📚 Year 12 WJEC Mathematics: A Complete Syllabus Breakdown | Year 12 WJEC 数学：课程大纲全面解析
Year 12 WJEC Mathematics covers the core AS-level syllabus, blending pure mathematics with applied modules in statistics and mechanics. Mastering the content requires a deep understanding of algebra, calculus, trigonometry, data analysis and Newtonian mechanics. This guide breaks down every major topic area, highlights key skills and offers essential revision pointers to help you build confidence and exam readiness.
Year 12 WJEC 数学涵盖 AS 阶段全部核心内容，将纯数学与应用模块（统计和力学）融会贯通。想要掌握这些知识，必须深刻理解代数、微积分、三角学、数据分析以及牛顿力学。本文将逐一拆解各大主题，指出关键技能并提供核心复习建议，帮助你建立自信，从容迎考。
1. Algebra and Functions | 代数与函数
You will extend your GCSE algebra by working with surds, laws of indices and quadratic functions, including completing the square and solving quadratic inequalities. The discriminant b² – 4ac is used to determine the nature of roots.
你需要拓展 GCSE 阶段的代数知识，处理无理数（surds）、指数律和二次函数，包括配方法（completing the square）以及求解二次不等式。判别式 b² – 4ac 用于判断方程根的性质。
Manipulating rational expressions and understanding function notation is central to the unit. Composite functions f(g(x)) and inverse functions f⁻¹(x) are tested regularly, alongside sketching graphs of functions and their transformations.
有理式的运算与函数符号的理解是本章核心。复合函数 f(g(x)) 和反函数 f⁻¹(x) 是常考内容，同时你还需要绘制函数图像并描述其变换。
2. Coordinate Geometry | 坐标几何
The equation of a straight line is often written as y – y₁ = m(x – x₁) or in the form ax + by + c = 0. You must be able to find midpoints, distances between two points and gradients of perpendicular lines (m₁m₂ = –1).
直线方程常写成 y – y₁ = m(x – x₁) 或 ax + by + c = 0 的形式。你必须能够求出中点、两点间的距离以及垂直直线的斜率关系（m₁m₂ = –1）。
For circles, the standard equation (x – a)² + (y – b)² = r² is used. Exam questions often ask you to complete the square, find tangents and chord lengths, or connect circle geometry with algebraic conditions.
对于圆，标准方程为 (x – a)² + (y – b)² = r²。考题经常要求你通过配方法求出圆心和半径、求切线方程和弦长，或者将圆的几何条件与代数方程结合。
3. Sequences and Series | 数列与级数
Arithmetic sequences use a common difference d. The nth term is uₙ = a + (n – 1)d and the sum of the first n terms is Sₙ = n/2[2a + (n – 1)d] or Sₙ = n/2(a + l).
等差数列具有公差 d。其通项公式为 uₙ = a + (n – 1)d，前 n 项和为 Sₙ = n/2[2a + (n – 1)d] 或 Sₙ = n/2(a + l)。
Geometric sequences rely on a common ratio r. You will learn to find the nth term uₙ = arⁿ⁻¹, the finite sum Sₙ = a(1 – rⁿ)/(1 – r), and the sum to infinity S∞ = a/(1 – r) for |r| < 1.
等比数列依赖公比 r。你将学习通项 uₙ = arⁿ⁻¹、有限和 Sₙ = a(1 – rⁿ)/(1 – r)，以及当 |r| < 1 时的无穷和 S∞ = a/(1 – r)。
4. Trigonometry | 三角学
The sine and cosine rules are essential for solving non-right-angled triangles. The area formula ½ab sin C is often used in coordinate and vector problems.
正弦定理和余弦定理是解非直角三角形的核心工具。面积公式 ½ab sin C 常用于坐标与向量问题。
You will study graphs of sin θ, cos θ and tan θ, and apply transformations to them. Solving trigonometric equations within given intervals and using identities such as tan θ ≡ sin θ / cos θ and sin²θ + cos²θ ≡ 1 are key skills.
你将学习 sin θ、cos θ 和 tan θ 的图像并进行变换。在给定区间内解三角方程，并运用恒等式，如 tan θ ≡ sin θ / cos θ 和 sin²θ + cos²θ ≡ 1，是本单元的关键技能。
5. Exponentials and Logarithms | 指数与对数
The exponential function y = eˣ and its inverse, the natural logarithm ln x, are introduced. Laws of logarithms – logₐ(xy) = logₐx + logₐy and logₐ(xⁿ) = n logₐx – are used to simplify expressions and solve exponential equations.
课程引入了指数函数 y = eˣ 及其反函数自然对数 ln x。对数运算律——如 logₐ(xy) = logₐx + logₐy 和 logₐ(xⁿ) = n logₐx——可用于化简表达式和求解指数方程。
Many modelling problems involve exponential growth and decay, often described by equations of the form A = A₀eᵏᵗ. You should be comfortable taking logarithms of both sides to find unknown constants.
许多建模问题涉及指数增长和衰减，通常用形如 A = A₀eᵏᵗ 的方程描述。你应当熟练地对等式两边取对数，以求解未知常数。
6. Differentiation | 微分
Differentiation finds the gradient of a curve. If y = xⁿ, then dy/dx = nxⁿ⁻¹. This rule is extended to sums and multiples, enabling differentiation of polynomials and rational expressions with negative powers.
微分可用于求曲线的斜率。若 y = xⁿ，则 dy/dx = nxⁿ⁻¹。该法则可推广到多项式以及带负指数的有理式。
You will use differentiation to find equations of tangents and normals, determine stationary points and classify them as maxima, minima or points of inflection using the second derivative d²y/dx².
你将利用微分求切线和法线方程、确定驻点，并通过二阶导数 d²y/dx² 判断驻点是极大值点、极小值点还是拐点。
7. Integration | 积分
Integration is the reverse of differentiation. The fundamental rule ∫ xⁿ dx = xⁿ⁺¹/(n + 1) + C (for n ≠ –1) allows you to find functions from their derivatives and calculate indefinite integrals.
积分是微分的逆运算。基本法则 ∫ xⁿ dx = xⁿ⁺¹/(n + 1) + C（n ≠ –1）可用于由导数求原函数以及计算不定积分。
Definite integrals ∫ₐᵇ f(x) dx give the area bounded by a curve and the x‑axis. You will also compute areas between two curves and solve simple kinematic problems using integration.
定积分 ∫ₐᵇ f(x) dx 可以计算曲线与 x 轴围成的面积。你还将计算两条曲线之间的面积，并用积分解决简单的运动学问题。
8. Vectors | 向量
Vectors describe quantities with both magnitude and direction. In two dimensions, vectors are represented as column vectors or using i, j notation. You will add, subtract and multiply vectors by scalars, and calculate magnitude |v| = √(x² + y²).
向量描述既有大小又有方向的量。在二维空间中，向量可用列向量或 i、j 形式表示。你将进行向量的加减和数乘，并计算模长 |v| = √(x² + y²)。
Position vectors and geometric applications appear frequently. Exam questions may require you to prove collinearity, find midpoints or solve problems involving vectors and geometry in a coordinate plane.
位置向量与几何应用频繁出现。考题可能要求你证明三点共线、求中点，或解决坐标平面内涉及向量的几何问题。
9. Statistics: Data and Probability | 统计学：数据与概率
Measures of location (mean, median, mode) and measures of spread (range, interquartile range, standard deviation) are revised and extended. You will calculate variance using the formula σ² = Σ(x – x̄)²/n or the equivalent computational form.
数据的集中趋势度量（均值、中位数、众数）和离散程度度量（极差、四分位距、标准差）将得到复习与拓展。你将使用公式 σ² = Σ(x – x̄)²/n 或等价的简便公式计算方差。
Probability concepts include mutually exclusive events, independent events and conditional probability calculated via P(A|B) = P(A ∩ B)/P(B). Tree diagrams and Venn diagrams support multi‑stage and set‑based probability questions.
概率概念包括互斥事件、独立事件以及条件概率，计算式为 P(A|B) = P(A ∩ B)/P(B)。树状图和韦恩图用于解决多阶段和基于集合的概率问题。
10. Statistical Distributions and Hypothesis Testing | 统计分布与假设检验
The binomial distribution B(n, p) models the number of successes in n independent trials. You will calculate probabilities using the formula P(X = r) = ⁿCᵣ pʳ(1 – p)ⁿ⁻ʳ and use cumulative tables or calculators.
二项分布 B(n, p) 用于描述 n 次独立试验中的成功次数。你需要利用公式 P(X = r) = ⁿCᵣ pʳ(1 – p)ⁿ⁻ʳ 计算概率，并会使用累积分布表或计算器。
The normal distribution N(μ, σ²) is introduced for continuous data, including standardisation to Z = (X – μ)/σ. You will also conduct hypothesis tests for binomial distributions: stating null and alternative hypotheses, identifying critical regions and interpreting p‑values.
正态分布 N(μ, σ²) 用于连续数据，包括标准化 Z = (X – μ)/σ。你还将进行二项分布的假设检验：提出零假设与备择假设，确定拒绝域并解释 p 值。
11. Mechanics: Kinematics | 力学：运动学
Kinematics describes motion using quantities displacement, velocity and acceleration. Constant acceleration problems are solved with SUVAT equations, such as v = u + at and s = ut + ½at².
运动学用位移、速度和加速度描述物体的运动。匀加速问题通过 SUVAT 方程求解，如 v = u + at 和 s = ut + ½at²。
You must be able to interpret displacement–time and velocity–time graphs, recognising that gradient gives velocity or acceleration, while the area under a velocity–time graph gives displacement.
你需要能够解读位移–时间图和速度–时间图，认识到斜率代表速度或加速度，而速度–时间图下的面积表示位移。
12. Mechanics: Forces and Newton’s Laws | 力学：力与牛顿定律
Newton’s three laws of motion underpin all mechanics problems. You will resolve forces, draw free‑body diagrams and apply F = ma in situations involving weight, tension, normal reaction and friction.
牛顿运动三定律是所有力学问题的基础。你将分解力、绘制受力分析图，并在涉及重力、张力、支持力和摩擦力的问题中应用 F = ma。
Connected particles, pulleys and inclined planes often appear in WJEC exams. You must form simultaneous equations by considering each mass separately and use limiting friction F ≤ μR for static problems.
连接体、滑轮和斜面常出现在 WJEC 考试中。你需要分别考虑每个质量并列出联立方程，并在静力问题中运用极限摩擦力关系 F ≤ μR。
Published by TutorHao | Mathematics Revision Series | aleveler.com
更多咨询请联系16621398022（同微信）
屏轩国际教育cambridge primary/secondary checkpoint, cat4, ukiset,ukcat,igcse,alevel,PAT,STEP,MAT, ibdp,ap,ssat,sat,sat2课程辅导，国外大学本科硕士研究生博士课程论文辅导