C ONTENTS FOREWORD iii 1. NUMBER SYSTEMS 1 1.1 Introduction 1 1.2 Irrational Numbers 5 1.3 Real Numbers and their Decimal Expansions 8 1.4 Representing Real Numbers on the Number Line 15 1.5 Operations on Real Numbers 18 1.6 Laws of Exponents for Real Numbers 24 1.7 Summary 27 2. POLYNOMIALS 28 2.1 Introduction 28 2.2 Polynomials in One Variable 28 2.3 Zeroes of a Polynomial 32 2.4 Remainder Theorem 35 2.5 Factorisation of Polynomials 40 2.6 Algebraic Identities 44 2.7 Summary 50 3. COORDINATE GEOMETRY 51 3.1 Introduction 51 3.2 Cartesian System 54 3.3 Plotting a Point in the Plane if its Coordinates are given 61 3.4 Summary 65 4. LINEAR EQUATIONS IN TWO VARIABLES 66 4.1 Introduction 66 4.2 Linear Equations 66 4.3 Solution of a Linear Equation 68 4.4 Graph of a Linear Equation in Two Variables 70 4.5 Equations of Lines Parallel to x-axis and y-axis 75 4.6 Summary 77 5. INTRODUCTION TO EUCLID’S GEOMETRY 78 5.1 Introduction 78 5.2 Euclid’s Definitions, Axioms and Postulates 80 5.3 Equivalent Versions of Euclid’s Fifth Postulate 86 5.4 Summary 88 6. LINES AND ANGLES 89 6.1 Introduction 89 6.2 Basic Terms and Definitions 90 6.3 Intersecting Lines and Non-intersecting Lines 92 6.4 Pairs of Angles 92 6.5 Parallel Lines and a Transversal 98 6.6 Lines Parallel to the same Line 101 6.7 Angle Sum Property of a Triangle 105 6.8 Summary 108 7. TRIANGLES 108 7.1 Introduction 109 7.2 Congruence of Triangles 109 7.3 Criteria for Congruence of Triangles 112 7.4 Some Properties of a Triangle 120 7.5 Some More Criteria for Congruence of Triangles 125 7.6 Inequalities in a Triangle 129 7.7 Summary 134 8. QUADRILATERALS 135 8.1 Introduction 135 8.2 Angle Sum Property of a Quadrilateral 136 8.3 Types of Quadrilaterals 137 8.4 Properties of a Parallelogram 139 8.5 Another Condition for a Quadrilteral to be a Parallelogram 145 8.6 The Mid-point Theorem 148 8.7 Summary 151 9. AREAS OF PARALLELOGRAMS AND TRIANGLES 152 9.1 Introduction 152 9.2 Figures on the same Base and Between the same Parallels 154 9.3 Parallelogramms on the same Base and between the same Parallels 156 9.4 Triangles on the same Base and between the same Parallels 160 9.5 Summary 167 10. CIRCLES 168 10.1 Introduction 168 10.2 Circles and its Related Terms : A Review 169 10.3 Angle Subtended by a Chord at a Point 171 10.4 Perpendicular from the Centre to a Chord 173 10.5 Circle through Three Points 174 10.6 Equal Chords and their Distances from the Centre 176 10.7 Angle Subtended by an Arc of a Circle 179 10.8 Cyclic Quadrilaterals 182 10.9 Summary 187 11. CONSTRUCTIONS 187 11.1 Introduction 188 11.2 Basic Constructions 189 11.3 Some Constructions of Triangles 191 11.4 Summary 196 12. HERON’S FORMULA 197 12.1 Introduction 197 12.2 Area of a Triangle – by Heron’s Formula 199 12.3 Application of Heron’s Formula in finding Areas of Quadrilaterals 203 12.4 Summary 207 13. SURFACE AREAS AND VOLUMES 208 13.1 Introduction 208 13.2 Surface Area of a Cuboid and a Cube 208 13.3 Surface Area of a Right Circular Cylinder 214 13.4 Surface Area of a Right Circular Cone 217 13.5 Surface Area of a Sphere 222 13.6 Volume of a Cuboid 226 13.7 Volume of a Cylinder 228 13.8 Volume of a Right Circular Cone 231 13.9 Volume of a Sphere 234 10.10 Summary 237 14. STATISTICS 238 14.1 Introduction 238 14.2 Collection of Data 239 14.3 Presentation of Data 240 14.4 Geographical Representation of Data 247 14.5 Measures of Central Tendency 261 14.6 Summary 270 15. PROBABILITY 271 15.1 Introduction 271 15.2 Probability – an Experimental Approach 272 15.3 Summary 285 APPENDIX – 1 PROOFS IN MATHEMATICS 286 A1.1 Introduction 286 A1.2 Mathematically Acceptable Statements 287 A1.3 Deductive Reasoning 290 A1.4 Theorems, Conjectures and Axioms 293 A1.5 What is a Mathematical Proof? 298 A1.6 Summary 305 APPENDIX – 2 INTRODUCTION TO MATHEMATICAL MODELLING 306 A2.1 Introduction 306 A2.2 Review of Word Problems 307 A2.3 Some Mathematical Models 311 A2.4 The Process of Modelling, its Advantages and Limitations 319 A2.5 Summary 322 ANSWERS/HINTS 325-350