[英语考试]ALEVEL数学大纲.pdf

syllabus cambridge international a the further development of mathematical skills including the use of applications of mathematics in the context of everyday situations and in other subjects that they may be studying the ability to analyse problems logically recognising when and how a situation may be represented mathematically the use of mathematics as a means of communication a solid foundation for further study the syllabus allows centres flexibility to choose from three different routes to as level mathematics pure mathematics only or pure mathematics and mechanics or pure mathematics and probability and statistics centres can choose from three different routes to a level mathematics depending on the choice of mechanics or probability and statistics or both in the broad area of applications 4 cambridge international a languages arts and humanities an a level counts as a double credit qualification and an as level as a single credit qualification within the cambridge aice award framework to be considered for an aice diploma a candidate must earn the equivalent of six credits by passing a combination of examinations at either double credit or single credit with at least one course coming from each of the three curriculum areas the examinations are administered in may june and october november sessions each year mathematics 9709 falls into group a mathematics and sciences learn more about aice at http www cie org uk qualifications academic uppersec aice 1 4 how can i find out more if you are already a cambridge centre you can make entries for this qualification through your usual channels e g cie direct if you have any queries please contact us at international cie org uk if you are not a cambridge centre you can find out how your organisation can become a cambridge centre email us at international cie org uk learn more about the benefits of becoming a cambridge centre at www cie org uk 5 cambridge international a mechanics units m1 and m2 probability and statistics units s1 and s2 centres and candidates may take all four advanced a level components in the same examination session for the full a level follow a staged assessment route to the a level by taking two advanced subsidiary as papers p1 take the advanced subsidiary as qualification only as level candidates take paper 1 pure mathematics 1 p1 1 hours about 10 shorter and longer questions 75 marks weighted at 60 of total plus one of the following papers paper 2 pure mathematics 2 p2 paper 4 mechanics 1 m1 paper 6 probability and statistics 1 s1 1 hours about 7 shorter and longer questions 50 marks weighted at 40 of total 1 hours about 7 shorter and longer questions 50 marks weighted at 40 of total 1 hours about 7 shorter and longer questions 50 marks weighted at 40 of total 6 cambridge international a otherwise the subject content for different units does not overlap although later units in each subject area assume knowledge of the earlier units availability this syllabus is examined in the may june examination session and the october november examination session this syllabus is available to private candidates centres in the uk that receive government funding are advised to consult the cie website www cie org uk for the latest information before beginning to teach this syllabus combining this with other syllabuses candidates can combine this syllabus in an examination session with any other cie syllabus except syllabuses with the same title at the same level 8 cambridge international a 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 context 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 where necessary select an appropriate mathematical method to solve the problem use mathematics as a means of communication with emphasis on the use of clear expression acquire the mathematical background necessary for further study in this or related subjects 3 2 assessment objectives the abilities assessed in the examinations cover a single area technique with application the examination will test the ability of candidates to understand relevant mathematical concepts terminology and notation recall accurately and use successfully appropriate manipulative techniques recognise the appropriate mathematical procedure for a given situation apply combinations of mathematical skills and techniques in solving problems present mathematical work and communicate conclusions in a clear and logical way 9 cambridge international a find the discriminant of a quadratic polynomial ax 2 bx c and use the discriminant e g to determine the number of real roots of the equation ax 2 bx c 0 solve quadratic equations and linear 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 x which are quadratic in some function of x e g x 4 5x 2 4 0 2 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 10 cambridge international a find the equation of a straight line given sufficient information e g the coordinates of two points on it or one point on it and its gradient understand and use the relationships between the gradients of parallel and perpendicular lines interpret and use linear equations particularly the forms y mx c and y y1 m x x1 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 including in simple cases the correspondence between a line being tangent to a curve and a repeated root of an equation 4 circular measure understand the definition of a radian and use the relationship between radians and degrees use the formulae s r and a 2 1 r 2 in solving problems concerning the arc length and sector area of a circle 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 exact values of the sine cosine and tangent of 30 45 60 and related angles e g cos 150 2 1 3 use the notations sin 1x cos 1x tan 1x to denote the principal values of the inverse trigonometric relations use the identities cos sini i tan and sin2 cos2 1 find all the solutions of simple trigonometrical equations lying in a specified interval general forms of solution are not included 11 cambridge international a carry out addition and subtraction of vectors and multiplication of a vector by a scalar and interpret these operations in geometrical terms use unit vectors displacement vectors and position vectors calculate the magnitude of a vector and the scalar product of two vectors use the scalar product to determine the angle between two directions and to solve problems concerning perpendicularity of vectors 7 series use the expansion of a b n where n is a positive integer knowledge of the greatest term and properties of the coefficients are not required but the notations r n and n should be known 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 8 differentiation understand the idea of the gradient of a curve and use the notations f x f x x y d d and 2 2 d d x y the technique of differentiation from first principles is not required use the derivative of xn for any rational n together with constant multiples sums 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 including connected rates of change locate stationary points and use information about stationary points in sketching graphs the ability to distinguish between maximum points and minimum points is required but identification of points of inflexion is not included 12 cambridge international a solve problems involving the evaluation of a constant of integration e g to find the equation of the curve through 1 2 for which x y d d 2x 1 evaluate definite integrals including simple cases of improper integrals such as xxxxdandd 0 1 2 1 2 1 3 yy use definite integration to find the area of a region bounded by a curve and lines parallel to the axes or between two curves a volume of revolution about one of the axes 13 cambridge international a divide a polynomial of degree not exceeding 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 e g to find factors solve polynomial equations or evaluate unknown coefficients 2 logarithmic and exponential functions understand the relationship between logarithms and indices and use the laws of logarithms excluding change of base understand the definition and properties of ex and in x including their relationship as inverse functions and their graphs use logarithms to solve equations of the form ax b and similar inequalities use logarithms to transform a given relationship to linear form and hence determine unknown constants by considering the gradient and or intercept 3 trigonometry understand the relationship of the secant cosecant and cotangent functions to cosine sine and tangent and use properties and graphs of all six trigonometric functions for angles of any magnitude use trigonometrical identities for the simplification and exact evaluation of expressions and in the course of solving equations and select an identity or identities appropriate to the context showing familiarity in particular with the use of sec2 1 tan2 and cosec2 1 cot2 the expansions of sin a b cos a b and tan a b the formulae for sin 2a cos 2a and tan 2a the expressions of a sin b cos in the forms r sin and r cos 14 cambridge international a differentiate products and quotients find and use the first derivative of a function which is defined parametrically or implicitly 5 integration extend the idea of reverse differentiation to include the integration of eax b bax 1 sin ax b cos ax b and sec2 ax b knowledge of the general method of integration by substitution is not required use trigonometrical relationships such as double angle formulae to facilitate the integration of functions such as cos2 x use the trapezium rule to estimate the value of a definite integral and use sketch graphs in simple cases to determine whether the trapezium rule gives an over estimate or an under estimate 6 numerical solution of equations locate approximately 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 approximations which converges to a root of an equation understand how a given simple iterative formula of the form xn 1 f xn 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 knowledge of the condition for convergence is not included but candidates should understand that an iteration may fail to converge 15 cambridge international a divide a polynomial of degree not exceeding 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 e g to find factors solve polynomial equations or evaluate unknown coefficients recall an appropriate form for expressing rational functions in partial fractions and carry out the decomposition in cases where the denominator is no more complicated than ax b cx d ex f ax b cx d 2 ax b x2 c2 and where the degree of the numerator does not exceed that of the denominator use the expansion of 1 x n where n is a rational number and x 1 finding a general term is not included but adapting the standard series to expand e g 2 2 1 x 1 is included 2 logarithmic and exponential functions understand the relationship between logarithms and indices and use the laws of logarithms excluding change of base understand the definition and properties of ex and in x including their relationship as inverse functions and their graphs use logarithms to solve equations of the form ax b and similar inequalities use logarithms to transform a given relationship to linear form and hence determine unknown constants by considering the gradient and or intercept 16 cambridge international a use trigonometrical identities for the simplification and exact evaluation of expressions and in the course of solving equations and select an identity or identities appropriate to the context showing familiarity in particular with the use of sec2 1 tan2 and cosec2 1 cot2 the expansions of sin a b cos a b and tan a b the formulae for sin 2a cos 2a and tan 2a the expressions of a sin b cos in the forms r sin and r cos 4 differentiation use the derivatives of ex in x sin x cos x tan x together with constant multiples sums differences and composites differentiate products and quotients find and use the first derivative of a function which is defined parametrically or implicitly 5 integration extend the idea of reverse differentiation to include the integration of eax b bax 1 sin ax b cos ax b and sec2 ax b use trigonometrical relationships such as double angle formulae to facilitate the integration of functions such as cos2 x integrate rational functions by means of decomposition into partial fractions restricted to the types of partial fractions specified in paragraph 1 above recognise an integrand of the form f f x x k and integrate for example 1 2 x x or tan x recognise when an integrand can usefully be regarded as a product and use integration by parts to integrate for example x sin 2x x2 ex or in x use a given substitution to simplify and evaluate either a definite or an indefinite integral use the trapezium rule to estimate the value of a definite integral and use sketch graphs in simple cases to determine whether the trapezium rule gives an over estimate or an under estimate 17 cambridge international a understand the idea of and use the notation for a sequence of approximations which converges to a root of an equation understand how a given simple iterative formula of the form xn 1 f xn 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 knowledge of the condition for convergence is not included but candidates should understand that an iteration may fail to converge 7 vectors understand the significance of all the symbols used when the equation of a straight line is expressed in the form r a tb determine whether two lines are parallel intersect or are skew find the angle between two lines and the point of intersection of two lines when it exists understand the significance of all the symbols used when the equation of a plane is expressed in either of the forms ax by cz d or r a n 0 use equations of lines and planes to solve problems concerning distances angles and intersections and in particular find the equation of a line or a plane given sufficient information determine whether a line lies in a plane is parallel to a plane or intersects a plane and find the point of intersection of a line and a plane when it exists find the line of intersection of two non parallel planes find the perpendicular distance from a point to a plane and from a point to a line find the angle between two planes and the angle between a line and a plane 18 cambridge international a 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 context of a problem being modelled by the equation 9 complex numbers understand the idea of a complex number recall the meaning of the terms real part imaginary part modulus argument conjugate and use the fact that two complex 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 complex numbers expressed in cartesian form x iy use the result that for a polynomial equation with real coefficients any non real roots occur in conjugate pairs represent complex numbers geometrically by means of an argand diagram carry out operations of multiplication and division of two complex numbers expressed in polar form r cos i sin r ei find the two square roots of a complex number understand in simple terms the geometrical effects of conjugating a complex number and of adding subtracting multiplying and dividing two complex numbers illustrate simple equations and inequalities involving complex numbers by means of loci in an argand diagram e g z a x1 or a related probability given the values of x1 finding a relationship between x1 and given the value of p x x1 or a related probability recall conditions under which the normal distribution can be used as an approximation to the binomial distribution n large enough to ensure that np 5 and nq 5 and use this approximation with a continuity corre