Year 13 CIE Further Mathematics: Formula & Theorem Quick Reference Handbook | Year 13 CIE 进阶数学：公式定理速查手册

📚 Year 13 CIE Further Mathematics: Formula & Theorem Quick Reference Handbook | Year 13 CIE 进阶数学：公式定理速查手册

This quick reference handbook brings together essential formulas and theorems from the Year 13 CIE Further Mathematics syllabus. It is designed for rapid revision, covering Further Pure, Mechanics, and Statistics components. Each section presents key results in a concise, bilingual format to reinforce understanding and retention ahead of your examinations.

本速查手册汇集了 Year 13 CIE 进阶数学课程的核心公式与定理，旨在帮助同学们快速复习进阶纯数、力学和统计部分。每个章节以简洁的中英双语呈现关键结论，强化理解，助力考前记忆。

1. Complex Numbers | 复数

De Moivre’s theorem: For any integer n, (cos θ + i sin θ)ⁿ = cos nθ + i sin nθ. This is the foundation for finding powers and roots of complex numbers.

棣莫弗定理： 对任意整数 n，(cos θ + i sin θ)ⁿ = cos nθ + i sin nθ。这是求复数乘方与开方的根本公式。

Exponential form: eⁱθ = cos θ + i sin θ, so z = reⁱθ. The modulus is r = |z| = √(x² + y²) and the argument satisfies tan θ = y/x.

指数形式： eⁱθ = cos θ + i sin θ，因此 z = reⁱθ。模长 r = |z| = √(x² + y²)，辐角满足 tan θ = y/x。

nth roots of unity: The solutions to zⁿ = 1 are z = e^(2kπi/n), k = 0, 1, …, n−1. They lie symmetrically on the unit circle.

单位根： 方程 zⁿ = 1 的根为 z = e^(2kπi/n), k = 0, 1, …, n−1。它们在单位圆上均匀分布。

Euler’s identities: sin θ = (eⁱθ − e⁻ⁱθ)/(2i), cos θ = (eⁱθ + e⁻ⁱθ)/2. These are useful for expressing powers of sin and cos as multiple angles.

欧拉恒等式： sin θ = (eⁱθ − e⁻ⁱθ)/(2i)，cos θ = (eⁱθ + e⁻ⁱθ)/2。用于将正弦、余弦的高次幂化为倍角形式。

Loci in the complex plane: |z − a| = r represents a circle centre a, radius r. |z − a| = |z − b| represents the perpendicular bisector of the line joining a and b.

复平面上的轨迹： |z − a| = r 表示以 a 为圆心、r 为半径的圆；|z − a| = |z − b| 表示连接 a 与 b 线段的垂直平分线。

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2. Matrices and Linear Transformations | 矩阵与线性变换

Determinant of a 3×3 matrix: For A = (aᵢⱼ), det(A) = a₁₁(a₂₂a₃₃ − a₂₃a₃₂) − a₁₂(a₂₁a₃₃ − a₂₃a₃₁) + a₁₃(a₂₁a₃₂ − a₂₂a₃₁). The determinant gives the scale factor of area/volume under transformation.

三阶行列式： 对于 A = (aᵢⱼ)，det(A) = a₁₁(a₂₂a₃₃ − a₂₃a₃₂) − a₁₂(a₂₁a₃₃ − a₂₃a₃₁) + a₁₃(a₂₁a₃₂ − a₂₂a₃₁)。行列式表示线性变换的面积/体积缩放因子。

Inverse of a 3×3 matrix: A⁻¹ = (1/det(A)) adj(A), where adj(A) is the transpose of the matrix of cofactors. The inverse exists only when det(A) ≠ 0.

三阶逆矩阵： A⁻¹ = (1/det(A)) adj(A)，其中 adj(A) 是余子式矩阵的转置。仅当 det(A) ≠ 0 时存在逆矩阵。

Eigenvalues and eigenvectors: For a square matrix A, satisfy A x = λ x. Eigenvalues are found from det(A − λI) = 0. They describe invariant directions.

特征值与特征向量： 对于方阵 A，满足 A x = λ x。特征值由 det(A − λI) = 0 解出，描述线性变换的不变方向。

Diagonalisation: If A has n linearly independent eigenvectors, then A = PDP⁻¹, where D is diagonal matrix of eigenvalues and P contains eigenvectors.

对角化： 若 A 有 n 个线性无关的特征向量，则 A = PDP⁻¹，其中 D 为特征值构成的对角阵，P 由特征向量构成。

Standard transformation matrices: Rotation by θ: [[cosθ, −sinθ], [sinθ, cosθ]]; reflection in the x-axis: [[1,0],[0,−1]]; shear parallel to x-axis: [[1,k],[0,1]].

标准变换矩阵： 绕原点旋转 θ：[[cosθ, −sinθ], [sinθ, cosθ]]；关于 x 轴反射：[[1,0],[0,−1]]；平行于 x 轴的剪切：[[1,k],[0,1]]。

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3. Polar Coordinates | 极坐标

Conversion: x = r cos θ, y = r sin θ, r² = x² + y², tan θ = y/x. Area element is (1/2) r² dθ.

转换关系： x = r cos θ，y = r sin θ，r² = x² + y²，tan θ = y/x。面积微元为 (1/2) r² dθ。

Area enclosed by a polar curve: A = (1/2) ∫[α, β] r² dθ. This formula integrates sectors of angle dθ.

极坐标曲线围成的面积： A = (1/2) ∫[α, β] r² dθ。将区域分割为扇形角的积分。

Arc length in polar coordinates: s = ∫[α, β] √(r² + (dr/dθ)²) dθ. Useful for cardioids and spirals.

极坐标下的弧长： s = ∫[α, β] √(r² + (dr/dθ)²) dθ。常用于心形线、螺线等曲线。

Tangents to polar curves: Gradient dy/dx = (r’ sin θ + r cos θ)/(r’ cos θ − r sin θ). Parallel to initial line when r’ cos θ − r sin θ = 0.

极坐标曲线的切线： 斜率 dy/dx = (r’ sin θ + r cos θ)/(r’ cos θ − r sin θ)。平行于极轴的条件是 r’ cos θ − r sin θ = 0。

Standard curves: Circle r = 2a cos θ (diameter a, through pole); cardioid r = a(1 + cos θ); rose curves r = a cos(nθ).

标准曲线： 圆 r = 2a cos θ（直径 a，过极点）；心形线 r = a(1 + cos θ)；玫瑰线 r = a cos(nθ)。

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4. Hyperbolic Functions | 双曲函数

Definitions: sinh x = (eˣ − e⁻ˣ)/2, cosh x = (eˣ + e⁻ˣ)/2, tanh x = sinh x / cosh x. They model catenaries and rapid growth.

定义： sinh x = (eˣ − e⁻ˣ)/2，cosh x = (eˣ + e⁻ˣ)/2，tanh x = sinh x / cosh x。它们用于悬链线及快速增长模型。

Fundamental identity: cosh² x − sinh² x = 1. Compare with the trigonometric cos² x + sin² x = 1.

基本恒等式： cosh² x − sinh² x = 1。与三角恒等式 cos² x + sin² x = 1 对比记忆。

Inverse hyperbolic functions: arsinh x = ln(x + √(x² + 1)); arcosh x = ln(x + √(x² − 1)), x ≥ 1; artanh x = (1/2) ln((1+x)/(1−x)), |x| < 1.

反双曲函数： arsinh x = ln(x + √(x² + 1))；arcosh x = ln(x + √(x² − 1))，x ≥ 1；artanh x = (1/2) ln((1+x)/(1−x))，|x| < 1。

Derivatives: d/dx sinh x = cosh x, d/dx cosh x = sinh x, d/dx tanh x = sech² x. Note the similar patterns to trig but with no negative sign for cosh.

导数： d/dx sinh x = cosh x，d/dx cosh x = sinh x，d/dx tanh x = sech² x。注意与三角函数类似，但 cosh 的导数没有负号。

Osborn’s rule: To obtain a hyperbolic identity from a trig identity, replace cos² by cosh² and sin² by −sinh². For example, cos 2x = 2cos² x − 1 becomes cosh 2x = 2cosh² x − 1.

奥斯本法： 将三角恒等式转换为双曲恒等式时，把 cos² 换成 cosh²，sin² 换成 −sinh²。如 cos 2x = 2cos² x − 1 变为 cosh 2x = 2cosh² x − 1。

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5. Differential Equations | 微分方程

First-order linear ODE: dy/dx + P(x)y = Q(x). Integrating factor μ(x) = e∫ P dx, solution: y μ = ∫ Q μ dx + C.

一阶线性微分方程： dy/dx + P(x)y = Q(x)。积分因子 μ(x) = e∫ P dx，通解为 y μ = ∫ Q μ dx + C。

Second-order homogeneous ODE with constant coefficients: a d²y/dx² + b dy/dx + c y = 0. Auxiliary equation: am² + bm + c = 0. If roots are real and distinct, y = A e^(m₁x) + B e^(m₂x); repeated root m, y = (A + Bx) e^(mx); complex α ± iβ, y = e^(αx)(A cos βx + B sin βx).

二阶常系数齐次微分方程： a d²y/dx² + b dy/dx + c y = 0。辅助方程 am² + bm + c = 0。两不等实根：y = A e^(m₁x) + B e^(m₂x)；重根：y = (A + Bx) e^(mx)；复根 α ± iβ：y = e^(αx)(A cos βx + B sin βx)。

Second-order non-homogeneous ODE: General solution = complementary function (CF) + particular integral (PI). PI forms: polynomial, exponential, trigonometric, or combinations.

二阶非齐次微分方程： 通解 = 余函数(CF) + 特解(PI)。PI 的形式根据右端项设为多项式、指数、三角函数或其组合。

Substitution methods: Use v = y/x or v = y’ to reduce order. For example, dy/dx = f(y/x) becomes separable after v = y/x.

代换法： 通过 v = y/x 或 v = y’ 降阶。例如 dy/dx = f(y/x) 可用 v = y/x 化为可分离变量方程。

Simple harmonic motion (linked to mechanics): d²x/dt² + ω² x = 0 → x = A cos ωt + B sin ωt or x = R sin(ωt + φ). Period T = 2π/ω.

简谐振动： d²x/dt² + ω² x = 0 → x = A cos ωt + B sin ωt，或 x = R sin(ωt + φ)。周期 T = 2π/ω。

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6. Further Vectors | 向量进阶

Scalar triple product: a · (b × c) = det of the 3×3 matrix formed by vectors. Gives the volume of the parallelepiped. If zero, vectors are coplanar.

标量三重积： a · (b × c) = 三向量构成的 3×3 行列式。其绝对值等于平行六面体体积。若为零，三向量共面。

Vector equation of a line: r = a + λ d, where a is a point on the line and d is direction vector. Two lines intersect if a₁ + λd₁ = a₂ + μd₂ has a solution.

直线的向量方程： r = a + λ d，a 为直线上一点，d 为方向向量。两直线相交要求 a₁ + λd₁ = a₂ + μd₂ 有解。

Vector equation of a plane: r · n = a · n = p, where n is normal vector. Alternatively, r = a + λ u + μ v for two direction vectors u, v lying in the plane.

平面的向量方程： r · n = a · n = p，n 为法向量。或 r = a + λ u + μ v，u 和 v 为平面内两个方向向量。

Distance from a point to a plane: If plane is r · n = p and point has position vector b, distance = |b · n − p| / |n|. For line r = a + λd, distance from point P to line = |(P−a) × d| / |d|.

点到平面的距离： 平面 r · n = p，点 b，距离 = |b · n − p| / |n|。点到直线 r = a + λd 的距离 = |(P−a) × d| / |d|。

Angles between lines and planes: Angle θ between two lines with directions d₁, d₂: cos θ = |d₁·d₂|/(|d₁||d₂|). Angle between line and plane: sin φ = |d·n|/(|d||n|).

线线角与线面角： 两方向向量为 d₁, d₂ 的直线夹角 θ：cos θ = |d₁·d₂|/(|d₁||d₂|)。直线与平面（法向量 n）夹角 φ：sin φ = |d·n|/(|d||n|)。

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7. Further Mechanics | 进阶力学

Constant acceleration equations (SUVAT): v = u + at, s = ut + ½at², s = ½(u+v)t, v² = u² + 2as. Apply only when acceleration is constant.

匀加速运动公式： v = u + at，s = ut + ½at²，s = ½(u+v)t，v² = u² + 2as。仅适用于加速度恒定的情况。

Impulse-momentum principle: Impulse J = change in momentum = m(v − u). For a variable force, J = ∫ F dt. Units: N s.

冲量-动量原理： 冲量 J = 动量变化 = m(v − u)。变力时 J = ∫ F dt。单位是牛顿秒 (N s)。

Work-energy principle: Work done by resultant force = change in kinetic energy: W = ½mv² − ½mu². Work by constant force: F d cos θ.

功能原理： 合力做的功 = 动能变化：W = ½mv² − ½mu²。恒力做功：F d cos θ。

Circular motion (horizontal): Centripetal force F = m v²/r = m r ω². Velocity v = rω, period T = 2π/ω. In vertical circles, energy conservation and radial force equation must be combined.

圆周运动： 向心力 F = m v²/r = m r ω²。线速度 v = rω，周期 T = 2π/ω。竖直面内圆周运动需结合能量守恒与径向力方程。

Center of mass: For discrete particles, position vector R = (Σ mᵢ rᵢ) / Σ mᵢ. For uniform lamina, use formulae for standard shapes. Composite bodies: X = Σ(mx)/Σm.

质心： 离散质点系位置矢量 R = (Σ mᵢ rᵢ) / Σ mᵢ。均匀薄板可套用标准形状公式；组合体用 X = Σ(mx)/Σm 计算。

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8. Further Probability & Statistics | 进阶概率统计

Poisson distribution: X ~ Po(λ), probability P(X = x) = e^(−λ) λˣ / x!. Mean and variance are both λ. Used for rare events in a fixed interval.

泊松分布： X ~ Po(λ)，概率为 P(X = x) = e^(−λ) λˣ / x!。期望和方差均为 λ。适用于固定区间内稀有事件发生次数。

Continuous random variables: Probability density function f(x) satisfies ∫ f(x) dx = 1 over domain. P(a < X < b) = ∫[a,b] f(x) dx. Cumulative distribution F(x) = P(X ≤ x) = ∫ f(t) dt.

连续随机变量： 概率密度函数 f(x) 满足在整个域上 ∫ f(x) dx = 1。P(a < X < b) = ∫[a,b] f(x) dx。累积分布函数 F(x) = P(X ≤ x) = ∫ f(t) dt。

Expectation and variance for continuous distributions: E(X) = ∫ x f(x) dx, Var(X) = E(X²) − [E(X)]² = ∫ x² f(x) dx − μ². For linear transformations, E(aX+b) = aE(X)+b, Var(aX+b) = a²Var(X).

连续分布的期望与方差： E(X) = ∫ x f(x) dx，Var(X) = E(X²) − [E(X)]² = ∫ x² f(x) dx − μ²。线性变换：E(aX+b) = aE(X)+b，Var(aX+b) = a²Var(X)。

Type I and Type II errors: Type I error: rejecting H₀ when true (α, significance level). Type II error: failing to reject H₀ when false (β). Power = 1 − β.

第一类与第二类错误： 第一类错误：H₀ 为真却被拒绝（概率 α，显著性水平）。第二类错误：H₀ 为假却未拒绝（概率 β）。功效 = 1 − β。

Chi-squared tests: Test statistic X² = Σ[(O − E)² / E], where O observed, E expected. Degrees of freedom = (rows−1)×(columns−1) for contingency tables. Goodness of fit: ν = k−1−p.

卡方检验： 统计量 X² = Σ[(O − E)² / E]，O 为观测频数，E 为期望频数。列联表自由度 = (行数−1)×(列数−1)。拟合优度自由度 ν = k−1−p。

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