📚 Year 13 AQA Mathematics: Core Knowledge Review | Year 13 AQA 数学：核心知识点梳理
This article reviews the essential topics in Year 13 AQA Mathematics, covering pure mathematics, statistics, and mechanics. Each section highlights key definitions, methods, and common pitfalls to help you consolidate your understanding for exam success.
本文梳理 Year 13 AQA 数学的核心内容，涵盖纯数学、统计与力学。每节突出关键定义、方法和常见错误，助你巩固理解，冲刺高分。
1. Proof | 证明
Year 13 requires you to apply formal proof techniques: deduction, exhaustion, and contradiction. Disproof by counterexample is just as important as proving a statement true. For instance, to prove that √2 is irrational, we use contradiction: assume √2 = p/q where p and q are coprime integers. Squaring gives p² = 2q², so p is even, which forces q to be even – a contradiction.
13 年级要求你运用正式的证明方法：演绎、穷举与反证法。使用反例进行反驳同样重要。例如，要证明 √2 是无理数，可采用反证法：假设 √2 = p/q（p、q 互质）。平方得 p² = 2q²，可见 p 为偶数，进而推出 q 也为偶数，矛盾。
Proof by exhaustion demands checking every possible case; deduction relies on logical steps from axioms. In contradiction, assume the negation and derive an impossibility. Be careful: incomplete cases or hidden assumptions can destroy a proof.
穷举证明必须检视所有情况；演绎则从公理出发进行逻辑推理。反证法中，先假设命题的否定，再导出不可能的结果。务必小心：情况不全或隐含假设都会毁掉证明。
2. Algebra and Functions | 代数与函数
Composite functions gf(x) = g(f(x)) and inverse functions f⁻¹(x) need careful domain and range analysis. Modulus functions |f(x)| create equations like |ax + b| = cx + d, often requiring squaring or piecewise consideration. Inequalities such as |x − 3| > 2 lead to x < 1 or x > 5.
复合函数 gf(x)=g(f(x)) 与反函数 f⁻¹(x) 要求仔细分析定义域和值域。模函数 |f(x)| 构造出类似 |ax+b|=cx+d 的方程，通常需要两边平方或分段讨论。不等式 |x−3|>2 的解为 x<1 或 x>5。
Graph transformations: y = f(x) → y = a f(b(x + c)) + d. The correct sequence is horizontal shift (x+c), then horizontal stretch (1/b), then reflection in y-axis if b<0, then vertical stretch (a), vertical reflection if a<0, and finally vertical translation +d. Rational functions are simplified into proper fractions and resolved into partial fractions of the forms A/(x+p), B/(x+p)², or (Cx+D)/(x²+q).
图像变换：y=f(x) → y=a f(b(x+c))+d。正确顺序为水平平移 (x+c)，然后水平拉伸 (1/b)，若 b<0 则关于 y 轴反射，接着垂直拉伸 (a)，若 a<0 垂直反射，最后垂直平移 +d。有理函数化简后分解为部分分式，形如 A/(x+p)、B/(x+p)² 或 (Cx+D)/(x²+q)。
3. Coordinate Geometry | 坐标几何
Parametric equations x = f(t), y = g(t) define curves. You must be able to convert to Cartesian form by eliminating t. For tangents and normals, find dy/dx via parametric differentiation: dy/dx = (dy/dt) ÷ (dx/dt). The equation of a tangent at (x₁, y₁) is y − y₁ = m(x − x₁).
参数方程 x=f(t), y=g(t) 描绘曲线。你必须能通过消去 t 将其化为笛卡尔形式。运用参数求导法则 dy/dx = (dy/dt) ÷ (dx/dt) 以求得切线与法线。切线方程在点 (x₁, y₁) 为 y−y₁ = m(x−x₁)。
Circles: (x − a)² + (y − b)² = r². A line can intersect a circle; use discriminant Δ = b² − 4ac to determine tangency, intersection, or no intersection. The perpendicular from the centre to a chord bisects it. Parametric form of a circle is x = a + r cos θ, y = b + r sin θ.
圆：(x−a)² + (y−b)² = r²。直线与圆相交，可利用判别式 Δ=b²−4ac 判断相切、相交或相离。圆心到弦的垂线平分该弦。圆的参数形式为 x=a+r cos θ, y=b+r sin θ。
4. Sequences and Series | 数列与级数
Arithmetic sequences have common difference d; sum of first n terms: Sₙ = n/2 [2a + (n−1)d]. Geometric sequences have common ratio r; sum Sₙ = a(1−rⁿ)/(1−r) for r≠1. The sum to infinity exists when |r|<1: S∞ = a/(1−r).
等差数列公差为 d，前 n 项和 Sₙ = n/2 [2a+(n−1)d]。等比数列公比为 r，和 Sₙ = a(1−rⁿ)/(1−r)（r≠1）。当 |r|<1 时存在无穷和 S∞ = a/(1−r)。
Binomial expansion: (1+x)ⁿ = 1 + nx + n(n−1)x²/2! + … for rational n, valid |x|<1. You can approximate by truncating the series. Sigma notation ∑ r, ∑ r², ∑ r³ are often used alongside term-by-term summation. Remember ∑ r² = n(n+1)(2n+1)/6, ∑ r³ = n²(n+1)²/4.
二项展开式：(1+x)ⁿ = 1 + nx + n(n−1)x²/2! + …，n 为有理数，有效范围 |x|<1。可通过截断级数进行近似。Sigma 符号 ∑ r, ∑ r², ∑ r³ 常与逐项求和同用。记住 ∑ r² = n(n+1)(2n+1)/6，∑ r³ = n²(n+1)²/4。
5. Trigonometry | 三角学
Radian measure: π radians = 180°, arc length s = rθ, sector area A = ½r²θ. Reciprocal functions sec θ = 1/cos θ, cosec θ = 1/sin θ, cot θ = 1/tan θ. Their graphs have vertical asymptotes. Key identities: 1 + tan²θ ≡ sec²θ, 1 + cot²θ ≡ cosec²θ.
弧度制：π 弧度 = 180°，弧长 s = rθ，扇形面积 A = ½r²θ。倒数函数 sec θ=1/cos θ、cosec θ=1/sin θ、cot θ=1/tan θ，其图像有垂直渐近线。关键恒等式：1+tan²θ≡sec²θ，1+cot²θ≡cosec²θ。
Addition formulas: sin(A±B) = sinA cosB ± cosA sinB, cos(A±B) = cosA cosB ∓ sinA sinB. Double angles: sin 2θ = 2 sinθ cosθ, cos 2θ = cos²θ − sin²θ. The harmonic form a cosθ + b sinθ ≡ R cos(θ ∓ α) or R sin(θ ± α) where R = √(a²+b²) and tan α = b/a. Small angle approximations: sin θ ≈ θ, cos θ ≈ 1 − θ²/2, tan θ ≈ θ for θ in radians.
和角公式：sin(A±B)=sinA cosB ± cosA sinB，cos(A±B)=cosA cosB ∓ sinA sinB。倍角：sin2θ=2 sinθ cosθ，cos2θ=cos²θ−sin²θ。调解形式 a cosθ+b sinθ≡R cos(θ∓α) 或 R sin(θ±α)，其中 R=√(a²+b²)，tan α=b/a。小角近似（弧度制）：sin θ≈θ，cos θ≈1−θ²/2，tan θ≈θ。
6. Exponentials and Logarithms | 指数与对数
The natural exponential eˣ has derivative eˣ and integral eˣ + c. The natural logarithm ln x is defined for x > 0, with derivative 1/x and integral x ln x − x + C. Laws of logs: ln(ab) = ln a + ln b, ln(a/b) = ln a − ln b, ln(aᵏ) = k ln a.
自然指数函数 eˣ 的导数为 eˣ，积分为 eˣ+c。自然对数 ln x 定义域 x>0，导数为 1/x，积分为 x ln x−x+C。对数定律：ln(ab)=ln a+ln b，ln(a/b)=ln a−ln b，ln(aᵏ)=k ln a。
Equations like e²ˣ = 5 are solved by taking ln: 2x = ln 5. The function aˣ can be written as eˣ ln a. Exponential modelling: y = A eᵏˣ for growth/decay; linearise by plotting ln y against x. Always check the domain of logs when solving equations.
对于 e²ˣ=5，两边取自然对数得 2x=ln5。aˣ 可写作 eˣ ln a。指数模型 y=Aeᵏˣ 表示增长或衰减，可通过画 ln y 对 x 的图来线性化。解方程时务必验证对数的定义域。
7. Differentiation | 微分
Chain rule: if y = f(u), u = g(x), then dy/dx = (dy/du)·(du/dx). Product rule: d/dx[uv] = u’ v + u v’. Quotient rule: d/dx[u/v] = (u’ v − u v’)/v². Derivatives of standard functions: d/dx(sin x)=cos x, d/dx(cos x)=−sin x, d/dx(tan x)=sec²x.
链式法则：若 y=f(u), u=g(x)，则 dy/dx=(dy/du)·(du/dx)。乘积法则：d/dx[uv]=u’ v+u v’。商法则：d/dx[u/v]=(u’ v−u v’)/v²。标准函数导数：d/dx(sin x)=cos x，d/dx(cos x)=−sin x，d/dx(tan x)=sec²x。
Implicit differentiation: differentiate both sides of an equation w.r.t. x, treat y as a function, multiply by dy/dx. For parametric equations x=f(t), y=g(t), dy/dx = (dy/dt)/(dx/dt). Second derivative: d²y/dx² = d/dx (dy/dx); for parametric, d²y/dx² = (d/dt[dy/dx])/(dx/dt). Applications: tangents, normals, stationary points, inflection, optimisation (maxima/minima).
隐函数微分：方程两边对 x 求导，视 y 为函数，每项乘以 dy/dx。参数方程 x=f(t), y=g(t)，则 dy/dx = (dy/dt)/(dx/dt)。二阶导数 d²y/dx² = d/dx(dy/dx)；对于参数形式，d²y/dx² = (d/dt[dy/dx])/(dx/dt)。应用包括切线、法线、驻点、拐点及最优化问题。
8. Integration | 积分
Reverse of differentiation; learn standard integrals: ∫ xⁿ dx = xⁿ⁺¹/(n+1) + C (n≠−1), ∫ 1/x dx = ln|x| + C, ∫ eˣ dx = eˣ + C, ∫ sin x dx = −cos x + C. By substitution, let u = g(x), replace dx. For definite integrals change limits.
积分是微分的逆运算；记住标准积分：∫ xⁿ dx = xⁿ⁺¹/(n+1)+C (n≠−1)，∫ 1/x dx = ln|x|+C，∫ eˣ dx = eˣ+C，∫ sin x dx = −cos x+C。换元法：令 u=g(x)，替换 dx。对于定积分，同时变换上下限。
Integration by parts: ∫ u dv = uv − ∫ v du; often used for ∫ x eˣ dx, ∫ ln x dx. Also trigger partial fractions to split rational integrands. Areas between curves: ∫ₐᵇ (top – bottom) dx. Volumes of revolution about x-axis: π∫ y² dx. Solving differential equations: separate variables, 1/g(y) dy = f(x) dx, integrate both sides. Trapezium rule for approximate area: ∫ₐᵇ f(x) dx ≈ h/2[y₀ + 2(y₁+…+yₙ₋₁) + yₙ] with h = (b−a)/n.
分部积分法：∫ u dv = uv − ∫ v du；常用于 ∫ x eˣ dx，∫ ln x dx。部分分式可拆分有理分式积分。曲线间面积：∫ₐᵇ (上曲线−下曲线) dx。绕 x 轴旋转体积：π∫ y² dx。解微分方程：分离变量，1/g(y) dy = f(x) dx，两边积分。梯形法则近似积分：∫ₐᵇ f(x) dx ≈ h/2[y₀+2(y₁+…+yₙ₋₁)+yₙ]，步长 h=(b−a)/n。
9. Numerical Methods | 数值方法
To locate roots of f(x)=0: show a sign change in a table. Iteration: rearrange to x = g(x) and use xₙ₊₁ = g(xₙ); cobweb/staircase diagrams illustrate convergence. Newton-Raphson formula: xₙ₊₁ = xₙ − f(xₙ)/f'(xₙ). It converges
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