📚 AQA Mathematics: End-of-Term Revision Checklist | AQA 数学：期末复习提纲
This revision checklist breaks down the core topics for AQA Mathematics end-of-term assessments. Work through each section systematically, practise past paper questions, and focus on understanding the reasoning behind every technique. The guide is bilingual to support both English and Chinese learners in consolidating knowledge before the exam.
本复习提纲系统梳理了 AQA 数学期末考试的核心主题。逐节复习，练习历年真题，并着重理解每种方法背后的原理。本指南采用中英双语，帮助中英文学习者在考前巩固知识。
1. Algebra and Functions | 代数与函数
Simplify algebraic expressions by expanding brackets, collecting like terms, and factorising quadratics and cubics using the factor theorem.
通过展开括号、合并同类项以及利用因式定理分解二次式和三次式来化简代数表达式。
Perform polynomial division and use the remainder theorem to find unknown coefficients or remainders when dividing by (x – a).
掌握多项式除法，并能运用余式定理求除以 (x – a) 时的未知系数或余数。
Simplify rational expressions by cancelling common factors, and solve equations involving algebraic fractions.
通过约去公因式化简有理表达式，并求解含有代数分式的方程。
Understand transformations of graphs: translations, stretches, and reflections. Apply these to y = f(x) + a, y = f(x + a), y = a f(x), and y = f(ax).
理解函数图像的变换：平移、伸缩和反射。将这些变换应用于 y = f(x) + a, y = f(x + a), y = a f(x) 和 y = f(ax)。
Work with functions including domain, range, inverse functions, and composite functions fg(x). Know the condition for an inverse to exist (one-to-one).
掌握函数相关概念，包括定义域、值域、反函数和复合函数 fg(x)。知道反函数存在的条件（一一映射）。
2. Coordinate Geometry | 坐标几何
Find the equation of a straight line using y – y₁ = m(x – x₁) and understand gradient m = (y₂ – y₁)/(x₂ – x₁). Recognise parallel and perpendicular gradients: m₁ = m₂ and m₁ × m₂ = -1.
利用点斜式 y – y₁ = m(x – x₁) 求直线方程，并理解斜率 m = (y₂ – y₁)/(x₂ – x₁)。掌握平行斜率 m₁ = m₂ 和垂直斜率 m₁ × m₂ = -1。
Calculate the midpoint of a line segment and the distance between two points using √((x₂ – x₁)² + (y₂ – y₁)²).
计算线段的中点坐标以及两点间的距离，使用公式 √((x₂ – x₁)² + (y₂ – y₁)²)。
Complete the square to find the centre and radius of a circle from x² + y² + 2gx + 2fy + c = 0. Centre = (-g, -f), radius = √(g² + f² – c).
通过配方法由一般式 x² + y² + 2gx + 2fy + c = 0 求圆的圆心和半径。圆心为 (-g, -f)，半径为 √(g² + f² – c)。
Determine whether a line intersects, touches, or misses a circle by solving simultaneous equations and examining the discriminant.
通过联立方程求解并考察判别式，判断直线与圆是相交、相切还是相离。
3. Sequences and Series | 数列与级数
Define arithmetic sequences using the nth term formula uₙ = a + (n – 1)d and sum to n terms Sₙ = n/2 [2a + (n – 1)d] or Sₙ = n/2 (a + l).
用通项公式 uₙ = a + (n – 1)d 定义等差数列，并掌握前 n 项和公式 Sₙ = n/2 [2a + (n – 1)d] 或 Sₙ = n/2 (a + l)。
Work with geometric sequences: uₙ = arⁿ⁻¹ and sum Sₙ = a(1 – rⁿ)/(1 – r) for |r| < 1. Understand the concept of a convergent series and sum to infinity S∞ = a/(1 - r), provided |r| < 1.
掌握等比数列：通项 uₙ = arⁿ⁻¹，前 n 项和 Sₙ = a(1 – rⁿ)/(1 – r)（当 |r| < 1）。理解收敛级数的概念，并会求无穷等比级数的和 S∞ = a/(1 - r)，条件为 |r| < 1。
Use sigma notation Σ to represent series and apply standard summation formulas for Σr, Σr², and Σr³ to solve problems.
熟练使用西格玛记号 Σ 表示级数，并能利用 Σr, Σr² 和 Σr³ 的标准求和公式解题。
4. Trigonometry | 三角学
Recall exact trigonometric values for angles 0°, 30°, 45°, 60°, 90°, and apply them to solve equations and find coordinates.
熟记 0°, 30°, 45°, 60°, 90° 角的精确三角值，并用于求解方程和确定坐标。
Use the identities tan θ = sin θ / cos θ and sin²θ + cos²θ = 1 to simplify expressions and solve trigonometric equations within a given interval.
运用恒等式 tan θ = sin θ / cos θ 和 sin²θ + cos²θ = 1 化简表达式，并在给定区间内求解三角方程。
Solve equations using the sine and cosine rules for non-right-angled triangles: a/sin A = b/sin B = c/sin C, and a² = b² + c² – 2bc cos A.
熟练运用正弦定理和余弦定理求解斜三角形：a/sin A = b/sin B = c/sin C 及 a² = b² + c² – 2bc cos A。
Work with radian measure: convert between degrees and radians (π rad = 180°), calculate arc length s = rθ and sector area A = ½r²θ.
掌握弧度制：进行度与弧度的换算（π rad = 180°），计算弧长 s = rθ 和扇形面积 A = ½r²θ。
5. Exponentials and Logarithms | 指数与对数
Understand the relationship between exponentials and logarithms: if aˣ = b, then x = logₐ b. Know the natural logarithm ln x = logₑ x.
理解指数与对数的关系：若 aˣ = b，则 x = logₐ b。熟记自然对数 ln x = logₑ x。
Apply logarithm laws: logₐ (xy) = logₐ x + logₐ y, logₐ (x/y) = logₐ x – logₐ y, and logₐ (xⁿ) = n logₐ x.
熟练运用对数运算法则：logₐ (xy) = logₐ x + logₐ y，logₐ (x/y) = logₐ x – logₐ y，以及 logₐ (xⁿ) = n logₐ x。
Solve equations of the form aˣ = b by taking logs of both sides, and use exponentials to model population growth and radioactive decay.
通过两边取对数求解形如 aˣ = b 的指数方程，并运用指数函数建立人口增长和放射性衰变模型。
Differentiate and integrate exponential functions: d/dx (eˣ) = eˣ, and ∫ eˣ dx = eˣ + C.
掌握指数函数的求导与积分：d/dx (eˣ) = eˣ，∫ eˣ dx = eˣ + C。
6. Differentiation | 微分
Differentiate polynomials using the power rule: if y = xⁿ, then dy/dx = nxⁿ⁻¹. Extend to sum/difference rules and constant multiples.
运用幂法则对多项式求导：若 y = xⁿ，则 dy/dx = nxⁿ⁻¹。进一步掌握和差法则及常数倍数法则。
Find the equation of a tangent and normal to a curve at a given point using the derivative as the gradient.
利用导数作为斜率，求出曲线在给定点处的切线和法线方程。
Determine stationary points by setting dy/dx = 0, and classify them (maximum, minimum, point of inflection) using the second derivative d²y/dx² or sign change.
通过令 dy/dx = 0 求驻点，并利用二阶导数 d²y/dx² 或符号变化判断其性质（极大值、极小值、拐点）。
Differentiate trigonometric functions: d/dx (sin x) = cos x, d/dx (cos x) = -sin x, and use chain, product, and quotient rules for composite functions.
掌握三角函数的导数：d/dx (sin x) = cos x, d/dx (cos x) = -sin x，并能运用链式法则、乘积法则和商法则对复合函数求导。
7. Integration | 积分
Integrate as the reverse process of differentiation: ∫ xⁿ dx = xⁿ⁺¹/(n+1) + C, for n ≠ -1. Understand indefinite and definite integrals.
将积分理解为微分的逆运算：∫ xⁿ dx = xⁿ⁺¹/(n+1) + C，其中 n ≠ -1。理解不定积分和定积分的概念。
Evaluate the area under a curve y = f(x) between limits a and b using ∫ₐᵇ f(x) dx. Recognise that areas below the x-axis are negative.
利用定积分 ∫ₐᵇ f(x) dx 计算曲线 y = f(x) 下方在界限 a 和 b 之间的面积。注意 x 轴下方的面积为负值。
Find areas bounded by a curve and a line by subtracting integrals, or by integrating with respect to y when appropriate.
通过积分相减求曲线与直线围成的面积，或在适当时对 y 进行积分。
Integrate simple expressions of the form eᵃˣ, 1/x, sin kx, and cos kx, and apply integration techniques such as substitution (reverse chain rule) in simple cases.
掌握形如 eᵃˣ, 1/x, sin kx 和 cos kx 的基本积分，并能在简单情况下应用换元积分法（逆链式法则）。
8. Vectors | 向量
Represent vectors in column or i, j notation, and calculate magnitude using √(x² + y²). Add and subtract vectors and multiply by scalars.
用列向量或 i, j 形式表示向量，并利用 √(x² + y²) 计算模长。掌握向量的加减及数乘运算。
Use position vectors and find the vector AB = b – a between two points. Solve geometric problems involving parallel vectors and collinearity.
使用位置向量，求出两点间的向量 AB = b – a。解决涉及平行向量和共线关系的几何问题。
Apply the dot product for two-dimensional vectors: a · b = |a||b| cos θ, and calculate the angle between two vectors. Recognise perpendicular vectors when a · b = 0.
应用二维向量的数量积：a · b = |a||b| cos θ，计算两向量夹角。当 a · b = 0 时，两向量垂直。
Solve kinematic problems using vectors for displacement, velocity, and acceleration, integrating and differentiating with respect to time.
利用位移、速度和加速度向量解决运动学问题，并对时间进行积分和微分。
9. Statistics | 统计学
Represent data using histograms, cumulative frequency diagrams, and box plots. Calculate and interpret measures of central tendency (mean, median, mode) and dispersion (range, interquartile range, standard deviation).
利用直方图、累积频率图和箱形图表示数据。计算并解释集中量数（平均数、中位数、众数）和离散量数（极差、四分位距、标准差）。
Calculate probabilities using Venn diagrams, tree diagrams, and the addition and multiplication rules. Understand mutually exclusive and independent events.
综合运用韦恩图、树状图以及加法和乘法法则计算概率。理解互斥事件和独立事件。
Use the binomial distribution B(n, p) when there are a fixed number of independent trials. Know its probability formula P(X = r) = ⁿCᵣ pʳ (1 – p)ⁿ⁻ʳ.
在固定次数的独立试验场景下，使用二项分布 B(n, p)。掌握概率公式 P(X = r) = ⁿCᵣ pʳ (1 – p)ⁿ⁻ʳ。
Conduct hypothesis tests for a binomial proportion: state null and alternative hypotheses, find the critical region, and interpret results in context.
对二项比例进行假设检验：陈述原假设和备择假设，找出拒绝域，并结合上下文解释结果。
10. Mechanics | 力学
Model motion with constant acceleration using SUVAT equations: v = u + at, s = ut + ½at², s = ½(u + v)t, v² = u² + 2as, and s = vt – ½at².
运用匀加速直线运动公式（SUVAT）建立运动模型：v = u + at, s = ut + ½at², s = ½(u + v)t, v² = u² + 2as, 以及 s = vt – ½at²。
Draw free-body force diagrams and resolve forces into components. Apply Newton’s second law F = ma to connected particles and systems.
绘制受力图并将力分解为分量。将牛顿第二定律 F = ma 应用于连接体系统。
Understand friction: the maximum static friction F ≤ μR and the kinetic friction F = μR, where R is the normal reaction.
理解摩擦力概念：最大静摩擦力 F ≤ μR，滑动摩擦力 F = μR，其中 R 为法向反作用力。
Analyse motion under gravity, including projectile motion: resolve initial velocity into horizontal and vertical components, and treat the vertical motion with acceleration g = 9.8 m s⁻².
分析重力作用下的运动，包括抛体运动：将初速度分解为水平和竖直分量，并处理竖直方向加速度 g = 9.8 m s⁻² 的运动。
Use time-varying forces and calculus: velocity is the derivative of displacement, acceleration is the derivative of velocity. Integrate acceleration to find velocity and displacement.
处理随时间变化的力并运用微积分：速度是位移的导数，加速度是速度的导数。对加速度积分可求得速度和位移。
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