📚 AQA A Level Maths Support Pack 1: Contents Overview | AQA A-Level 数学支持包1内容总览
This support pack maps the key content areas of the AQA A Level Mathematics specification. It is designed to help you check the main topics, common question types and the exact notation used in the exam. Work through each section and use the paired English-Chinese notes to consolidate your understanding before attempting past papers.
本支持包梳理了 AQA A Level 数学考试大纲中的关键内容领域。它旨在帮助你检查主要主题、常见题型以及考试中使用的准确符号。请逐节学习，并利用中英对照笔记在练习历年真题前巩固理解。
1. Pure Core: Algebra and Functions | 纯数学核心：代数与函数
The pure core starts with algebraic manipulation. You must be confident with quadratic equations, the discriminant b² - 4ac, polynomial division, the factor theorem and partial fractions. These tools are assumed in later topics such as integration and curve sketching.
纯数学核心从代数运算开始。你必须熟练掌握二次方程、判别式 b² - 4ac、多项式除法、因式定理以及部分分式。这些工具在后续的积分和曲线作图等主题中会默认使用。
For functions, check the domain and range, composition fg(x), and inverse functions f⁻¹(x). A one-to-one function is needed for an inverse to exist, so restricting a domain is often required before finding an inverse.
对于函数，请检查定义域和值域、复合函数 fg(x) 以及反函数 f⁻¹(x)。只有一一对应的函数才存在反函数，因此在求反函数之前通常需要限制定义域。
2. Coordinate Geometry | 坐标几何
ght line equation y - y₁ = m(x - x₁), and the distance formula √((x₂ - x₁)² + (y₂ - y₁)²).
坐标几何将代数与图像联系起来。你需要掌握斜率公式 m = (y₂ - y₁)/(x₂ - x₁)、直线方程 y - y₁ = m(x - x₁) 以及距离公式 √((x₂ - x₁)² + (y₂ - y₁)²)。
. Completing the square is often used to find the centre and radius from an expanded form.
圆的几何也在考试范围内。圆心为 (a, b)、半径为 r 的圆方程为 (x - a)² + (y - b)² = r²。常用配方法将圆的一般式化为标准式以求出圆心和半径。
3. Trigonometry | 三角学
in θ / cos θ. You must also know the sine and cosine rules for non-right-angled triangles.
AQA 要求能在角度制和弧度制下进行计算。记住 π 弧度 = 180°。关键的恒等式是 sin²θ + cos²θ = 1 和 tan θ = sin θ / cos θ。你还必须掌握适用于非直角三角形的正弦定理和余弦定理。
y = tan x helps avoid missing solutions, especially when the interval is given in radians.
对于 sin θ = 0.5 这类方程，要在给定区间内给出所有解。画出 y = sin x、y = cos x 和 y = tan x 的图像有助于避免漏解，尤其是在区间以弧度给出时。
4. Exponentials and Logarithms | 指数与对数
The exponential function eˣ and natural logarithm ln x are central. You must know that ln x is the inverse of eˣ, so ln(eˣ) = x and eln x = x. Any base a can be written as aˣ = ex ln a.
指数函数 eˣ 和自然对数 ln x 是核心内容。你必须知道 ln x 是 eˣ 的反函数，因此 ln(eˣ) = x 且 eln x = x。任何底数 a 都可以写成 aˣ = ex ln a。
, logₐ(x/y) = logₐx - logₐy and logₐ(xⁿ) = n logₐx. Use these to solve equations such as 2ˣ = 5 by taking logarithms of both sides.
对数法则会直接考查：logₐ(xy) = logₐx + logₐy、logₐ(x/y) = logₐx - logₐy 以及 logₐ(xⁿ) = n logₐx。要会用这些法则解如 2ˣ = 5 的方程，方法是对两边同时取对数。
5. Differentiation | 微分
Differentiation measures the rate of change. From first principles, the derivative is defined by the limit as h tends to 0 of (f(x + h) - f(x))/h. In practice, you need the power rule, product rule, quotient rule and chain rule.
微分用于度量变化率。从第一性原理出发，导数定义为当 h 趋于 0 时 (f(x + h) - f(x))/h 的极限。在实际计算中，你需要掌握幂函数求导法则、乘法法则、除法法则和链式法则。
tiation to find stationary points, and determine their nature by the second derivative d²y/dx².
对于 y = u/v，除法法则为 dy/dx = (v du/dx - u dv/dx)/v²。用微分求驻点，并通过二阶导数 d²y/dx² 判断其性质。
6. Integration | 积分
tegrals give the signed area between a curve and the x-axis, so areas below the axis must be treated carefully.
积分是微分的逆运算。基本结果是 ∫xⁿ dx = xⁿ⁺¹/(n + 1) + C，其中 n ≠ -1。定积分给出曲线与 x 轴之间的有向面积，因此 x 轴下方的区域需要谨慎处理。
rapezium rule can approximate a definite integral, but remember it gives an estimate, not the exact value.
还需要掌握换元积分法和分部积分法。分部积分法公式为 ∫u dv = uv - ∫v du。梯形法则可以近似计算定积分，但要记住它给出的是估计值而非精确值。
7. Sequences and Series | 数列与级数
a common ratio r: uₙ = arⁿ⁻¹, and the sum to n terms is Sₙ = a(1 - rⁿ)/(1 - r) for r ≠ 1.
等差数列有公差 d：uₙ = a + (n - 1)d，求和公式为 Sₙ = n/2 (2a + (n - 1)d)。等比数列有公比 r：uₙ = arⁿ⁻¹，前 n 项和为 Sₙ = a(1 - rⁿ)/(1 - r)，其中 r ≠ 1。
The binomial expansion for (1 + x)ⁿ is valid for |x| < 1 when n is not a positive integer. Use the sigma notation Σ to write series compactly and to identify general terms in exam questions.
当 n 不是正整数时，(1 + x)ⁿ 的二项展开在 |x| < 1 时成立。使用求和符号 Σ 可以简洁地表示级数，并在考试题中识别通项公式。
8. Statistics: Data and Probability | 统计：数据与概率
In statistics, calculate the mean, median, mode, range, interquartile range and standard deviation. A box plot gives a visual summary of the five-number summary: minimum, Q₁, median, Q₃ and maximum.
在统计中，要计算平均数、中位数、众数、极差、四分位距和标准差。箱线图可以直观地展示五数概括：最小值、Q₁、中位数、Q₃ 和最大值。
Probability uses the formula P(A|B) = P(A ∩ B)/P(B). Events A and B are independent if P(A ∩ ). Venn diagrams and tree diagrams are common tools for multi-stage probability questions.
概率中会用到公式 P(A|B) = P(A ∩ B)/P(B)。若 P(A ∩ B) = P(A)P(B)，则事件 A 与 B 相互独立。维恩图和树状图是解决多阶段概率问题的常用工具。
9. Mechanics: Kinematics and Forces | 力学：运动学与力
and s = ½(u + v)t. Always state the positive direction clearly before substituting values.
运动学描述匀加速运动。SUVAT 方程为 v = u + at、s = ut + ½at²、v² = u² + 2as 和 s = ½(u + v)t。代入数值前务必明确正方向。
Newton' s modelled as F ≤ μR, where R is the normal reaction and μ is the coefficient of friction.
牛顿第二定律为 F = ma。画出受力图，并将力沿斜面平行和垂直方向分解。摩擦力模型为 F ≤ μR，其中 R 是法向反作用力，μ 是摩擦系数。
10. Exam Technique and Support Pack Use | 考试技巧与支持包使用
Use this support pack as a checklist. After studying each section, attempt past-paper questions under timed conditions. Mark your answers against the official mark scheme and note the command words such as "show", "prove" and "hence".
请将本支持包作为检查清单使用。学完每一节后，在限时条件下尝试历年真题。对照官方评分标准批改答案，并注意 "show"、"prove" 和 "hence" 等指令词。
In the exam, show all working, give answers to 3 significant figures unless stated otherwise, and check units in mechanics. Spend the last few minutes reviewing your answers for arithmetic errors and missing solutions.
考试时要展示完整步骤，除非题目另有说明，答案保留 3 位有效数字，并检查力学题中的单位。最后留出几分钟检查计算错误和遗漏的解。
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