Pre-U Cambridge Mathematics: Quick Reference Handbook of Formulas and Theorems | Pre-U Cambridge 数学：公式定理速查手册

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

This quick reference handbook summarises the key formulas and theorems from the Cambridge Pre-U Mathematics syllabus, covering Pure Mathematics, Mechanics, and Probability & Statistics. It is designed to help students quickly locate essential results for revision and problem-solving.

这本速查手册总结了剑桥 Pre-U 数学大纲中的关键公式与定理，涵盖纯数学、力学和概率统计，旨在帮助学生快速定位复习与解题所需的重要结论。

1. Algebra and Functions | 代数与函数

This section covers quadratic equations, polynomial theorems, absolute value inequalities, and function manipulations that form the backbone of Pre-U algebraic work.

本节包括二次方程、多项式定理、绝对值不等式及函数变换等 Pre-U 代数核心内容。

Quadratic formula: For ax² + bx + c = 0, x = [−b ± √(b² − 4ac)] / (2a).

二次求根公式： 对于 ax² + bx + c = 0，x = [−b ± √(b² − 4ac)] / (2a)。

Discriminant: Δ = b² − 4ac determines nature of roots: Δ > 0 ⇒ two distinct real roots; Δ = 0 ⇒ one repeated real root; Δ < 0 ⇒ two complex conjugate roots.

判别式： Δ = b² − 4ac 决定根的性质：Δ > 0 ⇒ 两个不等实根；Δ = 0 ⇒ 一个重实根；Δ < 0 ⇒ 一对共轭复根。

Remainder Theorem: When polynomial f(x) is divided by (x − a), remainder = f(a). Factor Theorem: (x − a) is a factor of f(x) ⇔ f(a) = 0.

余数定理： 多项式 f(x) 除以 (x − a) 的余数为 f(a)。因式定理： (x − a) 为 f(x) 的因式 ⇔ f(a) = 0。

Absolute value inequalities: |f(x)| < k ⇔ −k < f(x) < k; |f(x)| > k ⇔ f(x) < −k or f(x) > k.

绝对值不等式： |f(x)| < k ⇔ −k < f(x) < k；|f(x)| > k ⇔ f(x) < −k 或 f(x) > k。

Completing the square: ax² + bx + c = a[(x + b/(2a))² − (b² − 4ac)/(4a²)].

配方法： ax² + bx + c = a[(x + b/(2a))² − (b² − 4ac)/(4a²)]。

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2. Trigonometry | 三角学

Essential trigonometric relationships, identities, and geometric applications are collected here, covering radian measure, triangle solutions, and compound angle formulas.

这里汇集了基本的三角函数关系、恒等式与几何应用，涵盖了弧度制、三角形求解及和差角公式。

Radian measure: π rad = 180°, arc length = rθ, sector area = ½ r²θ.

弧度制： π 弧度 = 180°，弧长 = rθ，扇形面积 = ½ r²θ。

Fundamental identities: sin²θ + cos²θ = 1; tanθ = sinθ/cosθ; 1 + tan²θ = sec²θ; 1 + cot²θ = cosec²θ.

基本恒等式： sin²θ + cos²θ = 1；tanθ = sinθ/cosθ；1 + tan²θ = sec²θ；1 + cot²θ = cosec²θ。

Compound angle formulas: sin(A ± B) = sinA cosB ± cosA sinB; cos(A ± B) = cosA cosB ∓ sinA sinB; tan(A ± B) = (tanA ± tanB)/(1 ∓ tanA tanB).

和差角公式： sin(A ± B) = sinA cosB ± cosA sinB；cos(A ± B) = cosA cosB ∓ sinA sinB；tan(A ± B) = (tanA ± tanB)/(1 ∓ tanA tanB)。

Double angle: sin 2θ = 2 sinθ cosθ; cos 2θ = cos²θ − sin²θ = 2 cos²θ − 1 = 1 − 2 sin²θ.

倍角公式： sin 2θ = 2 sinθ cosθ；cos 2θ = cos²θ − sin²θ = 2 cos²θ − 1 = 1 − 2 sin²θ。

Half-angle: sin²θ = ½(1 − cos 2θ); cos²θ = ½(1 + cos 2θ).

半角公式： sin²θ = ½(1 − cos 2θ)；cos²θ = ½(1 + cos 2θ)。

Sine & Cosine rules: a/sinA = b/sinB = c/sinC = 2R; a² = b² + c² − 2bc cosA. Area = ½ ab sinC.

正弦与余弦定理： a/sinA = b/sinB = c/sinC = 2R；a² = b² + c² − 2bc cosA。面积 = ½ ab sinC。

The R-form: a sinθ + b cosθ = R sin(θ + α) or R cos(θ − α) with R = √(a² + b²), tanα = b/a.

辅助角公式： a sinθ + b cosθ = R sin(θ + α) 或 R cos(θ − α)，其中 R = √(a² + b²)，tanα = b/a。

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3. Exponentials and Logarithms | 指数与对数

Properties of exponential and logarithmic functions and their derivatives and integrals are fundamental for modelling growth, decay, and solving transcendental equations.

指数函数与对数函数的性质及其导数、积分是建模增长、衰变以及求解超越方程的基础。

Exponential rules: aˣ aʸ = aˣ⁺ʸ; aˣ / aʸ = aˣ⁻ʸ; (aˣ)ʸ = aˣʸ; a⁻ˣ = 1/aˣ; a⁰ = 1; a¹ = a.

指数运算法则： aˣ aʸ = aˣ⁺ʸ；aˣ / aʸ = aˣ⁻ʸ；(aˣ)ʸ = aˣʸ；a⁻ˣ = 1/aˣ；a⁰ = 1；a¹ = a。

Logarithm rules: logₐ(MN) = logₐM + logₐN; logₐ(M/N) = logₐM − logₐN; logₐ(Mⁿ) = n logₐM; logₐ1 = 0; logₐa = 1.

对数运算法则： logₐ(MN) = logₐM + logₐN；logₐ(M/N) = logₐM − logₐN；logₐ(Mⁿ) = n logₐM；logₐ1 = 0；logₐa = 1。

Change of base: logₐx = log_b x / log_b a. In particular, logₐx = ln x / ln a.

换底公式： logₐx = log_b x / log_b a，特别是 logₐx = ln x / ln a。

The natural exponential: eˣ is its own derivative and integral; d/dx (eˣ) = eˣ, ∫ eˣ dx = eˣ + C.

自然指数函数： eˣ 的导数与积分均为其自身；d/dx (eˣ) = eˣ，∫ eˣ dx = eˣ + C。

Derivatives of logs: d/dx (ln x) = 1/x; d/dx (logₐx) = 1/(x ln a). ∫ (1/x) dx = ln |x| + C.

对数导数： d/dx (ln x) = 1/x；d/dx (logₐx) = 1/(x ln a)。∫ (1/x) dx = ln |x| + C。

Exponential and log equations: aˣ = b ⇔ x = logₐb. ln eˣ = x, eˡⁿ ˣ = x (x > 0).

指数与对数方程： aˣ = b ⇔ x = logₐb。ln eˣ = x，eˡⁿ ˣ = x（x > 0）。

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4. Calculus – Differentiation and Integration | 微积分- 微分与积分

This section presents standard derivatives, rules of differentiation, integration techniques, and applications to areas, volumes, and parametric equations.

本节列出标准导数、微分法则、积分技巧以及面积、体积和参数方程的应用。

f(x)	f'(x)	∫ f(x) dx
xⁿ (n ≠ −1)	n xⁿ⁻¹	xⁿ⁺¹/(n+1) + C
eˣ	eˣ	eˣ + C
aˣ	aˣ ln a	aˣ / ln a + C
ln x	1/x	x ln x − x + C
sin x	cos x	−cos x + C
cos x	−sin x	sin x + C
tan x	sec² x	ln |sec x| + C
sec x	sec x tan x	ln |sec x + tan x| + C
cosec x	−cosec x cot x	−ln |cosec x + cot x| + C
cot x	−cosec² x	ln |sin x| + C
arcsin x	1/√(1 − x²)	x arcsin x + √(1 − x²) + C
arccos x	−1/√(1 − x²)	x arccos x − √(1 − x²) + C
arctan x	1/(1 + x²)	x arctan x − ½ ln(1 + x²) + C

Differentiation rules: Product rule: (uv)’ = u’v + uv’; Quotient rule: (u/v)’ = (u’v − uv’)/v²; Chain rule: dy/dx = dy/du · du/dx.

微分法则： 乘积法则：(uv)’ = u’v + uv’；商法则：(u/v)’ = (u’v − uv’)/v²；链式法则：dy/dx = dy/du · du/dx。

Integration methods: Substitution: ∫ f(g(x)) g'(x) dx = ∫ f(u) du; Integration by parts: ∫ u dv = uv − ∫ v du.

积分方法： 换元法：∫ f(g(x)) g'(x) dx = ∫ f(u) du；分部积分法：∫ u dv = uv − ∫ v du。

Area and volume: Area between curves = ∫ₐᵇ |f(x) − g(x)| dx. Volume of revolution about x-axis: V = π ∫ₐᵇ y² dx.

面积与体积： 两曲线间面积 = ∫ₐᵇ |f(x) − g(x)| dx。绕 x 轴旋转体积：V = π ∫ₐᵇ y² dx。

Parametric differentiation: dy/dx = (dy/dt) / (dx/dt). Second derivative: d²y/dx² = d/dt(dy/dx) / (dx/dt).

参数方程求导： dy/dx = (dy/dt) / (dx/dt)。二阶导数：d²y/dx² = d/dt(dy/dx) / (dx/dt)。

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5. Vectors | 向量

Key results for vector algebra, dot and cross products, lines and planes, and distances are essential for both pure and applied work in Mechanics.

向量代数、点积与叉积、直线与平面以及距离的关键结果对于纯数学及力学应用都不可或缺。

Vector forms: Position vector r = x i + y j + z k. Magnitude |r| = √(x² + y² + z²). Unit vector = r / |r|.

向量表示： 位置向量 r = x i + y j + z k。模 |r| = √(x² + y² + z²)。单位向量 = r / |r|。

Dot product: a · b = |a||b| cosθ = a₁b₁ + a₂b₂ + a₃b₃. a · a = |a|². Vectors perpendicular if a · b = 0.

点积： a · b = |a||b| cosθ = a₁b₁ + a₂b₂ + a₃b₃。a · a = |a|²。垂直条件：a · b = 0。

Cross product (3D): a × b = |a||b| sinθ n̂, with determinant formula. Area of parallelogram = |a × b|.

叉积（三维）： a × b = |a||b| sinθ n̂，可由行列式计算。平行四边形面积 = |a × b|。

Line equations: Vector form r = a + t b, where a is a point on line, b is direction. Cartesian: (x − x₀)/b₁ = (y − y₀)/b₂ = (z − z₀)/b₃.

直线方程： 向量式 r = a + t b，a 为直线上一点，b 为方向向量。笛卡儿式：(x − x₀)/b₁ = (y − y₀)/b₂ = (z − z₀)/b₃。

Plane equations: Vector: r · n = a · n, where n is normal. Cartesian: n₁x + n₂y + n₃z = d.

平面方程： 向量式：r · n = a · n，n 为法向量。笛卡儿式：n₁x + n₂y + n₃z = d。

Distances: Point to plane = |ax₁ + by₁ + cz₁ − d| / √(a² + b² + c²). Point to line = |(b × (a − p))| / |b|.

距离： 点到平面 = |ax₁ + by₁ + cz₁ − d| / √(a² + b² + c²)。点到直线 = |(b × (a − p))| / |b|。

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6. Complex Numbers | 复数

Complex numbers extend the real number system and are central to solving polynomial equations, representing rotations, and performing algebraic operations in polar form.

复数扩展了实数系，是求解多项式方程、表示旋转以及进行极形式代数运算的核心工具。

Definition: z = a + bi, i² = −1. Real part a, imaginary part b. Complex conjugate: z̄ = a − bi.

定义： z = a + bi，i² = −1。实部 a，虚部 b。共轭复数：z̄ = a − bi。

Modulus: |z| = √(a² + b²). Argument: arg(z) = θ, where tanθ = b/a, adjusted for quadrant.

模： |z| = √(a² + b²)。辐角：arg(z) = θ，其中 tanθ = b/a，需按象限调整。

Polar form: z = r(cosθ + i sinθ) = r cis θ. Euler’s formula: eⁱθ = cosθ + i sinθ, so z = r eⁱθ.

极形式： z = r(cosθ + i sinθ) = r cis θ。欧拉公式：eⁱθ = cosθ + i sinθ，故 z = r eⁱθ。

Multiplication and division: z₁ z₂ = r₁ r₂ eⁱ(θ₁+θ₂); z₁/z₂ = (r₁/r₂) eⁱ(θ₁−θ₂).

乘法与除法： z₁ z₂ = r₁ r₂ eⁱ(θ₁+θ₂)；z₁/z₂ = (r₁/r₂) eⁱ(θ₁−θ₂)。

De Moivre’s theorem: (cosθ + i sinθ

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