1 Logarithm (JEE Mains & Advanced)

2 Sets, Relations and Functions

Sets

• Sets and their representations
• Types of sets (Empty, Finite, Infinite)
• Subsets, Power set
• Universal set
• Venn diagrams
• Operations on sets
• Laws of algebra of sets
• De Morgan’s Laws
• Cardinality of sets

Relations

• Ordered pairs
• Cartesian product of sets
• Relations and their types
• Domain and range of relations
• Types of relations
• Equivalence relations
• Partial order relations
• Number of relations

Functions

• Definition and notation
• Domain, co-domain and range
• Types of functions
• Composition of functions
• Inverse functions
• Binary operations
• Real-valued functions
• Even and odd functions
• Periodic functions

3 Complex Numbers and Quadratic Equations

Complex Numbers

• Definition and algebraic properties
• Argand plane representation
• Polar representation
• Euler’s form
• De Moivre’s theorem
• Cube roots of unity
• nth roots of unity
• Logarithm of a complex number

Quadratic Equations

• Standard form
• Nature of roots
• Relation between roots and coefficients
• Formation of quadratic equations
• Symmetric functions of roots
• Common roots
• Quadratic expressions and graphs
• Maximum and minimum values
• Location of roots

4 Matrices and Determinants

Matrices

• Types of matrices
• Equality of matrices
• Algebra of matrices
• Transpose of a matrix
• Symmetric and skew-symmetric
• Elementary transformations
• Rank of a matrix
• Inverse of a matrix
• Solution of linear equations

Determinants

• Properties of determinants
• Minors and cofactors
• Expansion of determinants
• Area of triangle using determinants
• Adjoint and inverse of matrix
• Cramer’s rule
• Consistency of linear equations
• Homogeneous system

5 Permutations and Combinations

• Fundamental principle of counting
• Permutations (Linear, Circular)
• Combinations (Basic, Restricted)
• Permutations of alike objects
• Division into groups
• Derangements
• Gap method
• Number of integral solutions

7 Binomial Theorem

• Statement and proof
• General term
• Middle term
• Greatest term
• Properties of binomial coefficients
• Multinomial theorem
• Applications in approximations
• Problems on divisibility

8 Sequences and Series

Sequences

• Definition and examples
• Arithmetic progression
• Geometric progression
• Harmonic progression
• Arithmetico-geometric progression
• Special sequences (Fibonacci)

Series

• Summation of series
• Sum to n terms of special series
• Method of differences
• Telescoping series
• Infinite series
• Convergence of series

9 Limits, Continuity and Differentiability

Limits

• Definition and algebra of limits
• Standard limits
• Evaluation of limits
• Limits using expansions
• L’Hôpital’s rule
• Sandwich theorem
• Limits at infinity

Continuity

• Definition and examples
• Algebra of continuous functions
• Types of discontinuities
• Intermediate value theorem
• Piecewise functions

Differentiability

• Definition and interpretation
• Derivatives of standard functions
• Rules of differentiation
• Composite functions
• Implicit differentiation
• Logarithmic differentiation
• Parametric differentiation
• Higher order derivatives

10 Integral Calculus

Indefinite Integrals

• Definition and standard integrals
• Methods of integration
• Integration of trigonometric functions
• Integration of irrational functions
• Integration of exponential functions

Definite Integrals

• Definition and properties
• Fundamental theorem of calculus
• Evaluation techniques
• Definite integrals as limit of sum
• Walli’s formula
• Leibnitz rule
• Reduction formulae

Applications of Integrals

• Area under curves
• Area between two curves
• Volume of revolution
• Length of curves
• Surface area of revolution

11 Differential Equations

• Order and degree
• Formation of differential equations
• Solution of first order equations
• Variable separable method
• Homogeneous equations
• Linear differential equations
• Solution of second order linear equations
• Applications in growth and decay
• Orthogonal trajectories

12 Coordinate Geometry

Straight Lines

• Various forms of equations
• Angle between lines
• Distance forms
• Concurrent lines
• Family of lines
• Reflection of points
• Distance of point from line

Circles

• Standard forms of equations
• Parametric equations
• Tangent and normal
• Chord of contact
• Power of point
• Radical axis
• Family of circles
• Orthogonal circles

Conic Sections

• Parabola (Standard forms, Properties)
• Ellipse (Standard forms, Properties)
• Hyperbola (Standard forms, Properties)
• Tangents and normals
• Chord of contact
• Director circle
• Asymptotes
• Rectangular hyperbola

13 Three Dimensional Geometry

• Direction ratios and cosines
• Equation of line in space
• Angle between lines
• Shortest distance
• Equation of plane
• Angle between planes
• Distance of point from plane
• Intersection of line and plane
• Coplanarity
• Sphere (Equation, Tangent plane)

14 Vector Algebra

• Types of vectors
• Addition and multiplication
• Scalar and vector products
• Triple products
• Applications in geometry
• Vector equations
• Differentiation of vectors

15 Statistics and Probability

Statistics

• Measures of central tendency
• Measures of dispersion
• Variance and standard deviation
• Analysis of frequency distributions
• Correlation and regression

Probability

• Random experiments
• Events and types
• Axiomatic approach
• Conditional probability
• Independent events
• Bayes’ theorem
• Random variables
• Probability distributions
• Bernoulli trials

16 Trigonometry

• Trigonometric functions and identities
• Compound angles
• Multiple and submultiple angles
• Transformation formulae
• Trigonometric equations
• Inverse trigonometric functions
• Properties of triangles
• Heights and distances

17 Mathematical Reasoning

• Statements and truth tables
• Logical operations
• Tautologies and contradictions
• Quantifiers
• Validity of statements
• Implications