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

引言

进阶数学（Further Mathematics）是 Pre-U 和 A-Level 课程中最具挑战性的学科之一。对于 WJEC 考试局的学生而言，掌握核心公式与定理是取得高分的关键。本文整理了 WJEC 进阶数学课程中最重要的公式和定理，涵盖纯数学、力学和统计三大模块，帮助学生高效复习与快速查阅。

Further Mathematics is one of the most challenging subjects in the Pre-U and A-Level curriculum. For students taking the WJEC examination board, mastering core formulas and theorems is the key to achieving top marks. This article compiles the most important formulas and theorems in the WJEC Further Mathematics syllabus, covering Pure Mathematics, Mechanics, and Statistics modules, to help students revise efficiently and reference quickly.

一、纯数学核心公式与定理 / Pure Mathematics Core Formulas and Theorems

1. 复数 / Complex Numbers

复数在 WJEC 进阶数学中占据重要地位。复数通常表示为 z = a + bi，其中 a 为实部，b 为虚部，i 是虚数单位，满足 i² = −1。复数的模定义为 |z| = √(a² + b²)，辐角为 θ = arctan(b/a)，需根据象限调整。

Complex numbers hold an important position in WJEC Further Mathematics. A complex number is typically expressed as z = a + bi, where a is the real part, b is the imaginary part, and i is the imaginary unit satisfying i² = −1. The modulus is defined as |z| = √(a² + b²), and the argument is θ = arctan(b/a), adjusted according to the quadrant.

棣莫弗定理（De Moivre’s Theorem）是复数的核心定理之一：(cos θ + i sin θ)ⁿ = cos(nθ) + i sin(nθ)。该定理在求解复数的幂和根时极为有用。例如，求解 zⁿ = 1 时，n 个 n 次单位根由公式 z_k = cos(2πk/n) + i sin(2πk/n)，k = 0, 1, …, n−1 给出。

De Moivre’s Theorem is one of the core theorems of complex numbers: (cos θ + i sin θ)ⁿ = cos(nθ) + i sin(nθ). This theorem is extremely useful for finding powers and roots of complex numbers. For example, when solving zⁿ = 1, the n nth roots of unity are given by z_k = cos(2πk/n) + i sin(2πk/n), k = 0, 1, …, n−1.

欧拉公式（Euler’s Formula）将指数函数与三角函数联系起来：e^(iθ) = cos θ + i sin θ。由此可推导出两个重要恒等式：cos θ = (e^(iθ) + e^(−iθ))/2，sin θ = (e^(iθ) − e^(−iθ))/(2i)。这些公式在简化三角函数表达式和求解微分方程时十分方便。

Euler’s Formula connects exponential functions with trigonometric functions: e^(iθ) = cos θ + i sin θ. From this, two important identities can be derived: cos θ = (e^(iθ) + e^(−iθ))/2, sin θ = (e^(iθ) − e^(−iθ))/(2i). These formulas are very convenient for simplifying trigonometric expressions and solving differential equations.

2. 矩阵与线性代数 / Matrices and Linear Algebra

矩阵是 WJEC 进阶数学的关键内容。对于一个 2×2 矩阵 A = [[a, b], [c, d]]，其行列式为 det(A) = ad − bc。逆矩阵（当 det(A) ≠ 0 时）为 A⁻¹ = (1/det(A)) × [[d, −b], [−c, a]]。3×3 矩阵的行列式可通过按行或按列展开计算：det(A) = a₁₁C₁₁ + a₁₂C₁₂ + a₁₃C₁₃，其中 Cᵢⱼ 为余子式。

Matrices are a key topic in WJEC Further Mathematics. For a 2×2 matrix A = [[a, b], [c, d]], the determinant is det(A) = ad − bc. The inverse matrix (when det(A) ≠ 0) is A⁻¹ = (1/det(A)) × [[d, −b], [−c, a]]. The determinant of a 3×3 matrix can be calculated by expansion along a row or column: det(A) = a₁₁C₁₁ + a₁₂C₁₂ + a₁₃C₁₃, where Cᵢⱼ is the cofactor.

特征值与特征向量满足方程 Av = λv，即 (A − λI)v = 0。特征值由特征方程 det(A − λI) = 0 求出。对于 2×2 矩阵，特征方程为 λ² − tr(A)λ + det(A) = 0。矩阵的对角化形式为 A = PDP⁻¹，其中 P 的列是特征向量，D 是对角特征值矩阵。

Eigenvalues and eigenvectors satisfy the equation Av = λv, that is, (A − λI)v = 0. Eigenvalues are found from the characteristic equation det(A − λI) = 0. For a 2×2 matrix, the characteristic equation is λ² − tr(A)λ + det(A) = 0. The diagonalization form of a matrix is A = PDP⁻¹, where the columns of P are eigenvectors and D is the diagonal matrix of eigenvalues.

3. 向量与三维几何 / Vectors and 3D Geometry

向量点积（标量积）：a · b = |a||b| cos θ = a₁b₁ + a₂b₂ + a₃b₃。叉积（向量积）：a × b = |a||b| sin θ n̂，其中 n̂ 是垂直于 a 和 b 的单位向量。在分量形式中，a × b = (a₂b₃ − a₃b₂)i − (a₁b₃ − a₃b₁)j + (a₁b₂ − a₂b₁)k。

The vector dot product (scalar product): a · b = |a||b| cos θ = a₁b₁ + a₂b₂ + a₃b₃. The cross product (vector product): a × b = |a||b| sin θ n̂, where n̂ is the unit vector perpendicular to both a and b. In component form, a × b = (a₂b₃ − a₃b₂)i − (a₁b₃ − a₃b₁)j + (a₁b₂ − a₂b₁)k.

直线的向量方程为 r = a + td，其中 a 是直线上一点的位置向量，d 是方向向量，t 是参数。平面的向量方程为 r · n = d，其中 n 是平面的法向量。两平面之间的夹角 = arccos(|n₁ · n₂| / (|n₁||n₂|))，直线与平面的夹角 = arcsin(|d · n| / (|d||n|))。

The vector equation of a line is r = a + td, where a is the position vector of a point on the line, d is the direction vector, and t is a parameter. The vector equation of a plane is r · n = d, where n is the normal vector of the plane. The angle between two planes = arccos(|n₁ · n₂| / (|n₁||n₂|)), and the angle between a line and a plane = arcsin(|d · n| / (|d||n|)).

4. 双曲函数 / Hyperbolic Functions

双曲函数是指数函数的组合，与三角函数有许多类似的性质。基本定义：sinh x = (eˣ − e⁻ˣ)/2，cosh x = (eˣ + e⁻ˣ)/2，tanh x = sinh x / cosh x = (eˣ − e⁻ˣ)/(eˣ + e⁻ˣ)。

Hyperbolic functions are combinations of exponential functions, with many properties analogous to trigonometric functions. Basic definitions: sinh x = (eˣ − e⁻ˣ)/2, cosh x = (eˣ + e⁻ˣ)/2, tanh x = sinh x / cosh x = (eˣ − e⁻ˣ)/(eˣ + e⁻ˣ).

双曲恒等式类比三角恒等式。最重要的恒等式是 cosh²x − sinh²x = 1（类比 cos²θ + sin²θ = 1，注意符号差异）。导数公式：d/dx(sinh x) = cosh x，d/dx(cosh x) = sinh x，d/dx(tanh x) = sech²x。反双曲函数在积分中十分有用：arsinh x = ln(x + √(x² + 1))，arcosh x = ln(x + √(x² − 1))（x ≥ 1），artanh x = ½ ln((1 + x)/(1 − x))（|x| < 1）。

Hyperbolic identities mirror trigonometric identities. The most important identity is cosh²x − sinh²x = 1 (compare with cos²θ + sin²θ = 1, note the sign difference). Derivative formulas: d/dx(sinh x) = cosh x, d/dx(cosh x) = sinh x, d/dx(tanh x) = sech²x. Inverse hyperbolic functions are very useful in integration: arsinh x = ln(x + √(x² + 1)), arcosh x = ln(x + √(x² − 1)) (x ≥ 1), artanh x = ½ ln((1 + x)/(1 − x)) (|x| < 1).

5. 极坐标 / Polar Coordinates

极坐标 (r, θ) 与直角坐标 (x, y) 之间的转换关系为：x = r cos θ，y = r sin θ，r = √(x² + y²)，θ = arctan(y/x)。极坐标曲线所围面积的公式为：A = ½ ∫ r² dθ（从 α 到 β）。极坐标曲线的弧长公式为：s = ∫ √(r² + (dr/dθ)²) dθ。

The conversion between polar coordinates (r, θ) and Cartesian coordinates (x, y) is: x = r cos θ, y = r sin θ, r = √(x² + y²), θ = arctan(y/x). The area enclosed by a polar curve is given by: A = ½ ∫ r² dθ (from α to β). The arc length of a polar curve is: s = ∫ √(r² + (dr/dθ)²) dθ.

6. 微分方程 / Differential Equations

一阶线性微分方程的标准形式为 dy/dx + P(x)y = Q(x)，其通解公式为 y × IF = ∫ Q(x) × IF dx + C，其中积分因子 IF = e^(∫P(x)dx)。

The standard form of a first-order linear differential equation is dy/dx + P(x)y = Q(x), with the general solution formula y × IF = ∫ Q(x) × IF dx + C, where the integrating factor IF = e^(∫P(x)dx).

二阶常系数齐次线性微分方程 ay” + by’ + cy = 0 的求解取决于特征方程 am² + bm + c = 0 的判别式。若 b² − 4ac > 0（两个不等实根）：y = Ae^(m₁x) + Be^(m₂x)。若 b² − 4ac = 0（重根）：y = (A + Bx)e^(mx)。若 b² − 4ac < 0（共轭复根 m = α ± iβ）：y = e^(αx)(A cos βx + B sin βx)。

The solution of a second-order homogeneous linear differential equation with constant coefficients ay” + by’ + cy = 0 depends on the discriminant of the characteristic equation am² + bm + c = 0. If b² − 4ac > 0 (two distinct real roots): y = Ae^(m₁x) + Be^(m₂x). If b² − 4ac = 0 (repeated root): y = (A + Bx)e^(mx). If b² − 4ac < 0 (complex conjugate roots m = α ± iβ): y = e^(αx)(A cos βx + B sin βx).

7. 级数与求和 / Series and Summation

麦克劳林级数（Maclaurin Series）：f(x) = f(0) + f'(0)x + f”(0)x²/2! + f”'(0)x³/3! + …。常见展开式包括：eˣ = ∑ xⁿ/n!（n = 0 到 ∞），sin x = ∑ (−1)ⁿx²ⁿ⁺¹/(2n+1)!，cos x = ∑ (−1)ⁿx²ⁿ/(2n)!，ln(1 + x) = ∑ (−1)ⁿ⁺¹xⁿ/n（|x| < 1），(1 + x)ⁿ = 1 + nx + n(n−1)x²/2! + ...（二项式展开，|x| < 1）。

The Maclaurin Series: f(x) = f(0) + f'(0)x + f”(0)x²/2! + f”'(0)x³/3! + … Common expansions include: eˣ = ∑ xⁿ/n! (n = 0 to ∞), sin x = ∑ (−1)ⁿx²ⁿ⁺¹/(2n+1)!, cos x = ∑ (−1)ⁿx²ⁿ/(2n)!, ln(1 + x) = ∑ (−1)ⁿ⁺¹xⁿ/n (|x| < 1), (1 + x)ⁿ = 1 + nx + n(n−1)x²/2! + ... (binomial expansion, |x| < 1).

标准求和公式：∑ k = n(n+1)/2（从 k=1 到 n），∑ k² = n(n+1)(2n+1)/6，∑ k³ = [n(n+1)/2]²。差分法（Method of Differences）用于求诸如 ∑ 1/(k(k+1)) 等裂项求和，通过部分分式展开和逐项抵消得出结果。

Standard summation formulas: ∑ k = n(n+1)/2 (from k=1 to n), ∑ k² = n(n+1)(2n+1)/6, ∑ k³ = [n(n+1)/2]². The Method of Differences is used for telescoping sums such as ∑ 1/(k(k+1)), yielding results through partial fraction decomposition and term-by-term cancellation.

二、力学核心公式 / Mechanics Core Formulas

1. 运动学 / Kinematics

匀加速运动公式（SUVAT 方程）：v = u + at，s = ut + ½at²，s = ½(u + v)t，v² = u² + 2as，s = vt − ½at²。其中 s = 位移，u = 初速度，v = 末速度，a = 加速度，t = 时间。

Uniform acceleration equations (SUVAT equations): v = u + at, s = ut + ½at², s = ½(u + v)t, v² = u² + 2as, s = vt − ½at². Where s = displacement, u = initial velocity, v = final velocity, a = acceleration, t = time.

变加速度下的公式：v = ∫ a dt，s = ∫ v dt。速度作为位移的函数时可使用链式法则：a = v(dv/ds)。投射运动：水平分量 x = u cos θ × t，竖直分量 y = u sin θ × t − ½gt²。

Formulas under variable acceleration: v = ∫ a dt, s = ∫ v dt. When velocity is expressed as a function of displacement, the chain rule gives: a = v(dv/ds). Projectile motion: horizontal component x = u cos θ × t, vertical component y = u sin θ × t − ½gt².

2. 动力学与牛顿定律 / Dynamics and Newton’s Laws

牛顿第二定律：F = ma（合外力 = 质量 × 加速度）。动量定义为 p = mv，冲量为 I = Ft = Δp = mv − mu。动量守恒定律：在无外力作用的系统中，碰撞前后总动量保持不变。

Newton’s Second Law: F = ma (resultant force = mass × acceleration). Momentum is defined as p = mv, and impulse is I = Ft = Δp = mv − mu. The Law of Conservation of Momentum: in a system with no external forces, the total momentum before and after a collision remains constant.

恢复系数（Coefficient of Restitution）e = (v₂ − v₁) / (u₁ − u₂)，其中 u₁、u₂ 为碰撞前速度，v₁、v₂ 为碰撞后速度。e = 1 为完全弹性碰撞，0 < e < 1 为非完全弹性碰撞，e = 0 为完全非弹性碰撞。

The Coefficient of Restitution e = (v₂ − v₁) / (u₁ − u₂), where u₁, u₂ are velocities before collision and v₁, v₂ are velocities after collision. e = 1 for perfectly elastic collisions, 0 < e < 1 for partially elastic collisions, and e = 0 for perfectly inelastic collisions.

3. 功、能与功率 / Work, Energy and Power

功（Work Done）：W = Fd cos θ（恒力做功）。动能（Kinetic Energy）：KE = ½mv²，重力势能（GPE）：GPE = mgh。功-能定理：合力做功 = 动能变化量。功率（Power）：P = Fv（恒力恒速条件下），P = dW/dt。

Work Done: W = Fd cos θ (work done by a constant force). Kinetic Energy: KE = ½mv², Gravitational Potential Energy: GPE = mgh. The Work-Energy Theorem: work done by the resultant force = change in kinetic energy. Power: P = Fv (under constant force and speed), P = dW/dt.

4. 圆周运动 / Circular Motion

匀速圆周运动中，角速度 ω = dθ/dt = v/r，向心加速度 a = v²/r = rω²，向心力 F = mv²/r = mrω²。在竖直圆周运动（例如过山车问题）中，需结合能量守恒来分析轨道顶部和底部受力。

In uniform circular motion, angular velocity ω = dθ/dt = v/r, centripetal acceleration a = v²/r = rω², centripetal force F = mv²/r = mrω². In vertical circular motion (e.g., roller coaster problems), energy conservation must be combined with force analysis at the top and bottom of the track.

5. 简谐运动 / Simple Harmonic Motion

简谐运动（SHM）满足微分方程 d²x/dt² = −ω²x。通解形式为 x = A cos(ωt) + B sin(ωt) 或 x = R sin(ωt + φ) 或 x = R cos(ωt − φ)。速度 v = ±ω√(A² − x²)，最大速度 v_max = ωA，最大加速度 a_max = ω²A。周期 T = 2π/ω。对于弹簧振子，T = 2π√(m/k)；对于单摆，T = 2π√(l/g)。

Simple Harmonic Motion (SHM) satisfies the differential equation d²x/dt² = −ω²x. The general solution is x = A cos(ωt) + B sin(ωt) or x = R sin(ωt + φ) or x = R cos(ωt − φ). Velocity v = ±ω√(A² − x²), maximum velocity v_max = ωA, maximum acceleration a_max = ω²A. Period T = 2π/ω. For a mass-spring system, T = 2π√(m/k); for a simple pendulum, T = 2π√(l/g).

三、统计核心公式 / Statistics Core Formulas

1. 概率与分布 / Probability and Distributions

条件概率公式：P(A|B) = P(A ∩ B) / P(B)。贝叶斯定理：P(A|B) = P(B|A) × P(A) / P(B)。对于独立事件 A 和 B：P(A ∩ B) = P(A) × P(B)，P(A|B) = P(A)。

Conditional probability formula: P(A|B) = P(A ∩ B) / P(B). Bayes’ Theorem: P(A|B) = P(B|A) × P(A) / P(B). For independent events A and B: P(A ∩ B) = P(A) × P(B), P(A|B) = P(A).

离散随机变量 X 的期望值 E(X) = ∑ x·P(X = x)，方差 Var(X) = E(X²) − [E(X)]² = E[(X − μ)²]。离散均匀分布：P(X = k) = 1/n，k = 1, 2, …, n。

For a discrete random variable X, the expected value E(X) = ∑ x·P(X = x), and the variance Var(X) = E(X²) − [E(X)]² = E[(X − μ)²]. Discrete uniform distribution: P(X = k) = 1/n, k = 1, 2, …, n.

二项分布：X ~ B(n, p)，P(X = k) = C(n,k) × p^k × (1−p)^(n−k)，E(X) = np，Var(X) = np(1−p)。泊松分布：X ~ Po(λ)，P(X = k) = e^(−λ) × λ^k / k!，E(X) = Var(X) = λ。正态分布：X ~ N(μ, σ²)，概率密度函数 f(x) = (1/(σ√(2π))) × e^((−(x−μ)²)/(2σ²))。标准化 Z = (X − μ)/σ ~ N(0, 1)。

Binomial distribution: X ~ B(n, p), P(X = k) = C(n,k) × p^k × (1−p)^(n−k), E(X) = np, Var(X) = np(1−p). Poisson distribution: X ~ Po(λ), P(X = k) = e^(−λ) × λ^k / k!, E(X) = Var(X) = λ. Normal distribution: X ~ N(μ, σ²), probability density function f(x) = (1/(σ√(2π))) × e^((−(x−μ)²)/(2σ²)). Standardization: Z = (X − μ)/σ ~ N(0, 1).

2. 假设检验 / Hypothesis Testing

假设检验的一般步骤：设定原假设 H₀ 和备择假设 H₁，选择显著性水平 α（通常为 5% 或 1%），计算检验统计量，确定临界值或 p 值，做出统计推断。对于均值检验：Z = (x̄ − μ₀) / (σ/√n)（方差已知时）。

The general steps for hypothesis testing: set up the null hypothesis H₀ and alternative hypothesis H₁, choose the significance level α (usually 5% or 1%), calculate the test statistic, determine the critical value or p-value, and make a statistical inference. For testing means: Z = (x̄ − μ₀) / (σ/√n) (when variance is known).

卡方检验（Chi-Squared Test）：用于检验分类数据的拟合优度和独立性。检验统计量 χ² = ∑ (Oᵢ − Eᵢ)² / Eᵢ，其中 Oᵢ 为观测频数，Eᵢ 为期望频数。自由度 = (行数−1) × (列数−1)（独立性检验）或 类别数 − 约束条件数（拟合优度检验）。

The Chi-Squared Test: used to test goodness of fit and independence for categorical data. The test statistic is χ² = ∑ (Oᵢ − Eᵢ)² / Eᵢ, where Oᵢ is the observed frequency and Eᵢ is the expected frequency. Degrees of freedom = (rows−1) × (columns−1) (independence test) or number of categories − number of constraints (goodness-of-fit test).

四、复习策略与应试技巧 / Revision Strategies and Exam Techniques

WJEC 进阶数学考试的关键在于对公式和定理的准确记忆与灵活运用。建议学生制作公式卡片（Flashcards），每日复习。重点关注公式的适用条件和边界情况，例如，二项式展开 (1 + x)ⁿ 仅在 |x| < 1 时收敛；使用 A⁻¹ 求解方程组时需确保 det(A) ≠ 0。

The key to the WJEC Further Mathematics examination lies in accurately memorizing and flexibly applying formulas and theorems. Students are advised to create formula flashcards and review them daily. Pay special attention to the conditions and boundary cases of formulas — for example, the binomial expansion (1 + x)ⁿ only converges when |x| < 1; when using A⁻¹ to solve systems of equations, ensure that det(A) ≠ 0.

在解答证明题时，结构化的 Reasoning 至关重要。每一步推导都应清晰标注所用定理或公式。力学题需先绘制受力分析图（Free Body Diagram），明确所有作用力后再列方程；统计题需明确写出 H₀ 和 H₁，展示完整的计算过程，最后用文字给出统计结论。

When answering proof questions, structured reasoning is crucial. Each step of the derivation should clearly label the theorem or formula used. For mechanics questions, draw a free body diagram first, clarify all acting forces, and then set up equations. For statistics questions, clearly state H₀ and H₁, show the complete calculation process, and provide a statistical conclusion in words.

最后，熟悉 WJEC 考试规范中的公式手册（Formula Booklet）内容。考试中提供的公式无需死记，重点记忆手册中未包含但常考的公式和推导过程。多做历年真题（Past Papers），总结命题规律，是提升成绩的最有效途径。

Finally, familiarize yourself with the content of the Formula Booklet provided in the WJEC examination specification. Formulas provided in the exam do not need to be memorized; focus on memorizing formulas and derivation processes that are not included in the booklet but frequently tested. Practicing past papers and summarizing question patterns is the most effective way to improve your score.