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LECTURES ON IIT JEE MATHEMATICS

Class XII Mathematics

Relations and functions, inverse trigonometry, matrices, determinants, differential and integral calculus, differential equations, vectors, three-dimensional geometry, probability and linear programming.

22 Chapters237 Planned Lectures

Table of Contents

Select a chapter to view its lectures. Lecture text, videos, practice problems and solutions will be published progressively.

01 Relations, Mappings and Invertibility7 Lectures
  1. Types of Relations
  2. Equivalence Relations and Equivalence Classes
  3. Types of Functions and Mapping Structure
  4. Composition and Inverse Functions — Introduction and Foundations
  5. Composition of Functions — Structural Results and Advanced Problems
  6. Invertibility, Inverse Functions and Involutions
  7. Counting Functions and Finite Mappings
02 Advanced Functions and Functional Equations7 Lectures
  1. Advanced Domains and Nested Functions
  2. Range of Functions — Algebraic Methods
  3. Range of Composite, Restricted and Piecewise Functions
  4. Graphs, Function Equations and Inequalities
  5. Iterated Functions, Fixed Points and Functional Structure
  6. Functional Equations I — Substitution, Symmetry and Mapping Methods
  7. Functional Equations II — Domain-Specific and Advanced Structures
03 Inverse Trigonometric Functions11 Lectures
  1. Principal Branches, Domains, Ranges and Graphs
  2. Principal Values, Allied-Angle Reduction and Composition with Trigonometric Functions
  3. Fundamental Identities and Conversion of Inverse Trigonometric Functions
  4. Trigonometric Functions of Inverse Trigonometric Arguments
  5. Sums and Differences of Inverse Trigonometric Functions
  6. Double-, Triple- and Multiple-Angle Inverse Trigonometric Formulae
  7. Equations Involving Inverse Trigonometric Functions
  8. Domains of Inverse Trigonometric and Composite Functions
  9. Ranges and Extremal Values of Inverse Trigonometric Expressions
  10. Inequalities Involving Inverse Trigonometric and Mixed Functions
  11. Limits of Inverse Trigonometric Functions — Left-Hand Limits, Right-Hand Limits, Continuity, Endpoint Behaviour and Advanced Applications
04 Matrices11 Lectures
  1. Matrices, Entries, Order and Equality
  2. Types and Special Forms of Matrices
  3. Algebra of Matrices
  4. Matrices as Transformations
  5. Transpose, Symmetric and Skew-Symmetric Matrices
  6. Powers and Special Algebraic Matrices
  7. Trace, Orthogonal Matrices and Related Structure
  8. Rank of a Matrix
  9. Adjoint, Inverse and Elementary Transformations
  10. Matrix Polynomials and the Cayley–Hamilton Idea
  11. Eigenvalues, Quadratic Forms and Classification
05 Determinants10 Lectures
  1. Determinants: Definition, Orders and Expansion
  2. Minors, Cofactors and General Expansion
  3. Properties and Elementary Transformations of Determinants
  4. Determinant Evaluation Strategies
  5. Special Determinants
  6. Factorisation and Parameter-Dependent Determinants
  7. Multiplication, Adjoint and Determinant Formulae
  8. Determinants and Geometry
  9. Differentiation, Integration and Summation of Determinants
  10. Determinants, Characteristic Polynomial and Bridge to Matrices
06 Systems of Linear Equations8 Lectures
  1. Linear Equations, Systems and Dependence
  2. Matrix Form and Inverse Method
  3. Cramer’s Rule
  4. Gaussian Elimination
  5. Rank and the Nature of Solutions
  6. Adjoint Criterion for Existence and Nature of Solutions
  7. Homogeneous Systems and Linear Dependence
  8. Parameter-Dependent Systems and Synthesis
07 Continuity and Differentiability9 Lectures
  1. Continuity at a Point: Limits, Function Values, and One-Sided Continuity
  2. Discontinuity: Removable, Jump, Infinite, and Oscillatory Behaviour
  3. Continuity on Intervals, Endpoints, and Domains
  4. Algebra and Composition of Continuous Functions; Special Functions
  5. Intermediate Value Theorem and Existence of Roots
  6. Derivative as a Limit and Limits Recognised as Derivatives
  7. Left and Right Derivatives; Differentiability at a Point
  8. Differentiability versus Continuity and Modes of Non-Differentiability
  9. Differentiability on Intervals and Advanced JEE Structures
08 Methods of Differentiation14 Lectures
  1. Standard Derivatives and Algebra of Differentiation
  2. Chain Rule and Differentiation of Composite Functions
  3. Exponential and Logarithmic Differentiation
  4. Logarithmic Differentiation and Variable Powers
  5. Inverse Trigonometric Functions and Principal-Value Structure
  6. Implicit Differentiation
  7. Parametric Differentiation
  8. Differentiation with Respect to Another Function and Inverse Functions
  9. Piecewise, Modulus and Special Real Functions
  10. Oscillatory and Highly Composite Functions
  11. Higher-Order and Successive Derivatives
  12. The nth Derivative and Periodic Derivative Patterns
  13. Leibniz Theorem and Structured Higher Derivatives
  14. Infinite Forms, Determinants and Introductory Partial Derivatives
09 Mean Value Theorems and Polynomial Roots9 Lectures
  1. Rolle’s Theorem: Hypotheses, Geometry, and Applicability
  2. Lagrange’s Mean Value Theorem and the Secant–Tangent Principle
  3. Consequences of the Mean Value Theorem: Constancy, Monotonicity, and Bounds
  4. Cauchy’s and Generalized Mean Value Theorems
  5. Mean Value Theorems in Inequalities and Comparison of Functions
  6. Polynomial Roots, Rolle Chains, and Interlacing of Zeros
  7. Repeated Roots, Multiplicity, and Derivative Structure of Polynomials
  8. Nature, Number, and Location of Real Roots; Cubics and Parameter Problems
  9. Fixed Points, Successive Approximation, and the Newton–Raphson Idea
10 Applications of Derivatives15 Lectures
  1. Derivative as a Rate of Change; Motion and Related Rates
  2. Differentials, Linear Approximation, and Errors
  3. Increasing and Decreasing Functions; Monotonicity
  4. Comparison of Functions and Inequalities Using Derivatives
  5. Critical Points, Stationary Points, and Local Extrema
  6. First-Derivative Test and Sign Analysis
  7. Second and Higher Derivative Tests; Concavity and Points of Inflection
  8. Absolute Extrema on Restricted, Piecewise, and Discontinuous Domains
  9. Optimization of Algebraic, Trigonometric, Exponential, and Logarithmic Expressions
  10. Geometric Optimization and Applications to Coordinate Geometry and Conics
  11. Tangents and Normals: General Theory and Geometric Conditions
  12. Tangents and Normals to Implicit and Parametric Curves
  13. Angle Between Curves, Orthogonal Intersection, Common Tangents and Common Normals
  14. Differential Applications to Circle, Parabola, Ellipse, and Hyperbola
  15. Subtangent, Subnormal, Tangent and Normal Lengths, and Related Geometric Properties
11 Curve Tracing and Geometry of Curves13 Lectures
  1. Transformations of Graphs and Structural Reading of a Curve
  2. Framework for Cartesian Curve Tracing: Domain, Symmetry, Intercepts, Branches, and Derivatives
  3. Vertical and Horizontal Asymptotes
  4. Oblique and Polynomial Asymptotes; Position Relative to an Asymptote
  5. Singular and Multiple Points; Tangents at Exceptional Points
  6. Nodes, Cusps, Isolated Points, and Local Branch Structure
  7. Complete Tracing of Explicit Cartesian Curves
  8. Complete Tracing of Implicit Algebraic Curves
  9. Parametric Curves: Orientation, Loops, Tangents, and Complete Tracing
  10. Polar Coordinates, Polar Derivatives, Symmetry, and Tangents
  11. Complete Tracing of Polar Curves
  12. Curvature and Radius of Curvature of Cartesian Curves
  13. Curvature of Parametric and Polar Curves; Centre of Curvature, Osculating Circle, and Evolute Ideas
12 Indefinite Integrals18 Lectures
  1. Antiderivatives and the Structure of the Indefinite Integral
  2. Standard Integrals and Recognition of Derivative Patterns
  3. Substitution: The Fundamental Change-of-Variable Method
  4. Algebraic Manipulation Before Integration
  5. Quadratic Denominators and Standard Rational Forms
  6. Radical Quadratics and Classical Trigonometric Substitutions
  7. Rational Functions and Partial Fractions — I
  8. Partial Fractions — II: Quadratic Factors and Advanced Decompositions
  9. Trigonometric Integrals — I: Powers and Products of Sine and Cosine
  10. Trigonometric Integrals — II: Powers of tan, sec, cot, and cosec
  11. Rational Trigonometric Integrals and the t = tan(x/2) Substitution
  12. Special Trigonometric Transformations and Derived Substitutions
  13. Integration by Parts: Principle, Choice, and Repeated Application
  14. Cyclic Integrals and Exponential–Trigonometric Products
  15. Algebraic Radical Integrals and Rationalising Substitutions
  16. Reciprocal, Symmetric, and Quartic Integrals
  17. Euler Substitutions and General Quadratic Radicals
  18. Reduction Formulae, Parameter Methods, and the Integration Strategy
13 Definite Integrals15 Lectures
  1. The Definite Integral: Meaning, Signed Area, and the Fundamental Theorem of Calculus
  2. Evaluation of Definite Integrals and Change of Variable
  3. Fundamental Algebraic Properties, Additivity, and Piecewise Integration
  4. Reflection and King’s Properties: Pairing an Integrand with Its Mirror Image
  5. Symmetric Limits, Even–Odd Functions, and Half-Interval Properties
  6. Special Symmetry Substitutions and Self-Reciprocal Integrals
  7. Periodic Functions and Integration over Complete and Partial Periods
  8. Modulus, Greatest Integer, Fractional Part, Signum, and Other Piecewise Integrands
  9. Ordering, Integral Inequalities, Bounds, and Mean Value Theorems
  10. Definite Integral as a Limit of a Sum: Riemann Sums and Recognition Techniques
  11. Integral Functions and Limits Involving Definite Integrals
  12. Differentiation Under the Integral Sign and the Leibniz Rule
  13. Improper Integrals: Infinite Limits and Unbounded Integrands
  14. Wallis Integrals and Definite Trigonometric Reduction Formulae
  15. Gamma and Beta Functions and Related Special Definite Integrals
14 Applications of Integrals14 Lectures
  1. Geometric Measure and the Construction of Area Integrals
  2. Vertical and Horizontal Strips: Integration with Respect to x and y
  3. Area Between Curves and Changing Boundaries
  4. Symmetry, Periodicity, and Area Reduction
  5. Regions Defined by Modulus, Piecewise Functions, and Inequalities
  6. Area Associated with Inverse Functions
  7. Parametric Curves and Area
  8. Polar Area: Sector Element and Regions Bounded by Polar Curves
  9. Areas of Standard Polar Curves and Area Between Polar Curves
  10. Area Functions, Moving Boundaries, and Inverse Area Problems
  11. Volumes from Known Cross-Sections
  12. Solids of Revolution: Discs, Washers, and Cylindrical Shells
  13. Arc Length of Cartesian, Parametric, and Polar Curves
  14. Surface Area of Revolution, Centroids, and the Unified Single-Integral Method
15 Differential Equations17 Lectures
  1. Differential Equations: Language, Order, Degree, Linearity, and Solutions
  2. Formation of Differential Equations by Elimination of Arbitrary Constants
  3. Variables Separable: Fundamental Method and Initial-Value Problems
  4. Equations Reducible to Variables Separable
  5. Homogeneous Functions and Homogeneous Differential Equations
  6. Reducible-to-Homogeneous Equations and Translation of Axes
  7. First-Order Linear Differential Equations and the Integrating Factor
  8. Bernoulli Differential Equations and Reduction to Linear Form
  9. Exact Differential Equations
  10. Integrating Factors and Equations Reducible to Exact Form
  11. Orthogonal and Constant-Angle Trajectories
  12. First-Order Higher-Degree Equations: Solvable for p
  13. First-Order Higher-Degree Equations: Solvable for x or y
  14. Clairaut’s Equation, General Solution, and Singular Solution
  15. Simple Higher-Order Differential Equations and Reduction of Order
  16. Geometrical Differential Equations and Tangent–Normal Conditions
  17. Mathematical Modelling with First-Order Differential Equations and Grand Mixed Drill
16 Vector Algebra and Geometric Representation8 Lectures
  1. Scalars, Vectors, and the Geometry of Directed Quantities
  2. Addition and Subtraction of Vectors: Geometric and Algebraic Structure
  3. Cartesian Components and Resolution of Vectors
  4. Position Vectors, Points, and Division of a Line Segment
  5. Direction Ratios, Direction Cosines, and Orientation in Space
  6. Linear Combinations and Geometric Span
  7. Linear Dependence, Independence, Collinearity, and Coplanarity
  8. Vector Methods in Plane Geometry
17 Products of Vectors and Projection Geometry10 Lectures
  1. Scalar Product: Definition, Algebra, and Angle Between Vectors
  2. Scalar Projection, Vector Projection, and Orthogonal Decomposition
  3. Dot Product Identities and Metric Geometry
  4. Vector Product: Orientation, Magnitude, and Algebra
  5. Area, Normal Vectors, and Perpendicular Distance Through Cross Products
  6. Lagrange’s Identity and Combined Dot–Cross Methods
  7. Scalar Triple Product and Oriented Volume
  8. Vector Triple Product and BAC–CAB Identity
  9. Identities Involving Products of Four Vectors
  10. Vector Equations and Geometric Synthesis
18 Three-Dimensional Geometry: Points, Lines, and Planes13 Lectures
  1. Cartesian Space, Coordinate Planes, Octants, and Position Vectors
  2. Direction Ratios, Direction Cosines, and Directions in Space
  3. Straight Line in Space: Vector and Parametric Equations
  4. Relative Position of Two Lines in Space
  5. Distance of a Point from a Line and Orthogonal Projection on a Line
  6. Shortest Distance Between Two Lines
  7. Plane: Normal Vector and Fundamental Equations
  8. Equations of a Plane from Geometric Data
  9. Angle Between Planes and Angle Between a Line and a Plane
  10. Distance from a Point to a Plane and Projection on a Plane
  11. Line–Plane Intersection and Relative Position
  12. Families of Planes and Angle-Bisector Planes
  13. Spatial Configurations of Lines and Planes
19 Sphere and Elementary Geometry of Surfaces in Space7 Lectures
  1. Sphere: Definition, Standard Equation, and General Equation
  2. Equations of Spheres from Geometric Data
  3. Plane Sections of a Sphere and Great Circles
  4. Tangency to a Sphere and Tangent Plane
  5. Two Spheres, Radical Plane, and Common Circle
  6. Cylinders as Loci Generated by Parallel Lines
  7. Sphere, Cylinder, and Spatial Loci
20 Conditional Probability and Independence6 Lectures
  1. Conditional Probability
  2. Multiplication Theorem of Probability
  3. Independent Events
  4. Independence of Three or More Events
  5. Law of Total Probability
  6. Bayes’ Theorem
21 Random Variables and Probability Distributions9 Lectures
  1. Random Variables
  2. Probability Distribution of a Discrete Random Variable
  3. Mathematical Expectation
  4. Variance and Standard Deviation
  5. Expectation of Sums and Indicator Variables
  6. Bernoulli Trials and the Binomial Distribution
  7. Binomial Probabilities
  8. Mean, Variance and Standard Deviation of a Binomial Distribution
  9. Properties and Mode of the Binomial Distribution
22 Linear Programming6 Lectures
  1. Linear Inequations in Two Variables and Half-Planes
  2. Systems of Linear Inequations and Feasible Regions
  3. Formulation of a Linear Programming Problem
  4. Graphical Solution — Corner Point Method
  5. Special Cases and Multiple Optimal Solutions
  6. Applications and Mathematical Modelling
About the lectures. Each lecture retains its theory, explanations, proofs and worked examples in the original sequence. End-of-lecture practice problems and their solutions will be presented separately when the lecture is published.