Source:** MOE 2019 O level Additional Maths Syllabus **

The syllabus prepares students adequately for A-Level H2 Mathematics, where a strong foundation in algebraic manipulation skills and mathematical 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 the content strand, the development of process skills, namely, reasoning, communication and connections, thinking skills and heuristics, and applications and modelling are also emphasised. The O-Level Additional Mathematics syllabus assumes knowledge of O-Level Mathematics.

A1 | 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 ax2 + 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 the roots and coefficients of a quadratic equation · Solving quadratic inequalities, and representing the solution on the number line | |||||||||||||||

A2 | Indices and surds | · Four operations on indices and surds, including rationalising the denominator · Solving equations involving indices and surds | |||||||||||||||

A3 | Polynomials and Partial Fractions | · Multiplication and division of polynomials · Use of remainder and factor theorems · Factorisation of polynomials · Use of: – a ^{3} + b^{3} = (a + b)(a^{2} – ab + b^{2})– a ^{3} – b^{3} = (a – b)(a^{2} + ab + b^{2})· Solving cubic equations · Partial fractions with cases where the denominator is no more complicated than: – (ax + b)(cx + d) – (ax + b)(cx + d) ^{2 }– (ax + b)(x^{2} + c^{2}) | |||||||||||||||

A4 | Binomial expansions | · Use of the Binomial Theorem for positive integer n · Use of the notations n! and (n) · Use of the general term (knowledge of the greatest term and properties of the coefficients is not required) | |||||||||||||||

A5 | Power, Exponential, Logarithmic, and Modulus functions | · Power functions y = ax^{n} where n is a simple rational number, and their graphs· Exponential and logarithmic functions a ^{x}, e^{x}, log_{a} x, ln x and their 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 their graphs · Solving simple equations involving exponential, logarithmic and modulus functions |

G1 | Trigonometric functions, identities and equations | · Six trigonometric functions for angles of any magnitude (in degrees or radians) · Principal values of sin ^{–1}x, cos^{–1}x, tan^{–1}x· Exact values of the trigonometric functions for special angles (30°, 45°, 60°) or ^{æ} ^{p} , ^{p} , ^{p} ^{ö }ç ÷_{è} 6 4 3 _{ø }· Amplitude, periodicity and symmetries related to the sine and cosine functions· Graphs of y = a sin (bx) + c, y = a sin ^{æ} ^{x }^{ö}^{ }+ c, y = a cos (bx) + c,ç ÷ è ^{b }øy = a cos ^{æ} ^{x }^{ö} + c and y = a tan (bx), where a is real, b is a positive integerç ÷ è ^{b }øand c is an integer. · Use of the following ^{sin }^{A }= tan A, ^{cos }^{A }= cot A, sin^{2 }A + cos^{2} A = 1, sec 2 A = 1+ tan2 A,– ^{cos }^{A }^{sin }^{A }cosec ^{2 }A = 1+ cot ^{2 }A– the expansions of sin(A ± B), cos(A ± B) and tan(A ± B) – the formulae for sin 2A, cos 2A and tan 2A – the expression for a cos q + b sin q in the form R cos (q ± a) or R sin (q ± a) · Simplification of trigonometric expressions · Solution of simple trigonometric equations in a given interval (excluding general solution) · Proofs of simple trigonometric identities | |||||||||||||||

G2 | Coordinate geometry in two dimensions | · Condition for two lines to be parallel or perpendicular · Midpoint of line segment · Area of rectilinear figure · Graphs of parabolas with equations in the form y ^{2} = kx· Coordinate geometry of circles in the form: – (x – a) ^{2} + (y – b)^{2} = r^{2 }– x^{2} + y^{2} + 2gx + 2fy + c = 0 (excluding problems involving 2 circles)· Transformation of given relationships, including y = ax ^{n} and y = kb^{x}, to linear form to determine the 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 theorem– tangent-chord theorem (alternate segment theorem) |

C1 | Differentiation and integration | · Derivative of f(x) as the gradient of the tangent to the graph of y = f(x) at a point · Derivative as rate of change · of standard notations f ^{¢}(x ), f^{¢} (x ), ^{d}^{y} , ^{d }^{y }^{é}= ^{ d } ^{æ} ^{d}^{y} ^{öù }2^{Use }_{ê}_{ }ç ÷_{ú }dx _{d}_{x }^{2 }_{êë}_{ }dx _{è} dx _{øû }· Derivatives of x^{n}, for any rational n, sin x, cos x, tan x, e^{x}, and ln x, together 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 the reverse of differentiation · Integration of x ^{n}, for any rational n, sin x, cos x, sec^{2} x and e^{x}, together 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 the area of a region bounded by a curve and line(s) (excluding area of region between two curves) · Finding areas of regions below the x-axis · Application of differentiation and integration to problems involving displacement, velocity and acceleration of a particle moving in a straight line |