Download Maths - Nanyang Technological University

R/ISU/13/02
NANYANG TECHNOLOGICAL UNIVERSITY
ENTRANCE EXAMINATION FOR FOREIGN APPLICANTS
SYLLABUS FOR MATHEMATICS 1
STRUCTURE OF EXAMINATION PAPER
1.
2.
3.
There will be one 3-hour paper consisting of 6 questions.
Each question carries 20 marks.
Candidates will be required to answer any 5 questions.
SYLLABUS
No.
1.
2.
3.
TOPICS
Elementary two-dimensional Cartesian
coordinate geometry.
Condition for two lines to be perpendicular.
Indices and surd notation; rationalising the
denominator.
Functions. Inverse of a one-one function.
Composition of functions.
Graphical illustration of the relationship
between a function and its inverse.
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)).
4.
5.
6.
7.
8.
9.
10.
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.
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.
Solution of quadratic inequalities.
The remainder and factor theorems.
Factors of polynomials.
Partial fractions.
Simultaneous equations, at least one linear,
in two unknowns.
Arithmetic and geometric progressions and
their sums to n terms.
Determination of unknown constants in a
relationship by plotting an appropriate
straight line graph.
Binomial expansion of (a  b) for positive
integral n and its use for simple
approximations.
n
1 ‘AO’ Maths
1
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.
Including the solution of a cubic equation.
Questions on the greatest term and on
properties of the coefficients will not be asked.
No.
11.
12.
13.
TOPICS
Simple properties and graphs of the
logarithmic and exponential functions.
Laws of logarithms.
Change of base.
Solution of ax = b.
Circular measure: arc length, area of a
sector of a circle.
The six trigonometric functions of angles of
any magnitude. The graphs of sine, cosine
and tangent.
Knowledge of the relationships
NOTES
Including ln x and ex. Their series expansions
are not required.
sin A
 tan A ,
cos A
cos A
 cot A ,
sin A
14.
15.
16.
17.
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.
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 .
Vectors in two dimensions:
magnitude of a vector, addition and
subtraction of vectors, multiplication by
scalars.
Position vectors. Unit vectors.
Derivatives of standard functions.
Derivative of a composite function.
Differentiation of sum, product and quotient
of functions and of simple functions defined
parametrically.
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.
Integration as the reverse process of
differentiation. Elementary properties of
integrals. Simple integration techniques.
The general solution of trigonometric
equations will not be required.
General solution excluded.
Questions may be set using any vector
notation including the unit vectors i and j.
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.
The integrals of (ax + b)n (including
n = – 1), eax + b, sin(ax + b), cos(ax + b).
Integration by simple substitution is included.
18.
19.
Definite integrals. Applications of integration
to plane areas; displacement, velocity and
acceleration.
Representation of a curve by means of a
pair of parametric equations.
Equations of tangent and normal.
Elementary permutations and
combinations.
Revised on Oct 2002 (‘AO’ Maths)
2
Single parameter only. Conversion from
parametric to Cartesian coordinates and from
Cartesian to parametric coordinates.