Year 10: Singapore O-Level Additional Mathematics 4049

Listed Mathematics sub-topics at O-Level based on the Singapore Examinations and Assessment Board (SEAB). Checked against the source on 2 October 2026.

Sub-topics from the Singapore–Cambridge GCE O-Level Additional Mathematics syllabus (4049) for the 2026 examination. Year 10 is Secondary 4, at the end of which the papers are sat. There are two papers of 90 marks each, both 2 hours 15 minutes and each worth half the grade. Knowledge of the O-Level Mathematics syllabus is assumed: it is not tested directly, but it may be needed in answering questions on other topics. Binomial coefficients, which the syllabus sets as a two-row bracket, are written inline here as C(n, r).

10sub-topics
55skills mapped
90%have practice

Coverage is the percentage of listed sub-topics with at least one linked practice skill; it does not measure how fully every part of an entry is practised. A skill can practise more than one sub-topic, so some appear more than once below.

Algebra

A1Quadratic functionsFinding the maximum or minimum value of a quadratic function using the method of completing the square • Conditions for y = ax² + bx + c to be always positive (or always negative) • Using quadratic functions as models
A2Equations and inequalitiesConditions 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 • Solving simultaneous equations in two variables by substitution, with one of the equations being a linear equation • Solving quadratic inequalities, and representing the solution on the number line
A3SurdsFour operations on surds, including rationalising the denominator • Solving equations involving surds
- Simplification of square and cubic roots
- Addition and subtraction of square roots
A4Polynomials and partial fractionsMultiplication and division of polynomials • Use of remainder and factor theorems, including factorising polynomials and solving cubic equations • Use of: − a³ + b³ = (a + b)(a² − ab + b²) − a³ − b³ = (a − b)(a² + ab + b²) • Partial fractions with cases where the denominator is no more complicated than: − (ax + b)(cx + d) − (ax + b)(cx + d)² − (ax + b)(x² + c²)
A5Binomial expansionsUse of the Binomial Theorem for positive integer n • Use of the notations n! and C(n, r) • Use of the general term C(n, r)aⁿ⁻ʳbʳ, 0 ⩽ r ⩽ n (knowledge of the greatest term and properties of the coefficients is not required)
No practice for this one yet.
A6Exponential and logarithmic functionsExponential and logarithmic functions aˣ, eˣ, logₐ x, ln x and their graphs, including − laws of logarithms − equivalence of y = aˣ and x = logₐ y − change of base of logarithms • Simplifying expressions and solving simple equations involving exponential and logarithmic functions • Using exponential and logarithmic functions as models

Geometry and trigonometry

G1Trigonometric functions, identities and equationsSix trigonometric functions for angles of any magnitude (in degrees or radians) • Principal values of sin⁻¹x, cos⁻¹x, tan⁻¹x • Exact values of the trigonometric functions for special angles (30°, 45°, 60°) or (π/6, π/4, π/3) • Amplitude, periodicity and symmetries related to sine and cosine functions • Graphs of y = a sin(bx) + c, y = a sin(x/b) + c, y = a cos(bx) + c, y = a cos(x/b) + c and y = a tan(bx), where a is real, b is a positive integer and c is an integer • Use of: − sin A/cos A = tan A, cos A/sin A = cot A, sin²A + cos²A = 1, sec²A = 1 + tan²A, cosec²A = 1 + cot²A − 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 of a cos θ + b sin θ in the form R cos(θ ± α) or R sin(θ ± α) • Simplification of trigonometric expressions • Solution of simple trigonometric equations in a given interval (excluding general solution) • Proofs of simple trigonometric identities • Using trigonometric functions as models
G2Coordinate geometry in two dimensionsCondition for two lines to be parallel or perpendicular • Midpoint of line segment • Area of rectilinear figure • Coordinate geometry of circles in the form: − (x − a)² + (y − b)² = r² − x² + y² + 2gx + 2fy + c = 0 (excluding problems involving two circles) • Transformation of given relationships, including y = axⁿ and y = kbˣ, to linear form to determine the unknown constants from a straight line graph
G3Proofs in plane geometryUse 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) (* These are properties learnt in O-Level Mathematics.)

Calculus

C1Differentiation and integrationDerivative of f(x) as the gradient of the tangent to the graph of y = f(x) at a point • Derivative as rate of change • Use of standard notations f′(x), f″(x), dy/dx, d²y/dx² [= d/dx(dy/dx)] • Derivatives of xⁿ, for any rational n, sin x, cos x, tan x, eˣ, and ln x together with constant multiples, sums and differences • Derivatives of products and quotients of functions • Use of Chain Rule • 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 • Apply differentiation to gradients, tangents and normals, connected rates of change and maxima and minima problems • Integration as the reverse of differentiation • Integration of xⁿ for any rational n, sin x, cos x, sec²x and eˣ, 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 2 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
- Differentiate simple polynomial functions

Other Year 10 skills

Year 10 skills not currently linked to an entry on this page. These may include revision, extension practice, or skills still awaiting a curriculum mapping.