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Prelims 3..16 Contents Preface to the First Edition xvii Preface to the Second Edition xix Preface to the Third Edition xx Preface to the Fourth Edition xxi Preface to the Fifth Edition xxii Hints on using the book xxiv Useful background information xxvi Part I: Foundation topics 1 Programme F.1 Arithmetic 3 Learning outcomes 3 Quiz F.1 4 Types of numbers ± The natural numbers, Numerals and place value, 5 Points on a line and order, The integers, Brackets, Addition and subtraction, Multiplication and division, Brackets and precedence rules, Basic laws ofarithmetic Revision summary 12 Revision exercise 13 Factors and prime numbers ± Factors, Prime numbers, Prime 14 factorization, Highest common factor (HCF), Lowest common multiple (LCM), Estimating, Rounding Revision summary 18 Revision exercise 18 Fractions ± Division of integers, Multiplying fractions, Of, Equivalent 20 fractions, Dividing fractions, Adding and subtracting fractions, Fractions on a calculator, Ratios, Percentages Revision summary 28 Revision exercise 28 Decimal numbers ± Division of integers, Rounding, Significant figures, 30 Decimal places, Trailing zeros, Fractions as decimals, Decimals as fractions, Unending decimals, Unending decimals as fractions, Rational, irrational and real numbers Revision summary 35 Revision exercise 35 Powers ± Raising a number to a power, The laws ofpowers, Powers on a 36 calculator, Fractional powers and roots, Surds, Multiplication and division by integer powers of10, Precedence rules, Standard form, Working in standard form, Using a calculator, Preferred standard form, Checking calculations, Accuracy Revision summary 49 Revision exercise 50 Number systems ± Denary, Binary, Octal, Duodecimal, Hexadecimal 51 Revision summary 55 Revision exercise 58 iv Contents Change of base ± Denary to binary, octal and duodecimal, Use ofoctals 60 as an intermediate step Reverse method 65 Revision summary 66 Revision exercise 66 Can You? Checklist F.1 67 Test exercise F.1 68 Further problems F.1 70 Programme F.2 Introduction to algebra 73 Learning outcomes 73 Quiz F.2 74 Algebraic expressions ± Symbols other than numerals, Constants, 76 Variables, Rules of algebra, Rules of precedence, Terms and coefficients, Collecting like items, Similar terms, Expanding brackets, Nested brackets Revision summary 82 Revision exercise 82 Powers ± Powers, Rules ofindices 84 Logarithms ± Powers, Logarithms, Rules oflogarithms, Base 10 and 86 base e, Change ofbase, Logarithmic equations Revision summary 93 Revision exercise 94 Multiplication of algebraic expressions of a single variable 95 Fractions ± Algebraic fractions, Additions and subtraction, 96 Multiplication and division Division of one expression by another 99 Revision summary 101 Revision exercise 101 Factorization of algebraic expressions ± Common factors, Common 103 factors by grouping, Useful products of two simple factors, Quadratic expressions as the product oftwo simple factors,Factorization ofa quadratic expression, Test for simple factors Revision summary 112 Revision exercise 112 Can You? Checklist F.2 114 Test exercise F.2 115 Further problems F.2 116 Programme F.3 Expressions and equations 119 Learning outcomes 119 Quiz F.3 120 Expressions and equations ± Evaluating expressions, Equations, 121 Evaluating independent variables, Transposition offormulas Revision summary 130 Revision exercise 131 The evaluation process ± Systems 132 Polynomial equations ± Polynomial expressions, Evaluation of 133 polynomials, Evaluation ofa polynomial by nesting Remainder theorem ± Factor theorem 135 Factorization of quartic polynomials 140 Contents v Revision summary 143 Revision exercise 144 Can You? Checklist F.3 145 Test exercise F.3 145 Further problems F.3 146 Programme F.4 Graphs 149 Learning outcomes 149 Quiz F.4 150 Graphs of equations ± Equations, Ordered pairs ofnumbers, Cartesian 151 axes, Drawing a graph Revision summary 158 Revision exercise 159 Using a spreadsheet ± Spreadsheets, Row and columns, Text and number 160 entry, Formulas, Clearing entries, Construction ofa Cartesian graph Revision summary 170 Revision exercise 170 Inequalities ± Less than or greater than 172 Revision summary 173 Revision exercise 173 Can You? Checklist F.4 174 Test exercise F.4 174 Further problems F.4 175 Programme F.5Linear equations and simultaneous 177 linear equations Learning outcomes 177 Quiz F.5 178 Linear equations ± Solution ofsimple equations 179 Simultaneous linear equations withtwo unknowns ± Solution by 182 substitution, Solution by equating coefficients Simultaneous linear equations withthreeunknowns ± 185 Pre-simplification Revision summary 190 Revision exercise 191 Can You? Checklist F.5 193 Test exercise F.5 193 Further problems F.5 194 Programme F.6 Polynomial equations 197 Learning outcomes 197 Quiz F.6 198 Polynomial equations ± Quadratic equations 199 Solution of cubic equations having at least one linear factor 205 Solution of quartic equations having at least two linear factors 209 Revision summary 213 Revision exercise 214 Can You? Checklist F.6 216 Test exercise F.6 216 Further problems F.6 217 vi Contents Programme F.7 Partial fractions 219 Learning outcomes 219 Quiz F.7 220 Partial fractions 221 Revision summary 227 Revision exercise 228 Denominators withrepeated and quadratic factors 229 Revision summary 237 Revision exercise 237 Can You? Checklist F.7 238 Test exercise F.7 238 Further problems F.7 239 Programme F.8 Trigonometry 241 Learning outcomes 241 Quiz F.8 242 Angles ± Rotation, Radians, Triangles, Trigonometric ratios, Reciprocal 243 ratios, Pythagoras' theorem, Special triangles, Halfequilateral Revision summary 256 Revision exercise 257 Trigonometric identities ± The fundamental identity, Two more 258 identities, Identities for compound angles Trigonometric formulas ± Sums and differences of angles, Double 263 angles, Sums and differences of ratios, Products of ratios Revision summary 264 Revision exercise 264 Can You? Checklist F.8 266 Test exercise F.8 266 Further problems F.8 267 Programme F.9 Binomial series 269 Learning outcomes 269 Quiz F.9 270 Factorials and combinations ± Factorials, Combinations, Three 271 properties of combinatorial coefficients Revision summary 278 Revision exercise 278 Binomial series ± Pascal's triangle, Binomial expansions, The general 279 term ofthe binomial expansion Revision summary 284 RevisionP exercise 285 The (sigma) notation ± General terms, The sum ofthe first n natural 286 numbers, Rules for manipulating sums The exponential number e 292 Revision summary 293 Revision exercise 294 Can You? Checklist F.9 295 Test exercise F.9 296 Further problems F.9 296 Contents vii Programme F.10 Differentiation 299 Learning outcomes 299 Quiz F.10 300 The gradient of a straight-line graph 301 The gradient of a curve at a given point 304 Algebraic determination of the gradient of a curve 306 Derivatives of powers of x ± Two straight lines, Two curves 308 Differentiation of polynomials 311 Derivatives ± an alternative notation 313 Revision summary 313 Revision exercise 314 sin Second derivatives ± Limiting value of as ! 0, Standard 315 derivatives Differentiation of products of functions 319 Differentiation of a quotient of two functions 320 Functions of a function ± Differentiation of a function of a 324 function Revision summary 328 Revision exercise 328 Can You? Checklist F.10 331 Test exercise F.10 331 Further problems F.10 332 Programme F.11 Integration 333 Learning outcomes 333 Quiz F.11 334 Integration ± Constant ofintegration 335 Standard integrals 336 Revision summary 337 Revision exercise 337 Integration of polynomial expressions 338 Functions of a linear function of x 340 Revision summary 341 Revision exercise 341 Integration by partial fractions 343 Revision summary 346 Revision exercise 346 Areas under curves 347 Revision summary 351 Revision exercise 351 Integration as a summation ± The area between a curve and an 352 intersecting line Revision summary 359 Revision exercise 359 Can You? Checklist F.11 363 Test exercise F.11 364 Further problems F.11 365 viii Contents Programme F.12 Functions 367 Learning outcomes 367 Quiz F.12 368 Processing numbers ± Functions are rules but not all rules are 370 functions, Functions and the arithmetic operations, Inverses of 3 1 functions, Graphs of inverses, The graph of y x , The graph of y x3, 3 1 The graphs of y x and y x3 plotted together Revision summary 381 Revision exercise 381 Composition ± `function of a function' ± Inverses ofcompositions 382 Trigonometric functions ± Rotation, The tangent, Period, Amplitude, 387 Phase difference, Inverse trigonometric functions, Trigonometric equations, Equations ofthe form a cos x b sin x c Revision summary 400 Revision exercise 401 Exponential and logarithmic functions ± Exponential functions, 403 Indicial equations Odd and even functions ± Odd and even parts, Odd and even parts of 407 the exponential function Revision summary 411 Revision exercise 411 Can You? Checklist F.12 413 Test exercise F.12 414 Further problems F.12 415 Part II 417 Programme 1 Complex numbers 1 419 Learning outcomes 419 Introduction ± Ideas and symbols 420 The symbol j ± Quadratic equations 420 Powers of j 422 Complex numbers 423 Revision exercise 427 Equal complex numbers 430 Graphical representation of a complex number 431 Graphical addition of complex numbers 434 Polar form of a complex number 436 Exponential form of a complex number 440 Revision summary 443 Can You? Checklist 1 444 Test exercise 1 445 Further problems 1 445 Programme 2 Complex numbers 2 447 Learning outcomes 447 Introduction
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