Mathematics (Syllabus 9758) - SEAB
Singapore–Cambridge General Certificate of EducationAdvanced Level Higher 2 (2022)Mathematics(Syllabus 9758) MOE & UCLES 2020
9758 MATHEMATICS GCE ADVANCED LEVEL H2 SYLLABUSCONTENTSPage3PREAMBLESYLLABUS AIMS3ASSESSMENT OBJECTIVES (AO)3USE OF A GRAPHING CALCULATOR (GC)4LIST OF FORMULAE AND STATISTICAL TABLES4INTEGRATION AND APPLICATION4SCHEME OF EXAMINATION PAPERS5CONTENT OUTLINE6ASSUMED KNOWLEDGE15MATHEMATICAL NOTATION172
9758 MATHEMATICS GCE ADVANCED LEVEL H2 SYLLABUSPREAMBLEMathematics is a basic and important discipline that contributes to the developments and understandings ofsciences and other disciplines. It is used by scientists, engineers, business analysts and psychologists, etc. tomodel, understand and solve problems in their respective fields. A good foundation in mathematics and theability to reason mathematically are therefore essential for students to be successful in their pursuit of variousdisciplines.H2 Mathematics is designed to prepare students for a range of university courses, including mathematics,sciences, engineering and related courses, where a good foundation in mathematics is required. It developsmathematical thinking and reasoning skills that are essential for further learning of mathematics. Throughapplications of mathematics, students also develop an appreciation of mathematics and its connections to otherdisciplines and to the real world.SYLLABUS AIMSThe aims of H2 Mathematics are to enable students to:(a) acquire mathematical concepts and skills to prepare for their tertiary studies in mathematics, sciences,engineering and other related disciplines(b) develop thinking, reasoning, communication and modelling skills through a mathematical approach toproblem-solving(c) connect ideas within mathematics and apply mathematics in the contexts of sciences, engineering andother related disciplines(d) experience and appreciate the nature and beauty of mathematics and its value 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, arguments and proofs.3
9758 MATHEMATICS GCE ADVANCED LEVEL H2 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 H2 Mathematics applications and contexts:Applications and contextsSome possible topics involvedKinematics and dynamics (e.g. free fall, projectilemotion, collisions)Functions; Calculus; VectorsOptimisation problems (e.g. maximising strength,minimising surface area)Inequalities; System of linear equations; CalculusElectrical circuitsComplex numbers; CalculusPopulation growth, radioactive decay, heating andcooling problemsDifferential equationsFinancial maths (e.g. banking, insurance)Sequences and series; Probability; SamplingdistributionsStandardised testingNormal distribution; ProbabilityMarket research (e.g. consumer preferences, productclaims)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
9758 MATHEMATICS GCE ADVANCED LEVEL H2 SYLLABUSSCHEME OF EXAMINATION PAPERSFor the examination in H2 Mathematics, there will be two 3-hour papers, each carrying 50% of the total mark,and each marked out of 100, as follows:PAPER 1 (3 hours)A paper consisting of 10 to 12 questions of different lengths and marks based on the Pure Mathematics sectionof the syllabus.There will be at least two questions on application of Mathematics in real-world contexts, including those fromsciences and engineering. Each question will carry at least 12 marks and may require concepts and skills frommore than one topic.Candidates will be expected to answer all questions.PAPER 2 (3 hours)A paper consisting of two sections, Sections A and B.Section A (Pure Mathematics – 40 marks) will consist of 4 to 5 questions of different lengths and marks basedon 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 in Section B on application of Mathematics in real-world contexts, includingthose from sciences and engineering. Each question will carry at least 12 marks and may require concepts andskills from more than one topic.Candidates will be expected to answer all questions.5
9758 MATHEMATICS GCE ADVANCED LEVEL H2 SYLLABUSCONTENT OUTLINEKnowledge of the content of the O-Level Mathematics syllabus and of some of the content of the O-LevelAdditional Mathematics syllabuses are assumed in the syllabus below and will not be tested directly, but it maybe required indirectly in response to questions on other topics. The assumed knowledge for O-Level AdditionalMathematics is appended after this section.Topic/Sub-topicsContentSECTION A: PURE MATHEMATICS1Functions and graphs1.1FunctionsInclude: concepts of function, domain and range use of notations such as f( x ) x 2 5 ,f : x x 2 5 , f 1( x ) , fg( x ) and f 2 ( x ) finding inverse functions and composite functionsconditions for the existence of inverse functionsand composite functionsdomain restriction to obtain an inverse functionrelationship between a function and its inverseExclude the use of the relation (fg) 1 g 1f 1 , andrestriction of domain to obtain a composite function.1.2Graphs and transformationsInclude: use of a graphing calculator to graph a givenfunction important characteristics of graphs such assymmetry, intersections with the axes, turningpoints and asymptotes of the following:x2 y 2 1 a2 b2x2a2 y2 1;b2ax by cx dy y2b2a2 1dx edetermining the equations of asymptotes, axes ofsymmetry, and restrictions on the possible valuesof x and/or yeffect of transformations on the graph of y f(x) asrepresented by y af(x), y f(x) a,y f(x a) and y f(ax), and combinations ofthese transformationsrelating the graphs of y f 1 (x ), y f (x ),( )6x2ax 2 bx cy f x , and y 1to the graph of y f(x)f (x )simple parametric equations and their graphs
9758 MATHEMATICS GCE ADVANCED LEVEL H2 SYLLABUS1.3Topic/Sub-topicsContentEquations and inequalitiesInclude: formulating an equation, a system of linearequations, or inequalities from a problem situation solving an equation exactly or approximately usinga graphing calculator solving a system of linear equations using agraphing calculatorf (x ) 0 where f(x) solving inequalities of the formg(x )and g(x) are linear expressions or quadraticexpressions that are either factorisable or alwayspositive concept of x , and use of relations x a b a b x a b and 2Sequences and series2.1Sequences and series x a b x a b or x a b, in thecourse of solving inequalitiessolving inequalities by graphical methodsInclude: concepts of sequence and series for finite andinfinite cases sequence as function y f(n) where n is a positiveinteger relationship between un (the nth term) and Sn (thesum to n terms) sequence given by a formula for the nth term use of Σ notation sum and difference of two series summation of series by the method of differences convergence of a series and the sum to infinity formula for the nth term and the sum of a finitearithmetic series formula for the nth term and the sum of a finitegeometric series condition for convergence of an infinite geometricseries formula for the sum to infinity of a convergentgeometric series7
9758 MATHEMATICS GCE ADVANCED LEVEL H2 SYLLABUSTopic/Sub-topics3Vectors3.1Basic properties of vectors in two and threedimensionsContentInclude: addition and subtraction of vectors, multiplication ofa vector by a scalar, and their geometricalinterpretations x x use of notations such as , y , x i y j , y z 3.2Scalar and vector products in vectorsx i y j zk , AB, aposition vectors, displacement vectors anddirection vectorsmagnitude of a vectorunit vectorsdistance between two pointscollinearityuse of the ratio theorem in geometrical applicationsInclude: concepts of scalar product and vector product ofvectors and their properties angle between two vectors geometrical meanings of a nˆ and a nˆ ,where n̂ is a unit vectorExclude triple products a b c and a b c .3.3Three-dimensional vector geometryInclude: vector and cartesian equations of lines and planes finding the foot of the perpendicular and distancefrom a point to a line or to a plane finding the angle between two lines, between a lineand a plane, or between two planes relationships between(i) two lines (coplanar or skew)(ii) a line and a plane(iii) two planesExclude: finding the shortest distance between two skewlines finding an equation for the common perpendicularto two skew lines8
9758 MATHEMATICS GCE ADVANCED LEVEL H2 SYLLABUSTopic/Sub-topicsContent4Introduction to Complex numbers4.1Complex numbers expressed in cartesianformInclude: extension of the number system from real numbersto complex numbers complex roots of quadratic equations conjugate of a complex number four operations of complex numbers equality of complex numbers conjugate roots of a polynomial equation with realcoefficients4.2Complex numbers expressed in polar formInclude: representation of complex numbers in the Arganddiagram complex numbers expressed in the formr(cos θ i sin θ), or reiθ where r 0 and–π θ π calculation of modulus (r) and argument (θ) of acomplex number multiplication and division of two complex numbersexpressed in polar form5Calculus5.1DifferentiationInclude: graphical interpretation of(i) f ′(x) 0, f ′(x) 0 and f ′(x) 0(ii) f ″(x) 0 and f ″(x) 0 relating the graph of y f ′(x) to the graph ofy f(x) differentiation of simple functions defined implicitlyor parametrically determining the nature of the stationary points(local maximum and minimum points and points ofinflexion) analytically, in simple cases, using thefirst derivative test or the second derivative test locating maximum and minimum points using agraphing calculator finding the approximate value of a derivative at agiven point using a graphing calculator finding equations of tangents and normals tocurves, including cases where the curve is definedimplicitly or parametrically local maxima and minima problems connected rates of change problemsExclude finding non-stationary points of inflexion andfinding second derivatives of functions definedparametrically.9
9758 MATHEMATICS GCE ADVANCED LEVEL H2 SYLLABUS5.2Topic/Sub-topicsContentMaclaurin seriesInclude: standard series expansion of (1 x)n for anyrational n, ex, sin x, cos x and In(1 x) derivation of the first few terms of the Maclaurinseries by–repeated differentiation, e.g. sec x–repeated implicit differentiation, e.g.y3 y2 y x2 – 2x–using standard series, e.g. ex cos 2x, 1 x In 1 x range of values of x for which a standard seriesconvergesconcept of “approximation”small angle approximations: sin x x,1cos x 1 x 2 , tan x x2Exclude derivation of the general term of the series.5.3Integration techniquesInclude: integration of f ′(x ) [f (x )] (including n –1), f ′(x)ef(x)nsin2 x, cos2 x, tan2 x,sin mx cos nx, cos mx cos nx and sin mx sin nx1111,,and22222a xx a2a2 x 2 a x integration by a given substitution integration by parts5.4Definite integralsInclude: concept of definite integral as a limit of sum 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, betweena curve and a line, or between two curves area below the x-axis finding the area under a curve definedparametrically finding the volume of revolution about the x- ory-axis finding the approximate value of a definite integralusing a graphing calculatorExclude finding the volume of revolution about thex-axis or y-axis where the curve is definedparametrically.10
9758 MATHEMATICS GCE ADVANCED LEVEL H2 SYLLABUS5.5Topic/Sub-topicsContentDifferential equationsInclude: solving for the general solutions and particularsolutions of differential equations of the formsdy f (x )(i)dxdy f (y )(ii)dxd2 y(iii) f (x )dx 2including those that can be reduced to (i) and (ii)by means of a given substitution formulating a differential equation from a problemsituation interpreting a differential equation and its solutionin terms of a problem situation11
9758 MATHEMATICS GCE ADVANCED LEVEL H2 SYLLABUSTopic/Sub-topicsContentSECTION B: PROBABILITY AND STATISTICS6Probability and Statistics6.1ProbabilityInclude: addition and multiplication principles for counting concepts of permutation (nPr) and combination(nCr) arrangements of objects in a line or in a circle,including cases involving repetition and restriction addition and multiplication of probabilities mutually exclusive events and independent events 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) 6.2Discrete random variablesP ( A B)P (B )Include: concept of discrete random variables,probability distributions, expectations andvariances concept of binomial distribution B(n, p) as anexample of a discrete probability distributionand use of B(n, p) as a probability model,including conditions under which the binomialdistribution is a suitable model use of mean and variance of binomialdistribution (without proof)Exclude finding cumulative distribution function of adiscrete random variable.12
9758 MATHEMATICS GCE ADVANCED LEVEL H2 SYLLABUS6.3Topic/Sub-topicsContentNormal distributionInclude: concept of a normal distribution as an example ofa continuous probability model and its mean andvariance; use of N(μ, σ 2) as a probability model 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 its properties finding a relationship between x1, μ, σ given thevalue of P(X x1), or a related probability solving problems involving the use of E(aX b)and Var (aX b) solving problems involving the use of E(aX bY)and Var (aX bY), where X and Y areindependentExclude normal approximation to binomial distribution.6.4SamplingInclude: concepts of population and simple random sample concept of the sample mean X as a randomσ2ndistribution of sample means from a normalpopulationuse of the Central Limit Theorem to treat samplemean as having normal distribution when thesample size is sufficiently large (e.g. n 30)calculation and use 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 ) 6.5Hypothesis 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 of knownvariance– a large sample from any population 1-tail and 2-tail tests interpretation of the results of a hypothesis test inthe context of the problemExclude the use of the term ‘Type I’ error, concept ofType II error and testing the difference between twopopulation means.13
9758 MATHEMATICS GCE ADVANCED LEVEL H2 SYLLABUS6.6Topic/Sub-topicsContentCorrelation 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 of alinear model to the scatter diagram finding and interpreting the product momentcorrelation coefficient (in particular, values close to 1, 0 and 1) concepts of linear regression and method of leastsquares to find the equation of the regression 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 regression model use of a square, reciprocal or logarithmictransformation to achieve linearityExclude: derivation of formulae relationship r 2 b1b2, where b1 and b2 areregression coefficients hypothesis tests14
9758 MATHEMATICS GCE ADVANCED LEVEL H2 SYLLABUSASSUMED KNOWLEDGEContent from O-Level Additional MathematicsALGEBRAA1Equations and inequalities conditions for a quadratic equation to have:(i) two real roots(ii) two equal roots(iii) no real roots conditions for ax2 bx c to be always positive (or always negative) solving simultaneous equations with at least one linear equation, by substitutionA2Indices and surds four operations on indices and surds rationalising the denominatorA3Polynomials and partial fractions multiplication and division of polynomials use of remainder and factor theorems partial fractions with cases where the denominator is not more complicated than:– (ax b)(cx d)– (ax b)(cx d)2– (ax b)(x2 c2)A4Power, Exponential, Logarithmic, and Modulus functions power functions y axn, where n is a simple rational number, and their graphs functions a x , e x , log a x , ln x and their graphs laws of logarithmsequivalence of y a x and x loga y change of base of logarithmsfunction x and graph of f(x) , where f(x) is linear, quadratic or trigonometricsolving simple equations involving exponential and logarithmic functionsGEOMETRY AND TRIGONOMETRYB5Coordinate geometry in two dimensions graphs of equations y2 kx coordinate geometry of the circle with the eq
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 .
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 .
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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Syllabus of Sixth Semester B. Pharm. 069 11. Syllabus of Seventh Semester B. Pharm. 081 12. Syllabus of Eight Semester B. Pharm. 091 B Ordianance and Rules (M. Pharm.) 101 1. Ordinance and Rules 102 2. Structure of Syllabus 107 C. Syllabus (Pharmaceutics) 115 D. Syllabus (
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SAMPLE CONTENT 1 Word Meaning bashfulness (n) shyness or discomfort with other people be a pussy-cat (phrase) here, laze around indoors beckoned (v) invited or guided someone with a gesture of a hand behold (v) see; witness betokening (v) be a sign of blanc-mange (n) almond flavoured milk pudding blunt (adj) here, saying something honestly without trying to be polite
