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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.