Year 13 CCEA Mathematics: Quick Reference Handbook of Formulas and Theorems | Year 13 CCEA 数学：公式定理速查手册

📚 Year 13 CCEA Mathematics: Quick Reference Handbook of Formulas and Theorems | Year 13 CCEA 数学：公式定理速查手册

This revision quick reference provides a concise collection of the most important formulas and theorems needed for Year 13 CCEA Mathematics. It covers pure maths, mechanics and statistics in a structured, easy‑to‑revise format.

本速查手册汇总了 Year 13 CCEA 数学中最核心的公式与定理，涵盖纯数学、力学与统计，结构清晰，方便考前快速查阅。

1. Algebra and the Binomial Theorem | 代数与二项式定理

The Binomial Theorem for rational n and |x| < 1 expands (1 + x)ⁿ as 1 + nx + n(n−1)x²/2! + n(n−1)(n−2)x³/3! + … . For positive integer n, (a + b)ⁿ = Σ (r=0 to n) nCr aⁿ⁻ʳ bʳ where nCr = n!/(r!(n−r)!).

对有理数 n 且 |x| < 1，二项式定理将 (1 + x)ⁿ 展开为 1 + nx + n(n−1)x²/2! + n(n−1)(n−2)x³/3! + … 。对正整数 n，(a + b)ⁿ = Σ(r=0 到 n) nCr aⁿ⁻ʳ bʳ，其中 nCr = n!/(r!(n−r)!)。

A rational function with distinct linear factors can be split into partial fractions such as (px+q)/((x−a)(x−b)) = A/(x−a) + B/(x−b). For repeated factors use A/(x−a) + B/(x−a)², and for irreducible quadratics include a linear numerator.

对于分母为不同线性因子的有理函数，可分解为部分分式，例如 (px+q)/((x−a)(x−b)) = A/(x−a) + B/(x−b)。有重因子时用 A/(x−a) + B/(x−a)²，含不可约二次因子时分子设为一次式。

The remainder theorem states that when a polynomial f(x) is divided by (x−a), the remainder is f(a); if f(a)=0 then (x−a) is a factor.

余数定理：多项式 f(x) 除以 (x−a) 的余数为 f(a)；若 f(a)=0，则 (x−a) 为因式。

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2. Trigonometry and Identities | 三角学与恒等式

Fundamental identities: sin²θ + cos²θ = 1, 1 + tan²θ = sec²θ, 1 + cot²θ = csc²θ. Compound-angle formulas: 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).

基本恒等式：sin²θ + cos²θ = 1，1 + tan²θ = sec²θ，1 + cot²θ = csc²θ。复合角公式：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 formulas: sin 2θ = 2 sin θ cos θ, cos 2θ = cos²θ − sin²θ = 2 cos²θ − 1 = 1 − 2 sin²θ, tan 2θ = 2 tan θ/(1 − tan²θ).

倍角公式：sin 2θ = 2 sin θ cos θ，cos 2θ = cos²θ − sin²θ = 2 cos²θ − 1 = 1 − 2 sin²θ，tan 2θ = 2 tan θ/(1 − tan²θ)。

The expression a sin θ + b cos θ can be written as R sin(θ + α) or R cos(θ − α) with R = √(a² + b²) and tan α = b/a.

表达式 a sin θ + b cos θ 可写为 R sin(θ + α) 或 R cos(θ − α)，其中 R = √(a² + b²)，tan α = b/a。

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3. Differentiation Rules and Standard Derivatives | 微分法则与标准导数

Standard derivatives: d/dx (xⁿ) = n xⁿ⁻¹, d/dx (sin x) = cos x, d/dx (cos x) = −sin x, d/dx (tan x) = sec²x, d/dx (eˣ) = eˣ, d/dx (ln x) = 1/x, d/dx (aˣ) = aˣ ln a.

标准导数：d/dx (xⁿ) = n xⁿ⁻¹，d/dx (sin x) = cos x，d/dx (cos x) = −sin x，d/dx (tan x) = sec²x，d/dx (eˣ) = eˣ，d/dx (ln x) = 1/x，d/dx (aˣ) = aˣ ln a。

Product rule: d/dx (u v) = u’ v + u v’. Quotient rule: d/dx (u/v) = (u’ v − u v’)/v². Chain rule: dy/dx = dy/du × du/dx.

乘积法则：d/dx (u v) = u’ v + u v’。商法则：d/dx (u/v) = (u’ v − u v’)/v²。链式法则：dy/dx = dy/du × du/dx。

For parametric equations x = f(t), y = g(t), dy/dx = (dy/dt)/(dx/dt). For implicit differentiation, differentiate both sides and apply the chain rule to terms involving y.

对于参数方程 x = f(t), y = g(t)，dy/dx = (dy/dt)/(dx/dt)。隐函数微分时，对方程两边求导并对包含 y 的项使用链式法则。

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4. Integration Techniques and Standard Integrals | 积分技巧与标准积分

Basic integrals: ∫ xⁿ dx = xⁿ⁺¹/(n+1) + C (n ≠ −1), ∫ 1/x dx = ln|x| + C, ∫ eˣ dx = eˣ + C, ∫ aˣ dx = aˣ/ln a + C, ∫ cos x dx = sin x + C, ∫ sin x dx = −cos x + C, ∫ sec²x dx = tan x + C.

基本积分：∫ xⁿ dx = xⁿ⁺¹/(n+1) + C (n ≠ −1)，∫ 1/x dx = ln|x| + C，∫ eˣ dx = eˣ + C，∫ aˣ dx = aˣ/ln a + C，∫ cos x dx = sin x + C，∫ sin x dx = −cos x + C，∫ sec²x dx = tan x + C。

Integration by parts: ∫ u dv = u v − ∫ v du. Choose u via LIATE (Log, Inverse trig, Algebraic, Trig, Exponential) order when possible.

分部积分法：∫ u dv = u v − ∫ v du。通常按 LIATE 顺序（对数、反三角、代数、三角、指数）选择 u。

Substitution: let u = g(x), then du = g'(x) dx, transform the integral and evaluate. Standard forms include ∫ f'(x)/f(x) dx = ln|f(x)| + C and ∫ f'(x) e^f(x) dx = e^f(x) + C.

换元积分：令 u = g(x)，则 du = g'(x) dx，转换积分后求值。常见形式包括 ∫ f'(x)/f(x) dx = ln|f(x)| + C 以及 ∫ f'(x) e^f(x) dx = e^f(x) + C。

Volumes of revolution: about the x‑axis V = π ∫ y² dx, about the y‑axis V = π ∫ x² dy, taken between appropriate limits.

旋转体体积：绕 x 轴 V = π ∫ y² dx，绕 y 轴 V = π ∫ x² dy，在相应的积分限之间计算。

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5. Exponential and Logarithmic Functions | 指数与对数函数

Exponential growth/decay models follow the form y = A e^(kt), where k > 0 for growth and k < 0 for decay. The logarithmic equivalent is ln y = ln A + k t.

指数增长/衰减模型为 y = A e^(kt)，其中 k > 0 表示增长，k < 0 表示衰减。对数形式为 ln y = ln A + k t。

Laws of logarithms: logₐ(xy) = logₐ x + logₐ y, logₐ(x/y) = logₐ x − logₐ y, logₐ(xⁿ) = n logₐ x. Natural logarithm ln x = logₑ x.

对数运算律：logₐ(xy) = logₐ x + logₐ y，logₐ(x/y) = logₐ x − logₐ y，logₐ(xⁿ) = n logₐ x。自然对数 ln x = logₑ x。

Exponential functions satisfy a⁰ = 1, a¹ = a, aˣ⁺ʸ = aˣ aʸ, aˣ⁻ʸ = aˣ/aʸ, (aˣ)ʸ = aˣʸ.

指数函数满足：a⁰ = 1，a¹ = a，aˣ⁺ʸ = aˣ aʸ，aˣ⁻ʸ = aˣ/aʸ，(aˣ)ʸ = aˣʸ。

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6. Complex Numbers | 复数

A complex number is expressed as z = a + bi where i² = −1. Its conjugate is z̄ = a − bi, modulus |z| = √(a² + b²), and argument arg(z) = θ where tan θ = b/a.

复数写为 z = a + bi，其中 i² = −1。其共轭为 z̄ = a − bi，模 |z| = √(a² + b²)，辐角 arg(z) = θ 满足 tan θ = b/a。

Polar form: z = r(cos θ + i sin θ) = r cis θ, with r = |z|. Euler’s formula gives z = r e^(iθ). de Moivre’s theorem: (cos θ + i sin θ)ⁿ = cos(nθ) + i sin(nθ).

极形式：z = r(cos θ + i sin θ) = r cis θ，其中 r = |z|。欧拉公式给出 z = r e^(iθ)。棣莫弗定理：(cos θ + i sin θ)ⁿ = cos(nθ) + i sin(nθ)。

Roots of complex numbers: the nth roots are given by r^(1/n) [cos((θ + 2kπ)/n) + i sin((θ + 2kπ)/n)] for k = 0, 1, …, n−1.

复数的 n 次方根：z 的 n 次方根为 r^(1/n) [cos((θ + 2k

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