# A Maths Syllabus 2012

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NANYANG TECHNOLOGICAL UNIVERSITY SYLLABUS FOR ENTRANCE EXAMINATION FOR INTERNATIONAL STUDENTS A-LEVEL MATHEMATICS STRUCTURE OF EXAMINATION PAPER 1. There will be one 2-hour paper consisting of 4 questions. 2. Each question carries 25 marks. 3. Candidates will be required to answer all 4 questions. The detailed syllabus is on the next page. Nanyang Technological University August 2012 CONTENT OUTLINE Knowledge of the content of the O Level Mathematics syllabus and of some of the content of t
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Nanyang Technological University  August 2012 NANYANG TECHNOLOGICAL UNIVERSITY SYLLABUS FOR ENTRANCE EXAMINATION FOR INTERNATIONAL STUDENTS  A-LEVEL MATHEMATICS STRUCTURE OF EXAMINATION PAPER 1. There will be one 2-hour paper consisting of 4 questions. 2. Each question carries 25 marks. 3. Candidates will be required to answer all 4 questions. The detailed syllabus is on the next page.    CONTENT OUTLINE Knowledge of the content of the O Level Mathematics   syllabus and   of some of the content of the O Level Additional Mathematics syllabus are assumed in the syllabus below and will not be tested directly, but it may be required indirectly in response to questions on other topics. The assumed knowledge for O Level Additional Mathematics is appended after this section. Topic/Sub-topics Content PURE MATHEMATICS   1 Functions and graphs 1.1 Functions, inverse functions and composite functions Include: ã   concepts of function, domain and range ã   use of notations such as 5)f(  2 +=  x  x  , 5:f   2 +  x  x   a , )(f   1  x  , )(fg  x  and )(f   2  x    ã   finding inverse functions and composite functions ã   conditions for the existence of inverse functions and composite functions ã   domain restriction to obtain an inverse function ã   relationship between a function and its inverse as reflection in the line  x y   =  Exclude the use of the relation 111 f g(fg)  =  1.2 Graphing techniques Include: ã   use of a graphic calculator to graph a given function ã   relating the following equations with their graphs 1 2222 =± by a x    d cx bax y  ++=   edx c bx ax y  +++= 2   ã   characteristics of graphs such as symmetry, intersections with the axes, turning points and asymptotes ã   determining the equations of asymptotes, axes of symmetry, and restrictions on the possible values of  x   and/or y    ã   effect of transformations on the graph of )f(  x y   =  as represented by )f(  x ay   = , a x y   +=  )f( , )f(  a x y   +=  and )f( ax y   = , and combinations of these transformations ã   relating the graphs of )(f   x y   = , )(f   x y   = , )f(1  x y   =  and )f( 2  x y   =  to the graph of )f(  x y   =   ã   simple parametric equations and their graphs 2    Topic/Sub-topics Content 1.3 Equations and inequalities   Include: ã   solving inequalities of the form 0)(g)(f  >  x  x   where )f(  x  and )g(  x  are quadratic expressions that are either factorisable or always positive ã   solving inequalities by graphical methods ã   formulating an equation or a system of linear equations from a problem situation ã   finding the numerical solution of equations (including system of linear equations) using a graphic calculator    2 Sequences and series 2.1 Summation of series Include: ã   concepts of sequence and series ã   relationship between n u   (the n th term) and n S   (the sum to n  terms) ã   sequence given by a formula for the n th term ã   sequence generated by a simple recurrence relation of the form )f( 1  nn  x  x   = +   ã   use of ∑  notation ã   summation of series by the method of differences ã   convergence of a series and the sum to infinity ã   binomial expansion of n  x  )1(  +  for any rational n   ã   condition for convergence of a binomial series ã   proof by the method of mathematical induction 2.2 Arithmetic and geometric series   Include: ã   formula for the n th term and the sum of a finite arithmetic series ã   formula for the n th term and the sum of a finite geometric series ã   condition for convergence of an infinite geometric series ã   formula for the sum to infinity of a convergent geometric series ã   solving practical problems involving arithmetic and geometric series 3 Vectors 3.1 Vectors in two and three dimensions Include: ã   addition and subtraction of vectors, multiplication of a vector by a scalar, and their geometrical interpretations ã   use of notations such as      y  x  ,      z y  x  ,  ji  y  x   + , k ji  z y  x   ++ ,    AB , a   ã   position vectors and displacement vectors ã   magnitude of a vector ã   unit vectors 3    Topic/Sub-topics Content ã   distance between two points ã   angle between a vector and the  x  -, y  - or z  -axis ã   use of the ratio theorem in geometrical applications 3.2 The scalar and vector products of vectors Include: ã   concepts of scalar product and vector product of vectors ã   calculation of the magnitude of a vector and the angle between two directions ã   calculation of the area of triangle or parallelogram ã   geometrical meanings of a.b  and ba × , where b  is a unit vector Exclude triple products ca.b ×  and cba  ××  3.3 Three-dimensional geometry Include: ã   vector and cartesian equations of lines and planes ã   finding the distance from a point to a line or to a plane ã   finding the angle between two lines, between a line and a plane, or between two planes ã   relationships between (i) two lines (coplanar or skew) (ii) a line and a plane (iii) two planes (iv) three planes ã   finding the intersections of lines and planes Exclude: ã   finding the shortest distance between two skew lines ã   finding an equation for the common perpendicular to two skew lines 4 Complex numbers 4.1 Complex numbers expressed in cartesian form Include: ã   extension of the number system from real numbers to complex numbers ã   complex roots of quadratic equations ã   four operations of complex numbers expressed in the form )i(  y  x   +   ã   equating real parts and imaginary parts ã   conjugate roots of a polynomial equation with real coefficients 4

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