Singapore Secondary 4 Additional Mathematics Syllabus (G2, G3)

This syllabus prepares students for A-Level H2 Mathematics. It builds upon the O-Level Mathematics syllabus and emphasizes algebraic manipulation and mathematical reasoning skills. The content is structured around three main strands: Algebra, Geometry and Trigonometry, and Calculus. Beyond these strands, the syllabus also stresses mathematical processes like reasoning, communication, application (including using models), which are also assessed.

Algebra

• A1 Quadratic Functions: This topic covers finding the maximum or minimum value of a quadratic function by completing the square, understanding conditions for a quadratic function to be always positive or always negative, and using quadratic functions in models.
• A2 Equations and Inequalities: This section focuses on the conditions for a quadratic equation to have two real roots, two equal roots, or no real roots. It also explores the related conditions for a line to intersect, be tangent to, or not intersect a given curve. Additionally, it covers solving simultaneous equations with one linear equation and one quadratic equation, and solving quadratic inequalities.
• A3 Surds: This topic includes performing four operations on surds, including rationalizing the denominator, and solving equations involving surds.
• A4 Polynomials and Partial Fractions: This section covers multiplication and division of polynomials, the remainder and factor theorems (including factorizing polynomials and solving cubic equations), and partial fractions with denominators up to the complexity of (ax + b)(cx + d), (ax + b)(cx + d)², and (ax + b)(x² + c²). Specific factorization identities are also included.

A5 Binomial Expansions: This topic covers the Binomial Theorem for positive integer n, the notations n! and nCr*, and the general term of a binomial expansion.

• A6 Exponential and Logarithmic Functions: This section explores exponential and logarithmic functions (aˣ, eˣ, logₐx, ln x) and their graphs, including laws of logarithms, the equivalence of y = aˣ and x = logₐy, and changing the base of logarithms. It also covers simplifying expressions, solving equations, and using these functions as models.

Geometry and Trigonometry

• G1 Trigonometric Functions, Identities, and Equations: This topic encompasses the six trigonometric functions for angles of any magnitude (in degrees or radians), principal values of inverse trigonometric functions, exact values for special angles, amplitude, periodicity, and symmetries of sine and cosine functions. Graphing trigonometric functions, trigonometric identities and expansions, simplifying expressions, solving equations in a given interval, proving simple identities, and using trigonometric functions as models are also included.
• G2 Coordinate Geometry in Two Dimensions: This section covers conditions for parallel and perpendicular lines, midpoints, area of rectilinear figures, and coordinate geometry of circles (excluding problems with two circles). It also includes transforming relationships like y = axⁿ and y = kbˣ to linear form.
• G3 Proofs in Plane Geometry: This topic involves using properties of parallel lines, perpendicular and angle bisectors, triangles, special quadrilaterals, circles, congruent and similar triangles, the midpoint theorem, and the tangent-chord theorem.

Calculus

• C1 Differentiation and Integration: This section covers the derivative as the gradient of the tangent and as a rate of change, standard notations for derivatives, derivatives of standard functions (including polynomial, trigonometric, exponential, and logarithmic functions), the chain rule, product and quotient rules, increasing and decreasing functions, stationary points, the second derivative test, applications to gradients, tangents, normals, connected rates of change, and maxima and minima problems. It also includes integration as the reverse of differentiation, integration of standard functions, definite integrals as areas under curves, evaluating definite integrals, finding areas bounded by curves and lines, and applications to displacement, velocity, and acceleration.