#### Topics

##### Relations and Functions

##### Relations and Functions

##### Algebra

##### Inverse Trigonometric Functions

##### Matrices

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

##### Calculus

##### Vectors and Three-dimensional Geometry

##### Determinants

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

##### Continuity and Differentiability

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

##### Linear Programming

##### Probability

##### Applications of Derivatives

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

##### Integrals

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

##### Sets

- Sets

##### Applications of the Integrals

##### Differential Equations

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

##### Vectors

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

##### Three - Dimensional Geometry

- Introduction of Three Dimensional Geometry
- Direction Cosines and Direction Ratios of a Line
- Relation Between Direction Ratio and Direction Cosines
- Equation of a Line in Space
- Angle Between Two Lines
- Shortest Distance Between Two Lines
- Three - Dimensional Geometry Examples and Solutions
- Equation of a Plane Passing Through Three Non Collinear Points
- 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
- Vector and Cartesian Equation of a Plane
- 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

##### Linear Programming

##### Probability

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

- Empty Relation
- Universal Relation
- Trivial Relations
- Identity relation
- Symmetric relation
- Transitive relation
- Equivalence Relation
- Antisymmetric relation
- Inverse relation
- One-One Relation (Injective)
- Many-one relation
- Into relation
- Onto relation (Surjective)

## Notes

A relation between elements of a set which is reflexive, symmetric, and transitive and which defines exclusive classes whose members bear the relation to each other and not to those in other classes.

1) Reflexive relations- A relation is said to be reflexive provided you take up any element from the given set say a∈A, then the ordered pair (a,a) belongs to that relation i.e for all (a,a) ∈R and this happens for each and every a present in the set.

Example- A={1,2,3} is the given set and we want the relation to be reflexive then,

(1,1)∈R, (2,2)∈R, (3,3)∈R should be a element in the relation, even if one does not lie then it will not be a reflexive relation.

(a,a)∈R can also be written as aRa.

2) Symmetric relations- For instance, a,b are the elements from the given set A i.e. a,b∈A and for any (a,b)∈R, if we want this relation to be symmetric then in that case it must imply that (b,a)∈R.

Example- A= {1,2,3} and if R{(1,2), (2,1)} then the relation is symmetric.

3) Transitive relations- For any a,b,c that belongs to set A, if (a,b)∈R and in continuity (b,c)∈R ⇒ (a,c)∈R then we say that the relation is transitive.

Example- A= {1,2,3} and if R= {(1,2), (2,3), (3,2)}

4) Not reflexive- For reflexive relation we write that ∀a∈A ⇒(a,a) ∈R, then for not reflexive it could be said that there exist atleast some element in A such that the ordered pair (a,a) does not belong R, it can be written as ∃a∈A (a,a)∉R, here ∀ means for all and ∃ means there exists.

5) Not symmetric- For not symmetric there exist some a and b belonging to the given set such that even your( a,b) belongs to R, but (b,a) does not belong to R. This can be expressed as ∃a,b∈A, (a,b)∈R but (b,a) ∉R.

6) Not transitive- There exist three elements in the set such that (a,b), (b,c)∈R, but (a,c) ∉R.

If relation is Reflexive, Symmetric and Transitive all together then the relations is said to be Equivalence Relation.