CAPE Pure Mathematics Syllabus, Specimen Papers, Mark .

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Macmillan Education4 Crinan Street, London, N1 9XWA division of Macmillan Publishers LimitedCompanies and representatives throughout the worldwww.macmillan-caribbean.comISBN 978-0-230-48247-0 Caribbean Examinations Council (CXC ) 2015www.cxc.orgwww.cxc-store.comThe author has asserted their right to be identified as the author of this work in accordance with theCopyright, Design and Patents Act 1988.First published 2014This revised version published 2015Permission to copyThe material in this book is copyright. However, the publisher grants permission for copies to bemade without fee. Individuals may make copies for their own use or for use by classes of which theyare in charge; institutions may make copies for use within and by the staff and students of thatinstitution. For copying in any other circumstances, prior permission in writing must be obtainedfrom Macmillan Publishers Limited. Under no circumstances may the material in this book be used,in part or in its entirety, for commercial gain. It must not be sold in any format.Designed by Macmillan Publishers LimitedCover design by Macmillan Publishers Limited and Red Giraffe

CAPE Pure Mathematics Free ResourcesLIST OF CONTENTSCAPE Pure Mathematics Syllabus Extract3CAPE Pure Mathematics Syllabus4CAPE Pure Mathematics Specimen Papers:Unit 1 Paper 01Unit 1 Paper 02Unit 1 Paper 032Unit 2 Paper 01Unit 2 Paper 02Unit 2 Paper 032687682869499CAPE Pure Mathematics Mark Schemes:Unit 1 Paper 01Unit 1 Paper 02Unit 1 Paper 032Unit 2 Paper 01Unit 2 Paper 02Unit 2 Paper 032103105118123125138CAPE Pure Mathematics Subject Reports:2004 Subject Report2005 Subject Report2006 Subject Report2007 Subject Report2008 Subject Report Trinidad and Tobago2008 Subject Report Rest of Caribbean2009 Subject Report2010 Subject Report2011 Subject Report2012 Subject Report2013 Subject Report2014 Subject Report143181213243275294313332355379402422

Pure MathematicsMathematics is one of the oldest and most universal means of creating, communicating,connecting and applying structural and quantitative ideas. Students doing this syllabuswill have already been exposed to Mathematics in some form mainly through coursesthat emphasise skills in using mathematics as a tool, rather than giving insight into theunderlying concepts.This syllabus will not only provide students with more advanced mathematical ideas,skills and techniques, but encourage them to understand the concepts involved, whyand how they "work" and how they are interconnected. It is also to be hoped that, in thisway, students will lose the fear associated with having to learn a multiplicity of seeminglyunconnected facts, procedures and formulae. In addition, the course should showthem that mathematical concepts lend themselves to generalisations, and that thereis enormous scope for applications to solving real problems. The course is thereforeintended to provide quality in selected areas rather than in a large number of topics.The syllabus is arranged into two (2) Units, each Unit consists of three Modules.Unit 1: Algebra, Geometry and CalculusModule 1–Basic Algebra and FunctionsModule 2–Trigonometry, Geometry and VectorsModule 3–Calculus IUnit 2: Complex Numbers, Analysis and MatricesModule 1–Complex Numbers and Calculus IIModule 2–Sequences, Series and ApproximationsModule 3–Counting, Matrices and Differential Equations

CARIBBEAN EXAMINATIONS COUNCILCaribbean Advanced Proficiency Examination CAPE PURE MATHEMATICSEffective for examinations from May–June 2013CXC A6/U2/12

Published by the Caribbean Examinations Council 2012, Caribbean Examinations CouncilAll rights reserved. No part of this publication may be reproduced, stored in a retrieval system, ortransmitted in any form, or by any means electronic, photocopying, recording or otherwise withoutprior permission of the author or publisher.Correspondence related to the syllabus should be addressed to:The Pro-RegistrarCaribbean Examinations CouncilCaenwood Centre37 Arnold Road, Kingston 5, JamaicaTelephone Number: 1(876) 630-5200Facsimile Number: 1(876) 967-4972E-mail Address: cxcwzo@cxc.orgWebsite: www.cxc.orgCopyright 2012 by Caribbean Examinations CouncilThe Garrison, St Michael BB14038, BarbadosCXC A6/U2/12

This document CXC A6/U2/12 replaces CXC A6/U2/07 issued in 2007.Please note that the syllabus has been amended and amendments are indicated by italics.First issued 1999Revised 2004Revised 2007Amended 2012Please check the website www.cxc.org for updates on CXC’s syllabuses.CXC A6/U2/12

ContentsContentsIntroductionINTRODUCTION . iRATIONALE . 1AIMS . 2SKILLS AND ABILITIES TO BE ASSESSED . 3PRE-REQUISITES OF THE SYLLABUS. 3STRUCTURE OF THE SYLLABUS . 3RECOMMENDED 2-UNIT OPTIONS. 4MATHEMATICAL MODELLING . 4UNIT 1: ALGEBRA, GEOMETRY AND CALCULUSMODULE 1 : BASIC ALGEBRA AND FUNCTIONS . 7MODULE 2 : TRIGONOMETRY, GEOMETRY AND VECTORS. 16MODULE 3 : CALCULUS I . 21UNIT 2: COMPLEX NUMBERS, ANALYSIS AND MATRICESMODULE 1 : COMPLEX NUMBERS AND CALCULUS II. 27MODULE 2 : SEQUENCES, SERIES AND APPROXIMATIONS . 33MODULE 3 : COUNTING, MATRICES AND DIFFERENTIAL EQUATIONS . 40OUTLINE OF ASSESSMENT . 46REGULATIONS FOR PRIVATE CANDIDATES . 54REGULATIONS FOR RE-SIT CANDIDATES . 54ASSESSMENT GRID . 55MATHEMATICAL NOTATION . 56CXC A6/U2/12

IntroductionThe Caribbean Advanced Proficiency Examination (CAPE) is designed to provide certification of theacademic, vocational and technical achievement of students in the Caribbean who, havingcompleted a minimum of five years of secondary education, wish to further their studies. Theexaminations address the skills and knowledge acquired by students under a flexible and articulatedsystem where subjects are organised in 1-Unit or 2-Unit courses with each Unit containing threeModules. Subjects examined under CAPE may be studied concurrently or singly.The Caribbean Examinations Council offers three types of certification. The first is the award of acertificate showing each CAPE Unit completed. The second is the CAPE Diploma, awarded tocandidates who have satisfactorily completed at least six Units including Caribbean Studies. The thirdis the CXC Associate Degree, awarded for the satisfactory completion of a prescribed cluster of sevenCAPE Units including Caribbean Studies and Communication Studies. For the CAPE Diploma and theCXC Associate Degree, candidates must complete the cluster of required Units within a maximumperiod of five years.Recognised educational institutions presenting candidates for CXC Associate Degree in one of thenine categories must, on registering these candidates at the start of the qualifying year, have themconfirm in the required form, the Associate Degree they wish to be awarded. Candidates will not beawarded any possible alternatives for which they did not apply.CXC A6/U2/12i

Pure Mathematics Syllabus RATIONALEMathematics is one of the oldest and most universal means of creating, communicating, connectingand applying structural and quantitative ideas. The discipline of Mathematics allows the formulationand solution of real-world problems as well as the creation of new mathematical ideas, both as anintellectual end in itself, as well as a means to increase the success and generality of mathematicalapplications. This success can be measured by the quantum leap that occurs in the progress made inother traditional disciplines once mathematics is introduced to describe and analyse the problemsstudied. It is therefore essential that as many persons as possible be taught not only to be able touse mathematics, but also to understand it.Students doing this syllabus will have already been exposed to Mathematics in some form mainlythrough courses that emphasise skills in using mathematics as a tool, rather than giving insight intothe underlying concepts. To enable students to gain access to mathematics training at the tertiarylevel, to equip them with the ability to expand their mathematical knowledge and to make properuse of it, it is necessary that a mathematics course at this level should not only provide them withmore advanced mathematical ideas, skills and techniques, but encourage them to understand theconcepts involved, why and how they "work" and how they are interconnected. It is also to behoped that, in this way, students will lose the fear associated with having to learn a multiplicity ofseemingly unconnected facts, procedures and formulae. In addition, the course should show themthat mathematical concepts lend themselves to generalisations, and that there is enormous scopefor applications to solving real problems.Mathematics covers extremely wide areas. However, students can gain more from a study ofcarefully selected, representative areas of Mathematics, for a "mathematical" understanding ofthese areas, rather than a superficial overview of a much wider field. While proper exposure to amathematical topic does not immediately make students into experts in it, proper exposure willcertainly give the students the kind of attitude which will allow them to become experts in othermathematical areas to which they have not been previously exposed. The course is thereforeintended to provide quality in selected areas rather than in a large number of topics.This syllabus will contribute to the development of the Ideal Caribbean Person as articulated by theCARICOM Heads of Government in the following areas: “demonstate multiple literacies, independentand critical thinking and innovative application of science and technology to problem solving. Such aperson should also demonstrate a positive work attitude and value and display creative imaginationand entrepreneurship”.CXC A6/U2/121

AIMSThe syllabus aims to:1.provide understanding of mathematical concepts and structures, their development and therelationships between them;2.enable the development of skills in the use of mathematical and information,communication and technology (ICT) tools;3.develop an appreciation of the idea of mathematical proof, the internal logical coherence ofMathematics, and its consequent universal applicability;4.develop the ability to make connections between distinct concepts in Mathematics, andbetween mathematical ideas and those pertaining to other disciplines;5.develop a spirit of mathematical curiosity and creativity, as well as a sense of enjoyment;6.enable the analysis, abstraction and generalisation of mathematical ideas;7.develop in students the skills of recognising essential aspects of concrete, real-worldproblems, formulating these problems into relevant and solvable mathematical problemsand mathematical modelling;8.develop the ability of students to carry out independent or group work on tasks involvingmathematical modelling;9.integrate ICT tools and skills;10.provide students with access to more advanced courses in Mathematics and its applicationsat tertiary institutions.CXC A6/U2/122