NANYANG TECHNOLOGICAL UNIVERSITY

ENTRANCE EXAMINATION FOR FOREIGN APPLICANTS

SYLLABUS FOR MATHEMATICS [1]

STRUCTURE OF EXAMINATION PAPER

  1. There will be one 3-hour paper consisting of 6 questions.
  2. Each question carries 20 marks.
  3. Candidates will be required to answer any 5 questions.

SYLLABUS

No. / TOPICS / NOTES
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. / 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 : . The set of values of f(x) is the range (image set) of f. The notation will be used for f(f(x)).
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.
Solution of quadratic inequalities. / 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.
6. / The remainder and factor theorems.
Factors of polynomials.
Partial fractions. / Including the solution of a cubic equation.
7. / Simultaneous equations, at least one linear, in 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 for positive integral n and its use for simple approximations. / Questions on the greatest term and on properties of the coefficients will not be asked.
11. / Simple properties and graphs of the logarithmic and exponential functions.
Laws of logarithms.
Change of base.
Solution of ax = b. / Including ln x and ex. Their series expansions are not required.
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
,
,
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. / The general solution of trigonometric equations will not be required.
14. / Addition Formulae,
sin(A ± B), cos(A ± B), tan(A ± B), and application to multiple angles.
Expression of as or and solution of . / General solution excluded.
15. / Vectors in two dimensions:
magnitude of a vector, addition and subtraction of vectors, multiplication by scalars.
Position vectors. Unit vectors. / Questions may be set using any vector notation including the unit vectors i and j.
16. / 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. / Both f(x) and will be used.
The derivatives of xn (for any rational n), sin x, cos x, tan x, ex, ln x and composite functions of these.
17. / Integration as the reverse process of differentiation. Elementary properties of integrals. Simple integration techniques.
Definite integrals. Applications of integration to plane areas; displacement, velocity and acceleration. / The integrals of (ax + b)n (including
n = – 1), eax + b, sin(ax + b), cos(ax + b).
Integration by simple substitution is included.
18. / Representation of a curve by means of a pair of parametric equations.
Equations of tangent and normal. / Single parameter only. Conversion from parametric to Cartesian coordinates and from Cartesian to parametric coordinates.
19. / Elementary permutations and combinations.

Revised on Oct 2002 (‘AO’ Maths)

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[1] ‘AO’ Maths