ADDITIONAL MATHEMATICS

ADDITIONAL MATHEMATICS GCE Ordinary Level (06) (Syllabus 4047) CONTENTS Page INTRODUCTION AIMS ASSESSMENT OBJECTIVES SCHEME OF ASSESSMENT 3 USE OF CALCULATORS 3 SUBJECT CONTENT 4 MATHEMATICAL FORMULAE 7 MATHEMATICAL NOTATION 8 Singapore Examinations and Assessment Board MOE & UCLES 04

4047 ADDITIONAL MATHEMATICS GCE ORDINARY LEVEL (06) INTRODUCTION The syllabus prepares students adequately for A Level H Mamatics, where a strong foundation in algebraic manipulation skills and mamatical reasoning skills are required. The content is organised into three strands, namely, Algebra, Geometry and Trigonometry, and Calculus. Besides conceptual understanding and skill proficiency explicated in content strand, development of process skills, namely, reasoning, communication and connections, thinking skills and heuristics, and applications and modelling are also emphasised. The O Level Additional Mamatics syllabus assumes knowledge of O Level Mamatics. AIMS The O Level Additional Mamatics syllabus aims to enable students who have an aptitude and interest in mamatics to: acquire mamatical concepts and skills for higher studies in mamatics and to support learning in or subjects, in particular, sciences develop thinking, reasoning and metacognitive skills through a mamatical approach to problemsolving connect ideas within mamatics and between mamatics and sciences through applications of mamatics appreciate abstract nature and power of mamatics. ASSESSMENT OBJECTIVES The assessment will test candidates' abilities to: AO understand and apply mamatical concepts and skills in a variety of contexts AO analyse information; formulate and solve problems, including those in real-world contexts, by selecting and applying appropriate techniques of solution; interpret mamatical results AO3 solve higher order thinking problems; make inferences; reason and communicate mamatically through writing mamatical explanation, arguments and proofs.

4047 ADDITIONAL MATHEMATICS GCE ORDINARY LEVEL (06) SCHEME OF ASSESSMENT Paper Duration Description Marks Weighting Paper h There will be 3 questions of varying marks and lengths. 80 44% Candidates are required to answer ALL questions. Paper h There will be 9 questions of varying marks and lengths. Candidates are required to answer ALL questions. 00 56% NOTES. Omission of essential working will result in loss of marks.. Some questions may integrate ideas from more than one topic of syllabus where applicable. 3. Relevant mamatical formulae will be provided for candidates. 4. Unless stated orwise within a question, three-figure accuracy will be required for answers. Angles in degrees should be given to one decimal place. 5. SI units will be used in questions involving mass and measures. Both -hour and 4-hour clock may be used for quoting times of day. In 4-hour clock, for example, 3.5 a.m. will be denoted by 03 5; 3.5 p.m. by 5 5. 6. Candidates are expected to be familiar with solidus notation for expression of compound units, e.g. 5 m/s for 5 metres per second. 7. Unless question requires answer in terms of π, calculator value for π or π = 3.4 should be used. USE OF CALCULATORS An approved calculator may be used in both Paper and Paper. 3

4047 ADDITIONAL MATHEMATICS GCE ORDINARY LEVEL (06) SUBJECT CONTENT Knowledge of content of O Level Mamatics syllabus is assumed in syllabus below and will not be tested directly, but it may be required indirectly in response to questions on or topics. Topic/Sub-topics Content ALGEBRA A Equations and inequalities Conditions for a quadratic equation to have: (i) two real roots (ii) two equal roots (iii) no real roots and 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 Conditions for ax + 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 roots and coefficients of a quadratic equation Solving quadratic inequalities, and representing solution on number line A Indices and surds Four operations on indices and surds, including rationalising denominator Solving equations involving indices and surds A3 A4 Polynomials and Partial Fractions Binomial expansions Multiplication and division of polynomials Use of remainder and factor orems Factorisation of polynomials Use of: a 3 + b 3 = (a + b)(a ab + b ) a 3 b 3 = (a b)(a + ab + b ) Solving cubic equations Partial fractions with cases where denominator is no more complicated than: (ax + b)(cx + d) (ax + b)(cx + d) (ax + b)(x + c ) Use of Binomial Theorem for positive integer n n Use of notations n! and r n n r r Use of general term a b r, 0, r Ğ n (knowledge of greatest term and properties of coefficients is not required) 4

4047 ADDITIONAL MATHEMATICS GCE ORDINARY LEVEL (06) Topic/Sub-topics Content A5 Power, Exponential, Logarithmic, and Modulus functions Power functions y = ax n where n is a simple rational number, and ir graphs Exponential and logarithmic functions a x, e x, log a x, ln x and ir graphs, including: laws of logarithms equivalence of y = a x and x = log a y change of base of logarithms Modulus functions x and f(x) where f(x) is linear, quadratic or trigonometric, and ir graphs Solving simple equations involving exponential, logarithmic and modulus functions GEOMETRY AND TRIGONOMETRY G Trigonometric functions, identities and equations Six trigonometric functions for angles of any magnitude (in degrees or radians) Principal values of sin x, cos x, tan x Exact values of trigonometric functions for special angles π π π (30, 45, 60 ) or,, 6 4 3 Amplitude, periodicity and symmetries related to sine and cosine functions x Graphs of y = a sin (bx) + c, y = a sin + c, y = a cos (bx) + c, b x y = a cos + c and y = a tan (bx), where a is real, b is a positive integer b and c is an integer. Use of following sin A cos A = tan A, = cot A, sin cos A sin A cosec A = + cot A A + cos A =, sec A = + tan expansions of sin(a 6 B), cos(a 6 B) and tan(a 6 B) formulae for sin A, cos A and tan A expression for a cos u + b sin u in form R cos (u 6 α) or R sin (u 6 α) Simplification of trigonometric expressions Solution of simple trigonometric equations in a given interval (excluding general solution) Proofs of simple trigonometric identities A, 5

4047 ADDITIONAL MATHEMATICS GCE ORDINARY LEVEL (06) G Topic/Sub-topics Coordinate geometry in two dimensions Content Condition for two lines to be parallel or perpendicular Midpoint of line segment Area of rectilinear figure Graphs of parabolas with equations in form y = kx Coordinate geometry of circles in form: (x a) + (y b) = r x + y + gx + fy + c = 0 (excluding problems involving circles) Transformation of given relationships, including y = ax n and y = kb x, to linear form to determine unknown constants from a straight line graph G3 Proofs in plane geometry Use of: properties of parallel lines cut by a transversal, perpendicular and angle bisectors, triangles, special quadrilaterals and circles congruent and similar triangles midpoint orem tangent-chord orem (alternate segment orem) Calculus C Differentiation and integration Derivative of f(x) as gradient of tangent to graph of y = f(x) at a point Derivative as rate of change Use of standard notations f ( x), f ( x), dy d y, = dx dx dy x dx Derivatives of x n, for any rational n, sin x, cos x, tan x, e x, and ln x, toger with constant multiples, sums and differences Derivatives of products and quotients of functions Derivatives of composite functions Increasing and decreasing functions Stationary points (maximum and minimum turning points and stationary points of inflexion) Use of second derivative test to discriminate between maxima and minima Applying differentiation to gradients, tangents and normals, connected rates of change and maxima and minima problems Integration as reverse of differentiation Integration of x n, for any rational n, sin x, cos x, sec x and e x, toger with constant multiples, sums and differences Integration of (ax + b) n, for any rational n, sin(ax + b), cos(ax + b), and e ax+b Definite integral as area under a curve Evaluation of definite integrals Finding area of a region bounded by a curve and line(s) (excluding area of region between two curves) Finding areas of regions below x-axis Application of differentiation and integration to problems involving displacement, velocity and acceleration of a particle moving in a straight line d d These are properties learnt in O Level Mamatics. 6

4047 ADDITIONAL MATHEMATICS GCE ORDINARY LEVEL (06) MATHEMATICAL FORMULAE. ALGEBRA Quadratic Equation For equation ax + bx + c = 0, x = b ± b 4ac a Binomial expansion n n n r n n n n n r r n ( a + b) = a + a b + a b +... + a b +... + b, where n is a positive integer and ( n )...( n r ) n n! n + r = = r! ( n r)! r!. TRIGONOMETRY Identities sin A + cos A = sec A = + tan A cosec A = + cot A sin(a ± B) = sin A cos B ± cos A sin B cos(a ± B) = cos A cos B sin A sin B tan A ± tan B tan( A ± B) = m tan A tan B sin A = sin A cos A cos A = cos A sin A = cos A = sin A tan A tana = tan A Formulae for ABC a b c = = sin A sin B sin C a = b + c bc cos A = bc sin A 7

4047 ADDITIONALL MATHEMATICSS GCE ORDINARY LEVEL (06) MA ATHEMATICAL NOTATION TheT list which follows summarises e notation used in Cambridge s Mamatics examinations. Although primp marily directed towardss A Level, listt also applies, wheree relevant, to examinations at all or levels.. Set Notationn {x{, x, } {x:{ } n(a) n A A Z Z + Q Q + Q + 0 R R + R + 0 R n `= ` [ a, b] [ a, b) ( a, b] ( a, b) is an element of is not an element of sett with elements x, x, sett off all x such that number of elements in set A emptyy set universal set complement of set A sett off integers, { 0, ±, ± ±, ±3, }} sett off positive integers, {,, 3, } sett off rational numbers sett off positive rational numbers, {x Q: x > 0} } sett off positive rational numbers anda zero, {xx Q: x ğ 0} sett off real numberss sett off positive real numbers, {xx R: x > 0} sett off positive real numbers and zero, {x R: x=ğğ 0} real n tuples sett off complex numbers is a subset of is a proper subset off is not a subset of is not a proper subset of o union intersection closed interval {xx R: a Y x Y b} interval {x R: a Y x < b} } interval {x R: a < x Y b} } open interval {x R: a < x < b} } 8

4047 ADDITIONAL MATHEMATICS GCE ORDINARY LEVEL (06). Miscellaneous Symbols = is equal to is not equal to is identical to or is congruent to is approximately equal to is proportional to < is less than Y; is less than or equal to; is not greater than > is greater than [; is greater than or equal to; is not less than infinity 3. Operations a + b a b a b, ab, a.b a a b,, a/b b a : b n a i i= a a a plus b a minus b a multiplied by b a divided by b ratio of a to b a + a +... + a n positive square root of real number a modulus of real number a n! n factorial for n Z + U {0}, (0! = ) n r n! binomial coefficient, for n, r Z + U {0}, 0 Y r Y n r!( n r)! n ( n )...( n r + ), for n Q, r Z + U {0} r! 9

4047 ADDITIONAL MATHEMATICS GCE ORDINARY LEVEL (06) 4. Functions f function f f(x) value of function f at x f: A B f is a function under which each element of set A has an image in set B f: x y function f maps element x to element y f g o f, gf inverse of function f composite function of f and g which is defined by (g o f)(x) or gf(x) = g(f(x)) lim f(x) limit of f(x) as x tends to a x a x ; dy dx d n y n δ x dx f'(x), f (x),, f (n) (x) an increment of x derivative of y with respect to x nth derivative of y with respect to x first, second, nth derivatives of f(x) with respect to x yd x b a yd x x&, & x&, indefinite integral of y with respect to x definite integral of y with respect to x for values of x between a and b first, second, derivatives of x with respect to time 5. Exponential and Logarithmic Functions e base of natural logarithms e x, exp x exponential function of x log a x logarithm to base a of x ln x natural logarithm of x lg x logarithm of x to base 0 6. Circular Functions and Relations sin, cos, tan, } circular functions cosec, sec, cot sin, cos, tan } cosec, sec, cot inverse circular functions 0

4047 ADDITIONAL MATHEMATICS GCE ORDINARY LEVEL (06) 7. Complex Numbers i square root of z a complex number, z = x + iy = r(cos θ + i sin θ ), r R + 0 = re iθ, r R + 0 Re z real part of z, Re (x + iy) = x Im z imaginary part of z, Im (x + iy) = y z modulus of z, x + iy = (x + y ), r (cos θ + i sinθ) = r arg z argument of z, arg(r(cos θ + i sin θ )) = θ, π < θ Ğ π z* complex conjugate of z, (x + iy)* = x iy 8. Matrices M M M T det M a matrix M inverse of square matrix M transpose of matrix M determinant of square matrix M 9. Vectors a vector a AB vector represented in magnitude and direction by directed line segment AB â a unit vector in direction of vector a i, j, k unit vectors in directions of cartesian coordinate axes a magnitude of a AB a.b apb magnitude of AB scalar product of a and b vector product of a and b

4047 ADDITIONAL MATHEMATICS GCE ORDINARY LEVEL (06) 0. Probability and Statistics A, B, C, etc. events A B A B P(A) union of events A and B intersection of events A and B probability of event A A' complement of event A, event not A P(A B) probability of event A given event B X, Y, R, etc. random variables x, y, r, etc. value of random variables X, Y, R, etc. x, x, f, f, p(x) observations frequencies with which observations, x, x occur value of probability function P(X = x) of discrete random variable X p, p probabilities of values x, x, of discrete random variable X f(x), g(x) F(x), G(x) E(X) E[g(X)] Var(X) B(n, p) value of probability density function of continuous random variable X value of (cumulative) distribution function P(X Y x) of random variable X expectation of random variable X expectation of g(x) variance of random variable X binominal distribution, parameters n and p Po(µ) Poisson distribution, mean µ N(µ, σ ) normal distribution, mean µ and variance σ µ population mean σ σ x s φ Φ ρ r population variance population standard deviation sample mean unbiased estimate of population variance from a sample, s = n ( x x) probability density function of standardised normal variable with distribution N (0, ) corresponding cumulative distribution function linear product-moment correlation coefficient for a population linear product-moment correlation coefficient for a sample