Pre-U AQA Mathematics: Formulas & Theorems Quick Reference Handbook | Pre-U AQA 数学：公式定理速查手册

📚 Pre-U AQA Mathematics: Formulas & Theorems Quick Reference Handbook | Pre-U AQA 数学：公式定理速查手册

This quick reference handbook is designed for students taking the AQA Pre-U Mathematics course. It compiles the essential formulas and theorems you will encounter across pure mathematics, statistics, and mechanics, presented in a clear bilingual format for rapid revision. Each entry is translated into Chinese to aid understanding and memorisation. Use this guide as your last-day refresher or as a pocket companion while tackling past papers.

本速查手册专为学习 AQA Pre-U 数学课程的学生设计，汇集了纯数学、统计和力学中必须掌握的关键公式与定理，以清晰的双语形式呈现，便于快速复习。每个条目均配有中文翻译，助力理解与记忆。可将本手册作为考前冲刺伴侣或刷历年真题时的随身参考。

1. Algebra and Functions | 代数与函数

The quadratic formula gives the solutions of ax² + bx + c = 0, where a ≠ 0, as x = (-b ± √(b² – 4ac)) / (2a).

二次公式给出方程 ax² + bx + c = 0（a ≠ 0）的解：x = (-b ± √(b² – 4ac)) / (2a)。

The discriminant Δ = b² – 4ac determines the nature of the roots: two distinct real roots if Δ > 0, a repeated real root if Δ = 0, and two complex conjugate roots if Δ < 0.

判别式 Δ = b² – 4ac 决定了根的性质：Δ > 0 时有两个不等实根，Δ = 0 时有重实根，Δ < 0 时有一对共轭复根。

Laws of indices: aᵐ × aⁿ = aᵐ⁺ⁿ, aᵐ ÷ aⁿ = aᵐ⁻ⁿ, (aᵐ)ⁿ = aᵐⁿ, a⁰ = 1 (a ≠ 0), a⁻ⁿ = 1/aⁿ.

指数运算法则：aᵐ × aⁿ = aᵐ⁺ⁿ，aᵐ ÷ aⁿ = aᵐ⁻ⁿ，(aᵐ)ⁿ = aᵐⁿ，a⁰ = 1（a ≠ 0），a⁻ⁿ = 1/aⁿ。

The logarithm logₐ x is the power to which a must be raised to give x. Key rules: logₐ (xy) = logₐ x + logₐ y, logₐ (x/y) = logₐ x – logₐ y, logₐ (xⁿ) = n logₐ x, and the change-of-base rule logₐ x = log_b x / log_b a.

对数 logₐ x 表示以 a 为底 x 的对数。重要规则：logₐ (xy) = logₐ x + logₐ y，logₐ (x/y) = logₐ x – logₐ y，logₐ (xⁿ) = n logₐ x，以及换底公式 logₐ x = log_b x / log_b a。

The binomial expansion for (1 + x)ⁿ, where n is rational and |x| < 1, is: (1 + x)ⁿ = 1 + n x + [n(n–1)/2!] x² + [n(n–1)(n–2)/3!] x³ + … . For expanding (a + b)ⁿ use (a + b)ⁿ = aⁿ (1 + b/a)ⁿ.

当 n 为有理数且 |x| < 1 时，(1 + x)ⁿ 的二项展开式为：(1 + x)ⁿ = 1 + n x + [n(n–1)/2!] x² + [n(n–1)(n–2)/3!] x³ + … 。展开 (a + b)ⁿ 时可写为 (a + b)ⁿ = aⁿ (1 + b/a)ⁿ。

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2. Trigonometry | 三角学

Fundamental identity: sin²θ + cos²θ = 1. Derived relationships: tan θ = sin θ / cos θ, and 1 + tan²θ = sec²θ, 1 + cot²θ = cosec²θ.

基本恒等式：sin²θ + cos²θ = 1。导出关系：tan θ = sin θ / cos θ，以及 1 + tan²θ = sec²θ，1 + cot²θ = cosec²θ。

Compound angle formulae: sin(A ± B) = sin A cos B ± cos A sin B, cos(A ± B) = cos A cos B ∓ sin A sin B, tan(A ± B) = (tan A ± tan B) / (1 ∓ tan A tan B).

和差角公式：sin(A ± B) = sin A cos B ± cos A sin B，cos(A ± B) = cos A cos B ∓ sin A sin B，tan(A ± B) = (tan A ± tan B) / (1 ∓ tan A tan B)。

Double angle formulae: sin 2A = 2 sin A cos A, cos 2A = cos²A – sin²A = 2 cos²A – 1 = 1 – 2 sin²A, tan 2A = 2 tan A / (1 – tan²A).

倍角公式：sin 2A = 2 sin A cos A，cos 2A = cos²A – sin²A = 2 cos²A – 1 = 1 – 2 sin²A，tan 2A = 2 tan A / (1 – tan²A)。

Sine rule: a / sin A = b / sin B = c / sin C. Cosine rule: a² = b² + c² – 2bc cos A. Area of triangle: Area = ½ ab sin C.

正弦定理：a / sin A = b / sin B = c / sin C。余弦定理：a² = b² + c² – 2bc cos A。三角形面积公式：面积 = ½ ab sin C。

Radian measure: π rad = 180°. Arc length s = rθ, sector area = ½ r²θ, where θ is in radians.

弧度制：π 弧度 = 180°。弧长 s = rθ，扇形面积 = ½ r²θ，其中 θ 以弧度为单位。

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3. Calculus | 微积分

Power rule for differentiation: d/dx (xⁿ) = n xⁿ⁻¹. Constant rule: d/dx (c) = 0. Sum rule: d/dx (u + v) = u’ + v’.

幂函数求导法则：d/dx (xⁿ) = n xⁿ⁻¹。常数法则：d/dx (c) = 0。和法则：d/dx (u + v) = u’ + v’。

Product rule: d/dx (uv) = u’v + uv’. Quotient rule: d/dx (u/v) = (u’v – uv’) / v².

乘法法则（乘积法则）：d/dx (uv) = u’v + uv’。除法法则（商法则）：d/dx (u/v) = (u’v – uv’) / v²。

Chain rule: if y = f(u) and u = g(x), then dy/dx = (dy/du) × (du/dx).

链式法则：若 y = f(u) 且 u = g(x)，则 dy/dx = (dy/du) × (du/dx)。

Basic integration: ∫ 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, ∫ cos x dx = sin x + C.

基本积分公式：∫ 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，∫ cos x dx = sin x + C。

The Fundamental Theorem of Calculus connects differentiation and integration: if F'(x) = f(x), then ∫ₐᵇ f(x) dx = F(b) – F(a).

微积分基本定理将微分与积分联系起来：若 F'(x) = f(x)，则 ∫ₐᵇ f(x) dx = F(b) – F(a)。

Volume of revolution about the x-axis: V = π ∫ₐᵇ y² dx. About the y-axis: V = π ∫ₐᵇ x² dy.

绕 x 轴旋转体体积：V = π ∫ₐᵇ y² dx。绕 y 轴旋转体体积：V = π ∫ₐᵇ x² dy。

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4. Vectors and Matrices | 向量与矩阵

Vector magnitude: for a = (x, y, z), |a| = √(x² + y² + z²). Unit vector in direction of a: â = a / |a|.

向量模长：对于 a = (x, y, z)，|a| = √(x² + y² + z²)。沿 a 方向的单位向量：â = a / |a|。

Dot (scalar) product: a · b = |a||b| cos θ, and in component form a · b = x₁x₂ + y₁y₂ + z₁z₂. Two vectors are perpendicular if a · b = 0.

点积（标量积）：a · b = |a||b| cos θ，在分量形式下 a · b = x₁x₂ + y₁y₂ + z₁z₂。若 a · b = 0，则两向量垂直。

Matrix multiplication: if A is m×n and B is n×p, the element (i,j) of AB is Σₖ aᵢₖ bₖⱼ. Matrix multiplication is not commutative.

矩阵乘法：若 A 为 m×n 矩阵，B 为 n×p 矩阵，则 AB 的第 (i,j) 元为 Σₖ aᵢₖ bₖⱼ。矩阵乘法不满足交换律。

For a 2×2 matrix A = [[a, b], [c, d]], the determinant is det A = ad – bc. The inverse is A⁻¹ = (1/(ad – bc)) [[d, –b], [–c, a]], provided det A ≠ 0.

对于 2×2 矩阵 A = [[a, b], [c, d]]，行列式为 det A = ad – bc。逆矩阵为 A⁻¹ = (1/(ad – bc)) [[d, –b], [–c, a]]，要求 det A ≠ 0。

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5. Complex Numbers | 复数

A complex number is written as z = x + iy, where i² = –1. The modulus is |z| = √(x² + y²), and the argument θ satisfies tan θ = y/x.

复数写作 z = x + iy，其中 i² = –1。模为 |z| = √(x² + y²)，辐角 θ 满足 tan θ = y/x。

Polar form: z = r (cos θ + i sin θ), often abbreviated as r cis θ. The exponential form is z = r eⁱᶿ.

极坐标形式：z = r (cos θ + i sin θ)，常缩写为 r cis θ。指数形式为 z = r eⁱᶿ。

De Moivre’s theorem: (cos θ + i sin θ)ⁿ = cos(nθ) + i sin(nθ) for integer n. It is used to find powers and roots of complex numbers.

棣莫弗定理：对于整数 n，(cos θ + i sin θ)ⁿ = cos(nθ) + i sin(nθ)。该定理用于求复数的乘方与方根。

Euler’s formula: eⁱᶿ = cos θ + i sin θ. This unifies exponential and trigonometric forms.

欧拉公式：eⁱᶿ = cos θ + i sin θ，它将指数形式与三角形式统一起来。

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6. Sequences and Series | 数列与级数

Arithmetic progression: nth term uₙ = a + (n–1)d, sum Sₙ = n/2 [2a + (n–1)d] = n/2 (a + l), where l is the last term.

等差数列：第 n 项 uₙ = a + (n–1)d，和 Sₙ = n/2 [2a + (n–1)d] = n/2 (a + l)，其中 l 为末项。

Geometric progression: nth term uₙ = a rⁿ⁻¹, sum Sₙ = a(1 – rⁿ)/(1 – r) for r ≠ 1. Sum to infinity S∞ = a/(1 – r) provided |r| < 1.

等比数列：第 n 项 uₙ = a rⁿ⁻¹，和 Sₙ = a(1 – rⁿ)/(1 – r)（r ≠ 1）。无穷项和 S∞ = a/(1 – r)，前提是 |r| < 1。

The Maclaurin series expansion of f(x) is f(0) + f'(0) x + f”(0)/2! x² + f”'(0)/3! x³ + … . Common series: eˣ = 1 + x + x²/2! + x³/3! + …, sin x = x – x³/3! + x⁵/5! – …, cos x = 1 – x²/2! + x⁴/4! – ….

麦克劳林级数展开：f(x) = f(0) + f'(0) x + f”(0)/2! x² + f”'(0)/3! x³ + … 。常用展开式：eˣ = 1 + x + x²/2! + x³/3! + …，sin x = x – x³/3! + x⁵/5! – …，cos x = 1 – x²/2! + x⁴/4! – …。

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7. Probability and Statistics | 概率与统计

Addition rule: P(A ∪ B) = P(A) + P(B) – P(A ∩ B). For mutually exclusive events, P(A ∩ B) = 0. Multiplication rule for independent events: P(A ∩ B) = P(A)P(B). Conditional probability: P(A|B) = P(A ∩ B) / P(B).

加法公式：P(A ∪ B) = P(A) + P(B) – P(A ∩ B)。对于互斥事件，P(A ∩ B) = 0。独立事件的乘法公式：P(A ∩ B) = P(A)P(B)。条件概率：P(A|B) = P(A ∩ B) / P(B)。

For a discrete random variable X, expectation E(X) = Σ x p(x). Variance Var(X) = E(X²) – [E(X)]² = Σ (x – μ)² p(x). Standard deviation σ = √Var(X).

对于离散随机变量 X，期望 E(X) = Σ x p(x)。方差 Var(X) = E(X²) – [E(X)]² = Σ (x – μ)² p(x)。标准差 σ = √Var(X)。

Binomial distribution X ~ B(n, p): P(X = r) = C(n,r) pʳ qⁿ⁻ʳ, where q = 1 – p. Mean μ = np, variance σ² = npq.

二项分布 X ~ B(n, p)：P(X = r) = C(n,r) pʳ qⁿ⁻ʳ，其中 q = 1 – p。均值 μ = np，方差 σ² = npq。

The normal distribution X ~ N(μ, σ²) has bell-shaped pdf. Standardisation: Z = (X – μ)/σ ~ N(0,1). Use the standard normal table for probabilities.

正态分布 X ~ N(μ, σ²) 具有钟形概率密度函数。标准化：Z = (X – μ)/σ ~ N(0,1)。使用标准正态分布表查找概率。

If a random sample of size n is drawn from a normal population N(μ, σ²), the sample mean X̄ ~ N(μ, σ²/n). The central limit theorem states that for large n, X̄ is approximately normal regardless of the population distribution.

若从正态总体 N(μ, σ²) 中抽取容量为 n 的随机样本，则样本均值 X̄ ~ N(μ, σ²/n)。中心极限定理指出，当 n 很大时，无论总体分布如何，X̄ 均近似服从正态分布。

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8. Mechanics | 力学

SUVAT equations for constant acceleration in a straight line: v = u + at, s = ut + ½ at², v² = u² + 2as, s = ½ (u+v)t, s = vt – ½ at².

匀变速直线运动中的 SUVAT 方程组：v = u + at，s = ut + ½ at²，v² = u² + 2as，s = ½ (u+v)t，s = vt – ½ at²。

Newton’s second law: F = ma, where F is the resultant force, m the mass, and a the acceleration. Weight: W = mg, where g = 9.8 m s⁻².

牛顿第二定律：F = ma，其中 F 为合力，m 为质量，a 为加速度。重力：W = mg，g = 9.8 m s⁻²。

Momentum: p = mv. The impulse-momentum principle: Ft = mv – mu. Law of conservation: total momentum before collision = total momentum after, provided no external force acts.

动量：p = mv。冲量-动量原理：Ft = mv – mu。动量守恒定律：若系统不受外力，碰撞前总动量等于碰撞后总动量。

Work done by a constant force: W = Fs cos θ, where θ is the angle between force and displacement. Kinetic energy: KE = ½ mv². Gravitational potential energy: GPE = mgh. Power = work / time = Fv.

恒力做功：W = Fs cos θ，其中 θ 为力与位移的夹角。动能：KE = ½ mv²。重力势能：GPE = mgh。功率 = 功 / 时间 = Fv。

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