Pre-U Cambridge Further Mathematics: Quick Reference Formula & Theorem Handbook | 剑桥Pre-U进阶数学：公式定理速查手册

📚 Pre-U Cambridge Further Mathematics: Quick Reference Formula & Theorem Handbook | 剑桥Pre-U进阶数学：公式定理速查手册

This concise handbook gathers the essential formulas, theorems, and techniques required for the Cambridge Pre-U Further Mathematics examination. It is designed as a rapid revision aid, covering pure mathematics core topics with clear English explanations immediately followed by precise Chinese translations. Use it to reinforce your memory and check key facts quickly.

本手册汇集了剑桥Pre-U进阶数学考试必备的核心公式、定理与技巧，旨在作为快速复习工具，覆盖纯数主要专题，每个要点先用英文阐述，紧接着配以准确的中文翻译，帮助巩固记忆并迅速查证关键内容。

1. Algebraic Identities & Binomial Theorem | 代数恒等式与二项式定理

Difference of squares: a² – b² = (a – b)(a + b).

平方差公式：a² – b² = (a – b)(a + b)。

Perfect squares: (a ± b)² = a² ± 2ab + b².

完全平方：(a ± b)² = a² ± 2ab + b²。

Sum and difference of cubes: a³ + b³ = (a + b)(a² – ab + b²) and a³ – b³ = (a – b)(a² + ab + b²).

立方和与立方差：a³ + b³ = (a + b)(a² – ab + b²)；a³ – b³ = (a – b)(a² + ab + b²)。

Binomial theorem for positive integer n: (a + b)ⁿ = Σₖ₌₀ⁿ ⁿCₖ aⁿ⁻ᵏ bᵏ, where ⁿCₖ = n!/(k!(n – k)!).

正整数指数的二项式定理：(a + b)ⁿ = Σₖ₌₀ⁿ ⁿCₖ aⁿ⁻ᵏ bᵏ，其中 ⁿCₖ = n!/(k!(n – k)!)。

General binomial expansion: For any real n, (1 + x)ⁿ = 1 + nx + [n(n–1)/2!]x² + [n(n–1)(n–2)/3!]x³ + … , valid for |x| < 1.

广义二项展开式：对于任意实数 n，(1 + x)ⁿ = 1 + nx + [n(n–1)/2!]x² + [n(n–1)(n–2)/3!]x³ + … ，收敛于 |x| < 1。

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2. Polynomials & Rational Functions | 多项式与有理函数

Remainder Theorem: When a polynomial P(x) is divided by (x – a), the remainder is P(a).

余数定理：多项式 P(x) 除以 (x – a) 所得余式为 P(a)。

Factor Theorem: (x – a) is a factor of P(x) if and only if P(a) = 0.

因子定理：(x – a) 是多项式 P(x) 的因子当且仅当 P(a) = 0。

Vieta’s formulas for quadratics: For x² + bx + c = 0 with roots α and β, α + β = –b and αβ = c. For ax² + bx + c = 0, sum = –b/a, product = c/a.

二次方程韦达定理：若 x² + bx + c = 0 有根 α, β，则 α + β = –b，αβ = c；对于 ax² + bx + c = 0，和为 –b/a，积为 c/a。

Partial fractions decomposition: Distinct linear factors: 1/[(x+a)(x+b)] = A/(x+a) + B/(x+b). Repeated linear factor: 1/[(x+a)²] = A/(x+a) + B/(x+a)². Irreducible quadratic: treat numerator as Bx + C.

部分分式分解：不同线性因子：1/[(x+a)(x+b)] = A/(x+a) + B/(x+b)。重复线性因子：1/[(x+a)²] = A/(x+a) + B/(x+a)²。不可约二次因子分子设为 Bx + C。

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3. Complex Numbers | 复数

Standard form: z = x + iy, conjugate z̄ = x – iy, modulus |z| = √(x² + y²).

标准形式：z = x + iy，共轭 z̄ = x – iy，模 |z| = √(x² + y²)。

Argument: arg(z) = θ where tan θ = y/x, taking the appropriate quadrant. Principal argument in (–π, π].

辐角：arg(z) = θ 满足 tan θ = y/x，需选取正确象限，主值范围为 (–π, π]。

Polar form: z = r(cos θ + i sin θ) where r = |z|.

极形式：z = r(cos θ + i sin θ)，其中 r = |z|。

Euler’s formula: e^(iθ) = cos θ + i sin θ, hence z = re^(iθ).

欧拉公式：e^(iθ) = cos θ + i sin θ，因此 z = re^(iθ)。

De Moivre’s theorem: (cos θ + i sin θ)ⁿ = cos(nθ) + i sin(nθ) for any integer n.

棣莫弗定理：(cos θ + i sin θ)ⁿ = cos(nθ) + i sin(nθ)，n 为任意整数。

Roots of a complex number: The nᵗʰ roots of z = re^(iθ) are w_k = r^(1/n) e^(i(θ+2πk)/n), k = 0,1,…,n–1.

复数的根：z = re^(iθ) 的 n 次方根为 w_k = r^(1/n) e^(i(θ+2πk)/n)，k = 0,1,…,n–1。

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4. Matrices & Determinants | 矩阵与行列式

Matrix multiplication: For A (m × n) and B (n × p), the product C = AB has entries c_ij = Σₖ₌₁ⁿ a_ik b_kj. Multiplication is not commutative.

矩阵乘法：若 A 为 m×n 阵，B 为 n×p 阵，则积 C = AB 的元 c_ij = Σₖ₌₁ⁿ a_ik b_kj。乘法不满足交换律。

Determinant of a 2×2 matrix: |a b; c d| = ad – bc.

二阶行列式：|a b; c d| = ad – bc。

Determinant of a 3×3 matrix: Expand by a row or column using cofactors. For A = [a b c; d e f; g h i], det(A) = a(ei – fh) – b(di – fg) + c(dh – eg).

三阶行列式：可沿一行或一列按余因子展开。对于 A = [a b c; d e f; g h i]，det(A) = a(ei – fh) – b(di – fg) + c(dh – eg)。

Inverse of a matrix: A⁻¹ = (1/det(A)) adj(A), where adj(A) is the transpose of the cofactor matrix. A⁻¹ exists iff det(A) ≠ 0.

逆矩阵：A⁻¹ = (1/det(A)) adj(A)，其中 adj(A) 为余子式矩阵的转置。det(A) ≠ 0 时逆存在。

Eigenvalues and eigenvectors: λ is an eigenvalue if det(A – λI) = 0. For each λ, solve (A – λI)v = 0 to find the eigenvector v.

特征值与特征向量：λ 为特征值若满足 det(A – λI) = 0。对此 λ 求解 (A – λI)v = 0 得特征向量 v。

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5. Vectors in 3D | 三维向量

Position vector and distance: For points A(x₁,y₁,z₁) and B(x₂,y₂,z₂), vector AB = (x₂–x₁)i + (y₂–y₁)j + (z₂–z₁)k. Distance |AB| = √((x₂–x₁)²+(y₂–y₁)²+(z₂–z₁)²).

位置向量与距离：点 A(x₁,y₁,z₁) 与 B(x₂,y₂,z₂) 间向量 AB = (x₂–x₁)i + (y₂–y₁)j + (z₂–z₁)k，距离 |AB| = √((x₂–x₁)²+(y₂–y₁)²+(z₂–z₁)²)。

Dot (scalar) product: a · b = |a||b| cos θ = a₁b₁ + a₂b₂ + a₃b₃. Used for angle, projection, and perpendicular test (a · b = 0).

点积（标量积）：a · b = |a||b| cos θ = a₁b₁ + a₂b₂ + a₃b₃。用于求夹角、投影以及垂直判定（a · b = 0）。

Cross (vector) product: a × b = |a||b| sin θ n̂, where n̂ is perpendicular to both a and b. In components, a × b = |i j k; a₁ a₂ a₃; b₁ b₂ b₃| = (a₂b₃ – a₃b₂)i – (a₁b₃ – a₃b₁)j + (a₁b₂ – a₂b₁)k.

叉积（矢量积）：a × b = |a||b| sin θ n̂，n̂ 垂直于 a 和 b 所在平面。坐标形式为 a × b = |i j k; a₁ a₂ a₃; b₁ b₂ b₃| = (a₂b₃ – a₃b₂)i – (a₁b₃ – a₃b₁)j + (a₁b₂ – a₂b₁)k。

Vector equation of a line: r = a + λb, where a is a point on the line and b is the direction vector.

空间直线向量方程：r = a + λb，其中 a 为直线上一点，b 为方向向量。

Equation of a plane: r · n = d, where n is normal to the plane, or scalar form ax + by + cz = d.

平面方程：r · n = d，n 为法向量，或坐标形式 ax + by + cz = d。

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6. 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.

定义：sinh x = (eˣ – e⁻ˣ)/2，cosh x

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