OAD/05/05
NANYANG TECHNOLOGICAL UNIVERSITY
ENTRANCE EXAMINATION SYLLABUS
FOR INTERNATIONAL STUDENTS
AO-LEVEL MATHEMATICS
STRUCTURE OF EXAMINATION PAPER
1.
2.
There will be one 2-hour paper consisting of 5 questions. Candidates will be required to answer
any 4 questions.
Each question carries 25 marks.
SYLLABUS
No.
TOPICS
1.
Elementary two-dimensional Cartesian
coordinate geometry.
Condition for two lines to be perpendicular.
2.
Indices and surd notation; rationalising the
denominator.
3.
Functions. Inverse of a one-one function.
Composition of functions.
Graphical illustration of the relationship
between a function and its inverse.
4.
The quadratic function x  ax2 + bx + c,
finding its maximum or minimum by any
method and hence sketching its graph or
determining its range for a given domain.
5.
The condition for the equation
ax2 + bx + c = 0 to have
(i) two real roots
(ii) two equal roots
(iii) no real roots,
and the solution of the equation for real roots.
NOTES
Including x  |f(x)|, where f(x) may be linear,
quadratic or trigonometric. A function will be
defined by giving its domain and rule e.g. f :
x  lg x, ( x  0) . The set of values of f(x) is the
range (image set) of f. The notation f 2 ( x ) will be
used for f(f(x)).
The condition for a given line to
(i) intersect a given curve,
(ii) be a tangent to a given curve,
(iii) not intersect a given curve.
Solution of quadratic inequalities.
6.
The remainder and factor theorems.
Factors of polynomials.
Including the solution of a cubic equation.
Partial fractions.
7.
Simultaneous equations, at least one linear, in
1
No.
TOPICS
NOTES
two unknowns.
8.
Arithmetic and geometric progressions and
their sums to n terms.
9.
Determination of unknown constants in a
relationship by plotting an appropriate straight
line graph.
10.
Binomial expansion of (a  b) n for positive
integral n and its use for simple
approximations.
11.
Simple properties and graphs of the logarithmic
and exponential functions.
Including ln x and ex. Their series expansions are
Laws of logarithms.
not required.
Change of base.
Questions on the greatest term and on properties
of the coefficients will not be asked.
Solution of ax = b.
12.
Circular measure: arc length, area of a sector
of a circle.
13.
The six trigonometric functions of angles of
any magnitude. The graphs of sine, cosine and
tangent.
Knowledge of the relationships
sin A
 tan A ,
cos A
cos A
 cot A ,
sin A
sin2A + cos2A = 1,
sec2A = 1 + tan2A,
cosec2A = 1 + cot2A.
Solution of simple trigonometric equations
involving any of the six trigonometric
functions and the above relationships between
them.
Simple identities.
14.
Addition Formulae,
sin(A ± B), cos(A ± B), tan(A ± B), and
application to multiple angles.
Expression of a cos  b sin as
R cos(   ) or R sin(   ) and solution of
a cos  b sin  c .
15.
The general solution of trigonometric equations
will not be required.
Vectors in two dimensions:
General solution excluded.
Questions may be set using any vector notation
including the unit vectors i and j.
magnitude of a vector, addition and
subtraction of vectors, multiplication by
scalars, scalar (dot) product.
2
No.
TOPICS
NOTES
Position vectors. Unit vectors.
16.
Derivatives of standard functions.
Derivative of a composite function.
Differentiation of sum, product and quotient
of functions and of simple functions defined
parametrically.
Both f(x) and dy will be used.
dx
The derivatives of xn (for any rational n), sin x,
cos x, tan x, ex, ln x and composite functions of
these.
Applications of differentiation to gradients,
tangents and normals, stationary points,
velocity and acceleration, connected rates of
change, small increments and approximations;
practical problems involving maxima and
minima.
17.
Integration as the reverse process of
differentiation. Elementary properties of
integrals. Simple integration techniques.
The integrals of (ax + b)n (including
n = – 1), eax + b, sin(ax + b), cos(ax + b).
Integration by simple substitution is included.
Definite integrals. Applications of integration
to plane areas; displacement, velocity and
acceleration.
18
Representation of a curve by means of a
pair of parametric equations.
Single parameter only. Conversion from
parametric to Cartesian coordinates and from
Cartesian to parametric coordinates.
Equations of tangent and normal.
19
Elementary permutations and combinations.
Sept 2005
3