a dipole, and why lines never cross.

8. Electric flux

   JEE Main Unit 11, electric flux: Δϕ = E·ΔS = EΔS cos θ, the area vector, sign of flux on closed surfaces, and flux through tilted plates and cubes in uniform fields.

9. Electric dipole

   JEE Main Unit 11, electric dipole: dipole moment, exact and far-field results on the axis and equator, the 2 : 1 ratio, 1/r^³ fall-off, and the point dipole.

10. Dipole in a uniform field

    JEE Main Unit 11, torque on a dipole in a uniform field: τ = pE sin θ, stable and unstable alignment, work to rotate a dipole, and the net force in a non-uniform field.

11. Continuous charge distribution

    JEE Main Unit 11, continuous charge distribution: λ, σ and ρ, the field by summing elements, and the field on the axis of a charged ring, E = kQx/(x^² + R^²)^3/2.

12. Gauss’s law

    JEE Main Unit 11, Gauss’s law: ϕ = q/ε_₀ for any closed surface, only enclosed charge counts, flux through one face of a cube (q/6ε_₀), and non-uniform fields through a cube.

13. Applications of Gauss’s law

    JEE Main Unit 11, applications of Gauss’s law: E = λ/2πε_₀r, E = σ/2ε_₀, the shell (zero inside, kq/r^² outside), two parallel sheets, and NCERT’s point-nucleus atom model.

Electrostatic Potential and Capacitance

NCERT Class 12 Physics, Chapter 2

1. Electrostatic potential energy

   JEE Main Unit 11, electric potential energy: work by external force and by field, path independence, zero at infinity, and energy conservation for a released charge.

2. Electrostatic potential

   JEE Main Unit 11, electric potential and potential difference: W = qΔV, U = qV, signs for negative charges, and energy in electron volts.

3. Potential due to a point charge

   JEE Main Unit 11, electric potential due to a point charge: V = kQ/r, V = Er for magnitudes, work between two distances, and potential energy in MeV.

4. Potential due to a dipole

   JEE Main Unit 11, potential due to an electric dipole: V = kp cos θ/r^², exact against approximate values, zero on the equator, and comparison with a point charge.

5. Potential due to a system of charges

   JEE Main Unit 11, potential of a system of charges: algebraic sums, points of zero potential, symmetric arrangements, and the shell’s constant potential inside.

6. Equipotential surfaces

   JEE Main Unit 11, equipotential surfaces: field normal to equipotentials, the field as the steepest fall of potential, E = V/d, and spacing of equipotentials.

7. Potential energy of a system of charges

   JEE Main Unit 11, potential energy of a system of point charges: one term per pair, signs, squares and triangles of charges, and work to rearrange charges.

8. Potential energy in an external field

   JEE Main Unit 11, potential energy in an external field: qV, mutual energy of two charges, U = −p·E for a dipole, work to rotate a dipole, and energy in eV.

9. Electrostatics of conductors

   JEE Main Unit 11, conductors in electrostatics: E = 0 inside, equipotential body, σ/ε_₀ at the surface, cavities and shielding, and fields of charged spheres.

10. Dielectrics and polarisation

    JEE Main Unit 11, dielectrics and electric polarisation: induced dipoles, P = ε_₀χ_ₑE, bound charge σp = P on a slab, and the reduced field inside a dielectric.

11. Capacitors and capacitance

    JEE Main Unit 11, capacitor and capacitance: C = Q/V, units, an isolated sphere as a capacitor, and breakdown limits set by dielectric strength.

12. The parallel plate capacitor

    JEE Main Unit 11, capacitance of a parallel plate capacitor: E = σ/ε_₀ between the plates, zero outside, V = Ed and C = ε_₀A/d, with numericals on plate size.

13. Effect of a dielectric on capacitance

    JEE Main Unit 11, capacitance with and without a dielectric: C = KC_₀, partly filled gaps, metal slabs, and battery connected or disconnected.

14. Combination of capacitors

    JEE Main Unit 11, combination of capacitors in series and parallel: equivalent capacitance, charge and voltage on each, and mixed networks.

15. Energy stored in a capacitor

    JEE Main Unit 11, energy stored in a capacitor: U = ½CV^², energy density ½ε_₀E^², energy lost when capacitors share charge, and dielectric effects.

Current Electricity

NCERT Class 12 Physics, Chapter 3

1. Electric current

   JEE Main Unit 12 starts from electric current: I = q/t and I = dQ/dt, counting electrons with n = I/e, and current direction against electron flow.

2. Electric currents in conductors

   JEE Main Unit 12, the picture behind electric current: free electrons in metals, random motion without a field, and the steady field a cell maintains.

3. Ohm’s law

   JEE Main Unit 12, Ohm’s law, resistance, resistivity and conductivity: R = ρl/A, current density, j = σE, and stretched-wire problems.

4. Experiment: resistance using Ohm’s law

   JEE Main Unit 20, Experimental Skills: the resistance of a given wire using Ohm’s law, from the slope of the V–I graph.

5. Drift of electrons

   JEE Main Unit 12, drift velocity, mobility and their relation with electric current: v_d = eEτ/m, I = neAv_d, σ = ne^²τ/m and μ = eτ/m.

6. Limitations of Ohm’s law

   JEE Main Unit 12, I–V characteristics of ohmic and non-ohmic conductors: straight lines through the origin against curved, one-way and folded graphs.

7. Resistivity of various materials

   JEE Main Unit 12, electrical resistivity and conductivity: the ranges for metals, semiconductors and insulators, and how resistivity sets resistance.

8. Temperature dependence of resistivity

   JEE Main Unit 12, temperature dependence of resistance: α from two readings, hot filaments, and why metals and semiconductors behave oppositely.

9. Electrical energy and power

   JEE Main Unit 12, electrical energy and power: P = IV = I^²R = V^²/R, heating, bulb ratings, and why power is transmitted at high voltage.

10. Combination of resistors

    JEE Main Unit 12, series and parallel combinations of resistors: equivalent resistance, voltage and current division, and reducing networks.

11. Cells, emf and internal resistance

    JEE Main Unit 12, internal resistance, potential difference and emf of a cell: V = ε − Ir, charging with V = ε + Ir, and maximum power.

12. Cells in series and in parallel

    JEE Main Unit 12, a combination of cells in series and parallel: equivalent emf and internal resistance, reversed cells, and which grouping gives more current.

13. Kirchhoff’s rules

    JEE Main Unit 12, Kirchhoff’s laws and their applications: sign conventions, multi-loop networks, symmetric networks and cells being charged.

14. Wheatstone bridge

    JEE Main Unit 12, Wheatstone bridge: the balance condition, removing the middle arm at balance, and unbalanced bridges solved by Kirchhoff’s laws.

15. Metre bridge

    JEE Main Unit 12 lists the metre bridge, and Unit 20 the resistivity of a wire with it: the balance length, end corrections and ρ = πd^²S/(4L).

Moving Charges and Magnetism

NCERT Class 12 Physics, Chapter 4

1. Magnetic force

   JEE Main Unit 13: the Lorentz force and the force on a current-carrying conductor, including bent wires and closed loops in a uniform field.

2. Motion in a magnetic field

   JEE Main Unit 13: circular and helical paths in a uniform field, radius and period, and comparing particles with the same speed, energy or accelerating voltage.

3. Biot–Savart law

   JEE Main Unit 13, Biot–Savart law: the vector form, the sin θ factor, directions from the cross product, and the field of short elements.

4. Field on the axis of a loop

   JEE Main Unit 13, Biot–Savart law applied to a circular loop: the axial field, the centre field, arcs and straight leads, and coils of N turns.

5. Ampere’s circuital law

   JEE Main Unit 13, Ampere’s law: its statement and sign rule, the long straight wire, fields inside and outside a thick wire, and enclosed current.

6. The solenoid

   JEE Main Unit 13, Ampere’s law applied to a solenoid: B = μ_₀nI inside, turns per unit length, several layers, and the field at the ends.

7. Force between parallel currents

   JEE Main Unit 13: the force between two parallel current-carrying conductors and the definition of the ampere, with three-wire and loop questions.

8. Torque on a current loop

   JEE Main Unit 13: torque on a current loop in a uniform field, the magnetic dipole moment, work in rotating a coil, and dipole fields on the axis and in the plane.

9. Moving coil galvanometer

   JEE Main Unit 13: the moving coil galvanometer, its current sensitivity, and its conversion to an ammeter and a voltmeter, with shunt and series resistances.

10. Experiment: galvanometer resistance

    JEE Main Unit 20 lists this experiment: the half-deflection circuit, G = RS/(R − S), the figure of merit k = ε/[(R + G)θ], and precautions.

Magnetism and Matter

NCERT Class 12 Physics, Chapter 5

1. The bar magnet

   JEE Main Unit 13: the bar magnet as an equivalent solenoid, its field on the axis and the equator, torque and potential energy in a uniform field.

2. Gauss’s law for magnetism

   JEE Main Unit 13 lists magnetic field lines; Gauss’s law for magnetism is not named there but explains them: closed lines, no monopoles, and spotting impossible field-line diagrams.

3. Magnetisation and magnetic intensity

   Not named in JEE Main Unit 13, but the susceptibility χ defined here is what sorts the para-, dia- and ferromagnetic substances that Unit 13 lists, with examples.

4. Magnetic properties of materials

   JEE Main Unit 13: para-, dia- and ferromagnetic substances with examples, and the effect of temperature on magnetic properties; know the χ and μr ranges of NCERT Table 5.2.

Electromagnetic Induction

NCERT Class 12 Physics, Chapter 6

1. Experiments of Faraday and Henry

   JEE Main Unit 14, electromagnetic induction: the experiments behind Faraday’s law, showing what induces a current, when it reverses and what makes it larger.

2. Magnetic flux

   JEE Main Unit 14, Faraday’s law: magnetic flux Φ = BA cos θ is the quantity whose rate of change gives the emf; know the area vector, the sign and the weber.

3. Faraday’s law of induction

   JEE Main Unit 14 names Faraday’s law and induced emf and current: rate of change of flux, N turns, average emf, current and charge, and flux-time graphs.

4. Lenz’s law

   JEE Main Unit 14 names Lenz’s law: the direction of induced currents in loops, coils and moving magnets, and why it follows from conservation of energy.

5. Motional emf

   JEE Main Unit 14, induced emf and current: motional emf Blv, rods on rails with force and power, rotating rods and wheels, and the direction of the emf.

6. Eddy currents

   Listed in JEE Main Unit 14; the 2026–27 NCERT book has no section on it and names it only as a transformer loss. Know Lenz-law braking, laminated cores and uses.

7. Inductance

   JEE Main Unit 14 names self and mutual inductance: solenoid and coil formulas, M12 = M21, back emf, stored energy ½LI^² and magnetic energy density.

8. AC generator

   JEE Main Unit 14 names the AC generator: emf NBAω sin ωt, peak emf, frequency, when the emf is largest, and the energy conversion.

Alternating Current

NCERT Class 12 Physics, Chapter 7

1. AC voltage applied to a resistor

   JEE Main Unit 14 names peak and rms values of alternating current and voltage: rms from peak, average power in a resistor, and in-phase current.

2. Phasors

   JEE Main Unit 14, ac circuits: phasor diagrams are the tool for reactance, impedance and LCR problems; read lead and lag and add phasors like vectors.

3. AC voltage applied to an inductor

   JEE Main Unit 14 lists reactance, power in AC circuits and wattless current: here XL = ωL, the π/2 lag of current in an inductor, its phasor diagram and zero average power.

4. AC voltage applied to a capacitor

   JEE Main Unit 14 lists reactance, power in AC circuits and wattless current: here XC = 1/ωC, the π/2 lead of current in a capacitor, its phasor diagram and zero average power.

5. Series LCR circuit

   JEE Main Unit 14 names impedance, the LCR series circuit and resonance: Z, tan φ, phasor sums of voltages, ω0 = 1/√LC and resonance numericals.

6. Power in ac circuits

   JEE Main Unit 14 names power in AC circuits and wattless current: P = VI cos φ, cos φ = R/Z, power and wattless components, power at resonance.

7. Transformers

   JEE Main Unit 14 names the transformer: Vs/Vp = Ns/Np, the inverse current ratio, efficiency, the four energy losses and high-voltage transmission.

Electromagnetic Waves

NCERT Class 12 Physics, Chapter 8

1. Displacement current

   JEE Main Unit 15, displacement current: id = ε_₀ dΦE/dt, the Ampere–Maxwell law, the field between the plates of a charging capacitor, and dΦE/dt and dV/dt problems.

2. Electromagnetic waves

   JEE Main Unit 15, electromagnetic waves: transverse E and B, E × B along the travel, c = 1/√(μ_₀ε_₀), B_₀ = E_₀/c, writing E and B for a plane wave, and energy density.

3. Electromagnetic spectrum

   JEE Main Unit 15, the electromagnetic spectrum: the order of the bands, NCERT’s wavelength ranges, how each band is produced, its uses, and photon energies E = hν.

Ray Optics and Optical Instruments

NCERT Class 12 Physics, Chapter 9

1. Spherical mirrors

   JEE Main Unit 16, spherical mirrors: the Cartesian sign convention, f = R/2, 1/v + 1/u = 1/f, m = −v/u, image nature for concave and convex mirrors, and image speed.

2. Refraction

   JEE Main Unit 16, refraction: Snell’s law, relative refractive index, n_₁₂ = 1/n_₂₁, lateral shift through a slab, and apparent depth h_₁ = h_₂/n with the slab shift t(1 − 1/n).

3. Experiment: glass slab

   JEE Main Unit 20 lists this experiment (item 15): three travelling-microscope readings, n = (a_₃ − a_₁)/(a_₃ − a_₂), the least count, the error in n and precautions.

4. Total internal reflection

   JEE Main Unit 16, total internal reflection: sin ic = 1/n, the two conditions, the circle of light from a source under water, and the acceptance angle of an optical fibre.

5. Lenses

   JEE Main Unit 16, lenses: n_₂/v − n_₁/u = (n_₂ − n_₁)/R, the lens maker’s formula, 1/v − 1/u = 1/f, m = v/u, power and lenses in contact, including a lens in a liquid.

6. Experiment: focal lengths

   JEE Main Unit 20 lists this experiment (item 13): focal lengths of a concave mirror, a convex mirror and a convex lens by the parallax method, with f = uv/(u + v).

7. Prism

   JEE Main Unit 16, the prism: δ = i + e − A, the δ–i graph, minimum deviation and n from it, the thin prism Dm = (n − 1)A, and emergence limits from total reflection.

8. Experiment: prism i–δ graph

   JEE Main Unit 20, item 14 (the i–δ plot for a prism): tracing rays with pins, the i–δ graph, reading Dm from it, and n = sin[(A + Dm)/2]/sin(A/2).

9. Optical instruments

   JEE Main Unit 16, microscopes and telescopes: m = 1 + D/f and D/f, m = (L/fo)(D/fe), m = fo/fe with tube length fo + fe, their near-point forms, and magnification against magnifying power.

Wave Optics

NCERT Class 12 Physics, Chapter 10

1. Huygens principle

   JEE Main Unit 16, wave optics: wavefront and Huygens principle, with spherical, plane and cylindrical wavefronts and rays normal to the wavefronts.

2. Refraction and reflection by Huygens principle

   JEE Main Unit 16, laws of reflection and refraction using Huygens principle: sin i/sin r = v1/v2, n1 sin i = n2 sin r, λ1/λ2 = v1/v2, frequency unchanged.

3. Coherent and incoherent addition

   JEE Main Unit 16, coherent sources and sustained interference: φ = (2π/λ)Δ, bright at nλ and dark at (n + 1/2)λ, I = 4I0 cos^²(φ/2), and Imax/Imin for unequal sources.

4. Young’s double-slit experiment

   JEE Main Unit 16, Young’s double-slit experiment and the expression for fringe width: path difference xd/D, β = λD/d, angular width λ/d, fringes in a liquid and with white light.

5. Diffraction

   JEE Main Unit 16, diffraction due to a single slit and the width of the central maximum: minima at a sin θ = nλ, angular width 2λ/a, linear width 2λD/a, double-slit envelope.

6. Polarisation

   JEE Main Unit 16, plane-polarised light and Polaroid: half the unpolarised intensity through one polaroid, Malus’ law, and a polaroid between crossed polaroids giving (I0/4) sin^²2θ.

7. Brewster’s law

   JEE Main Unit 16 lists Brewster’s law with plane-polarised light (not in the 2026–27 NCERT text): tan i_B = μ, reflected and refracted rays at right angles.

Dual Nature of Radiation and Matter

NCERT Class 12 Physics, Chapter 11

1. Electron emission

   JEE Main Unit 17, dual nature: work function, the electron volt and conversions between eV and joules, and the three kinds of electron emission.

2. Photoelectric effect

   JEE Main Unit 17, photoelectric effect: Hertz and Lenard’s observations, photoelectrons, the threshold frequency and photosensitive metals.

3. Experimental study of the photoelectric effect

   JEE Main Unit 17, photoelectric effect: I–V graphs for different intensities and frequencies, stopping potential, saturation current and the V0–ν graph.

4. Photoelectric effect and the wave theory

   JEE Main Unit 17, photoelectric effect: the failure of the wave theory (Kmax independent of intensity, threshold frequency, no time lag) and the need for light quanta.

5. Einstein’s photoelectric equation

   JEE Main Unit 17, Einstein’s photoelectric equation: Kmax = hν − φ0, ν0 = φ0/h, V0 = (h/e)ν − φ0/e, and finding h from the slope of the V0–ν graph.

6. Photon

   JEE Main Unit 17, particle nature of light: E = hν = hc/λ, p = h/λ, photons per second N = P/E, and the X-ray cut-off from eV = hν.

7. Wave nature of matter

   JEE Main Unit 17, matter waves: de Broglie relation λ = h/p = h/mv, λ = h/√(2mK) and 1.227/√V nm for electrons, and the photon case λ = c/ν.

Atoms

NCERT Class 12 Physics, Chapter 12

1. Rutherford’s nuclear model

   JEE Main Unit 18, alpha-particle scattering and Rutherford’s model: closest approach d = 2Ze^²/(4πε_₀K), impact parameter, and the energy of a hydrogen orbit.

2. Atomic spectra

   JEE Main Unit 18, hydrogen spectrum: emission and absorption line spectra, Balmer’s formula for the visible lines, and photon energy E = hc/λ for a line.

3. Bohr model

   JEE Main Unit 18, Bohr model: L = nh/2π, r = n^²a_₀, v ∝ 1/n, E = −13.6/n^² eV, excitation and ionisation energies, and hydrogen-like ions with charge Ze.

4. Line spectra of hydrogen

   JEE Main Unit 18, hydrogen spectrum: the Rydberg formula, Lyman, Balmer and Paschen series and their limits, the number of lines, and excitation by photons and electrons.

5. De Broglie and Bohr’s postulate

   JEE Main Unit 18, Bohr model: 2πr = nλ with λ = h/mv giving L = nh/2π, the de Broglie wavelength in the nth orbit, and why the model fails for helium.

Nuclei

NCERT Class 12 Physics, Chapter 13

1. Composition of the nucleus

   JEE Main Unit 18, composition and size of the nucleus: Z, N and A, the atomic mass unit, nucleon masses, and classifying isotopes, isobars and isotones.

2. Size of the nucleus

   JEE Main Unit 18, size of the nucleus: R = R_₀A^1/3, ratios of radii and volumes, and why nuclear density, about 2.3 × 10¹⁷ kg/m^³, does not depend on A.

3. Binding energy

   JEE Main Unit 18, mass defect, binding energy per nucleon and its variation with mass number: worked binding energies from atomic masses and the 931.5 MeV per u.

4. Nuclear force

   Not named in JEE Main Unit 18, but its short range explains the nearly constant binding energy per nucleon that Unit 18 lists; also charge independence and the minimum near 0.8 fm.

5. Nuclear energy

   JEE Main Unit 18, nuclear fission and fusion: Q values from masses, energy per fission, the Coulomb barrier and fusion temperature, and the proton–proton cycle.

Semiconductor Electronics: Materials, Devices and Simple Circuits

NCERT Class 12 Physics, Chapter 14

1. Metals, insulators and semiconductors

   JEE Main Unit 19, Electronic Devices, lists semiconductors; this page covers energy bands in conductors, insulators and semiconductors, and the band gap of each.

2. Intrinsic semiconductor

   JEE Main Unit 19, Electronic Devices, lists semiconductors; this page covers electrons and holes in pure Si and Ge, and why conductivity rises with temperature.

3. Extrinsic semiconductor

   JEE Main Unit 19, Electronic Devices, lists semiconductors; this page covers n-type and p-type doping, majority and minority carriers, and ne nh = ni^².

4. p-n junction

   JEE Main Unit 19, Electronic Devices, lists semiconductors and the diode; this page covers the p-n junction, diffusion and drift, the depletion region and the barrier.

5. Semiconductor diode

   JEE Main Unit 19 lists the semiconductor diode and its I-V characteristics in forward and reverse bias; Unit 20 has the experiment on the characteristic curves of a p-n junction diode.

6. Junction diode as a rectifier

   JEE Main Unit 19 lists the diode as a rectifier; this page covers half-wave and full-wave rectifiers, output frequency and the capacitor filter.

7. Zener diode

   JEE Main Unit 19 lists the Zener diode as a voltage regulator, and Unit 20 its characteristic curve (no NCERT textbook section): breakdown, the I-V curve and regulator currents.

8. LED, photodiode and solar cell

   JEE Main Unit 19 lists the I-V characteristics of the LED, the photodiode and the solar cell (no NCERT textbook section): band gap and colour, cut-off, V_OC and I_SC.

9. Experiment: identifying components

   JEE Main Unit 20, item 18: identifying a diode, LED, resistor and capacitor from a mixed collection with a multimeter; the NCERT textbook does not describe the method.

10. Logic gates

    JEE Main Unit 19 lists logic gates, OR, AND, NOT, NAND and NOR (no NCERT textbook section): truth tables, Boolean expressions, waveforms and NAND or NOR as universal gates.

Some Basic Concepts of Chemistry

NCERT Class 11 Chemistry, Chapter 1

1. Nature of matter

   Matter and its nature for JEE Main Unit 1: the three states, homogeneous and heterogeneous mixtures, elements as atoms or molecules, and why a compound has a fixed composition.

2. Laws of chemical combination

   Conservation of mass, definite and multiple proportions, Gay-Lussac’s volumes and Avogadro’s law, with the fix-one-element method and volume problems with an excess gas.

3. Dalton’s atomic theory

   Dalton’s postulates, how they give conservation of mass and definite and multiple proportions, the gas-volume failure, and the later changes: subatomic particles, isotopes and isobars.

4. Atomic and molecular masses

   Relative atomic mass, average atomic mass from isotope data and the reverse problem of finding abundances, molecular and formula masses, with 4 solved examples.

5. Mole concept

   Mole concept for JEE Main Unit 1: n = m/M = N/N_A, counting atoms, molecules and electrons, molar volume at 1 bar against 1 atm, and 5 solved examples.

6. Percentage composition

   Mass per cent, empirical formula from percentages with the multiply-to-whole-numbers step, molecular formula from molar mass, combustion analysis and gas density.

7. Stoichiometry

   Mole-ratio calculations, limiting reagent and excess left over, mass per cent, mole fraction, molarity, molality, interconversion with density, and dilution M_₁V_₁ = M_₂V_₂.

Structure of Atom

NCERT Class 11 Chemistry, Chapter 2

1. Developments leading to Bohr’s model

   c = νλ, E = hν, Einstein’s equation and the KE–frequency graph, work function, stopping potential, photons per second, and the Rydberg formula with the five hydrogen series.

2. Bohr’s model

   Bohr’s postulates and the derivation of r_ₙ and E_ₙ, Z^²/n^² scaling for He^⁺ and Li^²⁺, ΔE, frequency and wavenumber of lines, counting spectral lines, and the limitations.

3. Towards the quantum mechanical model

   λ = h/mv, λ from kinetic energy and accelerating voltage, 2πr = nλ for Bohr orbits, Δx × Δp ≥ h/4π, and electron versus proton wavelength ratios.

4. Quantum mechanical model

   Quantum number rules, radial and angular nodes, the (n + l) rule, configurations up to Kr, unpaired electrons, exchange-energy counts, Zeff trends and ions such as Fe^²⁺.

Classification of Elements and Periodicity in Properties

NCERT Class 11 Chemistry, Chapter 3

1. Modern periodic law

   Modern periodic law, Moseley’s √ν ∝ (Z − 1), period number as outer n, IUPAC groups 1–18 against the older A/B schemes, anomalous pairs (Ar/K, Co/Ni, Te/I, Th/Pa).

2. Naming elements above 100

   Systematic names and symbols from atomic numbers (nil, un, bi … enn + ium), the elision rules, and official names of elements 101–118.

3. Configurations and the periodic table

   Period number as outer n, period lengths from orbitals filled (5s 4d 5p; 6s 4f 5d 6p), locating elements from configurations, and series start points (Sc, Y, Ce, and after Ac).

4. s, p, d and f blocks

   Block configurations (ns^¹⁻², ns^²np^¹⁻⁶, (n − 1)d^¹⁻¹⁰ns^⁰⁻², (n − 2)f^¹⁻¹⁴(n − 1)d^⁰⁻¹ns^²), placing elements from configurations, Pd and Zn group anomalies.

5. Periodic trends

   Radius, ionisation and electron gain enthalpy trends with the B/Be, O/N, F/Cl and group 13 exceptions, isoelectronic sizes, Pauling values, diagonal relationship and oxide nature.

Chemical Bonding and Molecular Structure

NCERT Class 11 Chemistry, Chapter 4

1. Kössel–Lewis approach

   Lewis structures with charge counting, formal charge F.C. = V − L − ½B (O_₃, CO, CO_₃^²⁻, NH_₄^⁺, HNO_₃), incomplete, odd-electron and expanded octets.

2. Ionic bond

   Ionisation enthalpy, electron gain enthalpy and lattice enthalpy in ionic bond formation, the NaCl energy balance (+147.1, −788 kJ mol^⁻¹) and lattice enthalpy trends.

3. Bond parameters

   Bond order against length and enthalpy, mean bond enthalpy, resonance hybrids (O_₃, CO_₃^²⁻, CO_₂), dipole moments as vector sums, % ionic character and Fajans’ rules.

4. VSEPR theory

   Electron-pair counting ½(V + M − c + a), shapes from AB_₂E to AB_₄E_₂ and AB_₂E_₃, lone-pair positions in trigonal bipyramids (SF_₄, ClF_₃, XeF_₂), bond-angle order.

5. Valence bond theory

   Counting σ and π bonds, which orbital pairs can form σ or π bonds, and why pure orbitals fail for CH_₄, NH_₃ and H_₂O.

6. Hybridisation

   Steric-number method from sp to sp^³d^², s character and bond length, axial versus equatorial bonds in PCl_₅, changes of hybridisation in adducts.

7. Molecular orbital theory

   Bond order from bonding and antibonding electron counts, why B_₂ needs the N_₂-type order, species such as O_₂^⁺, N_₂^⁺, NO and CN^⁻, nodal planes.

8. Homonuclear diatomic molecules

   Bond order, magnetism and bond length for B_₂ to Ne_₂ and ions such as O_₂^²⁻, C_₂^²⁻ and Li_₂^⁺; why C_₂ has two π bonds; comparing species with equal bond order.

9. Hydrogen bonding

   Boiling points of H_₂O, HF and NH_₃, water against HF, o- and p-nitrophenol, KHF_₂, which molecules can donate or accept hydrogen bonds.

10. Metallic bonding

    Elementary idea of metallic bonding (JEE Main Unit 3): the electron sea model, bond strength and melting points, conductivity, lustre, malleability.

Thermodynamics

NCERT Class 11 Chemistry, Chapter 5

1. Thermodynamic terms

   Sign conventions (IUPAC against physics), state against path functions, adiabatic work and the first law for any process.

2. Work, enthalpy and heat capacity

   Reversible isothermal work 2.303 nRT log(Vf/Vi), counting Δng (gases only), ΔH against ΔU, Mayer’s relation.

3. Calorimetry

   From temperature rise to ΔU and ΔH per mole, converting bomb results with ΔngRT, calibration with a standard.

4. Reaction enthalpy and Hess’s law

   ΔrH from enthalpies of formation, Hess’s law by combining equations, reference states, phase-change steps and ΔU from ΔH.

5. Combustion, bond and lattice enthalpies

   Born–Haber cycles, ΔrH from bond enthalpies (gases only), mean bond enthalpy, ΔfH from combustion data, enthalpy of solution from lattice and hydration.

6. Enthalpy of neutralization

   The Unit 20 experiment: why a strong acid and a strong base always give about −57 kJ mol^⁻¹, the calorimeter constant, weak acids and bases, and the Lab Manual’s slips.

7. Experiment: enthalpy of solution of copper sulphate

   The Unit 20 experiment: why the blue crystals cool the water and the anhydrous salt warms it, the Hess cycle for forming the hydrate, the sign of ΔsolH and the Lab Manual’s slips.

8. Spontaneity, entropy and Gibbs energy

   Sign table of ΔH and ΔS against temperature, crossover temperature ΔH/ΔS, ΔG from ΔU, entropy of surroundings −ΔH/T.

9. Gibbs energy and equilibrium

   ΔG⊖ = −2.303 RT log K, ΔG against ΔG⊖, K from ΔH⊖ and ΔS⊖ at another temperature, the 5.71 kJ per factor of 10 rule.

Equilibrium

NCERT Class 11 Chemistry, Chapter 6

1. Equilibrium in physical processes

   What stays fixed at each physical equilibrium, vapour pressure against amount of liquid, Henry’s law with partial pressures. JEE Main’s gas–gas equilibria are not in NCERT Section 6.1.

2. Dynamic equilibrium in chemical processes

   Rates equal, concentrations constant but not equal; same atoms, volume and temperature give the same mixture; isotope scrambling as proof; tube areas as rate constants in NCERT’s activity.

3. Law of chemical equilibrium and equilibrium constant

   Kc expressions, 1/K for the reverse, K^ⁿ for multiples, K_₁K_₂ for added equations, and testing a mixture against K.

4. Homogeneous equilibria

   Kp–Kc conversion with Δn of gases only, R = 0.0831 bar L mol^⁻¹ K^⁻¹, units of K, and the square-root shortcut when the Kc expression is a perfect square.

5. Heterogeneous equilibria

   K with solids and liquids omitted, Δn of gases only, NH_₄HS-type total-pressure problems, and when the solid runs out so no equilibrium is set up.

6. Applications of equilibrium constants

   Q against K, ICE tables with quadratics and the perfect-square shortcut, choosing the sensible root, and why a large K says nothing about rate.

7. Equilibrium constant, reaction quotient and Gibbs energy

   ΔG = RT ln(Q/K), ΔG against ΔG⊖, reactions with positive ΔG⊖ that still go forward, coupled reactions, and the temperature at which K = 1 from ΔH⊖ and ΔS⊖.

8. Factors affecting equilibria

   Shifts explained with Q and K, Q = K × 2^Δn on halving the volume, inert gas at constant pressure, K against temperature, and worked shifts with numbers.

9. Ionic equilibrium in solution

   Strong against weak electrolytes, the dielectric constant, concentrations from the degree of ionisation, dissociation against ionisation and H_₃O^⁺.

10. Acids, bases and salts

    Conjugate pairs and charges, strong acid with weak conjugate base, amphiprotic species, and Lewis acids without protons (BF_₃, AlCl_₃, metal ions).

11. Ionization of acids and bases

    pH of strong, weak and very dilute acids, α against dilution, conjugate Ka and Kb, common ion and buffer-like mixtures, and hydrolysis pH; NCERT slips corrected.

12. Experiment: acid-base titration with an indicator

    The Unit 20 titration: primary and secondary standards, a1M1V1 = a2M2V2, choosing phenolphthalein or methyl orange from titration curves, and the Lab Manual’s slips.

13. Buffer solutions

    pH of acidic and basic buffers, ratio for a target pH, buffer action against added acid or alkali with numbers, and dilution.

14. Solubility equilibria of sparingly soluble salts

    Ksp = x^x y^y S^x+y, comparing salts of different formula types, Qsp after mixing, common ion suppression, and solubility of salts of weak acids against pH.

Redox Reactions

NCERT Class 11 Chemistry, Chapter 7

1. Classical idea of oxidation and reduction

   Classical oxidation and reduction, oxidising and reducing agents, and the metal hydride trap (2Na + H_₂ → 2NaH) that leads to the electron-transfer view.

2. Redox reactions as electron transfer

   Electron bookkeeping in half reactions, matching electrons before adding, predicting displacement from Zn > Cu > Ag, and why Co + Ni^²⁺ gives a mixture.

3. Oxidation number

   Oxidation numbers with their exceptions, disproportionation, balancing in acid and base by both methods, KMnO_₄ and iodometric titrations, with NCERT slips corrected.

4. Redox reactions and electrode processes

   Anode and cathode signs, electron against current flow, feasibility from E⊖ values, E⊖ of the cell as cathode minus anode, and why E⊖ does not scale with coefficients.

Organic Chemistry: Some Basic Principles and Techniques

NCERT Class 11 Chemistry, Chapter 8

1. Tetravalence of carbon and shapes

   Counting σ and π bonds, hybridisation carbon by carbon, s character against bond length and electronegativity, planarity and the allene trap.

2. Structural representations

   Converting between condensed and bond-line formulas, counting hydrogens at ends, bends and branch points, and reading wedge-dash drawings.

3. Classification of organic compounds

   Placing a compound in the scheme, aromatic against alicyclic rings, the general formulas of alkanes, alkenes and alkynes, and the CH_₂ (14 u) step between homologues.

4. IUPAC nomenclature of organic compounds

   Longest chain and lowest locants, alphabetical tie-breaks, which prefixes count in alphabetising, functional group priority, and o-, m-, p- and polysubstituted benzenes.

5. Isomerism

   Identifying the type of isomerism, counting alkane structural isomers (C_₄ to C_₇), and when an alkene shows cis–trans isomerism or a carbon gives optical isomers.

6. Organic reaction mechanism

   Intermediate stability orders, +I/−I and +R/−R groups, resonance-structure rules, hyperconjugation counts, and acid-strength orders from electronic effects.

7. Purification of organic compounds

   Choosing the right method for a mixture, why steam distillation boils below 373 K, fractionating columns, and Rf values and their meaning.

8. Qualitative analysis of organic compounds

   Lassaigne’s test chemistry: Prussian blue, nitroprusside and thiocyanate, why nitric acid and acetic acid are used, and the silver halide colours and solubilities.

9. Quantitative analysis

   Percentage formulas for each element, aqueous tension and STP correction, the Kjeldahl back titration, and empirical formulas from the results.

Hydrocarbons

NCERT Class 11 Chemistry, Chapter 9

1. Classification of hydrocarbons

   General formulas for each class, and the degree of unsaturation (rings + π bonds) from a molecular formula, with the alkene–cycloalkane trap.

2. Alkanes

   Free radical chlorination mechanism, monochloro product counts, carbon counts in Wurtz, decarboxylation and Kolbe, boiling points of isomers, and Newman projections.

3. Alkenes

   Markovnikov’s rule through carbocation stability, the peroxide effect (HBr only), cis-trans conditions, Lindlar versus Na/NH_₃, and working back from ozonolysis products.

4. Alkynes

   Acidity of terminal alkynes from s character, gem dihalides by Markovnikov addition, hydration to carbonyl compounds via the enol, and the ethanoic acid to benzene chain.

5. Aromatic hydrocarbons

   Hückel counts, the arenium-ion mechanism, nitronium ion formation, Friedel–Crafts rearrangement, directing and activating groups, and choosing the order of substitution steps.

Solutions

NCERT Class 12 Chemistry, Chapter 1

1. Types of solutions

   JEE Main Unit 5, Solutions, lists the different methods of expressing concentration; this page lays the groundwork: solvent, solute and the nine types of solution.

2. Concentration of solutions

   JEE Main Unit 5 lists molality, molarity, mole fraction and percentage by volume and mass; this page covers each, with conversions using density.

3. Solubility

   JEE Main names Henry’s law in Unit 6, Equilibrium (physical processes), not in Unit 5; this page covers solubility and Henry’s law with its uses, as in NCERT Section 1.3.

4. Vapour pressure of solutions

   JEE Main Unit 5 lists the vapour pressure of solutions and Raoult’s law, and vapour pressure–composition plots; this page covers both, with vapour composition.

5. Ideal and non-ideal solutions

   JEE Main Unit 5 lists ideal and non-ideal solutions and vapour pressure–composition plots for them; this page covers both, with the heat and volume of mixing.

6. Colligative properties

   JEE Main Unit 5 lists the four colligative properties of dilute solutions and finding molecular mass from them; this page covers all four, with Kb, Kf and Π = CRT.

7. Abnormal molar masses

   JEE Main Unit 5 lists the abnormal value of molar mass and the van ’t Hoff factor and its significance; this page covers both, with degrees of association and dissociation.

Electrochemistry

NCERT Class 12 Chemistry, Chapter 2

1. Electrochemical cells

   JEE Main Unit 7, Redox Reactions and Electrochemistry, lists electrolytic and galvanic cells; this page sets out both through the Daniell cell.

2. Galvanic cells

   JEE Main Unit 7 lists types of electrodes, standard electrode potential, half-cell and cell reactions, and the emf of a galvanic cell and its measurement.

3. Nernst equation

   JEE Main Unit 7 lists the Nernst equation and its applications and the relationship between cell potential and Gibbs energy change; both are here.

4. Conductance of electrolytic solutions

   JEE Main Unit 7 lists electrolytic and metallic conduction, conductance, molar conductivities and their variation with concentration, and Kohlrausch’s law and its applications.

5. Electrolytic cells and electrolysis

   JEE Main Unit 7 lists electrolytic and galvanic cells; this page covers electrolytic cells, Faraday’s laws and which products form at each electrode.

6. Batteries

   JEE Main Unit 7 lists the dry cell and lead accumulator; this page gives their construction and reactions, with the mercury and nickel-cadmium cells.

7. Fuel cells

   JEE Main Unit 7 lists fuel cells; this page covers the hydrogen–oxygen cell, its reactions and voltage, and why fuel cells are efficient.

Chemical Kinetics

NCERT Class 12 Chemistry, Chapter 3

1. Rate of a chemical reaction

   JEE Main Unit 8, Chemical Kinetics, starts with the rate of a chemical reaction; this page covers average and instantaneous rates and their units.

2. Factors influencing rate

   JEE Main Unit 8 lists elementary and complex reactions, order and molecularity, rate law, rate constant and its units; all are here.

3. Experiment: iodide and hydrogen peroxide

   A JEE Main Unit 20 experiment: the iodine clock, timing the blue colour with thiosulphate and starch, and how the rate depends on iodide concentration.

4. Integrated rate equations

   JEE Main Unit 8 lists the differential and integral forms of zero and first order reactions, their characteristics and half-lives; all are here.

5. Temperature dependence of rate

   JEE Main Unit 8 lists temperature and catalysts as factors, Arrhenius theory, and activation energy and its calculation; all are here.

6. Collision theory

   JEE Main Unit 8 lists collision theory of bimolecular gaseous reactions (no derivation): collision frequency, threshold energy, orientation and the steric factor are here.

The p-Block Elements

1. Group 13 elements

   JEE Main Unit 10 lists groups 13 to 18 of the p-block; here group 13: configuration, oxidation states and the inert pair effect, trends, properties and boron’s anomaly.

2. Group 14 elements

   JEE Main Unit 10 lists groups 13 to 18 of the p-block; here group 14: configuration, +4 and +2 states, trends, oxides, halides and carbon’s anomaly.

3. Group 15 elements

   JEE Main Unit 10 lists groups 13 to 18 of the p-block; here group 15: configuration, −3, +3 and +5 states, trends, hydrides, oxides, halides and nitrogen’s anomaly.

4. Group 16 elements

   JEE Main Unit 10 lists groups 13 to 18 of the p-block; here group 16: configuration, −2 to +6 states, trends, hydrides, oxides, halides and oxygen’s anomaly.

5. Group 17 elements

   JEE Main Unit 10 lists groups 13 to 18 of the p-block; here group 17: configuration, oxidation states, trends, oxidising power, hydrides, oxides, interhalogens and fluorine’s anomaly.

6. Group 18 elements

   JEE Main Unit 10 lists groups 13 to 18 of the p-block; here group 18: configuration, trends, low reactivity, the xenon fluorides and oxides, and helium.

The d- and f-Block Elements

NCERT Class 12 Chemistry, Chapter 4

1. Position in the periodic table

   JEE Main Unit 11 opens with a general introduction to the transition elements: their place in the table, the four series and the f-block are here.

2. Electronic configurations

   JEE Main Unit 11 lists the electronic configuration of the transition elements: the general pattern, the Cr and Cu exceptions and ion configurations are here.

3. Occurrence

   JEE Main Unit 11 lists occurrence with the transition elements; here: minerals and ores, native metals, and the ores of iron, copper, chromium and manganese with formulas.

4. General properties

   JEE Main Unit 11 lists these first-row trends: physical properties, ionisation enthalpy, oxidation states, radii, colour, catalysis, magnetism, complexes, interstitials and alloys.

5. Important compounds

   JEE Main Unit 11 lists the preparation, properties and uses of K_₂Cr_₂O_₇ and KMnO_₄; both are here with their half-reactions, plus the oxides of the 3d metals.

6. Experiment: KMnO₄ titrations

   A JEE Main Unit 20 experiment: KMnO_₄ standardised against oxalic acid (hot) and Mohr’s salt, as in the NCERT Laboratory Manual, Unit 6, with the calculations.

7. Lanthanoids

   JEE Main Unit 11 lists the lanthanoids’ electronic configuration, oxidation states and lanthanoid contraction; all three are here, with Ce^⁴⁺, Eu^²⁺ and Yb^²⁺.

8. Actinoids

   JEE Main Unit 11 lists the actinoids’ electronic configuration and oxidation states; both are here, with the actinoid contraction and the comparison with lanthanoids.

Coordination Compounds

NCERT Class 12 Chemistry, Chapter 5

1. Werner’s theory

   JEE Main Unit 12 lists Werner’s theory; this page covers primary and secondary valences, the AgCl and conductivity evidence, the postulates and double salts against complexes.

2. Experiment: double salts

   A JEE Main Unit 20 experiment: Mohr’s salt and potash alum prepared as double salts by crystallising equimolar mixtures, as in the NCERT Laboratory Manual, Unit 9.

3. Ligands and other terms

   JEE Main Unit 12 lists ligands, coordination number, denticity and chelation; this page adds ambidentate ligands, polyhedra, oxidation number and homoleptic complexes.

4. Nomenclature

   JEE Main Unit 12 lists IUPAC nomenclature of mononuclear coordination compounds; this page covers formula order, ligand names, -ate anions and oxidation states.

5. Isomerism

   JEE Main Unit 12 lists isomerism of coordination compounds; this page covers cis, trans, fac and mer, optical isomers, and the four structural types.

6. Bonding: VBT and CFT

   JEE Main Unit 12 lists bonding by the valence bond approach and basic ideas of crystal field theory, colour and magnetic properties; this page covers each.

7. Metal carbonyls

   Not named in JEE Main Unit 12, but it applies the bonding that Unit 12 lists to CO complexes: homoleptic carbonyls, their shapes, the zero oxidation state and synergic σ and π bonding.

8. Importance and applications

   JEE Main Unit 12 lists the importance of coordination compounds in qualitative analysis, extraction of metals and biological systems; this page covers each.

Haloalkanes and Haloarenes

NCERT Class 12 Chemistry, Chapter 6

1. Classification

   JEE Main Unit 14 lists classifying compounds “containing halogens”; Unit 16’s methods and reactions then turn on these classes: 1°, 2°, 3°, allylic, benzylic, vinylic and aryl halides.

2. Nomenclature

   JEE Main Unit 14 lists “Nomenclature (Trivial and IUPAC)”; for halogen compounds: common and IUPAC names, haloalkenes, o-, m-, p-, gem and vic dihalides, and the C_₅H_₁₁Br isomers.

3. Nature of C–X bond

   JEE Main Unit 16 lists the nature of the C–X bond; this page covers its polarity and the bond length, enthalpy and dipole trends from F to I.

4. Preparation of haloalkanes

   JEE Main Unit 16 lists general methods of preparation; for haloalkanes: alcohols with HX, PX_₃, PCl_₅ or SOCl_₂, alkanes, alkenes, Finkelstein and Swarts.

5. Preparation of haloarenes

   JEE Main Unit 16 lists general methods of preparation; for haloarenes: halogenation with Fe or FeCl_₃, why iodine and fluorine differ, and Sandmeyer’s reaction.

6. Physical properties

   JEE Main Unit 16 lists the properties of halogen compounds; this page: boiling point orders, branching, para isomers’ melting points, density and solubility.

7. Chemical reactions

   JEE Main Unit 16 lists reactions and “mechanisms of substitution reactions”; this page covers SN1 and SN2, inversion and racemisation, elimination, Grignard, Wurtz and haloarenes.

8. Polyhalogen compounds

   JEE Main Unit 16 lists the uses and environmental effects of chloroform, iodoform, freons and DDT; this page covers them with CH_₂Cl_₂ and CCl_₄.

Alcohols, Phenols and Ethers

NCERT Class 12 Chemistry, Chapter 7

1. Classification

   JEE Main Unit 17 covers alcohols, phenols and ethers; this page sorts them: mono- to polyhydric, 1°, 2°, 3°, allylic, benzylic, phenols and ethers.

2. Nomenclature

   JEE Main Unit 14 lists nomenclature and Unit 17 alcohols, phenols and ethers; here: -ol names, diols, cresols, benzenediols, alkoxy names of ethers.

3. Structures of functional groups

   JEE Main Unit 17 covers alcohols, phenols and ethers; this page explains their structures: bond lengths, angles at oxygen and phenol’s shorter C–O bond.

4. Alcohols and phenols

   JEE Main Unit 17 lists identifying 1°, 2° and 3° alcohols, the dehydration mechanism, and phenol’s acidity, halogenation, nitration, sulphonation and Reimer–Tiemann reaction.

5. Experiment: tests for –OH groups

   A JEE Main Unit 20 experiment; the NCERT book does not describe the method.

6. Commercially important alcohols

   JEE Main Unit 17 asks for the preparation, properties and uses of alcohols; this page covers the two commercial ones, methanol and ethanol.

7. Ethers

   JEE Main Unit 17 lists the preparation, properties and reactions of ethers and names their structure; this page covers Williamson synthesis and cleavage by HI.

Aldehydes, Ketones and Carboxylic Acids

NCERT Class 12 Chemistry, Chapter 8

1. Carbonyl group: names and structure

   JEE Main Unit 17 lists the nature of the carbonyl group; this page covers naming aldehydes and ketones and the planar, polar structure of C=O.

2. Preparation of aldehydes and ketones

   JEE Main Unit 17 lists general methods of preparation; this page covers the routes to aldehydes and ketones, including the named reactions.

3. Physical properties of aldehydes and ketones

   JEE Main Unit 17 lists properties of aldehydes and ketones; this page covers their boiling points, solubility and the reasons behind them.

4. Reactions of aldehydes and ketones

   JEE Main Unit 17 lists nucleophilic addition, relative reactivity, HCN and NH_₃ derivatives, Wolff–Kishner and Clemmensen, α-hydrogen acidity, aldol, Cannizzaro and haloform.

5. Experiment: tests for aldehydes and ketones

   A JEE Main Unit 20 experiment; the NCERT book has the theory (NCERT 8.4) but does not describe the method.

6. Experiment: preparation of iodoform

   A JEE Main Unit 20 experiment; the NCERT book has the theory (NCERT 8.4) but does not describe the method.

7. Uses of aldehydes and ketones

   JEE Main Unit 17 lists the uses of oxygen compounds; this page covers the uses of formaldehyde, acetaldehyde, benzaldehyde, acetone and others.

8. Carboxyl group: names and structure

   JEE Main Unit 17 covers carboxylic acids; this page covers their common and IUPAC names and the structure of the carboxyl group.

9. Preparation of carboxylic acids

   JEE Main Unit 17 lists general methods of preparation; this page covers the routes to carboxylic acids, including the Grignard and nitrile routes that add a carbon.

10. Physical properties of carboxylic acids

    JEE Main Unit 17 lists properties of carboxylic acids; this page covers their hydrogen-bonded dimers, boiling points and solubility, with NCERT data.

11. Chemical reactions of carboxylic acids

    JEE Main Unit 17 lists the acidic strength of carboxylic acids and the factors affecting it; this page covers both with NCERT’s pKa data, and the main reactions.

12. Experiment: tests for the carboxyl group

    A JEE Main Unit 20 experiment; the NCERT book has the theory (NCERT 8.9.1 and 8.9.2) but does not describe the method.

13. Uses of carboxylic acids

    JEE Main Unit 17 lists the uses of oxygen compounds; this page covers the uses of carboxylic acids, from vinegar and nylon-6,6 to food preservatives and soaps.

Amines

NCERT Class 12 Chemistry, Chapter 9

1. Structure of amines

   JEE Main Unit 18 lists the structure of amines; this page covers the sp^³ nitrogen, its lone pair and the pyramidal shape, with a note on the measured angle.

2. Classification of amines

   JEE Main Unit 18 lists the classification of amines; this page covers 1°, 2° and 3° amines, simple and mixed amines, and how they differ from alcohol classes.

3. Nomenclature of amines

   JEE Main Unit 18 lists the nomenclature of amines; this page covers common and IUPAC names, N-locants and the isomers of C_₄H_₁₁N.

4. Preparation of amines

   JEE Main Unit 18 lists general methods of preparation of amines; this page covers all six NCERT routes, with the carbon count of each and NCERT’s slips.

5. Physical properties of amines

   JEE Main Unit 18 lists properties of amines; this page covers their solubility and boiling points, primary > secondary > tertiary, with NCERT’s data.

6. Chemical reactions of amines

   JEE Main Unit 18 lists the basic character and identification of 1°, 2° and 3° amines; this page covers both, with NCERT’s pKb data and the class tests.

7. Experiment: tests for the amino group

   A JEE Main Unit 20 experiment; the NCERT book has the theory (NCERT 9.6 and 9.9) but does not describe the method.

8. Experiment: preparation of acetanilide and p-nitroacetanilide

   A JEE Main Unit 20 experiment; the NCERT book has the theory (NCERT 9.6) but does not describe the method.

9. Preparation of diazonium salts

   JEE Main Unit 18 lists the importance of diazonium salts in synthesis; this page covers how they are made, by diazotisation at 273–278 K.

10. Physical properties of diazonium salts

    JEE Main Unit 18 lists diazonium salts; this page covers their appearance, solubility and stability, which decide how they are handled.

11. Chemical reactions of diazonium salts

    JEE Main Unit 18 lists the importance of diazonium salts in synthesis; this page covers every NCERT replacement reaction and coupling.

12. Experiment: preparation of aniline yellow

    A JEE Main Unit 20 experiment; the NCERT book has the theory (NCERT 9.7 and 9.9) but does not describe the method.

13. Diazonium salts in the synthesis of aromatic compounds

    JEE Main Unit 18 lists the importance of diazonium salts in synthetic organic chemistry; this page covers NCERT Section 9.10 and Example 9.5.

Biomolecules

NCERT Class 12 Chemistry, Chapter 10

1. Carbohydrates

   JEE Main Unit 19 lists classification, aldoses and ketoses, glucose and fructose, and sucrose, lactose and maltose; all are on this page.

2. Proteins

   JEE Main Unit 19 lists α-amino acids, the peptide bond, polypeptides, the four levels of protein structure and denaturation; all are on this page.

3. Enzymes

   JEE Main Unit 19 lists enzymes under proteins; this page covers NCERT Section 10.3 and the mechanism of enzyme action.

4. Vitamins

   JEE Main Unit 19 lists vitamins: classification and functions; this page covers both, with NCERT Table 10.3.

5. Nucleic acids

   JEE Main Unit 19 lists the chemical constitution of DNA and RNA and the biological functions of nucleic acids; both are on this page.

6. Hormones

   JEE Main Unit 19 lists hormones (general introduction); this page covers it as in NCERT Section 10.6.

Practical Chemistry

1. Experiment: preparation of lyophilic and lyophobic sols

   A JEE Main Unit 20 experiment; the NCERT book does not describe the method.

2. Experiment: salt analysis, anions

   A JEE Main Unit 20 experiment; the NCERT book does not describe the method.

3. Experiment: salt analysis, cations

   A JEE Main Unit 20 experiment; the NCERT book has the theory (NCERT 5.7 and Class 11 NCERT 6.13) but does not describe the method.

Sets

NCERT Class 11 Mathematics, Chapter 1

1. Sets and their representations

   Roster and set-builder forms, converting between them, and reading conditions such as −3 ≤ x < 7 exactly, including which kind of number x may be.

2. The empty set

   Recognising empty sets from their conditions, including the type of number allowed, and why ϕ, {0} and {ϕ} are different sets.

3. Finite and infinite sets

   Deciding finiteness from a set-builder condition, counting distinct elements with n(S), and why the empty set is finite.

4. Equal sets

   Testing equality both ways, equal against equivalent sets, and solving for an unknown element in {2, x} = {2, 5}.

5. Subsets

   ∈ against ⊂ traps, counting subsets (2 to the power n), ⊂ against ⊆ conventions, and open, closed and half-open intervals with their lengths.

6. Power set

   The power set P(A), why it has 2 to the power n elements, counts of proper and non-empty proper subsets, and P(ϕ) = {ϕ}, as in JEE Main Unit 1.

7. Universal set

   Choosing U so that every set is a subset, and why the complement of a set changes when U changes, as in N against Z.

8. Venn diagrams

   Choosing between nested, separate and overlapping circles, and placing elements region by region, the base for union and intersection problems.

9. Operations on sets

   The laws of union and intersection, both distributive laws, A − B against B − A, and counting with n(A ∪ B) = n(A) + n(B) − n(A ∩ B).

10. Complement of a set

    Complements relative to U, De Morgan’s laws in both forms, and counting n(A′ ∩ B′) from n(U) − n(A ∪ B).

Relations and Functions

NCERT Class 11 Mathematics, Chapter 2

1. Cartesian product

   n(A × B) = pq, A × B against B × A, × over ∪, ∩ and −, and the n(A ∩ B) squared pairs that A × B and B × A share.

2. Relations

   Counting relations (2 to the power pq, and 2 to the power n^² on a set), range against codomain, and the reflexive, symmetric and transitive tests behind equivalence relations.

3. Functions

   Finding domains and ranges from formulas, the vertical line test, the greatest integer and signum functions, and the traps with [x] and cancelling factors.

Trigonometric Functions

NCERT Class 11 Mathematics, Chapter 3

1. Angles

   Converting between degrees and radians, arc length l = rθ with θ in radians, and angles beyond 360° or below 0°.

2. Trigonometric functions

   Unit-circle definitions, periodicity, signs by quadrant, domains and ranges of all six functions, and finding the other five functions from one.

3. Sum and difference formulas

   cos(x ± y), sin(x ± y) and tan(x ± y), double and triple angle formulas, half-angle signs, and sum-to-product and product-to-sum identities.

Complex Numbers and Quadratic Equations

NCERT Class 11 Mathematics, Chapter 4

1. Complex numbers

   i^² = −1, Re z and Im z, equating real and imaginary parts to find unknowns, and why complex numbers cannot be ordered.

2. Algebra of complex numbers

   The product and quotient in a + ib form, i to the power n by remainders, the √a √b trap for negative a and b, and expanding with identities.

3. Modulus and conjugate

   |z| and the conjugate, z times its conjugate equals |z|^², inverses and division by conjugates, and the modulus and conjugate rules for products and quotients.

4. The Argand plane

   z as the point (x, y), |z| and |z_₁ − z_₂| as distances, circles |z − z_₀| = r, and the conjugate as a reflection in the real axis.

5. Argument and polar form

   Modulus and argument, the principal argument in (−π, π] by quadrant, polar form r(cos θ + i sin θ), and multiplying and dividing in polar form.

6. Quadratic equations

   Solving quadratics with complex roots, the discriminant and the nature of roots, sum and product of roots, and forming an equation from given roots.

Permutations and Combinations

NCERT Class 11 Mathematics, Chapter 6

1. Fundamental principle of counting

   The multiplication principle, filling restricted places first, digits with and without repetition, and when to add cases rather than multiply.

2. Permutations

   nPr = n!/(n − r)!, arrangements with alike objects, block and gap methods, digits with 0, dictionary order, and rejecting impossible roots.

3. Combinations

   nCr = n!/(r!(n − r)!), nCr = nC(n−r), Pascal’s rule, choose-then-arrange problems, and at least / at most counting by cases or complements.

Binomial Theorem

NCERT Class 11 Mathematics, Chapter 7

1. Binomial theorem

   (a + b)^ⁿ with binomial coefficients ^ⁿC_ᵣ, n + 1 terms, (x − y)^ⁿ and (1 + x)^ⁿ, sums of coefficients 2^ⁿ and 0, approximations, and remainders such as 6^ⁿ − 5n on division by 25.

2. General and middle terms

   The general term T_ᵣ₊₁ = ^ⁿC_ᵣ a^ⁿ⁻ʳ b^ʳ and middle terms for even and odd n (JEE Main Unit 5), with simple applications: terms independent of x, given coefficients and ratios of neighbouring terms.

Sequences and Series

NCERT Class 11 Mathematics, Chapter 8

1. Sequences

   The nth term a_ₙ as a function of n, finding terms from a formula or a recurrence relation, and signs from (−1)^ⁿ⁻¹.

2. Series

   Sigma notation: reading the index and limits, counting n − m + 1 terms, splitting sums and taking constants out, and a constant added n times.

3. Arithmetic progression

   The A.P., its nth term, the sum of n terms, n(n + 1)/2, arithmetic means and inserting A.M.s (JEE Main 2027, Unit 6), with NCERT Miscellaneous Exercise Q4, Q12–Q14, Q16 and Q18 solved.

4. Geometric progression

   a_ₙ = ar^ⁿ⁻¹, S_ₙ for r ≠ 1 and r = 1, the sum to infinity a/(1 − r), three terms as a/r, a, ar, both signs of r, sums like 7 + 77 + 777, and inserting G.M.s.

5. A.M. and G.M.

   A ≥ G with equality only when a = b, the two numbers from A and G as roots of x^² − 2Ax + G^² = 0, and least values such as x + 1/x ≥ 2 for x > 0.

Straight Lines

NCERT Class 11 Mathematics, Chapter 9

1. Coordinate geometry basics

   Distance and section formulas (JEE Main 2027, Unit 10), external division, finding the ratio as k : 1, areas of triangles and polygons, and the collinearity condition.

2. Locus

   Locus and its equation (JEE Main 2027, Unit 10): the P(h, k) method, standard loci, parameters such as a sliding rod, and NCERT Miscellaneous Example 16 and Miscellaneous Exercise Q19.

3. Slope of a line

   Slope m = tan θ, m_₁ = m_₂ and m_₁m_₂ = −1, tan θ = |(m_₂ − m_₁)/(1 + m_₁m_₂)| with its two answers, vertical lines and collinearity by slopes (JEE Main 2027, Unit 10).

4. Forms of the equation of a line

   Every form of a line (JEE Main 2027, Unit 10), reading slope and intercepts from Ax + By + C = 0, parallel and perpendicular lines through a point, and the equal-intercepts trap.

5. Distance of a point from a line

   The distance of a point from a line (JEE Main 2027, Unit 10), the distance between parallel lines, the side of a line, the foot of the perpendicular and the image of a point.

6. Intersection of lines and triangle centres

   Intersecting and concurrent lines, L_₁ + λL_₂ = 0, centroid, circumcentre and orthocentre (JEE Main 2027, Unit 10), with NCERT Miscellaneous Examples 11, 12, 14, 15 and 11 exercise questions.

Conic Sections

NCERT Class 11 Mathematics, Chapter 10

1. Sections of a cone

   Which conic a plane cuts from the angles α and β, the degenerate sections through the vertex, and the eccentricity e = cos β / cos α (beyond NCERT).

2. Circle

   Standard and general forms x^² + y^² + 2gx + 2fy + c = 0 (JEE Main 2027, Unit 10), centre (−g, −f), radius √(g^² + f^² − c), point circles, and inside or outside.

3. Circle: diameter form and a line

   The circle on a given diameter and where a line meets a circle centred at the origin (JEE Main 2027, Unit 10): the discriminant test, tangents y = mx ± r√(1 + m^²) and chord length.

4. Parabola

   Standard parabolas (JEE Main 2027, Unit 10): reading focus, directrix and axis from an equation, latus rectum 4a, focal distance x + a and the point (at^², 2at).

5. Ellipse

   Standard ellipses (JEE Main 2027, Unit 10): which axis is major, a^² = b^² + c^², e = c/a, latus rectum 2b^²/a, focal distances a ± ex (a ± ey), and ellipses through two points.

6. Hyperbola

   Standard hyperbolas (JEE Main 2027, Unit 10): the positive term gives the transverse axis, c^² = a^² + b^², e > 1, latus rectum 2b^²/a; beyond NCERT, the asymptotes y = ±(b/a)x of x^²/a^² − y^²/b^² = 1.

Introduction to Three Dimensional Geometry

NCERT Class 11 Mathematics, Chapter 11

1. Coordinate axes and planes

   The start of JEE Main 2027 Unit 11: coordinate planes z = 0, x = 0 and y = 0, planes parallel to them, octant signs and mirror images in a coordinate plane.

2. Coordinates of a point in space

   Coordinates of a point in space (JEE Main 2027, Unit 11): the forms (x, 0, 0) and (0, y, z), feet of perpendiculars, mirror images and distances |x|, |y|, |z| from the planes.

3. Distance between two points

   Distance between two points in space (JEE Main 2027, Unit 11): PQ^² as a sum of three squares, distance from the origin and from an axis, loci, and the parallelogram test.

4. Section formula in 3D

   Section formula in space (JEE Main 2027, Unit 11): internal and external division, mid-point, centroid of a triangle and a tetrahedron, and the ratio in which a coordinate plane cuts a segment.

Limits and Derivatives

NCERT Class 11 Mathematics, Chapter 12

1. Intuitive idea of derivatives

   The idea behind every derivative in JEE Main 2027, Unit 7: (f(a + h) − f(a))/h as h → 0, read as an instantaneous rate of change and as the slope of a tangent.

2. Limits

   Limits for JEE Main 2027, Unit 7: one-sided limits, 0/0 forms by factorising, (x^ⁿ − a^ⁿ)/(x − a) → na^ⁿ⁻¹ for rational n, a > 0, and piecewise functions with a limit.

3. Limits of trigonometric functions

   Standard trigonometric limits for JEE Main 2027, Unit 7: (sin x)/x → 1, (1 − cos x)/x^² → 1/2, (tan x)/x → 1, radians only, and the sandwich theorem.

4. Derivatives

   Derivatives for JEE Main 2027, Unit 7: first principle, the product and quotient rules, standard derivatives of x^ⁿ and the six trigonometric functions, and where a derivative fails.

Statistics

NCERT Class 11 Mathematics, Chapter 13

1. Mean, median and mode

   Mean and median of raw and grouped data for JEE Main 2027, Unit 13, with the effect of shifting or scaling the data on the mean.

2. Mode

   Mode of ungrouped and grouped data for JEE Main 2027, Unit 13: the modal class, l + ((f_₁ − f_₀)/(2f_₁ − f_₀ − f_₂))h with its histogram construction, and the rule mode ≈ 3 median − 2 mean.

3. Measures of dispersion

   Measures of dispersion for JEE Main 2027, Unit 13: what each uses, why deviations cancel, and how shifting or scaling changes the range, MD, SD and variance (QD is not named in the syllabus).

4. Range

   The range comes under ‘measures of dispersion’ in JEE Main 2027, Unit 13: lists, frequency tables, grouped data (beyond NCERT), shifting or scaling, and its weakness with extreme values.

5. Mean deviation

   Mean deviation for JEE Main 2027, Unit 13: grouped and ungrouped data, the sum of distances least about the median, and the effect of shifting or scaling.

6. Variance and SD

   Variance and standard deviation for JEE Main 2027, Unit 13: mean of squares minus square of mean, step deviations, shifting and scaling, and correcting data.

Probability

NCERT Class 11 Mathematics, Chapter 14

1. Events

   Events underpin JEE Main 2027, Unit 13 (“Probability of an event”): sample spaces, ‘or’, ‘and’, ‘not’ and ‘neither’ as sets, and mutually exclusive and exhaustive events.

2. Axiomatic probability

   Probability of an event for JEE Main 2027, Unit 13: the axioms, counting equally likely outcomes, the addition rule for two and three events, and complements.

Relations and Functions (Class 12)

NCERT Class 12 Mathematics, Chapter 1

1. Types of relations

   JEE Main 2027 Unit 1: proving and disproving the three properties, equivalence classes as a partition, and counting reflexive, symmetric and equivalence relations.

2. Types of functions

   JEE Main 2027 Unit 1: one-one, into and onto functions, the horizontal line test, the codomain trap, and counting functions, one-one functions and bijections.

3. Composition and inverse

   JEE Main 2027 Units 1 and 7: composition of functions (Unit 1), inverse functions (Unit 7), domains of real composites, inverses of linear and rational functions, and (gof)^⁻¹.

Inverse Trigonometric Functions

NCERT Class 12 Mathematics, Chapter 2

1. Principal value branches

   JEE Main 2027 Units 7 and 14: principal ranges of all six inverse functions, negative inputs for cos^⁻¹, sec^⁻¹ and cot^⁻¹, and domains such as |x| ≥ 1 for sec^⁻¹.

2. Properties

   JEE Main 2027 Units 7 and 14: sin^⁻¹(sin x) outside the principal range, substitutions, sin^⁻¹ x + cos^⁻¹ x = π/2 and the tan^⁻¹ sum formula with its xy < 1 condition.

Matrices

NCERT Class 12 Mathematics, Chapter 3

1. Matrix and order

   JEE Main 2027 Unit 3: reading a_ᵢⱼ by row and column, counting the possible orders of a matrix, and constructing matrices from formulas for their elements.

2. Types of matrices

   JEE Main 2027 Unit 3: types of matrices, how identity, scalar and diagonal matrices nest, and finding unknowns from equal matrices, including equations that clash.

3. Operations on matrices

   JEE Main 2027 Unit 3: the order rule for AB, row-by-column products, AB ≠ BA, zero products of non-zero matrices, and matrix polynomials such as A^² = kA − 2I.

4. Transpose of a matrix

   JEE Main 2027 Unit 3: transposes, the reversal rule (AB)′ = B′A′ and its extension to three factors, and which products of transposes are defined.

5. Symmetric and skew symmetric

   JEE Main 2027 Unit 3: testing symmetry, the unique split into symmetric and skew parts, when AB is symmetric, and counting free entries of each type.

6. Invertible matrices

   JEE Main 2027 Unit 3 lists the inverse of a square matrix; this page: inverses from AB = I or from a matrix equation such as A^² − 6A + 8I = O, (AB)^⁻¹ = B^⁻¹A^⁻¹ and A′A = I.

Determinants

NCERT Class 12 Mathematics, Chapter 4

1. Determinant

   JEE Main 2027 Unit 3: evaluating 2 × 2 and 3 × 3 determinants quickly, the sign pattern, |kA| = k^ⁿ|A|, and the row and column properties used to simplify.

2. Area of a triangle

   JEE Main 2027 Unit 3: area of triangles using determinants, collinearity, finding an unknown vertex with both signs, and the line through two points.

3. Minors and cofactors

   JEE Main 2027 Unit 3: minors and cofactors, expansion by cofactors along any row or column, and the zero sum used for the adjoint.

4. Adjoint and inverse

   JEE Main 2027 Unit 3: adjoint and inverse of a square matrix, the test |A| ≠ 0, and the determinant rules for products, adjoints and inverses.

5. Solving linear systems

   JEE Main 2027 Unit 3: solving linear equations with matrices, testing consistency, and the singular case where (adj A)B = O can still give no solution.

Continuity and Differentiability

NCERT Class 12 Mathematics, Chapter 5

1. Continuity

   JEE Main 2027 Unit 7: continuity of functions, one-sided limits at joins, finding constants for continuity, and continuity of sums, products, quotients and composites.

2. Differentiability

   JEE Main 2027 Unit 7: differentiability, the chain rule, implicit differentiation and derivatives of inverse trigonometric functions, with the interval traps of substitutions.

3. Exponential and log functions

   JEE Main 2027 Unit 7: logarithmic and exponential functions and their derivatives, with the chain rule and the domains where log is defined.

4. Logarithmic differentiation

   JEE Main 2027 Unit 7: logarithmic differentiation of products, quotients and variable powers such as x^ˣ, and implicit equations like y^ˣ = x^ʸ.

5. Parametric derivatives

   Not named in JEE Main 2027 Unit 7, which lists composite functions: parametric dy/dx is the chain rule, used on circles, parabolas and cycloids.

6. Second order derivative

   JEE Main 2027 Unit 7 lists derivatives of order up to two. This page: relations such as y_₂ + y = 0, and second derivatives of inverse trigonometric and parametric functions.

Application of Derivatives

NCERT Class 12 Mathematics, Chapter 6

1. Rate of change

   JEE Main 2027 Unit 7: rate of change of quantities, related rates for areas, volumes and ladders, and units and signs of rates.

2. Increasing and decreasing

   JEE Main 2027 Unit 7 lists monotonic (increasing and decreasing) functions. This page: the sign of f′(x), intervals of increase and decrease, and parameter questions such as x^² + ax + 1 on [1, 2].

3. Maxima and minima

   JEE Main 2027 Unit 7: maxima and minima of functions of one variable, derivative tests, closed-interval extrema and optimisation problems.

Integrals

NCERT Class 12 Mathematics, Chapter 7

1. Anti derivatives

   JEE Main 2027 Unit 8: the integral as an anti derivative and the fundamental integrals of algebraic, trigonometric, exponential and logarithmic functions.

2. Substitution and identities

   JEE Main 2027 Unit 8 names integration by substitution and using trigonometric identities: choosing t, the log integrals of tan x and sec x, and interval traps.

3. Particular integrals

   JEE Main 2027 Unit 8 lists simple integrals of the types 1/(x^² + a^²), 1/(a^² − x^²), 1/√(x^² ± a^²), 1/√(a^² − x^²) and (px + q)/(ax^² + bx + c), with or without a root.

4. Partial fractions

   JEE Main 2027 Unit 8 names integration by partial fractions: dividing improper fractions first, repeated factors, and quadratic factors with a linear top.

5. Integration by parts

   JEE Main 2027 Unit 8 names integration by parts: choosing the first function, integrals that reappear, the e^ˣ[f + f′] form and √(x^² ± a^²).

6. Definite integral

   JEE Main 2027 Unit 8 lists the evaluation of definite integrals and areas bounded by simple curves. This page: F(b) − F(a), the limits and signed area.

7. Fundamental theorem

   JEE Main 2027 Unit 8 names the fundamental theorem of calculus. This page: F(b) − F(a), continuity on [a, b], and differentiating an integral with variable limits.

8. Substitution with limits

   JEE Main 2027 Unit 8 lists the evaluation of definite integrals. This page: substitution with new limits, limits that swap, and substitutions that break.

9. Properties of definite integrals

   JEE Main 2027 Unit 8 names the properties of definite integrals. This page: a + b − x, adding the integral to itself, even and odd functions, |…| integrands.

Application of Integrals

NCERT Class 12 Mathematics, Chapter 8

1. Area under curves

   JEE Main 2027 Unit 8 lists areas of regions bounded by simple curves in standard forms. This page: vertical and horizontal strips, parts below the axis, circles and ellipses.

2. Area between curves

   JEE Main 2027 Unit 8 lists areas of regions bounded by simple curves. The 2026–27 NCERT book has no section on the area between two curves. Top minus bottom, limits from intersections.

Differential Equations

NCERT Class 12 Mathematics, Chapter 9

1. Order and degree

   JEE Main 2027 Unit 9 lists ordinary differential equations, their order and degree. This page: when the degree is not defined, and radicals cleared first.

2. General and particular solutions

   JEE Main 2027 Unit 9 lists the solution of differential equations; each method there ends in a general solution, and a given condition fixes a particular one.

3. Solving first order equations

   JEE Main 2027 Unit 9 lists separation of variables and homogeneous and linear equations dy/dx + p(x)y = q(x). This page: all three, with integrating-factor traps.

Vector Algebra

NCERT Class 12 Mathematics, Chapter 10

1. Vectors: basic concepts

   JEE Main 2027 Unit 12 lists vectors and scalars. This page: magnitude, position vectors, and direction cosines with l^² + m^² + n^² = 1.

2. Types of vectors

   JEE Main 2027 Unit 12 lists vectors and scalars. This page: zero, unit, coinitial, collinear, equal and negative vectors.

3. Addition of vectors

   JEE Main 2027 Unit 12 lists the addition of vectors. This page: triangle and parallelogram laws, a − b, and polygon sums that close to zero.

4. Scalar multiple and components

   JEE Main 2027 Unit 12 lists components of a vector in two and three dimensions. This page: unit vectors, collinearity and the section formula.

5. Dot and cross products

   JEE Main 2027 Unit 12 lists scalar and vector products. This page: angles, projections, perpendicular unit vectors and areas of triangles and parallelograms.

Three Dimensional Geometry

NCERT Class 12 Mathematics, Chapter 11

1. Direction cosines and ratios

   JEE Main 2027 Unit 11 lists direction ratios and direction cosines. This page: l^² + m^² + n^² = 1, cosines from ratios, and collinear points.

2. Equation of a line in space

   JEE Main 2027 Unit 11 lists the equation of a line. This page: vector, parametric and Cartesian forms, and zero denominators.

3. Angle between two lines

   JEE Main 2027 Unit 11 lists the angle between two intersecting lines. This page: the acute angle, perpendicular and parallel conditions, and parameter questions.

4. Shortest distance between lines

   JEE Main 2027 Unit 11 lists skew lines and the shortest distance between them. This page: both formulas, and the test for lines that intersect.

5. Line of shortest distance

   JEE Main 2027 Unit 11 lists skew lines, the shortest distance between them and its equation. NCERT 2026–27 stops at the distance; this page finds the feet and the line.

Probability (Class 12)

NCERT Class 12 Mathematics, Chapter 13

1. Conditional probability

   JEE Main 2027 Unit 13 lists the multiplication theorem of probability and Bayes’ theorem, both built on conditional probability. This page: P(E|F) and its rules.

2. Multiplication theorem

   JEE Main 2027 Unit 13 lists the addition and multiplication theorems of probability. This page: the rule, three events, and draws without replacement.

3. Independent events

   JEE Main 2027 Unit 13 lists the multiplication theorem of probability, which for independent events is P(E ∩ F) = P(E) P(F). This page: tests and traps.

4. Bayes’ theorem

   JEE Main 2027 Unit 13 lists Bayes’ theorem and the probability distribution of a random variable. This page: partitions, total probability and Bayes’ theorem.

5. Probability distribution

   JEE Main 2027 Unit 13 lists the probability distribution of a random variable; NCERT 2026–27 defines a random variable only in a remark. This page: the table, mean and variance.