Contents 1. Foreword iii Sets 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 1.10 1.11 1.12 1 1 1 5 6 7 9 12 12 13 14 18 21 Introduction Sets and their Representations The Empty Set Finite and Infinite Sets Equal Sets Subsets Power Set Universal Set Venn Diagrams Operations on Sets Complement of a Set Practical Problems on Union and Intersection of Two Sets 2. Relations and Functions 2.1 Introduction 2.2 Cartesian Product of Sets 2.3 Relations 2.4 Functions 30 30 30 34 36 3. Trigonometric Functions 3.1 Introduction 3.2 Angles 3.3 Trigonometric Functions 3.4 Trigonometric Functions of Sum and Difference of Two Angles 3.5 Trigonometric Equations 49 49 49 55 63 74 4. Principle of Mathematical Induction 4.1 Introduction 4.2 Motivation 4.3 The Principle of Mathematical Induction 86 86 87 88 5. Complex Numbers and Quadratic Equations 5.1 Introduction 5.2 Complex Numbers 5.3 Algebra of Complex Numbers 5.4 The Modulus and the Conjugate of a Complex Number 5.5 Argand Plane and Polar Representation 5.6 Quadratic Equations 97 97 97 98 102 104 108 6. Linear Inequalities 6.1 Introduction 6.2 Inequalities 6.3 Algebraic Solutions of Linear Inequalities in One Variable and their Graphical Representation 6.4 Graphical Solution of Linear Inequalities in Two Variables 6.5 Solution of System of Linear Inequalities in Two Variables 116 116 116 7. Permutations and Combinations 7.1 Introduction 7.2 Fundamental Principle of Counting 7.3 Permutations 7.4 Combinations 134 134 134 138 148 8. Binomial Theorem 8.1 Introduction 8.2 Binomial Theorem for Positive Integral Indices 8.3 General and Middle Terms 160 160 160 167 9. Sequences and Series 9.1 Introduction 9.2 Sequences 9.3 Series 9.4 Arithmetic Progression (A.P.) 9.5 Geometric Progression (G.P.) 9.6 Relationship Between A.M. and G.M. 9.7 Sum to n terms of Special Series 177 177 177 179 181 186 191 194 10. Straight Lines 10.1 Introduction 10.2 Slope of a Line 10.3 Various Forms of the Equation of a Line 10.4 General Equation of a Line 10.5 Distance of a Point From a Line viii 118 123 127 203 203 204 212 220 225 11. Conic Sections 11.1 Introduction 11.2 Sections of a Cone 11.3 Circle 11.4 Parabola 11.5 Ellipse 11.6 Hyperbola 236 236 236 239 242 247 255 12. Introduction to Three Dimensional Geometry 12.1 Introduction 12.2 Coordinate Axes and Coordinate Planes in Three Dimensional Space 12.3 Coordinates of a Point in Space 12.4 Distance between Two Points 12.5 Section Formula 268 268 13. Limits and Derivatives 13.1 Introduction 13.2 Intuitive Idea of Derivatives 13.3 Limits 13.4 Limits of Trigonometric Functions 13.5 Derivatives 281 281 281 284 298 303 14. Mathematical Reasoning 14.1 Introduction 14.2 Statements 14.3 New Statements from Old 14.4 Special Words/Phrases 14.5 Implications 14.6 Validating Statements 321 321 321 324 329 335 339 15. Statistics 15.1 Introduction 15.2 Measures of Dispersion 15.3 Range 15.4 Mean Deviation 15.5 Variance and Standard Deviation 15.6 Analysis of Frequency Distributions 347 347 349 349 349 361 372 ix 269 269 271 273 16. Probability 16.1 Introduction 16.2 Random Experiments 16.3 Event 16.4 Axiomatic Approach to Probability 383 383 384 387 394 Appendix 1: Infinite Series A.1.1 Introduction A.1.2 Binomial Theorem for any Index A.1.3 Infinite Geometric Series A.1.4 Exponential Series A.1.5 Logarithmic Series 412 412 412 414 416 419 Appendix 2: Mathematical Modelling A.2.1 Introduction A.2.2 Preliminaries A.2.3 What is Mathematical Modelling 421 421 421 425 Answers 433 x