PURE MATHEMATICS Unit 1 - Macmillan Caribbean EBooks

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PURE MATHEMATICS Unit 1FOR CAPE EXAMINATIONSDIPCHAND BAHALLCAPE is a registered trade mark of the Caribbean Examinations Council (CXC). PureMathematics for CAPE Examinations Unit 1 is an independent publication and hasnot been authorised, sponsored, or otherwise approved by CXC.

Macmillan Education4 Crinan Street, London N1 9XWA division of Macmillan Publishers LimitedCompanies and representatives throughout the worldwww.macmillan-caribbean.comISBN 978-0-2304-6575-6 AERText Dipchand Bahall 2013Design and illustration Macmillan Publishers Limited 2013First published in 2013All rights reserved; no part of this publication may bereproduced, stored in a retrieval system, transmitted in anyform or by any means, electronic, mechanical, photocopying,recording, or otherwise, without the prior written permissionof the publishers.These materials may contain links for third party websites. We have no control over, andare not responsible for, the contents of such third party websites. Please use care whenaccessing them.Designed by Tech Type and Oxford Designers and IllustratorsTypeset and illustrated by MPS LimitedCover design by Clare WebberCover photo: Alamy/Science Photo Library0800022 FM.indd 26/26/13 2:31 PM

ContentsINTRODUCTIONxiiMATHEMATICAL MODELLINGxiiiMODULE 1 BASIC ALGEBRA AND FUNCTIONSCHAPTER 1CHAPTER 2REASONING AND LOGIC2Notation4Simple statement4Negation4Truth tables4Compound statements5Connectives6Conjunction6Disjunction (‘or’)7Conditional statements11Interpretation of p q12The contrapositive12Converse13Inverse13Equivalent propositions14Biconditional statements15Tautology and contradiction17Algebra of propositions18THE REAL NUMBER SYSTEM24Subsets of rational numbers25Real numbers26Operations26Binary 8Distributivity29Identity30Inverse31Constructing simple proofs in mathematics33Proof by exhaustion33Direct proof33Proof by contradiction35Proof by counter example36iii

CHAPTER 3PRINCIPLE OF MATHEMATICAL INDUCTIONSequences and seriesFinding the general term of a seriesSigma notationExpansion of a seriesStandard resultsSummation resultsMathematical inductionDivisibility tests and mathematical induction444545474748495357CHAPTER 4POLYNOMIALSReview of polynomialsDegree or order of polynomials626363Algebra of polynomialsEvaluating polynomialsRational expressionsComparing polynomialsRemainder theoremThe factor theoremFactorising polynomials and solving equationsFactorising xn yn63646465697477CHAPTER 5INDICES, SURDS AND LOGARITHMSIndicesLaws of indicesSurdsRules of surdsSimplifying surdsConjugate surdsRationalising the denominatorExponential functionsGraphs of exponential functionsThe number eExponential equationsLogarithmic functionsConverting exponential expressions tologarithmic expressionsChanging logarithms to exponents using thedefinition of logarithmProperties of 07Solving logarithmic equations108Equations involving exponents110Change of base formula (change to base b from base a)113Logarithms and exponents in simultaneous equations115

CHAPTER 6Application problems117Compound interest118Continuous compound interest120FUNCTIONS128Relations and functions129Describing a function130The vertical line test130One-to-one function (injective function)132Onto function (surjective function)134Bijective functions137Inverse functions139Graphs of inverse functionsOdd and even functions144Odd functions144Even functions144Periodic functions144The modulus function145Graph of the modulus functionComposite functions145146Relationship between inverse functionsIncreasing and decreasing functionsCHAPTER 7141149152Increasing functions152Decreasing functions152Transformations of graphs153Vertical translation153Horizontal translation154Horizontal stretch155Vertical stretch157Reflection in the x-axis158Reflection in the y-axis158Graphs of simple rational functions160Piecewise defined functions162CUBIC POLYNOMIALS171Review: Roots of a quadratic and the coefficient of thequadratic172Cubic equations173NotationFindingα3175 β3 γ3,using a formulaFinding a cubic equation, given the roots ofthe equation175176v

CHAPTER 8INEQUALITIES AND THE MODULUS FUNCTION185Theorems of inequalities186Quadratic inequalities186Sign table188Rational functions and inequalities191General results about the absolute value function196Square root ofx2201The triangle inequality201Applications problems for inequalities203MODULE 1 TESTS208MODULE 2 TRIGONOMETRY AND PLANE GEOMETRYCHAPTER 9TRIGONOMETRY212Inverse trigonometric functions and graphs213Inverse sine function213Inverse cosine function213Inverse tangent function214Solving simple trigonometric equationsGraphical solution of sin x k214Graphical solution of cos x k216Graphical solution of tan x k217Trigonometrical identitiesReciprocal identitiesPythagorean identitiesProving identitiesSolving trigonometric equations218218219220224Further trigonometrical identities229Expansion of sin (A B)229Expansion of cos (A B)230Expansion of tan (A B)234Double-angle formulae236Half-angle formulae238Proving identities using the addition theoremsand the double-angle formulae238The form a cos θ b sin θ241Solving equations of the form a cos θ b sin θ c244Equations involving double-angle or half-angle formulae249Products as sums and differences253Converting sums and differences to products254Solving equations using the sums and differences as productsvi214258

CHAPTER 10COORDINATE GEOMETRY266Review of coordinate geometry267The equation of a circle267Equation of a circle with centre (a, b) and radius r268General equation of the circle269Intersection of a line and a circle275Intersection of two circles276Intersection of two curves277Parametric representation of a curve278Cartesian equation of a curve given its parametric form279Parametric equations in trigonometric form280Parametric equations of a circle282Conic sections285Ellipses286Equation of an ellipse286Equation of an ellipse with centre (h, k)289Focus–directrix property of an ellipse291Parametric equations of ellipses291Equations of tangents and normals to an ellipse293ParabolasCHAPTER 11294Equation of a parabola295Parametric equations of parabolas296Equations of tangents and normals to a parabola296VECTORS IN THREE DIMENSIONS (ℝ3)303Vectors in 3D304Plotting a point in three dimensionsAlgebra of vectors304304Addition of vectors304Subtraction of vectors305Multiplication by a scalar305Equality of vectors305Magnitude of a vector306Displacement vectors306Unit vectors307Special unit vectors308Scalar product or dot product309Properties of the scalar product310Angle between two vectors310Perpendicular and parallel vectors312Perpendicular vectors312Parallel vectors313vii

Equation of a line316Finding the equation of a line given a point on a lineand the direction of the line316Finding the equation of a line given two points on the line317Vector equation of a line319Parametric equation of a line319Cartesian equation of a line320Finding the angle between two lines, given the equationsof the lines322Skew lines323Equation of a plane326Equation of a plane, given the distance from theorigin to the plane and a unit vector perpendicularto the plane327Equation of a plane, given a point on the plane and anormal to the plane328Cartesian equation of a plane330MODULE 2 TESTS338MODULE 3 CALCULUS ICHAPTER 12LIMITS AND CONTINUITY342Limits343The existence of a limit345Limit laws345Evaluating limits347Direct substitution347Factorising method349Conjugate method350Tending to infinityLimits at infinitySpecial limitsContinuityTypes of discontinuityInfinite discontinuityPoint discontinuityJump discontinuity351352354358359359359359Removable and non-removable discontinuityCHAPTER 13viii360DIFFERENTIATION 1366DifferentiationThe difference quotientExistence of a derivative367368368

CHAPTER 14Notation for derivativesInterpretations of derivativesFinding derivatives using first principlesDifferentiation of ag(x) where a is a constantDifferentiation of sums and differences of functionsFirst principle and sums and differences of functions of xRate of changeChain ruleProduct ruleQuotient ruleDifferentiation of trigonometric functionsHigher LICATIONS OF DIFFERENTIATION399Tangents and normalsEquations of tangents and normalsIncreasing and decreasing functionsStationary points/second derivativesMaximum and minimum valuesStationary pointsClassification of turning pointsFirst derivative testSecond derivative testInflexion pointsPractical maximum and minimum problemsParametric differentiationRate of changeCurve sketchingPolynomials, rational functions, trigonometricfunctionsGraph of a polynomialGraphs of functions of the form f(x) xn where n is aneven integerGraphs of functions of the form f(x) xn where n is anodd integer greater than 1Graphs of polynomialsZeros of a polynomialGraphing 0430434434435435438440Graphing functions with a table of values440Solving simultaneous equations graphically444Solving inequalities graphically447Review of trigonometrySine, cosine and tangent of 45 , 30 and 60 449449ix

xGraph of cosec x450Graph of sec x451Graph of cot x451Properties and graphs of trigonometric functions452Transformations of trigonometric functions456y a sin (bx) c and y a cos (bx) c459y a tan (bx) c460Graphs of rational functionsVertical asymptotesHorizontal asymptotesSketching graphs of rational functionsShape of a curve for large values of the independentvariable461462462463CHAPTER 15INTEGRATIONAnti-derivatives (integrations)The constant of integrationIntegrals of the form axnIntegration theoremsIntegration of polynomial functionsIntegration of a function involving a linear factorIntegration of trigonometric functionsIntegration of more trigonometric functionsIntegrating sin2 x and cos2 xIntegration of products of sines and cosinesThe definite integralIntegration by substitutionSubstituting with limitsThe equation of a CHAPTER 16APPLICATIONS OF INTEGRATIONApproximating the area under a curve, using rectanglesEstimating the area under a curve using n rectanglesof equal widthUsing integration to find the area under a curveArea between two curvesArea below the x-axisArea between the curve and the y-axisVolume of solids of revolutionRotation about the x-axisRotation about the y-axisVolume generated by the region bounded bytwo curves509510466512514516517519524524527530

CHAPTER 17DIFFERENTIAL EQUATIONSFamilies of curvesClassifying differential equationsLinear versus non-linear differential equationsPractical applications of differential equationsFirst order differential equationsSolutions of variable-separable differential equationsModelling problemsSecond order differential equations543544544544544545547549552MODULE 3 TESTS558UNIT 1—MULTIPLE CHOICE TESTS561INDEX575Answers are available online atwww.macmillan-caribbean.com/resourcesxi

IntroductionThese two volumes provide students with an understanding of pure mathematicsat the CAPE level taken from both a theoretical and an application aspect andencourage the learning of mathematics. They provide the medium throughwhich a student can find problems applied to different disciplines. The conceptsare developed step by step; they start from the basics (for those who did not doadditional mathematics) and move to the more advanced content areas, therebysatisfying the needs of the syllabus. Examination questions all seem to have answersthat are considered ‘nice’ whole numbers or small fractions that are easy to workwith; not all real-world problems have such answers and these books have avoidedthat to some extent. Expect any kind of numbers for your answers; there are nostrange or weird numbers.The objectives are outlined at the beginning of each chapter, followed by thekeywords and terms that a student should be familiar with for a better understandingof the subject. Every student should have a section of their work book for thelanguage of the subject. I have met many students who do not understand terms suchas ‘root’ and ‘factor’. A dictionary developed in class from topic to topic may assist thestudents in understanding the terms involved. Each objective is fulfilled throughoutthe chapters with examples clearly explained. Mathematical modelling is a conceptthat is developed throughout, with each chapter containing the relevant modellingquestions.The exercises at the end of each section are graded in difficulty and have adequateproblems so that a student can move on once they feel comfortable with the concepts.Additionally, review exercises give the student a feel for solving problems that arevaried in content. There are three multiple choice papers at the end of each Unit,and at the end of each module there are tests based on that module. For additionalpractice, the student can go to the relevant past papers and solve the problems given.After going through the questions in each chapter, a student should be able to do pastpaper questions from different examining boards for further practice.A checklist at the end of each chapter enables the student to note easily what isunderstood and to what extent. A student can identify areas that need work withproper use of this checklist. Furthermore, each chapter is summarised as far aspossible as a diagram. Students can use this to revise the content that was covered inthe chapter.The text provides all the material that is needed for the CAPE syllabus so thatteachers will not have to search for additional material. Both new and experiencedteachers will benefit from the text since it goes through the syllabus chapter bychapter and objective to objective. All objectives in the syllabus are dealt within detail and both students and teachers can work through the text, comfortablyknowing that the content of the syllabus will be covered.xii

PURE MATHEMATICS Unit 1 FOR CAPE EXAMINATIONS DIPCHAND BAHALL CAPE is a registered trade mark of the Caribbean Examinations Council (CXC). Pure Mathematics for CAPE Examinations Unit 1 is an independent publication and has not

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