CAPE Integrated Mathematics - Jhtm.nl
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Macmillan Education4 Crinan Street, London, N1 9XWA division of Macmillan Publishers LimitedCompanies and representatives throughout the worldwww.macmillan-caribbean.comISBN 978-0-230-49858-7 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 withthe Copyright, Design and Patents Act 1988.First 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 whichthey are in charge; institutions may make copies for use within and by the staff and students ofthat institution. For copying in any other circumstances, prior permission in writing must beobtained from Macmillan Publishers Limited. Under no circumstances may the material in this bookbe 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 Integrated Mathematics Free ResourcesLIST OF CONTENTSCAPE Integrated Mathematics Syllabus Extract4CAPE Integrated Mathematics Syllabus5CAPE Integrated Mathematics Specimen Papers and Mark Schemes/KeysPaper 01 Specimen Paper64Paper 02 Specimen Paper78Paper 032 Specimen Paper114Paper 01 Mark Schemes and Key77Paper 02 Mark Schemes and Key98Paper 032 Mark Schemes and Key124
Integrated MathematicsMathematics promotes intellectual development, is utilitarian and applicable to all disciplines.Additionally, its aesthetics and epistemological approaches provide solutions fit for any purpose.Therefore, Mathematics is the essential tool to empower people with the knowledge, competenciesand attitudes which are precursors for this dynamic world.This course is designed for all students pursuing CXC associate degree programme, with specialemphasis to those who do not benefit from the existing intermediate courses that cater primarilyfor mathematics career options. It will provide these students with the knowledge and skills setsrequired to model practical situations and provide workable solutions in their respective field ofstudy. These skills include critical and creative thinking, problem solving, logical reasoning,modelling ability, team work, decision making, research techniques, information communicationand technological competencies for life-long learning. Such holistic development becomes usefulfor the transition into industry as well as research and further studies required at tertiary levels.Moreover, the attitude and discipline which accompany the study of Mathematics also nurturedesirable character qualities.The Integrated Mathematics Syllabus comprises three Modules:Module 1-Foundations of MathematicsModule 2-StatisticsModule 3-Calculus
CARIBBEAN EXAMINATIONS COUNCILCar ib b e an Ad v an ce d Pr of icie n cy Ex am in at ion CAPE INTEGRATED MATHEMATICSSYLLABUSEffective for examinations from May–June 2016CXC A35/U1/15
Published by the Caribbean Examinations Council.All 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 2015 by Caribbean Examinations CouncilPrince Road, Pine Plantation Road, St Michael BB11091CXC A35/U1/15
ContentsINTRODUCTION . iRATIONALE . 1AIMS . 1SKILLS AND ABILITIES TO BE ASSESSED. 2PREREQUISITES OF THE SYLLABUS . 2STRUCTURE OF THE SYLLABUS . 3RESOURCES FOR ALL MODULES . 4MODULE 1: FOUNDATIONS OF MATHEMATICS . 5MODULE 2: STATISTICS . 11MODULE 3: CALCULUS . 17OUTLINE OF ASSESSMENT . 22REGULATIONS FOR PRIVATE CANDIDATES . 36REGULATIONS FOR RESIT CANDIDATES . 37ASSESSMENT GRID. 37APPENDIX I – GLOSSARY OF EXAMINATION TERMS . 38APPENDIX II – GLOSSARY OF MATHEMATICAL TERMS . 42SPECIMEN PAPERS . 55CXC A35/U1/15
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 at the CAPE level. The first isthe award of a certificate showing each CAPE Unit completed. The second is the CAPE Diploma,awarded to candidates who have satisfactorily completed at least six Units, including CaribbeanStudies. The third is the CXC Associate Degree, awarded for the satisfactory completion of a prescribedcluster of eight CAPE Units including Caribbean Studies, Communication Studies and IntegratedMathematics. Integrated Mathematics is not a requirement for the CXC Associate Degree inMathematics. The complete list of Associate Degrees may be found in the CXC Associate DegreeHandbook.For the CAPE Diploma and the CXC Associate Degree, candidates must complete the cluster of requiredUnits within a maximum period of five years. To be eligible for a CXC Associate Degree, the educationalinstitution presenting the candidates for the award, must select the Associate Degree of choice at thetime of registration at the sitting (year) the candidates are expected to qualify for the award.Candidates will not be awarded an Associate Degree for which they were not registered.iCXC A35/U1/15
Integrated Mathematics Syllabus RATIONALEThe Caribbean society is an integral part of an ever-changing world. The impact of globalisation on mostsocieties encourages this diverse Caribbean region to revisit the education and career opportunities of ourcurrent and future citizens. A common denominator is for Caribbean societies to create among its citizensa plethora of quality leadership with the acumen required to make meaningful projections and innovationsfor further development. Further, learning appropriate problem-solving techniques, inherent to the studyof mathematics, is vital for such leaders. Mathematics promotes intellectual development, is utilitarianand applicable to all disciplines. Additionally, its aesthetics and epistemological approaches providesolutions fit for any purpose. Therefore, Mathematics is the essential tool to empower people with theknowledge, competencies and attitudes which are required for academia as well as quality leadership forsustainability in this dynamic world.The Integrated Mathematics course of study will provide students with the knowledge and skills setsrequired to model practical situations and provide workable solutions in their respective field of study.These skills include critical and creative thinking, problem solving, logical reasoning, modelling ability,team work, decision making, research techniques, information communication and technologicalcompetencies for life-long learning. Such holistic development becomes useful for the transition intoindustry research and further studies required at tertiary levels. Moreover, the attitude and disciplinewhich accompany the study of Mathematics also nurture desirable character qualities.This syllabus will contribute to the development of the Ideal Caribbean Person as articulated by theCARICOM Heads of Government in the following areas: “demonstrate multiple literacies, independent andcritical thinking and innovative application of science and technology to problem solving. Such a personshould also demonstrate a positive work attitude and value and display creative imagination andentrepreneurship”. In keeping with the UNESCO Pillars of Learning, on completion of this course ofstudy, students will learn to do, learn to be and learn to transform oneself and society. AIMSThis syllabus aims to:1.improve on the mathematical knowledge, skills and techniques with an emphasis on accuracy;2.empower students with the knowledge, competencies and attitudes which are precursors foracademia as well as quality leadership for sustainability in the dynamic world;3.provide students with the proficiencies required to model practical situations and provideworkable solutions in their respective fields of work and study;CXC A35/U1/151
4.develop competencies in critical and creative thinking, problem solving, logical reasoning,modelling, team work, decision making, research techniques and information communication andtechnology for life-long learning;5.nurture desirable character qualities that include self-confidence, self-esteem, ethics andemotional security;6.make Mathematics interesting, recognisable and relevant to the students locally, regionally andglobally. SKILLS AND ABILITIES TO BE ASSESSEDThe skills and abilities that students are expected to develop on completion of this syllabus have beengroup under three headings:(a)Conceptual Knowledge;(b)Algorithmic Knowledge; and,(c)Reasoning.Conceptual KnowledgeThe examination will test candidates’ ability to recall, select and use appropriate facts, concepts andprinciples in a variety of contexts.Algorithmic KnowledgeThe examination will test candidates’ ability to manipulate mathematical expressions and proceduresusing appropriate symbols and language, logical deduction and inferences.ReasoningThe examination will test candidates’ ability to select appropriate strategy or select, use and evaluatemathematical models and interpret the results of a mathematical solution in terms of a given realworld problem and engage in problem-solving. PREREQUISITES OF THE SYLLABUSAny person with a good grasp of the contents of the syllabus of the Caribbean Secondary EducationCertificate (CSEC) General Proficiency course in Mathematics, or equivalent, should be able toundertake the course. However, successful participation in the course will also depend on thepossession of good verbal and written communication skills.CXC A35/U1/152
STRUCTURE OF THE SYLLABUSThe Integrated Mathematics Syllabus comprises three Modules, each requiring at least 50 hours. Studentswill develop the skills and abilities identified through the study of:Module 1-FOUNDATIONS OF MATHEMATICSModule 2-STATISTICSModule 3-CALCULUSCXC A35/U1/153
RESOURCES FOR ALL MODULESBackhouse, J.K., and Houldsworth, S.P.T.Pure Mathematics Book 1: A First Course. London: LongmanGroup Limited, 1981.Bostock, L., and Chandler, S.Core Maths for Advanced Level 3rd Edition. London: StanleyThornes (Publishers) Limited, 2000.Campbell, E.Pure Mathematics for CAPE: Volume 1, Kingston: LMHPublishing Limited. 2007.Dakin, A., and Porter, R.I.Elementary Analysis. London: Collins Educational, 1991.Hartzler, J.S., and Swetz, F.Mathematical Modelling in the Secondary School Curriculum:A Resource Guide of Classroom Exercises. Vancouver: NationalCouncil of Teachers of Mathematics, Incorporated, Reston,1991.Martin, A., Brown, K., Rigsby, P., and Advanced Level Mathematics Tutorials Pure Maths CDRidley, S.ROM (Trade Edition), Multi-User Version and Single Userversion. Cheltenham: Stanley Thornes (Publishers) Limited,2000.Stewart, J.Calculus 7th Edition. Belmont: Cengage Learning, 2011Talbert, J.F., and Heng, H.H.Additional Mathematics Pure and Applied 6th Edition.Singapore: Pearson Educational. 2010.Wolfram Mathematica (software)CXC A35/U1/154
MODULE 1: FOUNDATIONS OF MATHEMATICSGENERAL OBJECTIVESOn completion of this Module, students should:1.acquire competency in the application of algebraic techniques;2.appreciate the role of exponential or logarithm functions in practical modelling situations;3.understand the importance of relations functions and graphs in solving real-world problems;4.appreciate the difference between a sequence and a series and their applications;5.appreciate the need for accuracy in performing calculations;6.understand the usefulness of different types of numbers.SPECIFIC OBJECTIVES1.CONTENT & SKILLSNumbersStudents should be able to:1.1distinguish among the sets of numbers;Real and complex numbers;Identifying the set of complex number asthe superset of other numbers;Real and imaginary parts of a complexnumber 𝑧 𝑥 𝑖𝑦1.2solve problems involving the properties ofcomplex tion, subtraction, multiplication anddivision (realising the denominator).1.3represent complex numbers using the Arganddiagram;Represent complex numbers, the sum anddifference of two complex numbers.1.4find complex solutions, in conjugate pairs, toquadratic equations which has no realsolutions.Solving quadratic equations where thediscriminant is negative.CXC A35/U1/155
MODULE 1: FOUNDATIONS OF MATHEMATICS (cont’d)SPECIFIC OBJECTIVES2.CONTENT & SKILLSCoordinate GeometryStudents should be able to:2.1solve problems involving concepts ofcoordinate geometry;Application of: gradient; length and mid-point of aline segment; equation of a straight line.2.2relate the gradient of a straight line tothe angle it makes with the horizontalline.If 𝑦 𝑚𝑥 𝑐, then tan 𝜃 𝑚,where θ is the angle made with the positive x-axis.3.Functions, Graphs, Equations and InequalitiesStudents should be able to:3.1combine components of linear and Intercepts, gradient, minimum/maximum point andquadratic functions to sketch their roots.graphs;3.2determine the solutions of a pair of Graphical and algebraic solutions.simultaneous equations where one is Equations of the formlinear and the other is nonlinear;𝑎𝑥𝑦 𝑏𝑥 𝑐𝑦 𝑑 and 𝑒𝑥 2 𝑓𝑦 𝑔,where 𝑎, 𝑏, 𝑐, 𝑑, 𝑒, 𝑓, 𝑔 ℝ3.3apply solution techniques of equations Worded problems including quadratic equations,to solve real life problems;supply and demand functions and equations ofmotion in a straight line.3.4determine the solution set for linear and Graphical and algebraic solutions.quadratic inequalities;3.5solve equations and inequalities Equations and inequalities of type 𝑐 𝑏𝑐 𝑏involving absolute linear functions; 𝑎𝑥 𝑏 𝑐 𝑥 ,𝑎𝑎where 𝑎, 𝑏, 𝑐 \ℝ.Worded problems.3.6determine an invertible section of a Functions that are invertible for restricted domains.function;Quadratic functions and graphs.Domain and range of functions and their inverse.3.7evaluate the composition of functions Addition, subtraction, multiplication and division of𝑓(𝑥)for a given value of 𝑥.functions for example ℎ(𝑥) 𝑔(𝑥)Composite function:ℎ(𝑥) 𝑔[𝑓(𝑥)]Solving equations and finding function values.CXC A35/U1/156
MODULE 1: FOUNDATIONS OF MATHEMATICS (cont’d)SPECIFIC OBJECTIVES4.CONTENT & SKILLSLogarithms and ExponentsStudents should be able to:4.1apply the laws of indices to solve Equations of type 𝑎 𝑓(𝑥) 𝑏 𝑔(𝑥) , where 𝑓 𝑎𝑛𝑑 𝑔exponential equations in one unknown; are linear or quadratic .polynomials.4.2identify the properties of exponential Sketching the graphs of exponential (base e) andand logarithmic functions;logarithmic functions (bases 10 and e).4.3simplify logarithmic expressions using Laws of logarithm excluding𝑙𝑜𝑔𝑐 𝑏the laws of logarithm;𝑙𝑜𝑔𝑎 𝑏 𝑙𝑜𝑔𝑐 𝑎4.4identify the relationshipexponents and logarithms;4.5convert between the exponential andlogarithmic equations;4.6apply the laws of logarithms to solve Equations of the typeequationsinvolvinglogarithmic 𝑎 𝑙𝑜𝑔(𝑥 𝑏) 𝑙𝑜𝑔(𝑐) 𝑑expressions;4.7solve problems involving exponents and Equations of typelogarithms.𝑎 𝑥 𝑏 and 𝑙𝑜𝑔𝑎 𝑥 𝑏, where 𝑎 10 or 𝑒Converting equations to linear form:𝑦 𝑎𝑥 𝑏 𝑙𝑜𝑔 𝑦 𝑙𝑜𝑔 𝑎 𝑏 𝑙𝑜𝑔 𝑥𝑦 𝑎𝑏 𝑥 𝑙𝑜𝑔 𝑦 𝑙𝑜𝑔 𝑎 𝑥 𝑙𝑜𝑔 𝑏5.Remainder and Factor Theorembetween 𝑦 𝑙𝑜𝑔𝑎 𝑥 𝑎 𝑦 𝑥Students should be able to:5.1state the remainder and factor theorem; If 𝑓(𝑎) 0, then (𝑥 𝑎) is a factor of 𝑓.If 𝑓(𝑏) 0, then (𝑥 𝑏)leaves remainder 𝑟 ℝ when it divides 𝑓.5.2divide polynomials up to the third Methods of long division and inspection.degree by linear expressions;CXC A35/U1/157
MODULE 1: FOUNDATIONS OF MATHEMATICS (cont’d)SPECIFIC OBJECTIVESCONTENT & SKILLSRemainder and Factor Theorem (cont’d)Students should be able to:5.3solve problems involving the factor and If (𝑥 𝑎) is a factor of the polynomial 𝑓(𝑥),then 𝑓(𝑥) has a root at 𝑥 𝑎remainder theorems.Including finding coefficients of a polynomial given afactor or remainder when divided by a linearexpression.Factorising cubic polynomials where one factor canbe found by inspection.6.Sequences and SeriesStudents should be able to:6.1solve problems involving the binomial (𝑎 𝑏)𝑛 where 𝑛 is a positive integer not greaterexpansion;than 3 and 𝑎, 𝑏 ℝ.Problems involving finding the terms or thecoefficient of a term of an expansion of linearexpression such as (𝑎𝑥 𝑏)3 .While the students may become familiar with Pascal𝑛triangle and notations relating to ( ), it is sufficient𝑟to know the binomial coefficients 1-2-1 and 1-3-3-1for examination purposes.6.2identify arithmeticprogressions;6.3evaluate a term or the sum of a finite Problems including applications to simple andarithmetic or geometric series;compound interest, annually to quarterly6.4determine the sum to infinity for 1 𝑟 1geometric series;CXC A35/U1/15andgeometric Common ratio and common difference.8
MODULE 1: FOUNDATIONS OF MATHEMATICS (cont’d)SPECIFIC OBJECTIVES7.CONTENT & SKILLSMatrices and Systems of EquationsStudents should be able to:7.1perform thematrices;basicoperationson Addition, subtraction, multiplication, scalar multiple,equality of matrices.7.2represent data in matrix form;7.3evaluate the determinant of a 3x3matrix;7.4solve a system of three linear equations While the examination que
CAPE Integrated Mathematics Syllabus Extract 4 . Studies. The third is the CXC Associate Degree, awarded for the satisfactory completion of a prescribed . critical thinking and innovative application of science and technology to problem solving. Such a person
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