A-Level Mathematics (CAIE 9709)

Course Overview 课程介绍

CAIE 高级数学课程，分 AS 与 A2，理工科与经济专业核心基础。

A-Level Mathematics is a UK advanced secondary course split into AS (Year 1) and A2 (Year 2). It covers pure mathematics, mechanics, and probability & statistics, with strong emphasis on algebraic technique, function modelling and problem solving. A core subject for mathematics, engineering, physics, computer science, economics, finance and statistics applicants.

Key Facts 基本信息

Exam Board (考试局)	CAIE (Cambridge Assessment International Education)
CAIE Code (代码)	9709
Duration (学制)	2 years (AS + A2)
Grading (评分)	A* - E (A* only for full A-Level)
Exam Structure (CAIE)	AS: Paper 1/2 · A2: Paper 3/4/5
Coursework (课程作业)	No coursework, all external exams

AS Syllabus AS 阶段章节

AS1. Quadratics

Quadratic functions, discriminant, completing the square, quadratic formula, inequalities
AS2. Functions

Functions, domain and range, composite functions, inverse functions, graph transformations
AS3. Coordinate Geometry

Coordinate geometry, equation of a line, equation of a circle, distance and midpoint
AS4. Circular Measure

Circular measure, radian measure, arc length and sector area, radian-degree conversion
AS5. Trigonometry

Trigonometry, trigonometric identities, trigonometric equations, sine and cosine rules
AS6. Vectors

Vectors, 2D vectors, addition/subtraction and scalar multiplication, magnitude and unit vectors, dot product
AS7. Series

Series, arithmetic and geometric progressions, summation formulae, binomial expansion
AS8. Differentiation

Differentiation, derivative definition, power rule, tangents and normals, extrema and monotonicity
AS9. Integration

Integration, indefinite integrals, definite integrals, area under curve, volume of revolution
AS10. Forces and Equilibrium

Mechanics: forces and equilibrium, composition and resolution of forces, equilibrium conditions, friction
AS11. Kinematics of Motion

Mechanics: kinematics, displacement/velocity/acceleration, equations of uniform acceleration, calculus applications
AS12. Representation of Data and Probability

Statistics: data representation and probability, discrete and continuous data, permutations and combinations, probability fundamentals
AS13. Discrete Random Variables

Statistics: discrete random variables, probability distributions, expectation and variance, binomial distribution
AS14. Normal Distribution

Statistics: normal distribution, standardisation, probability from tables, inverse look-up

A2 Syllabus A2 阶段章节

A215. Algebra

Algebra (advanced), modulus and inequalities, partial fractions, higher-degree equations
A216. Logarithmic and Exponential Functions

Logarithmic and exponential functions, natural logarithm, change of base, exponential equations
A217. Trigonometry (Advanced)

Trigonometry (advanced), compound angle formulae, double and half angles, inverse trig functions, hyperbolic functions introduction
A218. Differentiation (Advanced)

Differentiation (advanced), chain rule, implicit and parametric differentiation, chain rule applications
A219. Integration (Advanced)

Integration (advanced), substitution, integration by parts, partial fraction integration, numerical integration
A220. Numerical Solutions of Equations

Numerical solutions of equations, bisection method, Newton-Raphson method, iterative convergence
A221. Vectors (Advanced)

Vectors (advanced), 3D vectors, cross product, equations of lines and planes
A222. Differential Equations

Differential equations, separable variables, first-order linear differential equations, modelling applications
A223. Complex Numbers

Complex numbers, imaginary unit, modulus and argument, polar form, de Moivre's theorem
A224. Forces and Equilibrium in 2D

Mechanics (advanced): forces and equilibrium in 2D, Newton's laws, connected particles, inclined planes
A225. Motion in 2D and Projectiles

Mechanics (advanced): motion in 2D, projectile motion, relative velocity, work and energy
A226. Continuous Random Variables

Statistics (advanced): continuous random variables, probability density functions, cumulative distributions
A227. Hypothesis Testing

Statistics (advanced): hypothesis testing, significance level, one- and two-tailed tests, sampling distributions

Exam Structure (CAIE 9709) 考试结构

Paper 1	Pure Mathematics 1 · 1h50m · 60% of AS / 30% of A-Level
Paper 2	Pure Mathematics 2 · 1h15m · 40% of AS / 20% of A-Level (AS route only)
Paper 3	Pure Mathematics 3 · 1h50m · 30% of A-Level
Paper 4	Mechanics · 1h15m · 10% of A-Level
Paper 5	Probability & Statistics 1 · 1h15m · 10% of A-Level

Suitable Majors 适用专业与衔接

数学、工程、经济、金融等专业基础科目，英国理工科普遍必修。

A-Level Mathematics is foundational for mathematics, engineering, physics, computer science, economics, finance, statistics, actuarial science, data science and operations research. Most UK STEM and economics programmes require A-Level Mathematics as mandatory. It provides essential prerequisite knowledge for university calculus, linear algebra and probability courses.