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1. Logarithm
- Definition of logarithm
- Fundamental properties of logarithms
- Change of base formula
- Logarithmic inequalities
- Graphs of logarithmic functions
- Domain and range of logarithmic expressions
- Simplification and evaluation of logarithmic expressions
- Logarithmic equations
- Application of logarithmic properties in solving algebraic problems
2. Sets, Relations and Functions
Sets
- Sets and their representations
- Types of sets (Empty, Finite, Infinite, Equal, Equivalent)
- Subsets, Power set
- Universal set
- Venn diagrams
- Operations on sets (Union, Intersection, Difference, Complement)
- 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 (Empty, Universal, Identity, Inverse)
- Equivalence relations
- Partial order relations
- Number of relations from one set to another
Functions
- Definition and notation
- Domain, co-domain and range
- Types of functions (One-one, Many-one, Onto, Into)
- 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 and geometric 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 their graphs
- Maximum and minimum values
- Location of roots
4. Matrices and Determinants
Matrices
- Types of matrices (Row, Column, Square, Diagonal, Scalar, Identity)
- Equality of matrices
- Algebra of matrices
- Transpose of a matrix
- Symmetric and skew-symmetric matrices
- Elementary transformations
- Rank of a matrix
- Inverse of a matrix
- Solution of system 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 of equations
5. Permutations and Combinations
- Fundamental principle of counting
- Permutations (Linear, Circular, Restricted)
- Combinations (Basic, Restricted)
- Permutations of alike objects
- Division into groups
- Derangements
- Gap method
- Number of integral solutions
6. Binomial Theorem
- Statement and proof
- General term
- Middle term
- Greatest term
- Properties of binomial coefficients
- Multinomial theorem
- Applications in approximations
- Problems on divisibility
7. Sequences and Series
Sequences
- Definition and examples
- Arithmetic progression
- Geometric progression
- Harmonic progression
- Arithmetico-geometric progression
- Special sequences (Fibonacci, etc.)
Series
- Summation of series
- Sum to n terms of special series
- Method of differences
- Telescoping series
- Infinite series
- Convergence of series
8. Limits, Continuity and Differentiability
Limits
- Definition and algebra of limits
- Standard limits
- Evaluation of limits (Factorisation, Rationalisation)
- Limits using expansions
- Limits using L’Hôpital’s rule
- Sandwich theorem
- Limits at infinity
Continuity
- Definition and examples
- Algebra of continuous functions
- Types of discontinuities
- Intermediate value theorem
- Problems on piecewise functions
Differentiability
- Definition and interpretation
- Derivatives of standard functions
- Rules of differentiation
- Derivatives of composite functions
- Implicit differentiation
- Logarithmic differentiation
- Parametric differentiation
- Higher order derivatives
9. Integral Calculus
Indefinite Integrals
- Definition and standard integrals
- Methods of integration (Substitution, By parts, Partial fractions)
- Integration of trigonometric functions
- Integration of irrational functions
- Integration of exponential functions
Definite Integrals
- Definition and properties
- Fundamental theorem of calculus
- Evaluation of definite integrals
- 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
10. Differential Equations
- Order and degree
- Formation of differential equations
- Solution of first order equations (Variable separable, Homogeneous, Linear)
- Solution of second order linear equations
- Applications in growth and decay
- Orthogonal trajectories
11. 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
12. 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)
13. Vector Algebra
- Types of vectors
- Addition and multiplication
- Scalar and vector products
- Triple products
- Applications in geometry
- Vector equations
- Differentiation of vectors
14. 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
15. Trigonometry
- Trigonometric functions and identities
- Compound angles
- Multiple and submultiple angles
- Transformation formulae
- Trigonometric equations
- Inverse trigonometric functions
- Properties of triangles
- Heights and distances
16. Mathematical Reasoning
- Statements and truth tables
- Logical operations
- Tautologies and contradictions
- Quantifiers
- Validity of statements
- Implications
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