#### Topics

##### Relations and Functions

##### Algebra

##### Calculus

##### Vectors and Three-dimensional Geometry

##### Linear Programming

##### Probability

##### Sets

##### Inverse Trigonometric Functions

##### Relations and Functions

##### Determinants

- Applications of Determinants and Matrices
- Elementary Transformations
- Inverse of a Square Matrix by the Adjoint Method
- Properties of Determinants
- Determinant of a Square Matrix
- Determinants of Matrix of Order One and Two
- Introduction of Determinant
- Area of a Triangle
- Minors and Co-factors
- Determinant of a Matrix of Order 3 × 3
- Rule A=KB

##### Matrices

- Introduction of Operations on Matrices
- Inverse of a Nonsingular Matrix by Elementary Transformation
- Multiplication of Two Matrices
- Negative of Matrix
- Properties of Matrix Addition
- Concept of Transpose of a Matrix
- Subtraction of Matrices
- Addition of Matrices
- Symmetric and Skew Symmetric Matrices
- Types of Matrices
- Proof of the Uniqueness of Inverse
- Invertible Matrices
- Elementary Operation (Transformation) of a Matrix
- Multiplication of Matrices
- Properties of Multiplication of Matrices
- Equality of Matrices
- Order of a Matrix
- Matrices Notation
- Introduction of Matrices
- Multiplication of a Matrix by a Scalar
- Properties of Scalar Multiplication of a Matrix
- Properties of Transpose of the Matrices

##### Continuity and Differentiability

- Derivative - Exponential and Log
- Concept of Differentiability
- Proof Derivative X^n Sin Cos Tan
- Infinite Series
- Higher Order Derivative
- Algebra of Continuous Functions
- Continuous Function of Point
- Mean Value Theorem
- Second Order Derivative
- Derivatives of Functions in Parametric Forms
- Logarithmic Differentiation
- Exponential and Logarithmic Functions
- Derivatives of Implicit Functions
- Derivatives of Inverse Trigonometric Functions
- Derivatives of Composite Functions - Chain Rule
- Concept of Continuity

##### Applications of Derivatives

- Maximum and Minimum Values of a Function in a Closed Interval
- Maxima and Minima
- Simple Problems on Applications of Derivatives
- Graph of Maxima and Minima
- Approximations
- Tangents and Normals
- Increasing and Decreasing Functions
- Rate of Change of Bodies Or Quantities
- Introduction to Applications of Derivatives

##### Applications of the Integrals

##### Differential Equations

- Linear Differential Equations
- Solutions of Linear Differential Equation
- Homogeneous Differential Equations
- Differential Equations with Variables Separable Method
- Formation of a Differential Equation Whose General Solution is Given
- General and Particular Solutions of a Differential Equation
- Order and Degree of a Differential Equation
- Basic Concepts of Differential Equation
- Procedure to Form a Differential Equation that Will Represent a Given Family of Curves

##### Integrals

- Definite Integrals Problems
- Indefinite Integral Problems
- Comparison Between Differentiation and Integration
- Geometrical Interpretation of Indefinite Integral
- Integrals of Some Particular Functions
- Indefinite Integral by Inspection
- Properties of Indefinite Integral
- Integration Using Trigonometric Identities
- Introduction of Integrals
- Evaluation of Definite Integrals by Substitution
- Properties of Definite Integrals
- Fundamental Theorem of Calculus
- Definite Integral as the Limit of a Sum
- Evaluation of Simple Integrals of the Following Types and Problems
- Methods of Integration - Integration by Parts
- Methods of Integration - Integration Using Partial Fractions
- Methods of Integration - Integration by Substitution
- Integration as an Inverse Process of Differentiation

##### Three - Dimensional Geometry

- Three - Dimensional Geometry Examples and Solutions
- Introduction of Three Dimensional Geometry
- Equation of a Plane Passing Through Three Non Collinear Points
- Relation Between Direction Ratio and Direction Cosines
- Intercept Form of the Equation of a Plane
- Coplanarity of Two Lines
- Distance of a Point from a Plane
- Angle Between Line and a Plane
- Angle Between Two Planes
- Angle Between Two Lines
- Vector and Cartesian Equation of a Plane
- Shortest Distance Between Two Lines
- Equation of a Line in Space
- Direction Cosines and Direction Ratios of a Line
- Equation of a Plane in Normal Form
- Equation of a Plane Perpendicular to a Given Vector and Passing Through a Given Point
- Plane Passing Through the Intersection of Two Given Planes

##### Vectors

- Concept of Direction Cosines
- Properties of Vector Addition
- Geometrical Interpretation of Scalar
- Scalar Triple Product of Vectors
- Vector (Or Cross) Product of Two Vectors
- Scalar (Or Dot) Product of Two Vectors
- Position Vector of a Point Dividing a Line Segment in a Given Ratio
- Multiplication of a Vector by a Scalar
- Addition of Vectors
- Introduction of Vector
- Magnitude and Direction of a Vector
- Basic Concepts of Vector Algebra
- Types of Vectors
- Components of a Vector
- Section formula
- Vector Joining Two Points
- Vectors Examples and Solutions
- Projection of a Vector on a Line
- Introduction of Product of Two Vectors

##### Linear Programming

##### Probability

- Variance of a Random Variable
- Probability Examples and Solutions
- Conditional Probability
- Multiplication Theorem on Probability
- Independent Events
- Baye'S Theorem
- Random Variables and Its Probability Distributions
- Mean of a Random Variable
- Bernoulli Trials and Binomial Distribution
- Introduction of Probability
- Properties of Conditional Probability

#### notes

Let X, A and B be matrices of, the same order such that X = AB. In order to apply a sequence of elementary row operations on the matrix equation X = AB, we will apply these row operations simultaneously on X and on the first matrix A of the product AB on RHS.

Similarly, in order to apply a sequence of elementary column operations on the matrix equation X = AB, we will apply, these operations simultaneously on X and on the second matrix B of the product AB on RHS.

In view of the above discussion, we conclude that if A is a matrix such that `A^(–1)` exists, then to find `A^(–1)` using elementary row operations, write A = IA and apply a sequence of row operation on A = IA till we get, I = BA. The matrix B will be the inverse of A. Similarly, if we wish to find `A^(–1)` using column operations, then, write A = AI and apply a sequence of column operations on A = AI till we get, I = AB.**Remark:** In case, after applying one or more elementary row (column) operations on A = IA (A = AI), if we obtain all zeros in one or more rows of the matrix A on L.H.S., then `A^(–1)` does not exist.