Mathematics GCE A Level (Junior College 1-2) Syllabus:


PURE MATHEMATICS

Functions, inverse functions and composite functions


Include:

  • concepts of function, domain and range

  • use of notations such as f( x ) = x 2 + 5 , f : x x 2 + 5 , f−1(x), fg(x)and f2(x)

  • finding inverse functions and composite functions

  • conditions for the existence of inverse functions and composite functions

  • domain restriction to obtain an inverse function

  • relationship between a function and its inverse as reflection in the line y = x


Exclude the use of the relation (fg)−1 = g−1f −1


Graphing techniques


Include:

  • use of a graphing calculator to graph a given function

  • relating the following equations with their graphs







  • characteristics of graphs such as symmetry, intersections with the axes, turning points and asymptotes

  • determining the equations of asymptotes, axes of symmetry, and restrictions on the possible values of x and/or y

  • effect of transformations on the graph of y = f(x) as represented by

and combinations of these transformations

  • relating the graphs of to the graph

  • simple parametric equations and their graphs


Equations and inequalities


Include:

  • solving inequalities of the form where and are quadratic expressions that are either factorable or always positive

  • solving inequalities by graphical methods

  • formulating an equation or a system of linear equations from a problem situation

  • finding the numerical solution of equations (including system of linear equations) using a graphing calculator


Summation of series


Include:

  • concepts of sequence and series

  • relationship between un (the n th term) and Sn (the sum to n terms)

  • sequence given by a formula for the n th term

  • sequence generated by a simple recurrence relation of the form xn+1 = f(xn )

  • use of ∑ notation

  • summation of series by the method of differences

  • convergence of a series and the sum to infinity

  • binomial expansion of (1+ x)n for any rational n

  • condition for convergence of a binomial series

  • proof by the method of mathematical induction


Arithmetic and geometric series


Include:

  • formula for the n th term and the sum of a finite arithmetic series

  • formula for the n th term and the sum of a finite geometric series

  • condition for convergence of an infinite geometric series

  • formula for the sum to infinity of a convergent geometric series

  • solving practical problems involving arithmetic and geometric series


Vectors in two and three dimensions


Include:

  • addition and subtraction of vectors, multiplication of a vector by a scalar, and their geometrical interpretations

  • use of notations such as a

  • position vectors and displacement vectors

  • magnitude of a vector

  • unit vectors

  • distance between two points

  • angle between a vector and the x-, y- or z-axis

  • use of the ratio theorem in geometrical applications


The scalar and vector products of vectors


Include:

  • concepts of scalar product and vector product of vectors

  • calculation of the magnitude of a vector and the angle between two directions

  • calculation of the area of triangle or parallelogram

  • geometrical meanings of |a.b| and |a × b| , where b is a unit vector


Exclude triple products a.b × c and a × b × c


Three-dimensional geometry


Include:

  • vector and cartesian equations of lines and planes

  • finding the distance from a point to a line or to a plane

  • finding the angle between two lines, between a line and a plane, or between two planes

  • relationships between

    1. –  two lines (coplanar or skew)

    2. –  a line and a plane

    3. –  two planes

    4. –  three planes

  • finding the intersections of lines and planes Exclude:

  • finding the shortest distance between two skew lines

  • finding an equation for the common perpendicular to two skew lines


Complex numbers expressed in cartesian form


Include:

  • extension of the number system from real numbers to complex numbers

  • complex roots of quadratic equations

  • four operations of complex numbers expressed in the form (x+iy)

  • equating real parts and imaginary parts

  • conjugate roots of a polynomial equation with real coefficients


Complex numbers expressed in polar form


Include:

  • complex numbers expressed in the form where r >0 and

  • calculation of modulus ( r ) and argument ( θ ) of a complex number

  • multiplication and division of two complex numbers expressed in polar form

  • representation of complex numbers in the Argand diagram

  • geometrical effects of conjugating a complex number and of adding, subtracting, multiplying, dividing two complex numbers

  • loci such as z − c r, |z – a| = |z – b| and arg(z − a) = α

  • use of de Moivre’s theorem to find the powers and n th roots of a complex number

Exclude:

  • loci such as |z – a| =k| z – b| , where k 1 and arg(z − a) − arg(z − b) = α

  • properties and geometrical representation of the n th roots of unity

  • use of de Moivre’s theorem to derive trigonometric identities


Differentiation


Include:

  • graphical interpretation of

    1. –  f(x)>0, f(x)=0 and f(x)<0

    2. –  f′′(x)>0 and f′′(x)<0

  • relating the graph of y = f(x) to the graph of y = f(x)

  • differentiation of simple functions defined implicitly or parametrically

  • finding the numerical value of a derivative at a given point using a graphing calculator

  • finding equations of tangents and normals to curves

  • solving practical problems involving differentiation

Exclude:

  • finding non-stationary points of inflexion

  • problems involving small increments and approximation


Maclaurin’s series


Include:

  • derivation of the first few terms of the series expansion of (1+x)n , ex , sin x , ln(1+ x), and other simple functions

  • finding the first few terms of the series expansions of sums and products of functions, e.g. excos2x, using standard series

  • summation of infinite series in terms of standard series

  • sinx ≈ x, cosx ≈ 1−, tanx ≈ x

  • concepts of ‘convergence’ and ‘approximation’

Integration techniques


Include:

  • integration of








  • integration by a given substitution

  • integration by parts


Definite integrals


Include:

  • concept of definite integral as a limit of sum

  • definite integral as the area under a curve

  • evaluation of definite integrals

  • finding the area of a region bounded by a curve and lines parallel to the coordinate axes, between a curve and a line, or between two curves

  • area below the x-axis

  • finding the area under a curve defined parametrically

  • finding the volume of revolution about the x- or y-axis

  • finding the numerical value of a definite integral using a graphing calculator


Differential equations


Include:

  • solving differential equations of the forms




  • formulating a differential equation from a problem situation

  • use of a family of solution curves to represent the general solution of a differential equation

  • use of an initial condition to find a particular solution

  • interpretation of a solution in terms of the problem situation


STATISTICS

Permutations and combinations


Include:

  • addition and multiplication principles for counting

  • concepts of permutation ( n ! or n Pr ) and combination (nCr )

  • arrangements of objects in a line or in a circle

  • cases involving repetition and restriction


Probability


Include:

  • addition and multiplication of probabilities

  • mutually exclusive events and independent events

  • use of tables of outcomes, Venn diagrams, and tree diagrams to calculate probabilities

  • calculation of conditional probabilities in simple cases

  • use of:

P(A) = 1−P(A)


P(A B) = P(A) + P(B) − P(A B)

P(A | B) =


Binomial and Poisson distributions


Include:

  • concepts of binomial distribution B(n,p) and Poisson distribution Po(μ); use of B(n,p) and Po(μ) as probability models

  • use of mean and variance of binomial and Poisson distributions (without proof)

  • solving problems involving binomial and Poisson variables

  • additive property of the Poisson distribution

  • Poisson approximation to binomial

Normal distribution


Include:

  • concept of a normal distribution and its mean and variance; use of N(μ,σ 2 ) as a probability model

  • standard normal distribution

  • finding the value of P( X < x1 ) given the values of μ,σ

  • use of the symmetry of the normal distribution

  • finding a relationship between x1, μ , σ given the value of P(X < x1)

  • solving problems involving normal variables

  • x1 ,

  • solving problems involving the use of E(aX + b) and Var (aX + b)

  • solving problems involving the use of E(aX + bY ) and Var (aX + bY ) , where X and Y are independent

  • normal approximation to binomial

  • normal approximation to Poisson


Sampling


Include:

  • concepts of population and sample

  • random, stratified, systematic and quota samples

  • advantages and disadvantages of the various sampling methods

  • distribution of sample means from a normal population

  • use of the Central Limit Theorem to treat sample means as having normal distribution when the sample size is sufficiently large

  • calculation of unbiased estimates of the population mean and variance from a sample

  • solving problems involving the sampling distribution


Hypothesis testing


Include:

  • concepts of null and alternative hypotheses, test statistic, level of significance and p-value

  • tests for a population mean based on:

    1. –  a sample from a normal population of known variance

    2. –  a sample from a normal population of unknown variance

    3. –  a large sample from any population

  • 1-tail and 2-tail tests

  • use of t-test Exclude testing the difference between two population means


Correlation coefficient and linear regression


Include:

  • concepts of scatter diagram, correlation coefficient and linear regression

  • calculation and interpretation of the product moment correlation coefficient and of the equation of the least squares regression line

  • concepts of interpolation and extrapolation

  • use of a square, reciprocal or logarithmic transformation to achieve linearity