Contents Foreword 1. 2. 3. Preface v Real Numbers 1 1.1 Introduction 1 1.2 Euclid’s Division Lemma 2 1.3 The Fundamental Theorem of Arithmetic 7 1.4 Revisiting Irrational Numbers 11 1.5 Revisiting Rational Numbers and Their Decimal Expansions 15 1.6 Summary 18 Polynomials 20 2.1 Introduction 20 2.2 Geometrical Meaning of the Zeroes of a Polynomial 21 2.3 Relationship between Zeroes and Coefficients of a Polynomial 28 2.4 Division Algorithm for Polynomials 33 2.5 Summary 37 Pair of Linear Equations in Two Variables 38 3.1 3.2 3.3 3.4 38 39 44 50 50 54 57 63 69 3.5 3.6 4. iii Introduction Pair of Linear Equations in Two Variables Graphical Method of Solution of a Pair of Linear Equations Algebraic Methods of Solving a Pair of Linear Equations 3.4.1 Substitution Method 3.4.2 Elimination Method 3.4.3 Cross-Multiplication Method Equations Reducible to a Pair of Linear Equations in Two Variables Summary Quadratic Equations 70 4.1 Introduction 70 4.2 Quadratic Equations 71 x 5. 6. 7. 8. 4.3 Solution of a Quadratic Equation by Factorisation 74 4.4 Solution of a Quadratic Equation by Completing the Square 76 4.5 Nature of Roots 88 4.6 Summary 91 Arithmetic Progressions 93 5.1 Introduction 93 5.2 Arithmetic Progressions 95 5.3 nth Term of an AP 100 5.4 Sum of First n Terms of an AP 107 5.5 Summary 116 Triangles 117 6.1 Introduction 117 6.2 Similar Figures 118 6.3 Similarity of Triangles 123 6.4 Criteria for Similarity of Triangles 129 6.5 Areas of Similar Triangles 141 6.6 Pythagoras Theorem 144 6.7 Summary 154 Coordinate Geometry 155 7.1 Introduction 155 7.2 Distance Formula 156 7.3 Section Formula 162 7.4 Area of a Triangle 168 7.5 Summary 172 Introduction to Trigonometry 173 8.1 Introduction 173 8.2 Trigonometric Ratios 174 8.3 Trigonometric Ratios of Some Specific Angles 181 8.4 Trigonometric Ratios of Complementary Angles 187 8.5 Trigonometric Identities 190 8.6 Summary 194 xi 9. Some Applications of Trigonometry 195 9.1 Introduction 195 9.2 Heights and Distances 196 9.3 Summary 205 10. Circles 206 10.1 Introduction 206 10.2 Tangent to a Circle 207 10.3 Number of Tangents from a Point on a Circle 209 10.4 Summary 215 11. Constructions 216 11.1 Introduction 216 11.2 Division of a Line Segment 216 11.3 Construction of Tangents to a Circle 220 11.4 Summary 222 12. Areas Related to Circles 223 12.1 Introduction 223 12.2 Perimeter and Area of a Circle — A Review 224 12.3 Areas of Sector and Segment of a Circle 226 12.4 Areas of Combinations of Plane Figures 231 12.5 Summary 238 13. Surface Areas and Volumes 239 13.1 Introduction 239 13.2 Surface Area of a Combination of Solids 240 13.3 Volume of a Combination of Solids 245 13.4 Conversion of Solid from One Shape to Another 248 13.5 Frustum of a Cone 252 13.6 Summary 258 14. Statistics 260 14.1 Introduction 260 14.2 Mean of Grouped Data 260 14.3 Mode of Grouped Data 272 xii 14.4 Median of Grouped Data 277 14.5 Graphical Representation of Cumulative Frequency Distribution 289 14.6 Summary 293 15. Probability 295 15.1 Introduction 295 15.2 Probability — A Theoretical Approach 296 15.3 Summary 312 Appendix A1 : Proofs in Mathematics 313 A1.1 Introduction 313 A1.2 Mathematical Statements Revisited 313 A1.3 Deductive Reasoning 316 A1.4 Conjectures, Theorems, Proofs and Mathematical Reasoning 318 A1.5 Negation of a Statement 323 A1.6 Converse of a Statement 326 A1.7 Proof by Contradiction 329 A1.8 Summary 333 Appendix A2 : Mathematical Modelling 334 A2.1 Introduction 334 A2.2 Stages in Mathematical Modelling 335 A2.3 Some Illustrations 339 A2.4 Why is Mathematical Modelling Important? 343 A2.5 Summary 344 Answers/Hints 345