Additional Mathematics GCE O Level (Secondary 3 – 4) Syllabus:


Set language and notation


  • use set language and set notation, and Venn diagrams, to describe sets and represent relationships between sets as follows:

  • Definition of sets, e.g.



A = {x : x is a natural number}

B =

C =

D = {a, b, c... }


Notation:


Union of A and B: A B

Intersection of A and B: A 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

Functions


  • understand the terms: function, domain, range (image set), one-one function, inverse function and composition of functions

  • use the notation f(x) = sin x, f: x 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

Equations and inequalities


  • Conditions for a quadratic equation to have:

two real roots, two equal roots, no real roots

  • and related conditions for a given line to:

intersect a given curve , be a tangent to a given curve, not intersect a given curve

  • Conditions for ax2 + bx + c to be always positive (or always negative)

  • Solving simultaneous equations in two variables with at least one linear equation, by substitution

  • Relationships between the roots and coefficients of a quadratic equation

  • Solving quadratic inequalities, and representing the solution on the number line


Quadratic functions


  • find the maximum or minimum value of the quadratic function f : x 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


Indices and surds


  • perform simple operations with indices and with surds, including rationalising the denominator


Factors of polynomials


  • know and use the remainder and factor theorems

  • find factors of polynomials

  • Use of:

○  a3 + b3 = (a + b)(a2 – ab + b2)

○  a3 – b3 = (a – b)(a2 + ab + b2)

  • solve cubic equations


Simultaneous equations


  • solve simultaneous equations in two unknowns with at least one linear equation

Logarithmic and exponential functions


  • know simple properties and graphs of the logarithmic and exponential functions including lnx 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


Straight line graphs


  • interpret the equation of a straight line graph in the form 
y = mx + c

  • transform given relationships, including y = ax n and y = Ab x, 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

Coordinate geometry in two dimensions


  • Condition for two lines to be parallel or perpendicular

  • Midpoint of line segment

  • Area of rectilinear figure

  • Graphs of parabolas with equations in the form y2 = kx

  • Coordinate geometry of circles in the form:

(x – a)2 + (y – b)2 = r2

x2 + y2 + 2gx + 2fy + c = 0 (excluding problems involving 2 circles)

  • Transformation of given relationships, including y = axn and y = kbx, to linear form to determine the unknown constants from a straight line graph


Circular measure


  • solve problems involving the arc length and sector area of a circle, including knowledge and use of radian measure


Trigonometry


  • know the six trigonometric functions of angles of any magnitude (sine, cosine, tangent, secant, cosecant, cotangent)

  • Principal values of sin–1x, cos–1x, tan–1x

  • Exact values of the trigonometric functions for special angles (30°,45°,60°)

  • understand amplitude and periodicity and the relationship between graphs of, e.g. sin x and sin 2x

  • draw and use the graphs of



where a and b are positive integers and c is an integer


  • use the relationships :




○  the expansions of sin(AB), cos(A B) and tan(AB)

○  the formulae for sin 2A, cos 2A and tan 2A

○  the expression for in the form Rcos(α) or Rsin(α)


  • 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


Permutations and combinations


  • recognise and distinguish between a permutation case and a combination case

  • know and use the notation n! (with 0! = 1), and the expressions for permutations and combinations of n items taken r 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)


Binomial expansions


  • use the Binomial Theorem for expansion of (a + b)n for positive integral

  • use the general term

  • (knowledge of the greatest term and properties of the coefficients is not required)


Vectors in 2 dimensions


  • use vectors in any form, e.g. , p, ai – bj

  • 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)


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


Differentiation and integration


  • understand the idea of a derived function

  • use the notations

  • use the derivatives of the standard functions
 x n (for any rational n), 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

  • integrate sums of terms in powers of x, excluding

  • integrate functions of the form (ax + b)n (excluding n = –1), 
eax + b, sin (ax + b), cos (ax + b)

  • 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-t and v-t graphs