AM SYLLABUS (2016) PURE MATHEMATICS AM 27 SYLLABUS
AM Syllabus (2016): Pure MathematicsAM SYLLABUS (2016)PURE MATHEMATICSAM 27SYLLABUS1
AM Syllabus (2016): Pure MathematicsPure Mathematics AM 27Syllabus1.(Available in September )Paper I(3hrs) Paper II(3hrs)AIMS To prepare students for further studies in Mathematics and related subjects.To extend the students’ range of mathematical techniques so as to apply them in moredifficult and unstructured problems.To develop in students the ability to read and understand a wider range of mathematicalarticles and arguments.To enable students to formulate a mathematical representation of a real life situation.To use appropriate technology such as computers and calculators as a mathematical tool.To encourage confidence, enjoyment and satisfaction through the development and use ofMathematics.The syllabus assumes a good knowledge of the subject at SEC level and coverage of the extensiontopics in Paper 2A. It aims at consolidating this knowledge and to extend it to include more advancedconcepts.2. ASSESSMENT OBJECTIVESCandidates are required to: demonstrate their knowledge of mathematical facts, concepts, theories and techniques indifferent contexts. construct mathematical arguments and proofs by means of precise statements, logicaldeduction and inference. recognise standard models and be able to apply them.3. SCHEME OF ASSESSMENTThe examination will consist of 2 papers of 3 hours each. Any examination question can test materialfrom more than one topic. Questions may be set on topics which are not explicitly mentioned in thesyllabus but such questions will contain suitable guidance so that candidates will be able to tacklethem with the mathematical knowledge they would have acquired during their studies of the materialin the syllabus. Knowledge of topics in Paper 1 is assumed and may be tested in Paper 2.Graphical calculators will not be allowed however scientific calculators could be used but allnecessary working must be shown. A booklet with mathematical formulae will be provided.Paper 1 will contain 10 questions, possibly of varying difficulty. Marks allotted to each question willbe shown. The total number of marks available in the paper is 100 and candidates will have to answerall the questions.Paper 2 will contain 10 questions and candidates will be asked to choose 7 questions. Each questionwill carry 15 marks.4. GRADE DESCRIPTION2
AM Syllabus (2016): Pure MathematicsGrade A: Candidates who are able to recall and select almost all concepts, techniques and theoriesrequired in different contexts. Candidates who use diagrams and sketches with a high level of accuracy and who are able toproceed logically in their proofs. Candidates who derive results to a high degree of accuracy.Grade C: Candidates who are able to recall and select most concepts, techniques and theories requiredin different contexts. Candidates who use diagrams and sketches with a reasonable level of accuracy and who areable to proceed logically in their proofs. Candidates who derive results to an appropriate degree of accuracy.Grade E: Candidates who are able to recall and select some concepts, techniques and theories requiredin different contexts. Candidates who use diagrams and sketches with some accuracy and who are able to proceedlogically in their proofs. Candidates who derive results to a fair degree of accuracy.5. SUBJECT CONTENTThe topics are not arranged in teaching order. The syllabus is not meant as a teaching scheme andteachers are free to adopt any teaching sequence that they deem to be suitable for their students.Pure Mathematics Paper 1Topics1.NotesSurds, Indices,Logarithms, PartialFractions and QuadraticsClassification of numbers: ℝ, ℂ, ℕ, ℚ and ℤ.Use and manipulation ofsurds.Positive and negativerational indicesTo include simplification and rationalisation of the! !!! !denominator of a fraction e.g. 15 4 27; !!!! !Properties of Indices i.e. Zero, negative and fractional.Applying the laws of indicesPowers of products and quotients3
AM Syllabus (2016): Pure MathematicsSimplifying expressions e.g.LogarithmsPartial FractionsRemainder and factortheorem! !!! ! !!(!!!)!!!(!!!).Definition of logarithms, the laws of logarithms.Common and natural logarithms.Change of base formula.Solution of equations involving indices and logarithmsInclude cases where the denominator is of the form:- (𝑎𝑥 𝑏)(𝑐𝑥 𝑑)(𝑒𝑥 𝑓)- (𝑎𝑥 𝑏) 𝑐𝑥 𝑑 2- (𝑎𝑥 𝑏)(𝑐 𝑥2 𝑑𝑥 𝑒)Include improper fractions.In these cases the degree of the denominator must not begreater than three.Finding the remainder and also factorizing cubic orquartic expressions.Sum and difference of two cubesPascal’s triangleQuadratic equations Simple inequalities in onevariableSolution of quadratic equations by factorizing orby completing the square. Locating the maximumor minimum value of a quadratic function.Sketching quadratic functions.Nature of roots of a quadratic equation.Knowledge of the relation between the roots𝛼 and 𝛽 and the coefficients of a quadraticequation. Forming new equations with rootsrelated to the original. Calculations of expressionsup to the third degree e.g. 𝛼 ! 𝛽 ! .Graphical or algebraic solution of: Linear inequalitiesQuadratic inequalitiesCubic inequalities, which can be factorizedin at least one linear factorInequalities involving modulus offunctions of the above typeRational inequalities reducible to the thirddegree4
AM Syllabus (2016): Pure Mathematics2.Sequences and SeriesArithmetic and GeometricseriesThe binomial expansion forrational indices3.4.Include: Definition of a sequence and a series The general term of an A.P. and a G.P. The sum of an A.P. and a G.P. Arithmetic and Geometric mean Use of notation Condition for convergence of an infinitegeometric series and its sum to infinityExpansion of 𝑎 𝑏𝑥 ! for any rational n ineither ascending or descending powers of x andcondition for convergence of a binomial seriesEnumeration andprobabilityAddition and multiplicationprinciples for countingProblems about selections, e.g. finding the number ofways in which a committee of 2 men and 3 women can beselected from a group of 10 men and 7 women.Simple counting problemsinvolving permutations andcombinations.Problems about arrangements of objects in a lineincluding those in which some objects are repeated andthose in which arrangement is restricted, e.g. by requiringthat two or more objects must, or must not, stand next toeach other.Applications to simpleproblems in probabilityThe knowledge of probability expected will be limited tothe calculation of probabilities arising from simpleproblems of enumeration of equally likely possibilities,including simple problems involving the probability ofcomplement of an event and of the union and intersection of two events.Graphic techniques andCoordinate GeometrySimple curve sketchingInclude: Curve sketching will be limited to polynomials upto three stationary points Effect of the simple transformations on the graphof 𝑦 𝑓(𝑥) as represented by 𝑦 𝑓(𝑥 𝑎),𝑦 𝑓 𝑥 𝑎, 𝑦 𝑓 𝑎𝑥 and 𝑦 𝑎𝑓(𝑥), andcombination of these transformations up to amaximum of three transformations5
AM Syllabus (2016): Pure Mathematics The relation of the equation of a graph to itssymmetriesStraight lineInclude: Distance between two points Mid-point of the line joining two points Various forms of equation of a line Condition for parallel and perpendicular lines Intersection and angle between two lines Perpendicular distance from a point to a lineLociFinding the equation of the locus of a point from a givendescription of the locusParametric coordinates of a point on a curveCircle5.Include: The two forms of the general equation of a circle Parametric coordinates of any point on a circle Equations of tangents to a circle External and internal contact of two circles Orthogonal circlesFunctionsFunctions, inverse functionsand composite functionsInclude: Concepts of function, domain and range One – one and onto functions Use of notations e.g. 𝑓 𝑥 𝑥 ! 3,𝑓: 𝑥 𝑥 ! 3, 𝑓 !! 𝑥 , 𝑓𝑔 𝑥 or 𝑓 𝑔 Domain restricted to obtain an inverse function Finding inverse functions for one – one functions Composition of two functions Condition for the existence of an inverse functionand composite function The relationship between a function and itsinverse as the reflection in the line y xExclude finding the domain and range of theinverse of composite functionsModulus of a function Use of the definition𝑥 𝑥, if 𝑥 0and 𝑥 𝑥, if 𝑥 06
AM Syllabus (2016): Pure Mathematics Sketching a modulus graphExclude the modulus of a function involving amodulus function e.g. 𝑥 3Rational functionsTypes of functionsThe exponential andlogarithmic functions6.The definition of a rational function and how to performlong division on rational functionsDefinition of odd, even and periodic functionsAn exponential function of the form 𝑓 𝑥 𝑎 ! ,where a 0 and x is real.The graphs of 𝑓 𝑥 𝑒 ! and 𝑔 𝑥 ln𝑥The idea that f and g are the inverse of each other.TrigonometryThe six trigonometricfunctionsInclude: Angles can be expressed in either degree or radianmeasure The inverse of these functions and identify thedomain for their existence. Their graphs The CAST RuleArc length, area of sectorand area of a segmentTrigonometric IdentitiesInclude: Fundamental identities Pythagorean identities Compound angle Identities Double and half angle identities Factor formulaeExclude on how to prove the compound angle identities.Also manipulative skills are expected but questionsrequiring lengthy manipulations will not be set.Solutions of simpletrigonometric equationsThe general solution!Knowledge of the values of cosine, sine and tangent of ! ,where k 1, 2, 3, 4, 6 in surd or rational formTransformation of theexpression 𝑎 cos 𝜃 𝑏 sin 𝜃into the forms such as𝑅 cos(𝜃 𝛼)Solution of equations of the form 𝑎 cos 𝜃 𝑏 sin 𝜃 c7
AM Syllabus (2016): Pure MathematicsSmall AnglesThe use of the approximations sin 𝑥 𝑥 tan 𝑥,and cos 𝑥 1 7.8.9.!!!Complex NumbersDefinition and basicproperties of ComplexnumbersAdd, subtract, multiply, divide and find the square root ofcomplex numbersConjugate complex numbers and solving quadraticequationsSimple examples of conjugate roots of polynomials, up toorder 3, with real coefficientsEquating real and imaginary partsThe Argand diagramComplex number is in the form of either 𝑎 𝑖𝑏 or𝑟(cos 𝜃 𝑖 sin 𝜃), where the argument 𝜃 satisfies– 𝜋 𝜃 𝜋 and the modulus r 0Properties of products and quotients of moduli andargumentsDifferentiationDefinition of the derivativeas a limitA rigorous treatment is not expectedDifferentiation of simplefunctions defined implicitlyor parametricallyDifferentiation of algebraic, trigonometric, exponentialand logarithmic functionsImplicit and parametric differentiationLogarithmic differentiationExclude differentiation of inverse trigonometric functionsDifferentiation RulesDifferentiation of sums, products, quotients andcomposition of functionsApplications ofDifferentiationInclude: Finding the equations of tangents and normal Finding stationary points and curve sketching Application of maximum or minimum to simplepractical problems Rates of changeIntegrationIntegration as the limit of asum and as the inverse ofA rigorous treatment is not expected8
AM Syllabus (2016): Pure MathematicsdifferentiationIntegration of simplefunctionsIntegration of algebraic, trigonometric, exponential andlogarithmic functionsIntegration RulesThe evaluation of integrals by means of: Standard forms Substitution or by sight Parts (A single integral cannot contain more than2 integration by parts) Partial Fractions Using trigonometric IdentitiesApplications of IntegrationDefinite integralsCalculating the area and the mean values of functions10.Differential EquationsFirst order differential equations of the separable type11.Vectors12.Vectors in two and threedimensionsInclude: Addition and subtraction of vectors, multiplicationof a vector by a scalar and their geometricinterpretation Use of the unit vectors i, j and k𝑎 Use of notations such as ai bj ck, 𝑏 , a, 𝐴𝐵𝑐 Position vectors Unit vector and magnitude of a vector Distance between two pointsThree-dimensionalgeometryVector, Cartesian and parametric equations of linesIntersection of two linesIntuitive understanding of skew lines in three dimensionsScalar productAngle between two vectorsAngle between two linesCondition for perpendicular linesMatricesThe algebra of matricesInclude: Definition of a matrix Special types of matrices i.e. zero, identity, square9
AM Syllabus (2016): Pure Mathematics and diagonal matricesAddition, subtraction and multiplicationCondition for equal matricesInverse of 2 2 matricesExclude finding the inverse of 3 3 matrices, but studentsshould be able to verify that two given 3 3 matrices areinverses of each other.Matrix PropertiesNon commutativity of multiplicationDistributivity of multiplication over additionAssociativityLinear transformations inthe planeFinding the matrix associated with a linear transformationand vice-versa. Rotation through an angle 𝜃 about theorigin, reflection in the line 𝑦 𝑥 tan 𝜃, magnification orstretching. Derivations are expected.Compound transformations in two dimensions.Exclude shear transformationPure Mathematics Paper 2Topics1.NotesSummation of SeriesMaclaurin’s SeriesFinding Maclaurin’s series of simple functions. Alsoinclude the general term in simple cases.Summation of simplefinite seriesInclude Using method of differences Using partial fractions Using standard results i.e.!!𝑟 ,!!!Summation of simpleinfinite series!!𝑟!𝑟 and!!!!!!If 𝑆! sum up to n terms, findlim 𝑆!! !10
AM Syllabus (2016): Pure MathematicsMethods one can use are 2.3.Method of MathematicalInductionMethod of differencesPartial fractionsComparison with standard power series i.e.binomial, logarithmic, exponential andtrigonometric series. One is expected to knowtheir region of convergence.Mathematical induction is a method of mathematicalproof typically used to establish a given statement for allnatural numbers. It is expected that the knowledge ofmathematical induction can be applied to simple problemsusing tools within the syllabus such as: De Moivre’s theorem (for positive integer only) Summation of series Inequalities Equations involving matrices Expressions involving a multiplicity or divisibilityproperty DifferentiationComplex NumbersDe Moivre’s Theorem forany rational indexInclude: Deriving trigonometric identities Finding the nth roots of a complex number oThe sum of these roots 0oAll the n nth roots of any complex number!oz lie on a circle of radius 𝑧 !!!Successive arguments differ by ! fromeach otherExponential form of acomplex numberEuler’s formula i.e. 𝑒 !" cos 𝜃 𝑖 sin 𝜃The exponential form for sin 𝜃 and cos 𝜃Loci of complex numbersLimited to Loci of the form 𝑧 𝑎 𝑐; 𝑧 𝑎 𝑘 𝑧 𝑏 ,where a and b are complex numbers, c and k arepositive real numbers Loci involving the Real or Imaginary part of anexpression11
AM Syllabus (2016): Pure MathematicsInequalities involving themodulus sign4.Sketching on the Argand diagram the regions defined by𝑧 𝑎 𝑐, 𝑧 𝑎 𝑘 𝑧 𝑏VectorsVector productDefinition of a vector product and how it is expressed indeterminant formIts properties i.e. Non-commutativity, its Distributivityover addition and non-associativityApplications of vectorsInclude: Equation of a plane in vector and Cartesian form Area of a triangle and a parallelogram Use of triple scalar product to find the volume of aparallelepiped and volume of a tetrahedronExclude the knowledge of the triple vector productGeometry of lines andplanes5.Include: Direction ratios and Direction vectors Angle between two planes, a line and a plane Intersection of two planes, a line and a plane Perpendicular distance from a point to a planeFurther curve sketchingCurve Sketching!! ! !!"!!Sketching rational functions of the form !! ! !!"!!Finding the range of values of 𝑦 𝑓(𝑥) so as to deducethe coordinates of the turning point(s)!Relating the graphs of 𝑦 𝑓(𝑥) , 𝑦 !(!) and𝑦 ! 𝑓(𝑥) to the graph of 𝑦 𝑓(𝑥)Linear asymptotes6.Include horizontal, vertical and oblique asymptotesFurther integrationApplications of integrationInclude: Volume of revolution for Cartesian or parametriccoordinates Arc length and area of surface of revolution forCartesian or parametric coordinatesExclude derivation of any formulae12
AM Syllabus (2016): Pure Mathematics7.Inverse trigonometricfunctionsDifferentiation and integration of inverse trigonometricfunctionsUse of trigonometric substitutions in integrationReduction formulaeFinding the reduction formulae for definite and indefiniteintegral.Exclude finding reduction formulae that involves morethan one variablePolar CoordinatesInclude: 8.Plotting of points in polar coordinatesConverting between polar and rectangularcoordinatesPolar curve sketching, including the symmetry forr being a function of cos 𝜃 only or of sin 𝜃 only.Curve sketching is limited to the form 𝑟 𝑓(𝜃)Intersection of polar curvesArea enclosed by a polar curveLocation of points at which tangents are eitherparallel to, or perpendicular to, the initial line.Further MatricesInverse of 3 3 matricesInclude: Definition of singular and non-singular matrices Definition of the determinant of a matrix Inverse of a 3 3 matrix using the adjoint methodand ERO methodSystem of three linearequationsSolving a system of linear equations using the inverse orthe ERO methodCondition for a system of equations to be consistent i.e.have a unique solution or infinite solution or inconsistenti.e. no solution at all. Also the geometric interpretation ofeach caseLinear transformations inthree dimensionsFinding the matrix associated with a linear transformationfor a Rotation through an angle 𝜃 about a coordinateaxis, Reflection in the planes x 0, y 0 or z 0, Enlargement (or reduction) in which the origin isthe centre of the enlargement (or reduction)13
AM Syllabus (2016): Pure MathematicsFinding the image of a point, line or plane under any ofthe above linear transformationsCompound transformations in three dimensionsDefinition of an invariant point, line or plane9.Further DifferentialEquations!"First orderSolving differential equations of the form !" 𝑃𝑦 𝑄,where P and Q are functions of x.Second orderSolving differential equations of the form!!!!"𝑎 !! ! 𝑏 !" 𝑐𝑦 𝑓(𝑥),where a, b and c are constants and 𝑓(𝑥) is p qx rx2,𝜆𝑒 !" or 𝑝 cos 𝑛𝑥 𝑞 sin 𝑛𝑥.The particular integral can be found by trial.Note that the trial solution will be given in problemsinvolving the failure case.10.11.Numerical MethodsLocation of rootsLocation of roots by considering changes in signThe Newton-Raphson method. This is limited to twoiterationsApproximate IntegrationThe trapezium and Simpson’s RuleIntegrating functions by using the first few terms of thecorresponding Maclaurin’s seriesOther series expansionsThe use of the logarithmic, exponential, binomial ortrigonometric series in finding an approximate value e.g.𝑒 !.! , ln1.2 etc.Further ProbabilityElementary probability.Calculation ofprobabilities of equallylikely eventsPermutations and combinationsAddition andmultiplication ofprobabilities.Mutually exclusive events.Use of Venn diagrams and tree diagrams to calculateprobabilities. Sampling with and without replacement.Understanding and use of P (A′) 1 P (A);14
AM Syllabus (2016): Pure MathematicsIndependent events.Conditional probability.Sum and product laws.Addition RuleP(A 𝐵) P(A) P(B) – P(A 𝐵)!(! !)and conditional probability defined as P (A B) !(!) .Two events A, B are independent if P(A 𝐵) P(A)P(B)Independence of a maximum of three events defined asfollows: 𝑃 𝐴 𝐵 𝐶 𝑃 𝐴 𝑃 𝐵 𝑃(𝐶) and any twoevents of A, B, C are independent.15
AM Syllabus (2016): Pure Mathematics 10 Matrix Properties Linear transformations in the plane and diagonal matrices Addition, subtraction and multiplication Condition for equal matrices Inverse of 2 2 matrices Exclude finding the inverse of 3 3 matrices, but students should be able to verify that two
Examinations syllabus for Cambridge International A & AS Level Mathematics 9709. The eight chapters of this book cover the pure mathematics in AS level. The series also contains a more advanced book for pure mathematics and one each for mechanics and statistics. These books are based on the highly successful series for the Mathematics in
The pure mathematics question in Papers I and II are based the core A-level Mathematics syllabus, with some minor additions, which is listed at the end of this book. The pure mathematics questions in Paper III are based on a ‘typical’ Further Mathematics mathematics A-level syllabus (at the time of writing,
Pure Mathematics and the Good My claim - pure mathematics itself embodies aspects of the good. 1. Validity in maths requires display of means of verification (proof, calculation) publicly and openly. Thus mathematics embodies the ethical values of openness and democracy 2. Mathematics grows through pure research --for its own sake -- based on .
Edexcel International A Level Mathematics Pure 3 Student Book 978 1 292244 92 1 21.00 Edexcel International A Level Mathematics Pure 3 Teacher Resource Pack 978 1 292244 93 8 100.00 Edexcel International A Level Mathematics Pure 4 Student Book 978 1 292245 12 6 21.00 Edexcel International A Level Mathematics Pure 4 Teacher Resource Pack
Pure vs. Applied Mathematics More Pure vs. Applied From [The Princeton Companion to Mathematics]: The word “pure” is more troublesome. As many have commented, there is no clear dividing line between pure and applied mathematics, and, just as a proper appreciation of modern mathematics
Centre: 91431 June 2019 - IAL Results Cand. No UCI Number Subject Code Subject Name Subject Grade Uniform Mark Maximum UMS 82 91431B190082T WMA11 PURE MATHEMATICS 1 (NEW) 100 100 82 91431B190082T WMA12 PURE MATHEMATICS 2 (NEW) 91 100 83 91431B190083A WMA11 PURE MATHEMATICS 1 (NEW) 99 100 83 91431B190083A WMA12 PURE MATHEMATICS 2 (NEW)
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 .
