Syllabus Cambridge International AS & A Level Mathematics 9709

Transcription

1 Syllabus Cambridge International AS & A Level Mathematics 9709 For eamination in June and November 00, 0 and 0. Also available for eamination in March 00, 0 and 0 for India only. Version

2 Why choose Cambridge? Cambridge Assessment International Education prepares school students for life, helping them develop an informed curiosity and a lasting passion for learning. We are part of the University of Cambridge. Our international qualifications are recognised by the world s best universities and employers, giving students a wide range of options in their education and career. As a not-for-profit organisation, we devote our resources to delivering high-quality educational programmes that can unlock learners potential. Our programmes and qualifications set the global standard for international education. They are created by subject eperts, rooted in academic rigour and reflect the latest educational research. They provide a strong platform for students to progress from one stage to the net, and are well supported by teaching and learning resources. We review all our syllabuses regularly, so they reflect the latest research evidence and professional teaching practice and take account of the different national contets in which they are taught. We consult with teachers to help us design each syllabus around the needs of their learners. Consulting with leading universities has helped us make sure our syllabuses encourage students to master the key concepts in the subject and develop the skills necessary for success in higher education. Our mission is to provide educational benefit through provision of international programmes and qualifications for school education and to be the world leader in this field. Together with schools, we develop Cambridge learners who are confident, responsible, reflective, innovative and engaged equipped for success in the modern world. Every year, nearly a million Cambridge students from schools in 60 countries prepare for their future with an international education from Cambridge International. We think the Cambridge curriculum is superb preparation for university. Christoph Guttentag, Dean of Undergraduate Admissions, Duke University, USA Quality management Our systems for managing the provision of international qualifications and education programmes for students aged 5 to 9 are certified as meeting the internationally recognised standard for quality management, ISO 900:008. Learn more at Cambridge Assessment International Education is part of the Cambridge Assessment Group. Cambridge Assessment is the brand name of the University of Cambridge Local Eaminations Syndicate (UCLES), which itself is a department of the University of Cambridge. UCLES retains the copyright on all its publications. Registered centres are permitted to copy material from this booklet for their own internal use. However, we cannot give permission to centres to photocopy any material that is acknowledged to a third party even for internal use within a centre.

3 Contents Why choose this syllabus?... Syllabus overview... 6 Aims 6 Content overview 7 Structure 8 Assessment overview 9 Assessment objectives 3 Subject content... Prior knowledge Pure Mathematics (for Paper ) Pure Mathematics (for Paper ) 6 3 Pure Mathematics 3 (for Paper 3) 9 4 Mechanics (for Paper 4) 4 5 Probability & Statistics (for Paper 5) 7 6 Probability & Statistics (for Paper 6) 30 4 Details of the assessment Relationship between components 33 Eamination information 33 Command words 35 5 List of formulae and statistical tables (MF9) What else you need to know...49 Before you start 49 Making entries 50 After the eam 5 How students, teachers and higher education can use the grades 5 Grade descriptions 5 Changes to this syllabus for 00, 0 and 0 53 Changes to this syllabus For information about changes to this syllabus for 00, 0 and 0, go to page 53.

4 Cambridge International AS & A Level Mathematics 9709 syllabus for 00, 0 and 0. Why choose this syllabus? Key benefits The best motivation for a student is a real passion for the subject they re learning. By offering students a variety of Cambridge International AS & A Levels, you can give them the greatest chance of finding the path of education they most want to follow. With over 50 subjects to choose from, students can select the ones they love and that they re best at, which helps motivate them throughout their studies. Following a Cambridge International AS & A Level programme helps students develop abilities which universities value highly, including: a deep understanding of their subjects higher order thinking skills analysis, critical thinking, problem solving presenting ordered and coherent arguments independent learning and research. Cambridge International AS & A Level Mathematics develops a set of transferable skills. These include the skill of working with mathematical information, as well as the ability to think logically and independently, consider accuracy, model situations mathematically, analyse results and reflect on findings. Learners can apply these skills across a wide range of subjects and the skills equip them well for progression to higher education or directly into employment. Our approach in Cambridge International AS & A Level Mathematics encourages learners to be: confident, using and sharing information and ideas, and using mathematical techniques to solve problems. These skills build confidence and support work in other subject areas as well as in mathematics. responsible, through learning and applying skills which prepare them for future academic studies, helping them to become numerate members of society. reflective, through making connections between different branches of mathematics and considering the outcomes of mathematical problems and modelling. innovative, through solving both familiar and unfamiliar problems in different ways, selecting from a range of mathematical and problem-solving techniques. engaged, by the beauty and structure of mathematics, its patterns and its many applications to real life situations. Cambridge students develop a deep understanding of subjects and independent thinking skills. Tony Hines, Principal, Rockledge High School, USA Back to contents page

5 Cambridge International AS & A Level Mathematics 9709 syllabus for 00, 0 and 0. Why choose this syllabus? Key concepts Key concepts are essential ideas that help students develop a deep understanding of their subject and make links between different aspects. Key concepts may open up new ways of thinking about, understanding or interpreting the important things to be learned. Good teaching and learning will incorporate and reinforce a subject s key concepts to help students gain: a greater depth as well as breadth of subject knowledge confidence, especially in applying knowledge and skills in new situations the vocabulary to discuss their subject conceptually and show how different aspects link together a level of mastery of their subject to help them enter higher education. The key concepts identified below, carefully introduced and developed, will help to underpin the course you will teach. You may identify additional key concepts which will also enrich teaching and learning. The key concepts for Cambridge International AS & A Level Mathematics are: Problem solving Mathematics is fundamentally problem solving and representing systems and models in different ways. These include: Algebra: this is an essential tool which supports and epresses mathematical reasoning and provides a means to generalise across a number of contets. Geometrical techniques: algebraic representations also describe a spatial relationship, which gives us a new way to understand a situation. Calculus: this is a fundamental element which describes change in dynamic situations and underlines the links between functions and graphs. Mechanical models: these eplain and predict how particles and objects move or remain stable under the influence of forces. Statistical methods: these are used to quantify and model aspects of the world around us. Probability theory predicts how chance events might proceed, and whether assumptions about chance are justified by evidence. Communication Mathematical proof and reasoning is epressed using algebra and notation so that others can follow each line of reasoning and confirm its completeness and accuracy. Mathematical notation is universal. Each solution is structured, but proof and problem solving also invite creative and original thinking. Mathematical modelling Mathematical modelling can be applied to many different situations and problems, leading to predictions and solutions. A variety of mathematical content areas and techniques may be required to create the model. Once the model has been created and applied, the results can be interpreted to give predictions and information about the real world. Back to contents page 3

6 Cambridge International AS & A Level Mathematics 9709 syllabus for 00, 0 and 0. Why choose this syllabus? Recognition and progression Every year thousands of students with Cambridge International AS & A Levels gain places at leading universities worldwide. Cambridge International AS & A Levels are accepted across 95 countries. They are valued by top universities around the world including those in the UK, US (including Ivy League universities), Europe, Australia, Canada and New Zealand. UK NARIC, the national agency in the UK for the recognition and comparison of international qualifications and skills, has carried out an independent benchmarking study of Cambridge International AS & A Level and found it to be comparable to the standard of AS & A Level in the UK. This means students can be confident that their Cambridge International AS & A Level qualifications are accepted as equivalent, grade for grade, to UK AS & A Levels by leading universities worldwide. Cambridge International AS Level Mathematics makes up the first half of the Cambridge International A Level course in mathematics and provides a foundation for the study of mathematics at Cambridge International A Level. Depending on local university entrance requirements, students may be able to use it to progress directly to university courses in mathematics or some other subjects. It is also suitable as part of a course of general education. Cambridge International A Level Mathematics provides a foundation for the study of mathematics or related courses in higher education. Equally it is suitable as part of a course of general education. For more information about the relationship between the Cambridge International AS Level and Cambridge International A Level see the Assessment overview section of the Syllabus overview. We recommend learners check the Cambridge recognitions database and the university websites to find the most up-to-date entry requirements for courses they wish to study. Learn more at The depth of knowledge displayed by the best A Level students makes them prime targets for America s Ivy League universities Yale University, USA 4 Back to contents page

7 Cambridge International AS & A Level Mathematics 9709 syllabus for 00, 0 and 0. Why choose this syllabus? Supporting teachers We provide a wide range of practical resources, detailed guidance, and innovative training and professional development so that you can give your learners the best possible preparation for Cambridge International AS & A Level. Teaching resources The School Support Hub Syllabus Scheme of work Learner guide Discussion forum Resource list Endorsed tetbooks and digital resources Training Face-to-face workshops around the world Online self-study training Online tutor-led training Cambridge Professional Development Qualifications Support for Cambridge International AS & A Level Eam preparation resources Question papers Mark schemes Eample candidate responses to understand what eaminers are looking for at key grades Eaminer reports to improve future teaching Community You can find useful information, as well as share your ideas and eperiences with other teachers, on our social media channels and community forums. Find out more at Cambridge International AS & A Levels prepare students well for university because they ve learnt to go into a subject in considerable depth. There s that ability to really understand the depth and richness and the detail of a subject. It s a wonderful preparation for what they are going to face at university. US Higher Education Advisory Council Back to contents page 5

8 Cambridge International AS & A Level Mathematics 9709 syllabus for 00, 0 and 0. Syllabus overview Aims The aims describe the purposes of a course based on this syllabus. The aims are to enable students to: develop their mathematical knowledge and skills in a way which encourages confidence and provides satisfaction and enjoyment develop an understanding of mathematical principles and an appreciation of mathematics as a logical and coherent subject acquire a range of mathematical skills, particularly those which will enable them to use applications of mathematics in the contet of everyday situations and of other subjects they may be studying develop the ability to analyse problems logically recognise when and how a situation may be represented mathematically, identify and interpret relevant factors and select an appropriate mathematical method to solve the problem use mathematics as a means of communication with emphasis on the use of clear epression acquire the mathematical background necessary for further study in mathematics or related subjects. Support for Cambridge International AS & A Level Mathematics Our School Support Hub provides Cambridge schools with a secure site for downloading specimen and past question papers, mark schemes, grade thresholds and other curriculum resources specific to this syllabus. The School Support Hub community offers teachers the opportunity to connect with each other and to ask questions related to the syllabus. 6 Back to contents page

9 Cambridge International AS & A Level Mathematics 9709 syllabus for 00, 0 and 0. Syllabus overview Content overview Content section Assessment component Topics included Pure Mathematics Paper. Quadratics. Functions.3 Coordinate geometry.4 Circular measure.5 Trigonometry.6 Series.7 Differentiation.8 Integration Pure Mathematics Paper. Algebra. Logarithmic and eponential functions.3 Trigonometry.4 Differentiation.5 Integration.6 Numerical solution of equations 3 Pure Mathematics 3 Paper 3 3. Algebra 3. Logarithmic and eponential functios 3.3 Trigonometry 3.4 Differentiation 3.5 Integration 3.6 Numerical solution of equations 3.7 Vectors 3.8 Differential equations 3.9 Comple numbers 4 Mechanics Paper 4 4. Forces and equilibrium 4. Kinematics of motion in a straight line 4.3 Momentum 4.4 Newton s laws of motion 4.5 Energy, work and power 5 Probability & Statistics Paper 5 5. Representation of data 5. Permutations and combinations 5.3 Probability 5.4 Discrete random variables 5.5 The normal distribution 6 Probability & Statistics Paper 6 6. The Poisson distribution 6. Linear combinations of random variables 6.3 Continuous random variables 6.4 Sampling and estimation 6.5 Hypothesis tests Back to contents page 7

10 Cambridge International AS & A Level Mathematics 9709 syllabus for 00, 0 and 0. Syllabus overview Structure There are si components that can be combined in specific ways (please see below for details): Paper : Pure Mathematics Paper 4: Mechanics Paper : Pure Mathematics Paper 5: Probability & Statistics Paper 3: Pure Mathematics 3 Paper 6: Probability & Statistics All AS Level candidates take two written papers. All A Level candidates take four written papers. AS Level Mathematics The Cambridge International AS Level Mathematics qualification offers three different options: Pure Mathematics only (Paper and Paper ) or Pure Mathematics and Mechanics (Paper and Paper 4) or Pure Mathematics and Probability & Statistics (Paper and Paper 5). Please note, the Pure-Mathematics-only option (Paper and Paper ) leads to an AS Level only and cannot be used as a staged route to a full A Level. Candidates who have taken Paper and Paper at AS Level and then wish to complete a full A Level would need to retake Paper alongside three other components. These should be chosen from the specific combinations below. A Level Mathematics The Cambridge International A Level Mathematics qualification offers two different options: Pure Mathematics, Mechanics and Probability & Statistics (Papers, 3, 4 and 5) or Pure Mathematics and Probability & Statistics (Papers, 3, 5 and 6). Please note, it is not possible to combine Paper 4 and Paper 6. This is because Paper 6 depends on prior knowledge of the subject content for Paper 5. See page 0 for a table showing all possible assessment routes. Structure of AS Level and A Level Mathematics AS Level Mathematics A Level Mathematics Paper and Paper Pure Mathematics only (No progression to A Level) Paper and Paper 4 Pure Mathematics and Mechanics Paper and Paper 5 Pure Mathematics and Probability & Statistics Paper, 3, 4 and 5 Pure Mathematics, Mechanics and Probability & Statistics Paper, 3, 5 and 6 Pure Mathematics and Probability & Statistics 8 Back to contents page

11 Cambridge International AS & A Level Mathematics 9709 syllabus for 00, 0 and 0. Syllabus overview Assessment overview Paper Paper 4 Pure Mathematics hour 50 minutes 75 marks 0 to structured questions based on the Pure Mathematics subject content Written eamination Eternally assessed 60% of the AS Level 30% of the A Level Compulsory for AS Level and A Level Mechanics hour 5 minutes 50 marks 6 to 8 structured questions based on the Mechanics subject content Written eamination Eternally assessed 40% of the AS Level 0% of the A Level Offered as part of AS Level or A Level Paper Paper 5 Pure Mathematics hour 5 minutes 50 marks 6 to 8 structured questions based on the Pure Mathematics subject content Written eamination Eternally assessed 40% of the AS Level Offered only as part of AS Level Probability & Statistics hour 5 minutes 50 marks 6 to 8 structured questions based on the Probability & Statistics subject content Written eamination Eternally assessed 40% of the AS Level 0% of the A Level Compulsory for A Level Paper 3 Paper 6 Pure Mathematics 3 hour 50 minutes 75 marks 9 to structured questions based on the Pure Mathematics 3 subject content Written eamination Eternally assessed 30% of the A Level only Compulsory for A Level Probability & Statistics hour 5 minutes 50 marks 6 to 8 structured questions based on the Probability & Statistics subject content Written eamination Eternally assessed 0% of the A Level only Offered only as part of A Level Back to contents page 9

12 Cambridge International AS & A Level Mathematics 9709 syllabus for 00, 0 and 0. Syllabus overview Three routes for Cambridge International AS & A Level Mathematics Candidates may combine components as shown below to suit their particular interests. Route AS Level only (Candidates take the AS components in the same series) Either Paper Pure Mathematics Paper Pure Mathematics Paper 3 Pure Mathematics 3 Not available for AS Level Paper 4 Mechanics Or Or Note this option in Route cannot count towards A Level Paper 5 Probability & Statistics Paper 6 Probability & Statistics Not available for AS Level Route A Level (staged over two years) Paper Pure Mathematics Paper Pure Mathematics Paper 3 Pure Mathematics 3 Paper 4 Mechanics Paper 5 Probability & Statistics Paper 6 Probability & Statistics Either Year AS Level Year Complete the A Level Or Year AS Level Year Complete the A Level Not available for A Level Or Year AS Level Year Complete the A Level Route 3 A Level (Candidates take the A Level components in the same series) Paper Pure Mathematics Paper Pure Mathematics Paper 3 Pure Mathematics 3 Paper 4 Mechanics Paper 5 Probability & Statistics Paper 6 Probability & Statistics Either Not available for Or A Level 0 Back to contents page

13 Cambridge International AS & A Level Mathematics 9709 syllabus for 00, 0 and 0. Syllabus overview Assessment objectives The assessment objectives (AOs) are: AO Knowledge and understanding Show understanding of relevant mathematical concepts, terminology and notation Recall accurately and use appropriate mathematical manipulative techniques AO Application and communication Recognise the appropriate mathematical procedure for a given situation Apply appropriate combinations of mathematical skills and techniques in solving problems Present relevant mathematical work, and communicate corresponding conclusions, in a clear and logical way Weighting for assessment objectives The approimate weightings (± 5%) allocated to each of the assessment objectives (AOs) are summarised below. Assessment objectives as an approimate percentage of each component Assessment objective Weighting in components % Paper Paper Paper 3 Paper 4 Paper 5 Paper 6 AO Knowledge and understanding AO Application and communication Assessment objectives as an approimate percentage of each qualification Assessment objective Weighting in AS Level % Weighting in A Level % AO Knowledge and understanding 55 5 AO Application and communication Back to contents page

14 Cambridge International AS & A Level Mathematics 9709 syllabus for 00, 0 and 0. 3 Subject content The mathematical content for each component is detailed below. You can teach the topics in any order you find appropriate. Information about calculator use and information about the relationships between syllabus components can be found in 4 Details of the assessment. are included to clarify the subject content. Please note that these are eamples only and eamination questions may differ from the eamples given. Prior knowledge Knowledge of the content of the Cambridge IGCSE Mathematics 0580 (Etended curriculum), or Cambridge International O Level (404/409), is assumed. Candidates should be familiar with scientific notation for compound units, e.g. 5 m s for 5 metres per second. In addition, candidates should: 6 be able to carry out simple manipulation of surds (e.g. epressing as 3 and as 3 ), know the shapes of graphs of the form y = k n, where k is a constant and n is an integer (positive or negative) or!. Pure Mathematics (for Paper ). Quadratics carry out the process of completing the square for a quadratic polynomial a + b + c and use a completed square form find the discriminant of a quadratic polynomial a + b + c and use the discriminant solve quadratic equations, and quadratic inequalities, in one unknown solve by substitution a pair of simultaneous equations of which one is linear and one is quadratic recognise and solve equations in which are quadratic in some function of. e.g. to locate the verte of the graph of y = a + b + c or to sketch the graph e.g. to determine the number of real roots of the equation a + b + c = 0. Knowledge of the term repeated root is included. By factorising, completing the square and using the formula. e.g. + y + = 0 and + y = 5, + 3y = 7 and 3 = 4 + 4y. e.g = 0, 6+ = 0, tan = + tan. Back to contents page

15 Cambridge International AS & A Level Mathematics 9709 syllabus for 00, 0 and 0. Subject content Pure Mathematics. Functions understand the terms function, domain, range, one-one function, inverse function and composition of functions identify the range of a given function in simple cases, and find the composition of two given functions determine whether or not a given function is one-one, and find the inverse of a one-one function in simple cases illustrate in graphical terms the relation between a one-one function and its inverse understand and use the transformations of the graph of y = f() given by y = f() + a, y = f( + a), y = af(), y = f(a) and simple combinations of these. e.g. range of f : 7 for H and range of g: 7 + for! R. Including the condition that a composite function gf can only be formed when the range of f is within the domain of g. e.g. finding the inverse of 3 h: 7 ^ + 3h 4 for. Sketches should include an indication of the mirror line y =. Including use of the terms translation, reflection and stretch in describing transformations. Questions may involve algebraic or trigonometric functions, or other graphs with given features..3 Coordinate geometry find the equation of a straight line given sufficient information interpret and use any of the forms y = m + c, y y = m( ), a + by + c = 0 in solving problems understand that the equation ( a) + (y b) = r represents the circle with centre (a, b) and radius r use algebraic methods to solve problems involving lines and circles understand the relationship between a graph and its associated algebraic equation, and use the relationship between points of intersection of graphs and solutions of equations. e.g. given two points, or one point and the gradient. Including calculations of distances, gradients, midpoints, points of intersection and use of the relationship between the gradients of parallel and perpendicular lines. Including use of the epanded form + y + g + fy + c = 0. Including use of elementary geometrical properties of circles, e.g. tangent perpendicular to radius, angle in a semicircle, symmetry. Implicit differentiation is not included. e.g. to determine the set of values of k for which the line y = + k intersects, touches or does not meet a quadratic curve. Back to contents page 3

16 Cambridge International AS & A Level Mathematics 9709 syllabus for 00, 0 and 0. Subject content Pure Mathematics.4 Circular measure understand the definition of a radian, and use the relationship between radians and degrees use the formulae s = riand A = r i in solving problems concerning the arc length and sector area of a circle. Including calculation of lengths and angles in triangles and areas of triangles..5 Trigonometry sketch and use graphs of the sine, cosine and tangent functions (for angles of any size, and using either degrees or radians) use the eact values of the sine, cosine and tangent of 30, 45, 60, and related angles use the notations sin, cos, tan to denote the principal values of the inverse trigonometric relations sin i use the identities cos / tan i and i sin i+ cos i / find all the solutions of simple trigonometrical equations lying in a specified interval (general forms of solution are not included). Including e.g. y = 3 sin, y = cos, y = tan c +. 4 rm e.g. cos50 3 c = 3, sin 4 r =. No specialised knowledge of these functions is required, but understanding of them as eamples of inverse functions is epected. e.g. in proving identities, simplifying epressions and solving equations. e.g. solve 3sin + = 0for r r, 3sin i 5cos i = 0for 0c G i G 360c..6 Series use the epansion of (a + b) n, where n is a positive integer recognise arithmetic and geometric progressions use the formulae for the nth term and for the sum of the first n terms to solve problems involving arithmetic or geometric progressions use the condition for the convergence of a geometric progression, and the formula for the sum to infinity of a convergent geometric progression. Including the notations fp n and n! r Knowledge of the greatest term and properties of the coefficients are not required. Including knowledge that numbers a, b, c are in arithmetic progression if b = a + c (or equivalent) and are in geometric progression if b = ac (or equivalent). Questions may involve more than one progression. 4 Back to contents page

17 Cambridge International AS & A Level Mathematics 9709 syllabus for 00, 0 and 0. Subject content Pure Mathematics.7 Differentiation understand the gradient of a curve at a point as the limit of the gradients of a suitable sequence of chords, and use the notations d y d f (), f (),, and d for first and second dy derivatives use the derivative of n (for any rational n), together with constant multiples, sums and differences of functions, and of composite functions using the chain rule apply differentiation to gradients, tangents and normals, increasing and decreasing functions and rates of change locate stationary points and determine their nature, and use information about stationary points in sketching graphs. Only an informal understanding of the idea of a limit is epected. e.g. includes consideration of the gradient of the chord joining the points with coordinates and ( + h) on the curve y = 3. Formal use of the general method of differentiation from first principles is not required. d y e.g. find, given y 3 = + 5 d. Including connected rates of change, e.g. given the rate of increase of the radius of a circle, find the rate of increase of the area for a specific value of one of the variables. Including use of the second derivative for identifying maima and minima; alternatives may be used in questions where no method is specified. Knowledge of points of infleion is not included..8 Integration understand integration as the reverse process of differentiation, and integrate (a + b) n (for any rational n ecept ), together with constant multiples, sums and differences solve problems involving the evaluation of a constant of integration e.g. 3 ;^ 5+ h d, > + 3 d. ^ h e.g. to find the equation of the curve through (, ) d y for which = +. d evaluate definite integrals Including simple cases of improper integrals, such as ; dand ; d use definite integration to find the area of a region bounded by a curve and lines parallel to the aes, or between a curve and a line or between two curves a volume of revolution about one of the aes. A volume of revolution may involve a region not bounded by the ais of rotation, e.g. the region between y = 9 and y = 5 rotated about the -ais. Back to contents page 5

18 Cambridge International AS & A Level Mathematics 9709 syllabus for 00, 0 and 0. Subject content Pure Mathematics (for Paper ) Knowledge of the content for Paper : Pure Mathematics is assumed, and candidates may be required to demonstrate such knowledge in answering questions.. Algebra understand the meaning of, sketch the graph of y = a + b and use relations such as a = b a = b and a < b a b < < a + b when solving equations and inequalities divide a polynomial, of degree not eceeding 4, by a linear or quadratic polynomial, and identify the quotient and remainder (which may be zero) use the factor theorem and the remainder theorem. Graphs of y = f() and y = f( ) for non-linear functions f are not included. e.g. 3 = + 7, + 5 < + e.g. to find factors and remainders, solve polynomial equations or evaluate unknown coefficients. Including factors of the form (a + b) in which the coefficient of is not unity, and including calculation of remainders.. Logarithmic and eponential functions understand the relationship between logarithms and indices, and use the laws of logarithms (ecluding change of base) understand the definition and properties of e and ln, including their relationship as inverse functions and their graphs use logarithms to solve equations and inequalities in which the unknown appears in indices use logarithms to transform a given relationship to linear form, and hence determine unknown constants by considering the gradient and/or intercept. Including knowledge of the graph of y = e k for both positive and negative values of k. e.g. 5, 3 3 # 5, 3 = 4 + e.g. y = k n gives ln y = ln k + n ln which is linear in ln and ln y y = k (a ) gives ln y = ln k + ln a which is linear in and ln y Back to contents page

19 Cambridge International AS & A Level Mathematics 9709 syllabus for 00, 0 and 0. Subject content Pure Mathematics.3 Trigonometry understand the relationship of the secant, cosecant and cotangent functions to cosine, sine and tangent, and use properties and graphs of all si trigonometric functions for angles of any magnitude use trigonometrical identities for the simplification and eact evaluation of epressions, and in the course of solving equations, and select an identity or identities appropriate to the contet, showing familiarity in particular with the use of sec i/ + tan i and cosec i/ + cot i the epansions of sin(a ± B), cos(a ± B) and tan(a ± B) the formulae for sin A, cos A and tan A the epression of asini+ bcos i in the forms R sin^i! ah and R cos^i! ah. e.g. simplifying cos^ 30ch 3 sin^ 60ch. e.g. solving tani+ cot i = 4, sec i- tan i = 5, 3cosi+ sin i =..4 Differentiation use the derivatives of e, ln, sin, cos, tan, together with constant multiples, sums, differences and composites differentiate products and quotients e.g. find and use the first derivative of a function which is defined parametrically or implicitly , ln, e l. e.g. = t e t, y = t + e t. e.g. + y = y + 7. Including use in problems involving tangents and normals..5 Integration etend the idea of reverse differentiation to include the integration of e a + b, a + b, sin(a + b), cos(a + b) and sec (a + b) use trigonometrical relationships in carrying out integration understand and use the trapezium rule to estimate the value of a definite integral. Knowledge of the general method of integration by substitution is not required. e.g. use of double-angle formulae to integrate sin or cos (). Including use of sketch graphs in simple cases to determine whether the trapezium rule gives an overestimate or an under-estimate. Back to contents page 7

20 Cambridge International AS & A Level Mathematics 9709 syllabus for 00, 0 and 0. Subject content Pure Mathematics.6 Numerical solution of equations locate approimately a root of an equation, by means of graphical considerations and/or searching for a sign change understand the idea of, and use the notation for, a sequence of approimations which converges to a root of an equation understand how a given simple iterative formula of the form n + = F( n ) relates to the equation being solved, and use a given iteration, or an iteration based on a given rearrangement of an equation, to determine a root to a prescribed degree of accuracy. e.g. finding a pair of consecutive integers between which a root lies. Knowledge of the condition for convergence is not included, but an understanding that an iteration may fail to converge is epected. 8 Back to contents page

21 Cambridge International AS & A Level Mathematics 9709 syllabus for 00, 0 and 0. Subject content 3 Pure Mathematics 3 (for Paper 3) Knowledge of the content of Paper : Pure Mathematics is assumed, and candidates may be required to demonstrate such knowledge in answering questions. 3. Algebra understand the meaning of, sketch the graph of y = a + b and use relations such as a = b a = b and a < b a b < < a + b when solving equations and inequalities divide a polynomial, of degree not eceeding 4, by a linear or quadratic polynomial, and identify the quotient and remainder (which may be zero) use the factor theorem and the remainder theorem recall an appropriate form for epressing rational functions in partial fractions, and carry out the decomposition, in cases where the denominator is no more complicated than (a + b)(c + d)(e + f ) (a + b)(c + d) (a + b)(c + d) use the epansion of ( + ) n, where n is a rational number and. Graphs of y = f() and y = f( ) for non-linear functions f are not included. e.g. 3 = + 7, + 5 < +. e.g. to find factors and remainders, solve polynomial equations or evaluate unknown coefficients. Including factors of the form (a + b) in which the coefficient of is not unity, and including calculation of remainders. Ecluding cases where the degree of the numerator eceeds that of the denominator Finding the general term in an epansion is not included. Adapting the standard series to epand e.g. ` j is included, and determining the set of values of for which the epansion is valid in such cases is also included. Back to contents page 9

22 Cambridge International AS & A Level Mathematics 9709 syllabus for 00, 0 and 0. Subject content 3 Pure Mathematics 3 3. Logarithmic and eponential functions understand the relationship between logarithms and indices, and use the laws of logarithms (ecluding change of base) understand the definition and properties of e and ln, including their relationship as inverse functions and their graphs use logarithms to solve equations and inequalities in which the unknown appears in indices use logarithms to transform a given relationship to linear form, and hence determine unknown constants by considering the gradient and/or intercept. Including knowledge of the graph of y = e k for both positive and negative values of k. e.g. 5, 3 3 # < 5, 3 = 4 + e.g. y = k n gives ln y = ln k + n ln which is linear in ln and ln y. y = k (a ) gives ln y = ln k + ln a which is linear in and ln y Trigonometry understand the relationship of the secant, cosecant and cotangent functions to cosine, sine and tangent, and use properties and graphs of all si trigonometric functions for angles of any magnitude use trigonometrical identities for the simplification and eact evaluation of epressions, and in the course of solving equations, and select an identity or identities appropriate to the contet, showing familiarity in particular with the use of sec i/ + tan iand cosec i/ + cot i the epansions of sin(a ± B), cos(a ± B) and tan(a ± B) the formulae for sin A, cos A and tan A the epression of asini+ bcos i in the forms R sin^i! ah and R cos^i! ah. e.g. simplifying cos( 30 ) 3 sin( 60 ). e.g. solving tani+ cot i = 4, sec i tan i = 5, 3cosi+ sin i =. 0 Back to contents page

23 Cambridge International AS & A Level Mathematics 9709 syllabus for 00, 0 and 0. Subject content 3 Pure Mathematics Differentiation use the derivatives of e, ln, sin, cos, tan, tan, together with constant multiples, sums, differences and composites differentiate products and quotients e.g. find and use the first derivative of a function which is defined parametrically or implicitly. Derivatives of sin and cos are not required , ln, e. e.g. = t e t, y = t + e t. e.g. + y = y + 7. Including use in problems involving tangents and normals. 3.5 Integration etend the idea of reverse differentiation to include the integration of e a + b, a + b, sin(a + b), cos(a + b), sec (a + b) and a + use trigonometrical relationships in carrying out integration integrate rational functions by means of decomposition into partial fractions kf l^h recognise an integrand of the form, and f^h integrate such functions recognise when an integrand can usefully be regarded as a product, and use integration by parts use a given substitution to simplify and evaluate either a definite or an indefinite integral. Including eamples such as. + 3 e.g. use of double-angle formulae to integrate sin or cos (). Restricted to types of partial fractions as specified in topic 3. above. e.g. integration of, tan. + e.g. integration of sin, e, ln, tan. e.g. to integrate sin cos using the substitution u = sin. Back to contents page

24 Cambridge International AS & A Level Mathematics 9709 syllabus for 00, 0 and 0. Subject content 3 Pure Mathematics Numerical solution of equations locate approimately a root of an equation, by means of graphical considerations and/or searching for a sign change understand the idea of, and use the notation for, a sequence of approimations which converges to a root of an equation understand how a given simple iterative formula of the form n + = F( n ) relates to the equation being solved, and use a given iteration, or an iteration based on a given rearrangement of an equation, to determine a root to a prescribed degree of accuracy. e.g. finding a pair of consecutive integers between which a root lies. Knowledge of the condition for convergence is not included, but an understanding that an iteration may fail to converge is epected. 3.7 Vectors use standard notations for vectors, i.e. f p, i + yj, fp, y i + yj + zk, AB, a y z carry out addition and subtraction of vectors and multiplication of a vector by a scalar, and interpret these operations in geometrical terms calculate the magnitude of a vector, and use unit vectors, displacement vectors and position vectors understand the significance of all the symbols used when the equation of a straight line is epressed in the form r = a + tb, and find the equation of a line, given sufficient information determine whether two lines are parallel, intersect or are skew, and find the point of intersection of two lines when it eists use formulae to calculate the scalar product of two vectors, and use scalar products in problems involving lines and points. e.g. OABC is a parallelogram is equivalent to OB = OA + OC. The general form of the ratio theorem is not included, but understanding that the midpoint of AB has position vector _ OA + OBi is epected. In or 3 dimensions. e.g. finding the equation of a line given the position vector of a point on the line and a direction vector, or the position vectors of two points on the line. Calculation of the shortest distance between two skew lines is not required. Finding the equation of the common perpendicular to two skew lines is also not required. e.g. finding the angle between two lines, and finding the foot of the perpendicular from a point to a line; questions may involve 3D objects such as cuboids, tetrahedra (pyramids), etc. Knowledge of the vector product is not required. Back to contents page

25 Cambridge International AS & A Level Mathematics 9709 syllabus for 00, 0 and 0. Subject content 3 Pure Mathematics Differential equations formulate a simple statement involving a rate of change as a differential equation find by integration a general form of solution for a first order differential equation in which the variables are separable use an initial condition to find a particular solution interpret the solution of a differential equation in the contet of a problem being modelled by the equation. The introduction and evaluation of a constant of proportionality, where necessary, is included. Including any of the integration techniques from topic 3.5 above. Where a differential equation is used to model a real-life situation, no specialised knowledge of the contet will be required. 3.9 Comple numbers understand the idea of a comple number, recall the meaning of the terms real part, imaginary part, modulus, argument, conjugate, and use the fact that two comple numbers are equal if and only if both real and imaginary parts are equal carry out operations of addition, subtraction, multiplication and division of two comple numbers epressed in Cartesian form + iy use the result that, for a polynomial equation with real coefficients, any non-real roots occur in conjugate pairs represent comple numbers geometrically by means of an Argand diagram carry out operations of multiplication and division of two comple numbers epressed in polar form r cosi+ i sin i / re i i ^ h Notations Re z, Im z, z, arg z, z* should be known. The argument of a comple number will usually refer to an angle i such that r i G r, but in some cases the interval 0 G i r may be more convenient. Answers may use either interval unless the question specifies otherwise. For calculations involving multiplication or division, full details of the working should be shown. e.g. in solving a cubic or quartic equation where one comple root is given. Including the results z z = z z and arg_ zz arg z arg z i= _ i+ _ i, and corresponding results for division. find the two square roots of a comple number e.g. the square roots of 5 + i in eact Cartesian form. Full details of the working should be shown. understand in simple terms the geometrical effects of conjugating a comple number and of adding, subtracting, multiplying and dividing two comple numbers illustrate simple equations and inequalities involving comple numbers by means of loci in an Argand diagram e.g. z a < k, z a = z b, arg(z a) = α. Back to contents page 3

26 Cambridge International AS & A Level Mathematics 9709 syllabus for 00, 0 and 0. Subject content 4 Mechanics (for Paper 4) Questions set will be mainly numerical, and will aim to test mechanical principles without involving difficult algebra or trigonometry. However, candidates should be familiar in particular with the following trigonometrical results: sin sin^90c - ih / cos i, cos^90c - ih / sin i, tan i / i cos, sin i+ cos i /. i Knowledge of algebraic methods from the content for Paper : Pure Mathematics is assumed. This content list refers to the equilibrium or motion of a particle. Eamination questions may involve etended bodies in a realistic contet, but these etended bodies should be treated as particles, so any force acting on them is modelled as acting at a single point. Vector notation will not be used in the question papers. 4. Forces and equilibrium identify the forces acting in a given situation understand the vector nature of force, and find and use components and resultants use the principle that, when a particle is in equilibrium, the vector sum of the forces acting is zero, or equivalently, that the sum of the components in any direction is zero understand that a contact force between two surfaces can be represented by two components, the normal component and the frictional component use the model of a smooth contact, and understand the limitations of this model understand the concepts of limiting friction and limiting equilibrium, recall the definition of coefficient of friction, and use the relationship F = nr or F G n R, as appropriate e.g. by drawing a force diagram. Calculations are always required, not approimate solutions by scale drawing. Solutions by resolving are usually epected, but equivalent methods (e.g. triangle of forces, Lami s Theorem, where suitable) are also acceptable; these other methods are not required knowledge, and will not be referred to in questions. Terminology such as about to slip may be used to mean in limiting equilibrium in questions. use Newton s third law. e.g. the force eerted by a particle on the ground is equal and opposite to the force eerted by the ground on the particle. 4 Back to contents page

27 Cambridge International AS & A Level Mathematics 9709 syllabus for 00, 0 and 0. Subject content 4 Mechanics 4. Kinematics of motion in a straight line understand the concepts of distance and speed as scalar quantities, and of displacement, velocity and acceleration as vector quantities Restricted to motion in one dimension only. The term deceleration may sometimes be used in the contet of decreasing speed. sketch and interpret displacement time graphs and velocity time graphs, and in particular appreciate that the area under a velocity time graph represents displacement, the gradient of a displacement time graph represents velocity, the gradient of a velocity time graph represents acceleration use differentiation and integration with respect to time to solve simple problems concerning displacement, velocity and acceleration use appropriate formulae for motion with constant acceleration in a straight line. Calculus required is restricted to techniques from the content for Paper : Pure Mathematics. Questions may involve setting up more than one equation, using information about the motion of different particles. 4.3 Momentum use the definition of linear momentum and show understanding of its vector nature use conservation of linear momentum to solve problems that may be modelled as the direct impact of two bodies. For motion in one dimension only. Including direct impact of two bodies where the bodies coalesce on impact. Knowledge of impulse and the coefficient of restitution is not required. Back to contents page 5

28 Cambridge International AS & A Level Mathematics 9709 syllabus for 00, 0 and 0. Subject content 4 Mechanics 4.4 Newton s laws of motion apply Newton s laws of motion to the linear motion of a particle of constant mass moving under the action of constant forces, which may include friction, tension in an inetensible string and thrust in a connecting rod If any other forces resisting motion are to be considered (e.g. air resistance) this will be indicated in the question. use the relationship between mass and weight W = mg. In this component, questions are mainly numerical, and use of the approimate numerical value 0 (m s ) for g is epected. solve simple problems which may be modelled as the motion of a particle moving vertically or on an inclined plane with constant acceleration solve simple problems which may be modelled as the motion of connected particles. Including, for eample, motion of a particle on a rough plane where the acceleration while moving up the plane is different from the acceleration while moving down the plane. e.g. particles connected by a light inetensible string passing over a smooth pulley, or a car towing a trailer by means of either a light rope or a light rigid towbar. 4.5 Energy, work and power understand the concept of the work done by a force, and calculate the work done by a constant force when its point of application undergoes a displacement not necessarily parallel to the force understand the concepts of gravitational potential energy and kinetic energy, and use appropriate formulae understand and use the relationship between the change in energy of a system and the work done by the eternal forces, and use in appropriate cases the principle of conservation of energy use the definition of power as the rate at which a force does work, and use the relationship between power, force and velocity for a force acting in the direction of motion solve problems involving, for eample, the instantaneous acceleration of a car moving on a hill against a resistance. W = Fd cos i; Use of the scalar product is not required. Including cases where the motion may not be linear (e.g. a child on a smooth curved slide ), where only overall energy changes need to be considered. Including calculation of (average) power as Work done. Time taken P = Fv. 6 Back to contents page

29 Cambridge International AS & A Level Mathematics 9709 syllabus for 00, 0 and 0. Subject content 5 Probability & Statistics (for Paper 5) Questions set will be mainly numerical, and will test principles in probability and statistics without involving knowledge of algebraic methods beyond the content for Paper : Pure Mathematics. Knowledge of the following probability notation is also assumed: P(A), P ^A, Bh, P^A+ Bh, P(A B) and the use of A to denote the complement of A. 5. Representation of data select a suitable way of presenting raw statistical data, and discuss advantages and/or disadvantages that particular representations may have draw and interpret stem-and-leaf diagrams, boand-whisker plots, histograms and cumulative frequency graphs understand and use different measures of central tendency (mean, median, mode) and variation (range, interquartile range, standard deviation) Including back-to-back stem-and-leaf diagrams. e.g. in comparing and contrasting sets of data. use a cumulative frequency graph e.g. to estimate medians, quartiles, percentiles, the proportion of a distribution above (or below) a given value, or between two values. calculate and use the mean and standard deviation of a set of data (including grouped data) either from the data itself or from given totals / and /, or coded totals / ^ ah and / ^ ah, and use such totals in solving problems which may involve up to two data sets. 5. Permutations and combinations understand the terms permutation and combination, and solve simple problems involving selections solve problems about arrangements of objects in a line, including those involving repetition (e.g. the number of ways of arranging the letters of the word NEEDLESS ) restriction (e.g. the number of ways several people can stand in a line if two particular people must, or must not, stand net to each other). Questions may include cases such as people sitting in two (or more) rows. Questions about objects arranged in a circle will not be included. Back to contents page 7

30 Cambridge International AS & A Level Mathematics 9709 syllabus for 00, 0 and 0. Subject content 5 Probability & Statistics 5.3 Probability evaluate probabilities in simple cases by means of enumeration of equiprobable elementary events, or by calculation using permutations or combinations use addition and multiplication of probabilities, as appropriate, in simple cases understand the meaning of eclusive and independent events, including determination of whether events A and B are independent by comparing the values of P ^A+ Bh and P^Ah# P^Bh calculate and use conditional probabilities in simple cases. e.g. the total score when two fair dice are thrown. e.g. drawing balls at random from a bag containing balls of different colours. Eplicit use of the general formula P^A, Bh= P^Ah+ P^Bh P^A+ Bh is not required. e.g. situations that can be represented by a sample space of equiprobable elementary events, or a tree P ^A+ Bh diagram. The use of P^ABh = may be P^Bh required in simple cases. 5.4 Discrete random variables draw up a probability distribution table relating to a given situation involving a discrete random variable X, and calculate E(X) and Var(X) use formulae for probabilities for the binomial and geometric distributions, and recognise practical situations where these distributions are suitable models use formulae for the epectation and variance of the binomial distribution and for the epectation of the geometric distribution. Including the notations B(n, p) and Geo(p). Geo(p) denotes the distribution in which p r = p( p) r for r =,, 3,. Proofs of formulae are not required. 8 Back to contents page

31 Cambridge International AS & A Level Mathematics 9709 syllabus for 00, 0 and 0. Subject content 5 Probability & Statistics 5.5 The normal distribution understand the use of a normal distribution to model a continuous random variable, and use normal distribution tables solve problems concerning a variable X, where X + N_ nv, i, including finding the value of P_ X > i, or a related probability, given the values of, n, v. finding a relationship between, n and v given the value of P_ X > i or a related probability recall conditions under which the normal distribution can be used as an approimation to the binomial distribution, and use this approimation, with a continuity correction, in solving problems. Sketches of normal curves to illustrate distributions or probabilities may be required. For calculations involving standardisation, full details of the working should be shown. _ X - ni e.g. Z = v n sufficiently large to ensure that both np > 5 and nq > 5. Back to contents page 9

32 Cambridge International AS & A Level Mathematics 9709 syllabus for 00, 0 and 0. Subject content 6 Probability & Statistics (for Paper 6) Knowledge of the content of Paper 5: Probability & Statistics is assumed, and candidates may be required to demonstrate such knowledge in answering questions. Knowledge of calculus within the content for Paper 3: Pure Mathematics 3 will also be assumed. 6. The Poisson distribution use formulae to calculate probabilities for the distribution Po^h m use the fact that if X + Po^h m then the mean and variance of X are each equal to m understand the relevance of the Poisson distribution to the distribution of random events, and use the Poisson distribution as a model use the Poisson distribution as an approimation to the binomial distribution where appropriate use the normal distribution, with continuity correction, as an approimation to the Poisson distribution where appropriate. Proofs are not required. The conditions that n is large and p is small should be known; n > 50 and np < 5, approimately. The condition that m is large should be known; m 5, approimately. 6. Linear combinations of random variables use, when solving problems, the results that E(aX + b) = ae(x) + b and Var(aX + b) = a Var(X) E(aX + by) = ae(x) + be(y) Var(aX + by) = a Var(X) + b Var(Y) for independent X and Y if X has a normal distribution then so does ax + b if X and Y have independent normal distributions then ax + by has a normal distribution if X and Y have independent Poisson distributions then X + Y has a Poisson distribution. Proofs of these results are not required Back to contents page

33 Cambridge International AS & A Level Mathematics 9709 syllabus for 00, 0 and 0. Subject content 6 Probability & Statistics 6.3 Continuous random variables understand the concept of a continuous random variable, and recall and use properties of a probability density function use a probability density function to solve problems involving probabilities, and to calculate the mean and variance of a distribution. For density functions defined over a single interval only; the domain may be infinite, e.g. 34 for H. Including location of the median or other percentiles of a distribution by direct consideration of an area using the density function. Eplicit knowledge of the cumulative distribution function is not included. 6.4 Sampling and estimation understand the distinction between a sample and a population, and appreciate the necessity for randomness in choosing samples eplain in simple terms why a given sampling method may be unsatisfactory recognise that a sample mean can be regarded as a random variable, and use the facts that v E_ X i = n and that Var_ X i = n Including an elementary understanding of the use of random numbers in producing random samples. Knowledge of particular sampling methods, such as quota or stratified sampling, is not required. use the fact that _ X i has a normal distribution if X has a normal distribution use the Central Limit Theorem where appropriate Only an informal understanding of the Central Limit Theorem (CLT) is required; for large sample sizes, the distribution of a sample mean is approimately normal. calculate unbiased estimates of the population mean and variance from a sample, using either raw or summarised data determine and interpret a confidence interval for a population mean in cases where the population is normally distributed with known variance or where a large sample is used determine, from a large sample, an approimate confidence interval for a population proportion. Only a simple understanding of the term unbiased is required, e.g. that although individual estimates will vary the process gives an accurate result on average. Back to contents page 3

34 Cambridge International AS & A Level Mathematics 9709 syllabus for 00, 0 and 0. Subject content 6 Probability & Statistics 6.5 Hypothesis tests understand the nature of a hypothesis test, the difference between one-tailed and two-tailed tests, and the terms null hypothesis, alternative hypothesis, significance level, rejection region (or critical region), acceptance region and test statistic Outcomes of hypothesis tests are epected to be interpreted in terms of the contets in which questions are set. formulate hypotheses and carry out a hypothesis test in the contet of a single observation from a population which has a binomial or Poisson distribution, using direct evaluation of probabilities a normal approimation to the binomial or the Poisson distribution, where appropriate formulate hypotheses and carry out a hypothesis test concerning the population mean in cases where the population is normally distributed with known variance or where a large sample is used understand the terms Type I error and Type II error in relation to hypothesis tests calculate the probabilities of making Type I and Type II errors in specific situations involving tests based on a normal distribution or direct evaluation of binomial or Poisson probabilities. 3 Back to contents page

35 Cambridge International AS & A Level Mathematics 9709 syllabus for 00, 0 and 0. 4 Details of the assessment Relationship between components Candidates build their knowledge of the mathematics content as they progress through the course. Paper : Pure Mathematics is the foundation for all other components. Paper : Pure Mathematics and Paper 3: Pure Mathematics 3 build on the subject content for Paper : Pure Mathematics. Paper 4: Mechanics and Paper 5: Probability & Statistics components assume prior knowledge of the Paper : Pure Mathematics content. The probability and statistics content in the qualification also builds sequentially through the components, so Paper 5: Probability & Statistics is the foundation for studying Paper 6: Probability & Statistics. The subject content assessed in the different eamination components does not normally overlap. Only Paper has a significant overlap with Paper 3, as the Pure Mathematics subject content is largely a subset of the Pure Mathematics 3 subject content. Candidates may not take both Paper and Paper 3 in the same eamination series. Paper and Paper 3 are taken in alternative routes through the qualification Paper is for AS Level only, and Paper 3 is for A Level. Eamination information All components are assessed by written eaminations which are eternally marked. Sample assessment materials are available on our website at showing the question style and level of the eamination papers. Application of mathematical techniques As well as demonstrating the appropriate techniques, candidates need to apply their knowledge in solving problems. Individual eamination questions may involve ideas and methods from more than one section of the subject content for that component. The main focus of eamination questions will be the AS & A Level Mathematics subject content. However, in eamination questions, candidates may need to make use of prior knowledge and mathematical techniques from previous study, as listed in section 3 of this syllabus. Structure of the question paper All questions in the eamination papers are compulsory. An approimate number of questions for each paper is given in the Assessment overview in section of this syllabus. Questions are of varied lengths and often contain several parts, labelled (a), (b), (c), which may have sub-parts (i), (ii), (iii), as needed. Some questions might require candidates to sketch graphs or diagrams, or draw accurate graphs. Answer space Candidates answer on the question paper. All working should be shown neatly and clearly in the spaces provided for each question. New questions often start on a fresh page, so more answer space may be provided than is needed. If additional space is required, candidates should use the lined page at the end of the question paper, where the question number or numbers must be clearly shown. Back to contents page 33

36 Cambridge International AS & A Level Mathematics 9709 syllabus for 00, 0 and 0. Details of the assessment Degrees of accuracy Candidates should give non-eact numerical answers correct to three significant figures (or one decimal place for angles in degrees) unless a different level of accuracy is specified in the question. To earn accuracy marks, candidates should avoid rounding figures until they have their final answer. Additional materials for eaminations Candidates are epected to have the following equipment in eaminations: a ruler a scientific calculator (see the following section). Note: a protractor and a pair of compasses are not required. A list of formulae and statistical tables (MF9) is supplied in eaminations for the use of candidates. A copy of the list of formulae and tables is given for reference in section 5 of this syllabus. Note that MF9 is a combined formulae list for AS & A Level Mathematics (9709) and AS & A Level Further Mathematics (93). Some formulae in the list are not needed for this syllabus, and are only for Further Mathematics (93); these are listed in separate sections labelled Further Pure Mathematics, Further Mechanics, and Further Probability & Statistics. Calculators It is epected that candidates will have a calculator with standard scientific functions available for use in all the eaminations. Computers, graphical calculators and calculators capable of symbolic algebraic manipulation or symbolic differentiation or integration are not permitted. The General Regulations concerning the use of calculators are contained in the Cambridge Handbook at Candidates are epected to show all necessary working; no marks will be given for unsupported answers from a calculator. Mathematical notation The list of mathematical notation that may be used in eaminations for this syllabus is available on our website at Back to contents page

37 Cambridge International AS & A Level Mathematics 9709 syllabus for 00, 0 and 0. Details of the assessment Command words The table below includes command words used in the assessment for this syllabus. The use of the command word will relate to the subject contet. Command word Calculate Describe Determine Evaluate Eplain Identify Justify Show (that) Sketch State Verify What it means work out from given facts, figures or information state the points of a topic / give characteristics and main features establish with certainty judge or calculate the quality, importance, amount, or value of something set out purposes or reasons / make the relationships between things evident / provide why and/or how and support with relevant evidence name/select/recognise support a case with evidence/argument provide structured evidence that leads to a given result make a simple freehand drawing showing the key features epress in clear terms confirm a given statement/result is true Back to contents page 35

38 Cambridge International AS & A Level Mathematics 9709 syllabus for 00, 0 and 0. 5 List of formulae and statistical tables (MF9) Mensuration Volume of sphere = 4 3 π r 3 Surface area of sphere = 4π r PURE MATHEMATICS Volume of cone or pyramid = base area height 3 Area of curved surface of cone = π r slant height Arc length of circle = rθ (θ in radians) Area of sector of circle = (θ in radians) r θ Algebra For the quadratic equation For an arithmetic series: For a geometric series: a + b + c = 0 : b ± b 4ac = a u = a+ ( n ) d, S = na ( + l) = n{ a+ ( n ) d} n n n n un = ar a( r ), Sn = ( r ) r Binomial series: n n n ( a+ b) = a + a b+ a b + a b + K + b 3 n n n n n 3 3 n n n! and = r r!( n r)! a r, S = ( r < ), where n is a positive integer n nn ( ) nn ( )( n ) 3 ( + ) = + n + + +K, where n is rational and <! 3! 36 Back to contents page

39 Cambridge International AS & A Level Mathematics 9709 syllabus for 00, 0 and 0. List of formulae and statistical tables (MF9) Trigonometry cos θ + sin θ, sinθ tanθ cosθ + tan θ sec θ, sin( A± B) sin Acos B± cos Asin B cos( A± B) cos Acos Bm sin Asin B cot θ + cosec θ tan A± tan B tan( A± B) m tan Atan B sin A sin Acos A cos A cos A sin A cos A sin A tan A tan A tan A Principal values: π sin π, 0 cos π, tan π < < π Differentiation f( ) f( ) n ln e sin cos n n e cos sin tan sec sec sec tan cosec cosec cot cot tan uv If = f( t) and y = g( t) then dy = dy d d dt dt u v cosec + du dv v + u d d du dv v u d d v Back to contents page 37

40 Cambridge International AS & A Level Mathematics 9709 syllabus for 00, 0 and 0. List of formulae and statistical tables (MF9) Integration (Arbitrary constants are omitted; a denotes a positive constant.) f( ) n e sin cos sec a dv du u d = uv v d d d f( ) d = ln f ( ) f( ) + a a f( ) d n+ n + ln e cos sin tan tan a a a ln a + a a+ ln a a ( n ) ( > a) ( < a) Vectors a= ai+ a j+ a k and b= bi+ bj+ b3k then If 3 ab. = ab + ab + ab = a b cosθ Back to contents page

41 Cambridge International AS & A Level Mathematics 9709 syllabus for 00, 0 and 0. List of formulae and statistical tables (MF9) Algebra Summations: FURTHER PURE MATHEMATICS n r= nn ( + ), r= n r = nn ( + )( n+ ), 6 r= n r= 3 = ( + ) 4 r n n Maclaurin s series: r ( r) f( ) = f(0) + f (0) + f (0) + K+ f (0) + K! r! r e = ep( ) = K+ + K (all )! r! 3 r+ ln( + ) = + K+ ( ) + K ( < ) 3 r 3 5 r+ r sin = + K+ ( ) + K (all ) 3! 5! (r + )! 4 r r cos = + K+ ( ) + K (all )! 4! ( r)! 3 5 r+ r tan = + K+ ( ) + K ( ) 3 5 r r+ sinh = K+ + K (all ) 3! 5! (r + )! 4 r cosh = K+ + K (all )! 4! ( r)! tanh 3 5 r+ r = K+ + K ( < < ) 3 5 r + Trigonometry If t = tan then: t sin = and + t t cos = + t Hyperbolic functions cosh sinh, sinh sinh cosh, ( ) sinh = ln + + ( ) cosh = ln + ( ) + tanh = ln ( ) < cosh cosh + sinh Back to contents page 39

42 Cambridge International AS & A Level Mathematics 9709 syllabus for 00, 0 and 0. List of formulae and statistical tables (MF9) Differentiation f( ) f( ) sin cos sinh cosh tanh sinh cosh tanh cosh sinh sech + Integration (Arbitrary constants are omitted; a denotes a positive constant.) f( ) f( ) d sec ln sec tan ln tan( ) + = + π ( < π ) 4 cosec ln cosec + cot = ln tan ( ) (0 < <π ) sinh cosh cosh sinh sech a a a + sin cosh sinh tanh a a a ( < a) ( > a) 40 Back to contents page

43 Cambridge International AS & A Level Mathematics 9709 syllabus for 00, 0 and 0. List of formulae and statistical tables (MF9) MECHANICS Uniformly accelerated motion v= u+ at, s = ( u+ vt ), s = ut + at, v = u + as Motion of a projectile Equation of trajectory is: FURTHER MECHANICS g y = tanθ V cos θ Elastic strings and springs λ T =, l λ E = l Motion in a circle For uniform circular motion, the acceleration is directed towards the centre and has magnitude ω r or v r Centres of mass of uniform bodies Triangular lamina: along median from verte 3 Solid hemisphere of radius r: 3 r from centre 8 Hemispherical shell of radius r: r from centre Circular arc of radius r and angle α: r sinα from centre α Circular sector of radius r and angle α: r sin α from centre 3α Solid cone or pyramid of height h: 3 h from verte 4 Back to contents page 4

44 Cambridge International AS & A Level Mathematics 9709 syllabus for 00, 0 and 0. List of formulae and statistical tables (MF9) Summary statistics For ungrouped data: PROBABILITY & STATISTICS For grouped data: Σ =, standard deviation n Σ( ) Σ = = n n Σf =, standard deviation Σ f Σ( ) f Σ f = = Σf Σf Discrete random variables E( X ) = Σ p, Var( X) = Σ p {E( X)} For the binomial distribution B( n, p ) : n r pr = p ( p) r n r For the geometric distribution Geo(p): p = p( p) r, r, µ = np, σ = np( p) µ = p For the Poisson distribution Po( λ ) p r r λ λ = e, µ = λ, r! σ = λ Continuous random variables E( X ) = f( ) d, Var( X) = f( ) d {E( X)} Sampling and testing Unbiased estimators: Σ =, n s = = Σ n n n Σ( ) ( Σ ) Central Limit Theorem: σ X ~ N µ, n Approimate distribution of sample proportion: p( p) N p, n 4 Back to contents page

45 Cambridge International AS & A Level Mathematics 9709 syllabus for 00, 0 and 0. List of formulae and statistical tables (MF9) Sampling and testing Two-sample estimate of a common variance: FURTHER PROBABILITY & STATISTICS s Σ( ) + Σ( ) = n + n Probability generating functions X G X ( t ) = E( t ), E( X ) = G (), X Var( X ) = G () + G () {G ()} X X X Back to contents page 43

46 Cambridge International AS & A Level Mathematics 9709 syllabus for 00, 0 and 0. List of formulae and statistical tables (MF9) If Z has a normal distribution with mean 0 and variance, then, for each value of z, the table gives the value of Φ(z), where Φ(z) = P(Z z). For negative values of z, use Φ( z) = Φ(z). THE NORMAL DISTRIBUTION FUNCTION z ADD If Z has a normal distribution with mean 0 and variance, then, for each value of p, the table gives the value of z such that P(Z z) = p. Critical values for the normal distribution p z Back to contents page

47 Cambridge International AS & A Level Mathematics 9709 syllabus for 00, 0 and 0. List of formulae and statistical tables (MF9) CRITICAL VALUES FOR THE t-distribution If T has a t-distribution with ν degrees of freedom, then, for each pair of values of p and ν, the table gives the value of t such that: P(T t) = p. p ν = Back to contents page 45

48 Cambridge International AS & A Level Mathematics 9709 syllabus for 00, 0 and 0. List of formulae and statistical tables (MF9) If X has a CRITICAL VALUES FOR THE χ -DISTRIBUTION χ -distribution with ν degrees of freedom then, for each pair of values of p and ν, the table gives the value of such that P(X ) = p. p ν = Back to contents page