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Contents

Foreword iii

1. Sets 1

1.1 Introduction 1

1.2 Sets and their Representations 1

1.3 The Empty Set 5

1.4 Finite and Infinite Sets 6

1.5 Equal Sets 7

1.6 Subsets 9

1.7 Power Set 12

1.8 Universal Set 12

1.9 Venn Diagrams 13

1.10 Operations on Sets 14

1.11 Complement of a Set 18

1.12 Practical Problems on Union and Intersection of Two Sets 21

2. Relations and Functions 30

2.1 Introduction 30

2.2 Cartesian Product of Sets 30

2.3 Relations 34 2.4 Functions 36

3. Trigonometric Functions 49

3.1 Introduction 49

3.2 Angles 49

3.3 Trigonometric Functions 55

3.4 Trigonometric Functions of Sum and Difference of Two Angles 63

3.5 Trigonometric Equations 74

4. Principle of Mathematical Induction 86

4.1 Introduction 86

4.2 Motivation 87

4.3 The Principle of Mathematical Induction 88

5. Complex Numbers and Quadratic Equations 97

5.1 Introduction 97

5.2 Complex Numbers 97

5.3 Algebra of Complex Numbers 98

5.4 The Modulus and the Conjugate of a Complex Number 102

5.5 Argand Plane and Polar Representation 104

6. Linear Inequalities 116

6.1 Introduction 116

6.2 Inequalities 116

6.3 Algebraic Solutions of Linear Inequalities in One Variable and their Graphical                Representation 118

6.4 Graphical Solution of Linear Inequalities in Two Variables 123

6.5 Solution of System of Linear Inequalities in Two Variables 127

7. Permutations and Combinations 134

7.1 Introduction 134

7.2 Fundamental Principle of Counting 134

7.3 Permutations 138

7.4 Combinations 148

8. Binomial Theorem 160

8.1 Introduction 160

8.2 Binomial Theorem for Positive Integral Indices 160

8.3 General and Middle Terms 167

9. Sequences and Series 177

9.1 Introduction 177

9.2 Sequences 177

9.3 Series 179

9.4 Arithmetic Progression (A.P.) 181

9.5 Geometric Progression (G.P.) 186

9.6 Relationship Between A.M. and G.M. 191

9.7 Sum to n terms of Special Series 194

10. Straight Lines 203

10.1 Introduction 203

10.2 Slope of a Line 204

10.3 Various Forms of the Equation of a Line 212

10.4 General Equation of a Line 220

10.5 Distance of a Point From a Line 225

11. Conic Sections 236

11.1 Introduction 236

11.2 Sections of a Cone 236

11.3 Circle 239

11.4 Parabola 242

11.5 Ellipse 247

11.6 Hyperbola 255

12. Introduction to Three Dimensional Geometry 268

12.1 Introduction 268

12.2 Coordinate Axes and Coordinate Planes in Three Dimensional Space 269

12.3 Coordinates of a Point in Space 269

12.4 Distance between Two Points 271

12.5 Section Formula 273

13. Limits and Derivatives 281

13.1 Introduction 281

13.2 Intuitive Idea of Derivatives 281

13.3 Limits 284

13.4 Limits of Trigonometric Functions 298

13.5 Derivatives 303

14. Mathematical Reasoning 321

14.1 Introduction 321

14.2 Statements 321

14.3 New Statements from Old 324

14.4 Special Words/Phrases 329

14.5 Implications 335

14.6 Validating Statements 339

15. Statistics 347

15.1 Introduction 347

15.2 Measures of Dispersion 349

15.3 Range 349

15.4 Mean Deviation 349

15.5 Variance and Standard Deviation 361

15.6 Analysis of Frequency Distributions 372

16. Probability 383

16.1 Introduction 383

16.2 Random Experiments 384

16.3 Event 387

16.4 Axiomatic Approach to Probability 394

Appendix 1: Infinite Series 412

A.1.1 Introduction 412

A.1.2 Binomial Theorem for any Index 412

A.1.3 Infinite Geometric Series 414

A.1.4 Exponential Series 416

A.1.5 Logarithmic Series 419

Appendix 2: Mathematical Modelling 421

A.2.1 Introduction 421

A.2.2 Preliminaries 421

A.2.3 What is Mathematical Modelling 425