GENERAL MATHEMATICS-1 (INTRODUCTION TO ALGEBRA)

 Unit # CONTENTS Page # 1 1. INTRODUCTION TO MATHEMATICS 1 2 2. REAL NUMBER SYSTEMS 2.1  Real Numbers 7 2.2  Properties of Real Numbers 11 2.3  Rational and Irrational Numbers 16 2.4 Decimal Presentation of Rational and Irrational    Numbers 18 PRACTICE EXERCISE – 2.1 21 2.5 HCF & LCM 29 PRACTICE EXERCISE – 2.2 32 3 3. COMPLEX NUMBERS 3.1  Definition of Complex Numbers 35 3.2  Operations on Complex Numbers 37 3.3  Properties of the Complex Numbers 39 3.4    Geometrical Representation of Complex Numbers 42 3.5    Graphical Addition and Subtraction of  Complex Numbers 44 PRACTICE EXERCISE – 3 45 4 4. INTRODUCTION TO SETS 4.1  Basic Definitions of Sets 54 4.2  Representation of Sets in Set Theory 56 4.3  Types of Sets 57 4.4  Subsets and its Types with Examples 58 4.5  Operations on Sets 60 4.6  Properties of Sets 62 PRACTICE EXERCISE – 4 63 5 5. FUNCTIONS 5.1  Definition of Function 75 5.2  How to Evaluate the Function 76 5.3  Function, Domain and Range 78 5.4  SET- Builder Notation for a Function 79 5.5  Types of Function 80 5.6  Linear and Quadratic Function 86 5.7  Inverse of Function 88 PRACTICE EXERCISE – 5 89 6 6. INTRODUCTION TO MATRIX 6.1  Basic Introduction 99 6.2  Important Formulas for Matrices 100 6.3  Types of Matrices 101 6.4  Operations on Matrices 103 6.5  Properties of Matrix 105 6.6  Transpose of Matrix with Examples 106 6.7  Special Types of Matrices 108 6.8  Echelon and Reduced Echelon Forms of Matrices 110 6.9  Invertible Matrix 112 PRACTICE EXERCISE – 6 114 7 7. DETERMINANTS 7.1  Determinants of Square Matrix 126 7.2  Types of Determinants 127 7.3  Properties of Determinants 130 7.4  Adjoint of a Matrix 136 7.5   Application of Determinants and Matrices 138 PRACTICE EXERCISE – 7 140 8 8. SYSTEMS OF LINEAR EQUATIONS 8.1  Basic Introduction 149 8.2  Methods to Solve Systems of Linear Equation 150 8.3  Systems of Linear Equations in Two Variables 153 8.4  Cramer’s Rule 161 8.5  Application of Systems of Linear Equations 163 PRACTICE EXERCISE – 8 164 9 9. SOLUTION OF QUADRATIC EQUATIONS 9.1  Basic Introduction 174 9.2  Solution of Quadratic Equations with         Examples 174 9.3  Derivation of the Quadratic Formula 180 PRACTICE EXERCISE – 9.1 184 9.4  Qualitative Analysis of Roots of Quadratic Equations 189 PRACTICE EXERCISE – 9.2 192 9.6  Solution of Equations Reducible to the             Quadratic Equation with Examples 194 PRACTICE EXERCISE – 9.3 203 9.7  Cube Root of Unity 204 9.8  Properties of Cube Root of Unity 209 9.9  Four Fourth Roots of Unity 212 PRACTICE EXERCISE – 9.4 213 9.10 Relations Between the Roots and the    Coefficients of a Quadratic Equation 216 PRACTICE EXERCISE – 9.5 219 10 10. SEQUENCES & SERIES 10.1  Introduction to Sequence and Series 226 10.2  Types of Sequence & Series 228 10.3  Sequence and Series Formulas 228 PRACTICE EXERCISE – 10.1 232 10.4  Arithmetic Progression / Sequence 236 PRACTICE EXERCISE – 10.2 240 10.5   Arithmetic Mean (A.M) 242 PRACTICE EXERCISE – 10.3 243 10.6   Arithmetic Series 244 PRACTICE EXERCISE – 10.4 247 10.8   Geometric Progression / Sequence 251 PRACTICE EXERCISE – 10.5 254 10.9   Geometric Means (G.M) 256 PRACTICE EXERCISE – 10.6 258 10.10  Geometric Series 260 PRACTICE EXERCISE – 10.7 264 10.11  Application of Geometric Sequence &   Series 266 10.12  Harmonic Progression / Sequence 268 10.13  Harmonic Mean 270 10.14  Application of Harmonic Sequence 272 PRACTICE EXERCISE – 10.8 273 11 11.    MATHEMATICAL INDUCTION 11.1  Principles of Mathematical Induction 282 11.2  Principles of Extended Mathematical Induction 284 PRACTICE EXERCISE – 11 286 12 12.    BINOMIAL THEOREM 12.1  Introduction to Binomial Theorem 291 PRACTICE EXERCISE – 12.1 296 12.2  Application of Binomial Theorem 303 PRACTICE EXERCISE – 12.2 308 13 13. TRIGONOMETRY 13.1  Fundamentals of Trigonometry 313 Practice Exercise   13.1 320 13.2  Trigonometric Functions 323 Practice Exercise  13.2 327 13.3  The Values of Trigonometric Functions 331 Practice Exercise  13.3 334 13.4  Fundamental Law of Trigonometry 344 13.5  Trigonometric Ratios of Allied Angles 344 13.6  Applications of Basic Identities 346 Practice Exercise  13.4 349 13.7  Half, Double and Triple Angle Identities 352 Practice Exercise  13.5 354 14 Important Definitions 361 15 Paper Pattern and Guideline for Mid and Final Exam 373 16 Question Papers 374 17 References 382

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