Mathematics (Syllabus 8865) - SEAB
Singapore–Cambridge General Certificate of EducationAdvanced Level Higher 1 (2022)Mathematics(Syllabus 8865) MOE & UCLES 2020
8865 MATHEMATICS GCE ADVANCED LEVEL H1 SYLLABUSCONTENTSPage3PREAMBLESYLLABUS AIMS3ASSESSMENT OBJECTIVES (AO)3USE OF A GRAPHING CALCULATOR (GC)4LIST OF FORMULAE AND STATISTICAL TABLES4INTEGRATION AND APPLICATION4SCHEME OF EXAMINATION PAPERS5CONTENT OUTLINE5MATHEMATICAL NOTATION102
8865 MATHEMATICS GCE ADVANCED LEVEL H1 SYLLABUSPREAMBLEThe applications of mathematics extend beyond the sciences and engineering domains. A basic understanding ofmathematics and statistics, and the ability to think mathematically and statistically are essential for an educatedand informed citizenry. For example, social scientists use mathematics to analyse data, support decisionmaking, model behaviour, and study social phenomena.H1 Mathematics provides students with a foundation in mathematics and statistics that will support theirbusiness or social sciences studies at the university. It is particularly appropriate for students without O-LevelAdditional Mathematics because it offers an opportunity for them to learn important mathematical concepts andskills in algebra and calculus that were taught in Additional Mathematics. Students will also learn basicstatistical methods that are necessary for studies in business and social sciences.SYLLABUS AIMSThe aims of H1 Mathematics are to enable students to:(a) acquire mathematical concepts and skills to support their tertiary studies in business and the socialsciences(b) develop thinking, reasoning, communication and modelling skills through a mathematical approach toproblem-solving(c) connect ideas within mathematics and apply mathematics in the context of business and social sciences(d) experience and appreciate the value of mathematics in life and other disciplines.ASSESSMENT OBJECTIVES (AO)There are three levels of assessment objectives for the examination.The assessment will test candidates’ abilities to:AO1Understand and apply mathematical concepts and skills in a variety of problems, including those thatmay be set in unfamiliar contexts, or require integration of concepts and skills from more than onetopic.AO2Formulate real-world problems mathematically, solve the mathematical problems, interpret andevaluate the mathematical solutions in the context of the problems.AO3Reason and communicate mathematically through making deductions and writing mathematicalexplanations and arguments.3
8865 MATHEMATICS GCE ADVANCED LEVEL H1 SYLLABUSUSE OF A GRAPHING CALCULATOR (GC)The use of an approved GC without computer algebra system will be expected. The examination papers will beset with the assumption that candidates will have access to GC. As a general rule, unsupported answersobtained from GC are allowed unless the question states otherwise. Where unsupported answers from GC arenot allowed, candidates are required to present the mathematical steps using mathematical notations and notcalculator commands. For questions where graphs are used to find a solution, candidates should sketch thesegraphs as part of their answers. Incorrect answers without working will receive no marks. However, if there iswritten evidence of using GC correctly, method marks may be awarded.Students should be aware that there are limitations inherent in GC. For example, answers obtained by tracingalong a graph to find roots of an equation may not produce the required accuracy.LIST OF FORMULAE AND STATISTICAL TABLESCandidates will be provided in the examination with a list of formulae and statistical tables.INTEGRATION AND APPLICATIONNotwithstanding the presentation of the topics in the syllabus document, it is envisaged that some examinationquestions may integrate ideas from more than one topic, and that topics may be tested in the contexts ofproblem solving and application of mathematics.Possible list of H1 Mathematics applications and contexts:Applications and contextsSome possible topics involvedOptimisation problems (e.g. maximising profits,minimising costs)Inequalities; System of linear equations; CalculusPopulation growth, radioactive decayExponential and logarithmic functionsFinancial maths (e.g. profit and cost analysis,demand and supply, banking, insurance)Equations and inequalities; Probability; Samplingdistributions; Correlation and regressionGames of chance, electionsProbabilityStandardised testingNormal distribution; ProbabilityMarket research (e.g. consumer preferences,product claims)Sampling distributions; Hypothesis testing;Correlation and regressionClinical research (e.g. correlation studies)Sampling distributions; Hypothesis testing;Correlation and regressionThe list illustrates some types of contexts in which the mathematics learnt in the syllabus may be applied, and isby no means exhaustive. While problems may be set based on these contexts, no assumptions will be madeabout the knowledge of these contexts. All information will be self-contained within the problem.4
8865 MATHEMATICS GCE ADVANCED LEVEL H1 SYLLABUSSCHEME OF EXAMINATION PAPERSFor the examination in H1 Mathematics, there will be one 3-hour paper marked out of 100 as follows:Section A (Pure Mathematics – 40 marks) will consist of about 5 questions of different lengths and marksbased on the Pure Mathematics section of the syllabus.Section B (Probability and Statistics – 60 marks) will consist of 6 to 8 questions of different lengths and marksbased on the Probability and Statistics section of the syllabus.There will be at least two questions, with at least one in each section, on the application of Mathematics inreal-world contexts, including those from business and the social sciences. Each question will carry at least 12marks and may require concepts and skills from more than one topic.Candidates will be expected to answer all questions.CONTENT OUTLINETopics/Sub-topicsContentSECTION A: PURE MATHEMATICS1Functions and Graphs1.1Exponential and logarithmic functions andGraphing techniquesInclude: concept of function as a rule or relationshipwhere for every input there is only one output use of notations such as f(x) x2 5 functions ex and ln x and their graphs exponential growth and decay logarithmic growth equivalence of y ex and x In y laws of logarithms use of a graphing calculator to graph a givenfunction characteristics of graphs such as symmetry,intersections with the axes, turning points andasymptotes (horizontal and vertical)Exclude: use of the terms domain and range use of notation f : x x2 5 change of base of logarithms5
8865 MATHEMATICS GCE ADVANCED LEVEL H1 SYLLABUSTopics/Sub-topicsContent1.2Equations and inequalitiesInclude: conditions for a quadratic equation to have(i) two real roots, (ii) two equal roots, and(iii) no real roots conditions for ax2 bx c to be always positive(or always negative) solving simultaneous equations, one linear andone quadratic, by substitution solving quadratic equations and inequalities inone unknown analytically solving inequalities by graphical methods formulating an equation or a system of linearequations from a problem situation finding the approximate solution of an equationor a system of linear equations using agraphing calculator2Calculus2.1DifferentiationInclude: derivative of f(x) as the gradient of the tangentto the graph of y f(x) at a pointdy use of standard notations f′(x) anddx derivatives of xn for any rational n, ex, In x,together with constant multiples, sums anddifferences use of chain rule graphical interpretation of f′(x) 0, f′(x) 0and f′(x) 0 use of the first derivative test to determine thenature of the stationary points (local maximumand minimum points and points of inflexion) insimple cases locating maximum and minimum points using agraphing calculator finding the approximate value of a derivative ata given point using a graphing calculator finding equations of tangents to curves local maxima and minima problems connected rates of change problemsExclude: differentiation from first principles derivatives of products and quotients offunctionsdy1 use ofdx dxdy differentiation of functions defined implicitly orparametrically finding non-stationary points of inflexion relating the graph of y f′(x) to the graph ofy f(x)6
8865 MATHEMATICS GCE ADVANCED LEVEL H1 ude: integration as the reverse of differentiation integration of xn for any rational n, and ex,together with constant multiples, sums anddifferences integration of (ax b)n for any rational n, ande(ax b) definite integral as the area under a curve evaluation of definite integrals finding the area of a region bounded by a curveand lines parallel to the coordinate axes,between a curve and a line, or between twocurves finding the approximate value of a definiteintegral using a graphing calculatorExclude: definite integral as a limit of sum approximation of area under a curve using thetrapezium rule area below the x-axisSECTION B: PROBABILITY AND STATISTICS3Probability and Statistics3.1ProbabilityInclude: addition and multiplication principles forcounting concepts of permutation (nPr) and combination(nCr) arrangements of distinct objects in a lineincluding cases involving restriction addition and multiplication of probabilities mutually exclusive events and independentevents use of tables of outcomes, Venn diagrams, treediagrams, and permutations and combinationstechniques to calculate probabilities calculation of conditional probabilities in simplecases use of:P ( A' ) 1 P ( A )P ( A B ) P ( A ) P (B ) P ( A B )P ( A B) 7P ( A B)P (B )
8865 MATHEMATICS GCE ADVANCED LEVEL H1 SYLLABUSTopics/Sub-topicsContent3.2Binomial distributionInclude: knowledge of the binomial expansion of (a b)nfor positive integer n binomial random variable as an example of adiscrete random variable concept of binomial distribution B(n, p) and useof B(n, p) as a probability model, includingconditions under which the binomial distributionis a suitable model use of mean and variance of a binomialdistribution (without proof)3.3Normal distributionInclude: concept of a normal distribution as an exampleof a continuous probability model and its meanand variance; use of N(μ, σ 2) as a probabilitymodel standard normal distribution finding the value of P(X x1) or a relatedprobability given the values of x1, μ, σ symmetry of the normal curve and itsproperties finding a relationship between x1, μ, σ given thevalue of P(X x1) or a related probability solving problems involving the use ofE (aX b) and Var (aX b) solving problems involving the use ofE (aX bY) and Var (aX bY), where X and Yare independentExclude normal approximation to binomialdistribution.3.4SamplingInclude: concepts of population and simple randomsample. concept of the sample mean X as a randomσ2ndistribution of sample means from a normalpopulationuse of the Central Limit Theorem to treatsample mean as having normal distributionwhen the sample size is sufficiently large(e.g. n 30)calculation of unbiased estimates of thepopulation mean and variance from a sample,including cases where the data are given insummarised form Σx and Σx2, or Σ(x – a) andΣ(x – a)2( )variable with E X μ and Var ( X ) 8
8865 MATHEMATICS GCE ADVANCED LEVEL H1 SYLLABUS3.5Topics/Sub-topicsContentHypothesis testingInclude: concepts of null hypothesis (H0) and alternativehypotheses (H1), test statistic, critical region,critical value, level of significance and p-value formulation of hypotheses and testing for apopulation mean based on:– a sample from a normal population ofknown variance– a large sample from any population 1-tail and 2-tail tests interpretation of the results of a hypothesis testin the context of the problemExclude the use of the term ‘Type I’ error, conceptof Type II error and testing the difference betweentwo population means.3.6Correlation and Linear regressionInclude: use of scatter diagram to determine if there is aplausible linear relationship between the twovariables correlation coefficient as a measure of the fit ofa linear model to the scatter diagram finding and interpreting the product momentcorrelation coefficient (in particular, valuesclose to 1, 0 and 1) concepts of linear regression and method ofleast squares to find the equation of theregression line concepts of interpolation and extrapolation use of the appropriate regression line to makeprediction or estimate a value in practicalsituations, including explaining how well thesituation is modelled by the linear regressionmodelExclude: derivation of formulae relationship r 2 b1b2, where b1 and b2 areregression coefficients hypothesis tests use of a square, reciprocal or logarithmictransformation to achieve linearity9
8865 MATHEMATICS GCE ADVANCED LEVEL H1 SYLLABUSMATHEMATICAL NOTATIONThe list which follows summarises the notation used in Cambridge’s Mathematics examinations. Althoughprimarily directed towards A-Level, the list also applies, where relevant, to examinations at all other levels.1. Set Notation is an element of is not an element of{x1, x2, }the set with elements x1, x2, {x: }the set of all x such thatn(A)the number of elements in set A the empty setuniversal setA′the complement of the set Aℤthe set of integers, {0, 1, 2, 3, }ℤ the set of positive integers, {1, 2, 3, }ℚthe set of rational numbersℚ the set of positive rational numbers, {x ℚ: x 0} ℚ0ℝthe set of positive rational numbers and zero, {x ℚ: x 0}ℝ the set of positive real numbers, {x ℝ: x 0}the set of real numbers ℝ0ℝnthe set of positive real numbers and zero, {x ℝ: x 0}ℂthe set of complex numbers is a subset of is a proper subset of is not a subset of is not a proper subset of union intersection[a, b]the closed interval {x ℝ: a x b}the real n-tuples(a, b]the interval {x ℝ: a x b}(a, b)the open interval {x ℝ: a x b}[a, b)the interval {x ℝ: a x b}10
8865 MATHEMATICS GCE ADVANCED LEVEL H1 SYLLABUS2. Miscellaneous Symbols is equal to is not equal to is identical to or is congruent to is approximately equal to is proportional to is less than ; is less than or equal to; is not greater than is greater than ; is greater than or equal to; is not less thaninfinity 3. Operationsa ba plus ba–ba minus ba b, ab, a.ba multiplied by ba b,a:bab, a/ba divided by bthe ratio of a to bn ai 1aia1 a2 . anthe positive square root of the real number aathe modulus of the real number an!n factorial for n ℤ {0}, (0! 1) n r the binomial coefficientn!, for n, r ℤ {0}, 0 r nr! (n r )!n(n 1).(n r 1), for n ℚ, r ℤ {0}r!11
8865 MATHEMATICS GCE ADVANCED LEVEL H1 SYLLABUS4. Functionsfthe function ff(x)the value of the function f at xf: A Bf is a function under which each element of set A has an image in set Bf: x ythe function f maps the element x to the element yf –1the inverse of the function fg o f, gfthe composite function of f and g which is defined by(g o f)(x) or gf(x) g(f(x))lim f(x)the limit of f(x) as x tends to ax aΔx ;δxdyan increment of xthe derivative of y with respect to xdxdn ydx nthe nth derivative of y with respect to xf'(x), f''(x), , f (n)(x)the first, second, nth derivatives of f(x) with respect to x y dxindefinite integral of y with respect to x bay dxx , x , the definite integral of y with respect to x for values of x between a and bthe first, second, derivatives of x with respect to time5. Exponential and Logarithmic Functionsebase of natural logarithmsex, exp xexponential function of xlog a xlogarithm to the base a of xln xnatural logarithm of xlg xlogarithm of x to base 106. Circular Functions and Relationssin, cos, tan,cosec, sec, cotsin–1, cos–1, tan–1cosec–1, sec–1, cot–1} the circular functions} the inverse circular functions12
8865 MATHEMATICS GCE ADVANCED LEVEL H1 SYLLABUS7. Complex Numbersithe square root of –1za complex number, z x iy r(cos θ i sin θ ), r ℝ 0 reiθ, r ℝ 0 Re zthe real part of z, Re (x iy) xIm zthe imaginary part of z, Im (x iy) yzarg zz*the modulus of z, x iy x 2 y 2 , r (cosθ i sinθ ) rthe argument of z, arg(r(cos θ i sin θ )) θ , –π θ πthe complex conjugate of z, (x iy)* x – iy8. MatricesMa matrix MM–1the inverse of the square matrix MMTthe transpose of the matrix Mdet Mthe determinant of the square matrix M9. Vectorsathe vector aABthe vector represented in magnitude and direction by the directed line segment ABâa unit vector in the direction of the vector ai, j, kunit vectors in the directions of the cartesian coordinate axesathe magnitude of aABthe magnitude of ABa.bthe scalar product of a and ba bthe vector product of a and b13
8865 MATHEMATICS GCE ADVANCED LEVEL H1 SYLLABUS10. Probability and StatisticsA, B, C, etc.eventsA Bunion of events A and BA Bintersection of the events A and BP(A)probability of the event AA'complement of the event A, the event ‘not A’P(A B)probability of the event A given the event BX, Y, R, etc.random variablesx, y, r, etc.value of the random variables X, Y, R, etc.x1, x2, observationsf1, f2, frequencies with which the observations, x1, x2, occurp(x)the value of the probability function P(X x) of the discrete random variable Xp1, p2, probabilities of the values x1, x2, of the discrete random variable Xf(x), g(x) the value of the probability density function of the continuous random variable XF(x), G(x) E(X)the value of the (cumulative) distribution function P(X x) of the random variable XE[g(X)]expectation of g(X)Var(X)variance of the random variable XB(n, p)binomial distribution, parameters n and pPo(μ)Poisson distribution, mean μN(μ, σ2)normal distribution, mean μ and variance σ2μpopulation meanσ2population varianceσpopulation standard deviationxsample means2expectation of the random variable Xunbiased estimate of population variance from a sample,s2 12 ( x x )n 1φprobability density function of the standardised normal variable with distribution N (0, 1)Φcorresponding cumulative distribution functionρlinear product-moment correlation coefficient for a populationrlinear product-moment correlation coefficient for a sample14
advanced level higher 1 (2022) mathematics (syllabus 8865) 8865 mathematics gce advanced level h1 syllabus . 2 . contents page preamble 3 syllabus aims 3 . assessment objectives (ao) 3 . use of a graphing calculator (gc) 4 . list of formulae and statistical tables 4 . integration and application 4 .
9758 MATHEMATICS GCE ADVANCED LEVEL H2 SYLLABUS . 6 . CONTENT OUTLINE . Knowledge of the content of the O-Level Mathematics syllabus and of some of the content of the O-Level Additional Mathematics syllabuses are assumed in the syllabus below and will not be tested directly, but it may be required indirectly in response to questions on other .
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graphing calculator finding the approximate value of a derivative at a given point using a graphing calculator finding equations of tangents to curves local maxima and minima problems connected rates of change problems Exclude: differentiation from first principles derivatives of products and quotients of functions
IBDP MATHEMATICS: ANALYSIS AND APPROACHES SYLLABUS SL 1.1 11 General SL 1.2 11 Mathematics SL 1.3 11 Mathematics SL 1.4 11 General 11 Mathematics 12 General SL 1.5 11 Mathematics SL 1.6 11 Mathematic12 Specialist SL 1.7 11 Mathematic* Not change of base SL 1.8 11 Mathematics SL 1.9 11 Mathematics AHL 1.10 11 Mathematic* only partially AHL 1.11 Not covered AHL 1.12 11 Mathematics AHL 1.13 12 .
as HSC Year courses: (in increasing order of difficulty) Mathematics General 1 (CEC), Mathematics General 2, Mathematics (‘2 Unit’), Mathematics Extension 1, and Mathematics Extension 2. Students of the two Mathematics General pathways study the preliminary course, Preliminary Mathematics General, followed by either the HSC Mathematics .
2. 3-4 Philosophy of Mathematics 1. Ontology of mathematics 2. Epistemology of mathematics 3. Axiology of mathematics 3. 5-6 The Foundation of Mathematics 1. Ontological foundation of mathematics 2. Epistemological foundation of mathematics 4. 7-8 Ideology of Mathematics Education 1. Industrial Trainer 2. Technological Pragmatics 3.
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the principles of English etymology, than as a general introduction to Germanic philology. The Exercises in translation will, it is believed, furnish all the drill necessary to enable the student to retain the forms and constructions given in the various chapters. The Selections for Reading relate to the history and literature of King Alfred’s day, and are sufficient to give the student a .
