#### Topics

##### Relations and Functions

##### Relations and Functions

##### Inverse Trigonometric Functions

##### Algebra

##### 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 Transformations
- 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

##### Calculus

##### Vectors and Three-dimensional Geometry

##### 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

##### Linear Programming

##### 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

##### Probability

##### 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

##### Sets

##### 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

##### 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

##### Vectors

- 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
- Vectors and Their Types
- 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

##### 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

##### Linear Programming

##### Probability

- Variance of a Random Variable
- Probability Examples and Solutions
- Conditional Probability
- Multiplication Theorem on Probability
- Independent Events
- Bayes’ 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

The equation is `x^2 + y^2 + 2x – 4y + 4 = 0` ... (1)

represents a circle having centre at (–1, 2) and radius 1 unit.

Differentiating equation (1) with respect to x, we get

`(dy)/(dx) = (x + 1)/(2 - y)` (y ≠ 2) ...(2)

which is a differential equation. You will find later on that this equation represents the family of circles and one member of the family is the circle given in equation (1).

Let us consider the equation:

`x^2 + y^2 = r^2 `... (3)

By giving different values to r, we get different members of the family e.g. `x^2 + y^2 = 1`, `x^2 + y^2 = 4`, `x^2 + y^2 = 9` etc Fig .

Thus, equation (3) represents a family of concentric circles centered at the origin and having different radii.

We are interested in finding a differential equation that is satisfied by each member of the family. The differential equation must be free from r because r is different for different members of the family. This equation is obtained by differentiating equation (3) with respect to x, i.e.,

2x + 2y `(dy)/(dx)` = 0 or x + y `(dy)/(dx)` =0 ...(4)

which represents the family of concentric circles given by equation (3).