📚 AQA Math: Last-Minute Revision Notes | AQA 数学：考前冲刺笔记
Master key concepts and avoid common pitfalls with these concise, exam-focused revision notes for AQA A-level Mathematics. Covering pure, statistics and mechanics, this guide highlights the essentials you need to recall under time pressure.
掌握核心概念，避开常见陷阱，本指南为AQA A-level数学提供精炼的考前冲刺笔记。涵盖纯数、统计和力学，助你在考场上快速回忆关键要点。
1. Algebraic Techniques & Quadratic Equations | 代数技巧与二次方程
Quadratic formula: the solutions of ax²+bx+c=0 are x = (-b ± √(b²-4ac)) / (2a). Remember to check the discriminant Δ = b²-4ac for the nature of roots.
二次方程求根公式：ax²+bx+c=0 的解为 x = (-b ± √(b²-4ac)) / (2a)。记住通过判别式 Δ = b²-4ac 判断根的性质。
Completing the square: rewrite x²+bx as (x + b/2)² – (b/2)². This is vital for finding vertices of parabolas and for integration.
配方法：将 x²+bx 改写为 (x + b/2)² – (b/2)²。这对求抛物线顶点和积分至关重要。
Inequalities: when multiplying or dividing by a negative number, reverse the inequality sign. Sketch graphs to solve quadratic inequalities.
不等式：乘或除以负数时，要反转不等号。画草图可帮助解二次不等式。
2. Functions and Graphs | 函数与图像
A function must map each input to exactly one output. Use the vertical line test for graphs. Learn domain and range notation.
函数必须将每个输入映射到唯一输出。对图像使用垂直线检验。掌握定义域和值域的表示法。
Transformations: f(x)+a (vertical shift), f(x+a) (horizontal shift opposite sign), af(x) (vertical stretch), f(ax) (horizontal stretch by factor 1/a). Combined transformations: apply inside the bracket first.
图像变换：f(x)+a 纵向平移，f(x+a) 横向平移（符号相反），af(x) 纵向伸缩，f(ax) 横向伸缩（因子 1/a）。复合变换时优先处理括号内的变换。
Inverse functions: reflect in y=x. Only one-to-one functions have inverses without domain restriction. Remember f⁻¹(f(x)) = x.
反函数：关于直线 y=x 对称。只有一一映射函数在未限制定义域时才有反函数。牢记 f⁻¹(f(x)) = x。
3. Trigonometry Essentials | 三角学要点
Exact values: know sin, cos, tan of 0°, 30°, 45°, 60°, 90°. Use special triangles or the unit circle. Radian measure: 180° = π rad.
精确值：牢记 0°、30°、45°、60°、90° 的 sin、cos、tan 值。利用特殊三角形或单位圆。弧度制：180° = π 弧度。
Trig identities: sin²θ + cos²θ = 1; tanθ = sinθ/cosθ. Double-angle: sin2θ = 2sinθcosθ, cos2θ = cos²θ-sin²θ = 2cos²θ-1 = 1-2sin²θ.
三角恒等式：sin²θ + cos²θ = 1；tanθ = sinθ/cosθ。倍角公式：sin2θ = 2sinθcosθ，cos2θ = cos²θ-sin²θ = 2cos²θ-1 = 1-2sin²θ。
Solving trig equations: use the CAST diagram to determine sign in each quadrant. For sinθ = k, general solutions θ = nπ + (-1)ⁿα (radians). Always check the domain.
解三角形方程：利用 CAST 图确定各象限的符号。sinθ = k 的通解为 θ = nπ + (-1)ⁿα（弧度）。务必检查定义域。
4. Differentiation Rules and Applications | 微分法则与应用
Basic rules: d/dx (xⁿ) = nxⁿ⁻¹. Product rule: (uv)’ = u’v + uv’. Quotient rule: (u/v)’ = (u’v – uv’) / v². Chain rule: dy/dx = dy/du · du/dx.
基本法则：d/dx (xⁿ) = nxⁿ⁻¹。乘法法则：(uv)’ = u’v + uv’。除法法则：(u/v)’ = (u’v – uv’) / v²。链式法则：dy/dx = dy/du · du/dx。
Second derivative: d²y/dx² tests the nature of stationary points. f”(x) > 0 → minimum, < 0 → maximum. Also used for convexity/concavity.
二阶导数：d²y/dx² 可判断驻点性质。f”(x) > 0 为极小值，< 0 为极大值。也可判断凹凸性。
Differentiation from first principles: f'(x) = lim(h→0) [f(x+h)-f(x)]/h. Be prepared to derive basic results like the derivative of x².
从第一性原理求导：f'(x) = lim(h→0) [f(x+h)-f(x)]/h。考纲可能要求推导 x² 等基本函数的结果。
5. Integration Techniques | 积分技巧
Reverse power rule: ∫xⁿ dx = xⁿ⁺¹/(n+1) + C, n≠-1. Definite integral: ∫ₐᵇ f(x)dx = F(b)-F(a). The definite integral gives the signed area.
幂函数反求导：∫xⁿ dx = xⁿ⁺¹/(n+1) + C，n≠-1。定积分：∫ₐᵇ f(x)dx = F(b)-F(a)。定积分求出带符号的面积。
Integration by substitution: use u = g(x) and change limits for definite integrals. Common for (ax+b)ⁿ and composite trig functions.
换元积分法：设 u = g(x)，定积分需同步换限。常用于 (ax+b)ⁿ 和复合三角函数的积分。
Integration by parts: ∫u dv = uv – ∫v du. Follow the LIATE rule (Log, Inverse trig, Algebraic, Trig, Exponential) to choose u.
分部积分法：∫u dv = uv – ∫v du。按 LIATE 优先顺序（对数、反三角、代数、三角、指数）选取 u。
Area between curves: top curve minus bottom curve, integrate w.r.t. x or y. For parametric curves: ∫y dx = ∫ y(t) x'(t) dt.
两曲线间面积：上减下，对 x 或 y 积分。参数方程下：∫y dx = ∫ y(t) x'(t) dt。
6. Vectors in Pure Maths | 纯数中的向量
Vector basics: i, j, k unit vectors. Magnitude |v| = √(x²+y²+z²). Dot product: a·b = |a||b|cosθ = a₁b₁+a₂b₂+a₃b₃.
向量基础：单位向量 i, j, k。模长 |v| = √(x²+y²+z²)。点积：a·b = |a||b|cosθ = a₁b₁+a₂b₂+a₃b₃。
Angle between vectors: cosθ = (a·b)/(|a||b|). Perpendicular if a·b=0. Equation of a line: r = a + λb.
向量夹角：cosθ = (a·b)/(|a||b|)。垂直向量满足 a·b=0。直线方程：r = a + λb。
Intersection of lines: solve simultaneous parametric equations. In 3D, lines can be skew (not intersecting, not parallel).
直线交点：联立参数方程求解。在三维空间中，直线可能为异面直线（不平行也不相交）。
7. Exponentials and Logarithms | 指数与对数
Natural base e: e = lim(n→∞)(1+1/n)ⁿ ≈ 2.718. d/dx(eˣ)=eˣ, ∫eˣ dx = eˣ + C. Derivative of ln x is 1/x.
自然常数 e：e ≈ 2.718。d/dx(eˣ)=eˣ，∫eˣ dx = eˣ + C。ln x 的导数为 1/x。
Log laws: ln(ab) = ln a + ln b; ln(a/b) = ln a – ln b; ln(aᵏ) = k ln a. Exponential growth/decay: N = N₀ eᵏᵗ. Sketch transformed log and exponential graphs.
对数运算法则：ln(ab) = ln a + ln b；ln(a/b) = ln a – ln b；ln(aᵏ) = k ln a。指数增长/衰减模型：N = N₀ eᵏᵗ。能画出变形后的对数和指数图像。
8. Statistical Distributions | 统计分布
Binomial distribution X~B(n,p): P(X=r) = ⁿCᵣ pʳ (1-p)ⁿ⁻ʳ. Mean = np, variance = np(1-p). Use tables or calculator for cumulative probabilities.
二项分布 X~B(n,p)：P(X=r) = ⁿCᵣ pʳ (1-p)ⁿ⁻ʳ。均值 = np，方差 = np(1-p)。用表格或计算器求累积概率。
Normal distribution X~N(μ,σ²): standardise Z = (X-μ)/σ. Use percentage-point tables for inverse normal. Remember the 68-95-99.7 rule.
正态分布 X~N(μ,σ²)：标准化 Z = (X-μ)/σ。用百分位表求逆。掌握 68-95-99.7 经验规则。
Hypothesis testing: null H₀, alternative H₁. Significance level, p-value, critical region. For binomial, use exact test; normal approximation applies when np > 5, nq > 5 with continuity correction.
假设检验：原假设 H₀，备择假设 H₁。显著性水平、p 值、临界区域。二项分布用精确检验；当 np > 5, nq > 5 时可用正态近似并带连续性校正。
9. Mechanics: Kinematics and Forces | 力学：运动学与力
SUVAT equations (constant acceleration): v = u+at, s = ut+½at²
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