11. Three Dimensional Geography

11.1. Introduction

11.2. Direction Cosines Direction Ratios of a Line

11.2. Direction Cosines Direction Ratios of a Line

11.2. Direction Cosines Direction Ratios of a Line

11.2.1. Relation between the Direction Cosines of a Line

11.2.2. Direction Cosines of a Line Passing through Two Points

11.3. Equation of a Line in Space

11.3. Equation of a Line in Space

11.3. Equation of a Line in Space

11.3.1. Equation of a Line through a Given Point and Parallel to a Given Vector

11.3.2. Equation of a Line Passing through Two Given Points

11.4. Angle between Two Lines

11.4. Angle between Two Lines

11.4. Angle between Two Lines

11.5. Shortest Distance between Two Lines

11.5.1. Distance between Two Skew Lines

11.5.2. Distance between Parallel Lines

11.6. Plane

11.6.1. Equation of a Plane in a Normal Form

11.6.1. Equation of a Plane in a Normal Form

11.6.1. Equation of a Plane in a Normal Form

11.6.2. Equation of a Plane Perpendicular to a Given Vector and Passing through a Given Point

11.6.3. Equation of a Plane Passing through Three Non Collinear Points

11.6.4. Intercept Form of the Equation of a Plane

11.6.5. Plane Passing through the Intersection of Two Given Planes

11.7. Coplanarity of Two Lines

11.8. Angle between Two Planes

11.8. Angle between Two Planes

11.8. Angle between Two Planes

11.9. Distance of a Point from a Plane

11.10. Angle between a Line and a Plane

11.10. Angle between a Line and a Plane

11.10. Angle between a Line and a Plane

11.10. Angle between a Line and a Plane

1. Relations and Functions

1.1 Introduction

1.2. Types of Relations

1.2. Types of Relations

1.2. Types of Relations

1.2. Types of Relations

1.3. Types of Functions

1.3. Types of Functions

1.3. Types of Functions

1.4. Composition of Functions and Invariable Functions

1.4. Composition of Functions and Invariable Functions

1.4. Composition of Functions and Invariable Functions

1.5. Binary Operations

2. Inverse Trigonometric Functions

2.1. Introduction

2.2. Basic Concepts

2.3. Properties of Inverse Trigonometric Functions

2.3. Properties of Inverse Trigonometric Functions

2.3. Properties of Inverse Trigonometric Functions

2.3. Properties of Inverse Trigonometric Functions

2.3. Properties of Inverse Trigonometric Functions

2.3. Properties of Inverse Trigonometric Functions

3.Matrices

3.1. Introduction

3.2. Matrix

3.2. Matrix

3.2. Matrix

3.2.1. Order of Matrix

3.3. Types of Matrices

3.3. Types of Matrices

3.3. Types of Matrices

3.3.1. Equality of Matrices

3.3.1. Equality of Matrices

3.3.1. Equality of Matrices

3.4. Operations in Matrices

3.4.1. Addition of Matrices

3.4.2. Multiplication of Matrix by a Scalar

3.4.3. Properties of Matrix Addition

3.4.4. Properties of Scalar Multiplication of a Matrix

3.4.5. Multiplication of Matrices

3.4.5. Multiplication of Matrices

3.4.6. Properties of Multiplication of Matrices

3.5. Transpose of a Matrix

3.5.1. Properties of Transpose of the Matrix

3.6. Symmetric and Skew Symmetric Matrices

3.7. Elementary Operation (Transformation) of a Matrix

3.8. Invertible Matrices

3.8.1. Inverse of a Matrix by Elementary Operations

3.8.1. Inverse of a Matrix by Elementary Operations

3.8.1. Inverse of a Matrix by Elementary Operations

4. Determinants

4.1. Introduction

4.2. Determinant

4.2.1. Determinant of a Matrix of Order One

4.2.2. Determinant of a Matrix of Order Two

4.2.3. Determinant of a Matrix of Order 3 x 3

4.3. Properties of Determinants

4.3. Properties of Determinants

4.3. Properties of Determinants

4.4. Area of Triangle

4.5. Minors and Co-Factors

4.6. Adjoint and Inverse of a Matrix

4.6.1. Adjoint of a Matrix

4.7. Applications of Determinants and Matrices

4.7.1. Solution of System of Linear Equations Using Inverse of a Matrix

5. Continuity and Differentiability

5.1. Introduction

5.2. Continuity

5.2. Continuity

5.2. Continuity

5.2. Continuity

5.2. Continuity

5.2. Continuity

5.2. Continuity

5.2.1. Algebra of Continuous Functions

5.3. Differentiability

5.3. Differentiability

5.3. Differentiability

5.3. Differentiability

5.3. Differentiability

5.3. Differentiability

5.3. Differentiability

5.3.1. Derivatives of Composite Functions

5.3.1. Derivatives of Composite Functions

5.3.1. Derivatives of Composite Functions

5.3.2. Derivatives of Implicit Functions

5.3.3. Derivatives of Inverse Trigonometric Functions

5.4. Exponential and Logarithmic Functions

5.5. Logarithmic Differentiation

5.6. Derivatives of Functions in Parametric Forms

5.7. Second Order Derivative

5.7. Second Order Derivative

5.7. Second Order Derivative

5.8. Mean Value Theorem

6. Application of Derivatives

6.1. Introduction

6.2. Rate of Change of Quantities

6.2. Rate of Change of Quantities

6.2. Rate of Change of Quantities

6.3. Increasing and Decreasing

6.3. Increasing and Decreasing

6.3. Increasing and Decreasing

6.3. Increasing and Decreasing

6.3. Increasing and Decreasing

6.4. Tangents and Normals

6.5. Approximations

6.5. Approximations

6.5. Approximations

6.5. Approximations

6.5. Approximations

6.5. Approximations

6.5. Approximations

6.6. Maxima and Minima

6.6. Maxima and Minima

6.6. Maxima and Minima

6.6.1. Maximum and Minimum Values of a Function in a Closed Interval

7. Integrals

7.1. Introduction

7.1. Introduction

7.1. Introduction

7.2. Integration as an Inverse Process of Differentiation

7.2.1. Geometrical Interpretation of Indefinite Integral

7.2.2. Some Properties of Indefinite Integral

7.2.2. Some Properties of Indefinite Integral

7.2.3. Comparison between Differentiation and Integration

7.3. Methods of Integration

7.3.1. Integration by Substitution

7.3.1. Integration by Substitution

7.3.1. Integration by Substitution

7.3.2. Integration using Trigonometric Identities

7.4. Integrals of Some Particular Functions

7.5. Integration by Partial Fractions

7.5. Integration by Partial Fractions

7.5. Integration by Partial Fractions

7.6. Integration by Parts

7.6. Integration by Parts

7.6. Integration by Parts

7.6.1. Integral of the Type

7.6.2. Integrals of Some More Types

7.7. Definite Integral

7.7.1. Definite Integral as the Limit of a Sum

7.7.1. Definite Integral as the Limit of a Sum

7.7.1. Definite Integral as the Limit of a Sum

7.8. Fundamental Theorem of Calculus

7.8.1. Area of Function

7.8.2. First Fundamental Theorem of Integral Calculus

7.8.3. Second Fundamental Theorem of Integral Calculus

7.9 Evolution of Definite Integrals by Substitution

7.10. Some Properties of Definite Integrals

7.10. Some Properties of Definite Integrals

7.10. Some Properties of Definite Integrals

8. Application of Integrals

8.1. Introduction

8.2. Area Under Simple Curves

8.2.1. The Area of the Region bounded by Curve and a Line

8.3. Area between Two Curves

9. Differential Equations

9.1. Introduction

9.2. Basic Concepts

9.2.1. Order of Differential Equation

9.2.2. Degree of a Differential Equation

9.3. General and Particular Solutions of a Differential Equation

9.4. Formation of a Differential Equation Whose General Solution is Given

9.4.1. Procedure to form a Differential Equation that will represent a given Family of Curves

9.5. Methods of Solving First Order, First Degree Differential Equations

9.5.1. Differential Equations with Variables Separable

9.5.2. Homogeneous Differential Equations

9.5.3. Linear Differential Equations

10.Vector Algebra

10.1. Introduction

10.2. Some Basic Concepts

10.2. Some Basic Concepts

10.2. Some Basic Concepts

10.2. Some Basic Concepts

10.2. Some Basic Concepts

10.3. Types of Vectors

10.4. Addition of Vectors

10.5. Multiplication of Vector by a Scalar

10.5. Multiplication of Vector by a Scalar

10.5.1. Components of a Vector

10.5.2. Vector Joining Two Points

10.5.3. Section Formula

10.6. Product of Two Vectors

10.6.1. Scalar (or dot) Product of Two Vectors

10.6.1. Scalar (or dot) Product of Two Vectors

10.6.1. Scalar (or dot) Product of Two Vectors

10.6.2. Projection of a Vector on a Line

10.6.3. Vector (or cross) Product of Two Vectors

12. Linear Programming

12.1. Introduction

12.2. Linear Programming Problem and its Mathematical Formulation

12.2.1. Mathematical Formulation of the Problem

12.2.2. Graphical Method of solving Linear Programming Problems

12.3. Different Types of Linear Programming Problems

13. Probability

13.1. Introduction

13.2. Conditional Probability

13.2. Conditional Probability

13.2. Conditional Probability

13.2.1. Properties of Conditional Probability

13.2.1. Properties of Conditional Probability

13.2.1. Properties of Conditional Probability

13.3. Multiplication Theorem on Probability

13.4. Independent Events

13.4. Independent Events

13.4. Independent Events

13.5. Bayes' Theorem

13.5.1. Partition of a Sample Space

13.5.2. Theorem of Total Probability

13.6. Random Variables and its Probability Distributions

13.6. Random Variables and its Probability Distributions

13.6. Random Variables and its Probability Distributions

13.6.1. Probability Distribution of a Random Variable

13.6.2. Mean of a Random Variable

13.6.3. Variance of Random Variable

13.7. Bernoulli Trials and Binomial Distribution

13.7. Bernoulli Trials and Binomial Distribution

13.7. Bernoulli Trials and Binomial Distribution

13.7.1. Bernoulli Trials

13.7..2. Binomial Distribution