Igcse mathematics additional standards

 Igcse mathematics additional standards

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  IGCSE MATHEMATICS –ADDITIONAL STANDARDS Aims consolidate and extend their elementary mathematical skills, and use these in the context of more advanced techniques further develop their knowledge of mathematical concepts and principles, and use this knowledge for problem solving appreciate the interconnectedness of mathematical knowledge acquire a suitable foundation in mathematics for further study in the subject or in mathematics related subjects devise mathematical arguments and use and present them precisely and logically integrate information technology (IT) to enhance the mathematical experience develop the confidence to apply their mathematical skills and knowledge in appropriate situations develop creativity and perseverance in the approach to problem solving derive enjoyment and satisfaction from engaging in mathematical pursuits, and gain an appreciation of the beauty, power and usefulness of mathematics. Assessment objectives recall and use manipulative technique interpret and use mathematical data, symbols and terminology comprehend numerical, algebraic and spatial concepts and relationships recognise the appropriate mathematical procedure for a given situation formulate problems into mathematical terms and select and apply appropriate techniques of solution. Syllabus content 1. Set language and notation use set language and notation, and Venn diagrams to describe sets and represent relationships between sets as follows: A = {x: x is a natural number} B = {(x,y): y = mx + c} C = {x: a Y x Y b} D = {a, b, c, …} understand and use the following notation: Union of A and B A ? B Intersection of A and B A n B Number of elements in set A n(A) “…is an element of…” ? “…is not an element of…” ? Complement of set A A’ The empty set Ø Universal set A is a subset of B A ? B A is a proper subset of B A ? B A is not a subset of B A ? B A is not a proper subset of B A ?B 2. Functions understand the terms: function, domain, range (image set), oneone function, inverse function and composition of functions use the notation f(x) = sin x, f: x a lg x (x > 0), f–1(x) and f2(x) [= f (f(x))] understand the relationship between y = f(x) and y = f(x), where f(x) may be linear, quadratic or trigonometric explain in words why a given function is a function or why it does not have an inverse find the inverse of a one-one function and form composite functions use sketch graphs to show the relationship between a function and its inverse 3. Quadratic functions find the maximum or minimum value of the quadratic function f : x a ax2 + bx + c by any method use the maximum or minimum value of f(x) to sketch the graph or determine the range for a given domain know the conditions for f(x) = 0 to have: (i) two real roots, (ii) two equal roots, (iii) no real roots and the related conditions for a given line to (i) intersect a given curve, (ii) be a tangent to a given curve, (iii) not intersect a given curve solve quadratic equations for real roots and find the solution set for quadratic inequalities 4. Indices and surds perform simple operations with indices and with surds, including rationalising the denominator 5. Factors of polynomials know and use the remainder and factor theorems find factors of polynomials solve cubic equations 6. Simultaneous equations solve simultaneous equations in two unknowns with at least one linear equation
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  7. Logarithmic andexponential functions know simple properties and graphs of the logarithmic and exponential functions including ln x and ex (series expansions are not required) know and use the laws of logarithms (including change of base of logarithms) solve equations of the form ax = b 8. Straight line graphs interpret the equation of a straight line graph in the form y = mx + c transform given relationships, including y = axn and y = Abx, to straight line form and hence determine unknown constants by calculating the gradient or intercept of the transformed graph solve questions involving mid-point and length of a line know and use the condition for two lines to be parallel or perpendicular 9. Circular measure solve problems involving the arc length and sector area of a circle, including knowledge and use of radian measure 10. Trigonometry know the six trigonometric functions of angles of any magnitude (sine, cosine, tangent, secant, cosecant, cotangent) understand amplitude and periodicity and the relationship between graphs of e.g. sin x and sin 2x draw and use the graphs of y = a sin (bx) + c y = a cos (bx) + c y = a tan (bx) + c where a, b are positive integers and c is an integer use the relationships and solve simple trigonometric equations involving the six trigonometric functions and the above relationships (not including general solution of trigonometric equations) prove simple trigonometric identities 11. Permutations and combinations recognise and distinguish between a permutation case and a combination case know and use the notation ! (with 0! = 1), and the expressions for permutations and combinations of items taken at a time answer simple problems on arrangement and selection (cases with repetition of objects, or with objects arranged in a circle or involving both permutations and combinations, are excluded) 12. Binomial expansions use the Binomial Theorem for expansion of (a + b) for potive integral use the general term (knowledge of the greatest term and properties of the coefficients is not required) 13. Vectors in 2 dimensions use vectors in any form know and use position vectors and unit vectors find the magnitude of a vector; add and subtract vectors and multiply vectors by scalars compose and resolve velocities use relative velocity, including solving problems on interception (but not closest approach) 14. Matrices display information in the form of a matrix of any order and interpret the data in a given matrix solve problems involving the calculation of the sum and product (where appropriate) of two matrices and interpret the results calculate the product of a scalar quantity and a matrix use the algebra of 2 × 2 matrices (including the zero and identity matrix) calculate the determinant and inverse of a non-singular 2 × 2 matrix and solve simultaneous linear equations 15. Differentiation and integration understand the idea of a derived function use the notations use the derivatives of the standard functions x (for any rational ), sin x, cos x, tan x, ex, ln x, together with constant multiples, sums and composite functions of these differentiate products and quotients of functions apply differentiation to gradients, tangents and normals, stationary points, connected rates of change, small increments and approximations and practical maxima and minima problems discriminate between maxima and minima by any method understand integration as the reverse process of differentiation
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  integrate sums ofterms in powers of x, excluding 1/x integrate functions of the form evaluate definite integrals and apply integration to the evaluation of plane areas apply differentiation and integration to kinematics problems that involve displacement, velocity and acceleration of a particle moving in a straight line with variable or constant acceleration, and the use of x - and -h