A-Level WJEC Maths: Formula Compendium | A-Level WJEC 数学：公式汇总手册

📚 A-Level WJEC Maths: Formula Compendium | A-Level WJEC 数学：公式汇总手册

This article provides a structured summary of the essential formulae needed for the WJEC A-Level Mathematics specification, covering pure mathematics, mechanics, and statistics. Each section is presented with concise explanations and paired English–Chinese paragraphs to help you master the content and perform confidently in the exam.

本文系统梳理了 WJEC A-Level 数学考试所需的核心公式，涵盖纯数学、力学和统计。每个部分均配备精简说明和中英对照段落，帮助你掌握内容，自信应考。

1. Basic Algebraic Rules and Laws of Indices | 基本代数规则与指数律

any real numbers a, b and integers m, n, aᵐ × aⁿ = aᵐ⁺ⁿ, aᵐ ÷ aⁿ = aᵐ⁻ⁿ, and (aᵐ)ⁿ = aᵐⁿ.

理解指数是处理代数表达式的基础。对于任意实数 a, b 和整数 m, n，有 aᵐ × aⁿ = aᵐ⁺ⁿ，aᵐ ÷ aⁿ = aᵐ⁻ⁿ，以及 (aᵐ)ⁿ = aᵐⁿ。

Negative and fractional indices follow the definitions a⁻ⁿ = 1/aⁿ and a¹/ⁿ = ⁿ√a. The general fractional index aᵐ/ⁿ = (ⁿ√a)ᵐ = ⁿ√aᵐ.

负指数和分数指数分别定义为 a⁻ⁿ = 1/aⁿ 以及 a¹/ⁿ = ⁿ√a。一般分数指数 aᵐ/ⁿ = (ⁿ√a)ᵐ = ⁿ√aᵐ。

The binomial expansion for positive integer n is (a + b)ⁿ = Σ (nCᵣ) aⁿ⁻ʳ bʳ. For rational functions, the expansion (1 + x)ⁿ ≈ 1 + nx + n(n−1)/2! x² + ... holds for |x| < 1.

正整数指数下的二项展开式为 (a + b)ⁿ = Σ (nCᵣ) aⁿ⁻ʳ bʳ。对于有理函数，当 |x| < 1 时有展开式 (1 + x)ⁿ ≈ 1 + nx + n(n−1)/2! x² + …。

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2. Quadratics and the Discriminant | 二次方程与判别式

atic equation is ax² + bx + c = 0, with solutions given by the quadratic formula x = (−b ± √b² − 4ac) / 2a. The expression Δ = b² − 4ac is the discriminant.

一般二次方程为 ax² + bx + c = 0，其解由求根公式 x = (−b ± √b² − 4ac) / 2a 给出。表达式 Δ = b² − 4ac 称为判别式。

s one repeated real root; and when Δ < 0 there are no real roots (two complex conjugates).

当 Δ > 0 时，有两个不等的实根；当 Δ = 0 时，有一个重实根；当 Δ < 0 时，无实根（一对共轭复根）。

Completing the square transforms the quadratic into the form a(x + p)² + q, where the vertex is (−p, q). The sum of roots is −b/a and product is c/a.

配方法可将二次式转化为 a(x + p)² + q 的形式，其中顶点坐标为 (−p, q)。根之和为 −b/a，根之积为 c/a。

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3. Sequences and Series | 数列与级数

the first n terms is Sₙ = n/2 [2a + (n−1)d] or Sₙ = n/2 (a + l) where l is the last term.

等差数列首项为 a，公差为 d：uₙ = a + (n−1)d。前 n 项和为 Sₙ = n/2 [2a + (n−1)d] 或 Sₙ = n/2 (a + l)，其中 l 为末项。

sequence has first term a and common ratio r: uₙ = arⁿ⁻¹. Its sum to infinity exists when |r| < 1: S∞ = a / (1 − r). Sum of first n terms: Sₙ = a(1 − rⁿ) / (1 − r) for r ≠ 1.

等比数列首项为 a，公比为 r：uₙ = arⁿ⁻¹。当 |r| < 1 时，无穷和为 S∞ = a / (1 − r)。前 n 项和：Sₙ = a(1 − rⁿ) / (1 − r)，r ≠ 1。

The binomial expansion for rational n uses (1 + x)ⁿ = 1 + nx + n(n−1)/2! x² + n(n−1)(n−2)/3! x³ + … , valid for |x| < 1. The rth term of (a + bx)ⁿ is nCᵣ aⁿ⁻ʳ (bx)ʳ when n ∈ ℕ.

有理指数 n 的二项展开式为 (1 + x)ⁿ = 1 + nx + n(n−1)/2! x² + n(n−1)(n−2)/3! x³ + …，要求 |x| < 1。当 n 为正整数时，(a + bx)ⁿ 的第 r 项为 nCᵣ aⁿ⁻ʳ (bx)ʳ。

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

p/adj. For any angle, sin² θ + cos² θ = 1, and sec² θ = 1 + tan² θ, cosec² θ = 1 + cot² θ.

基本定义：sin θ = 对边/斜边，cos θ = 邻边/斜边，tan θ = 对边/邻边。恒有 sin² θ + cos² θ = 1，以及 sec² θ = 1 + tan² θ，cosec² θ = 1 + cot² θ。

Compound angle formulae: 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).

和差角公式：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)。

θ; cos 2θ = cos² θ − sin² θ = 2cos² θ − 1 = 1 − 2sin² θ; tan 2θ = 2 tan θ / (1 − tan² θ).

倍角公式：sin 2θ = 2 sin θ cos θ；cos 2θ = cos² θ − sin² θ = 2cos² θ − 1 = 1 − 2sin² θ；tan 2θ = 2 tan θ / (1 − tan² θ)。

b/sin B = c/sin C (sine rule); a² = b² + c² − 2bc cos A (cosine rule). Area = ½ ab sin C.

任意三角形的正弦与余弦定理：a/sin A = b/sin B = c/sin C（正弦定理）；a² = b² + c² − 2bc cos A（余弦定理）。面积 = ½ ab sin C。

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5. Exponentials and Logarithms | 指数函数与对数函数

The exponential function y = eˣ has derivative dy/dx = eˣ and integral ∫ eˣ dx = eˣ atural logarithm ln x is the inverse, defined for x > 0 with derivative d/dx (ln x) = 1/x.

指数函数 y = eˣ 的导数为 dy/dx = eˣ，积分为 ∫ eˣ dx = eˣ + C。自然对数 ln x 是其反函数，定义域 x > 0，导数为 d/dx (ln x) = 1/x。

a. The change-of-base formula is logₐ b = ln b / ln a. For solving equations, take logs: aˣ = b ⇒ x = ln b / ln a.

对数运算律：ln(ab) = ln a + ln b，ln(a/b) = ln a − ln b，ln(aᵏ) = k ln a。换底公式为 logₐ b = ln b / ln a。解指数方程时取对数：aˣ = b ⇒ x = ln b / ln a。

he form N = N₀ eᵏᵗ, where k > 0 indicates growth and k < 0 decay. The half-life is (ln 2)/|k|.

指数增长与衰减模型为 N = N₀ eᵏᵗ，其中 k > 0 表示增长，k < 0 表示衰减。半衰期为 (ln 2)/|k|。

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6. Differentiation Rules and Techniques | 微分法则与技巧

For f(x) = xⁿ, f'(x) = n xⁿ⁻¹ (n real). The chain rule: if y = f(g(x)), then dy/dx = f'(g(x)) g'(x). Product rule: d/dx [uv] = u'v + uv'. Quotient rule: d/dx [u/v] = (u'v − uv') / v².

对于 f(x) = xⁿ，有 f'(x) = n xⁿ⁻¹（n 为实数）。链式法则：若 y = f(g(x))，则 dy/dx = f'(g(x)) g'(x)。乘法法则：d/dx [uv] = u'v + uv'。除法法则：d/dx [u/v] = (u'v − uv') / v²。

ns: d/dx sin x = cos x, d/dx cos x = −sin x, d/dx tan x = sec² x; d/dx ln x = 1/x; d/dx eᵏˣ = k eᵏˣ.

常用函数导数：d/dx sin x = cos x，d/dx cos x = −sin x，d/dx tan x = sec² x；d/dx ln x = 1/x；d/dx eᵏˣ = k eᵏˣ。

ltiplying by dy/dx for each y term. Parametric differentiation: dy/dx = (dy/dt) / (dx/dt).

隐函数求导用于 y 由隐式定义的情形，逐项求导并对含 y 的项乘以 dy/dx。参数求导：dy/dx = (dy/dt) / (dx/dt)。

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7. Integration and Its Applications | 积分及其应用

ration is the reverse of differentiation: ∫ xⁿ dx = xⁿ⁺¹/(n+1) + C (n ≠ −1), ∫ 1/x dx = ln |x| + C. For linear functions, ∫ f(ax+b) dx can be approached using reverse chain rule.

不定积分是微分的逆运算：∫ xⁿ dx = xⁿ⁺¹/(n+1) + C (n ≠ −1)，∫ 1/x dx = ln |x| + C。对于线性函数，∫ f(ax+b) dx 可用逆链式法则处理。

Area = ∫ₐᵇ f(x) dx. Area between two curves y = f(x) and y = g(x) is ∫ₐᵇ [f(x) − g(x)] dx.

定积分给出曲线在 a 到 b 之间的面积：面积 = ∫ₐᵇ f(x) dx。两条曲线 y = f(x) 和 y = g(x) 之间的面积为 ∫ₐᵇ [f(x) − g(x)] dx。

Integration by substitution: let u = g(x), then ∫ f(g(x)) g' ∫ f(u) du. Integration by parts: ∫ u dv = uv − ∫ v du, often chosen using the LIATE rule.

换元积分法：令 u = g(x)，则 ∫ f(g(x)) g'(x) dx = ∫ f(u) du。分部积分法：∫ u dv = uv − ∫ v du，通常按 LIATE 规则选取 u。

finite integral: ∫ₐᵇ f(x) dx ≈ h/2 [y₀ + 2(y₁ + y₂ + ... + yₙ₋₁) + yₙ], where h = (b−a)/n.

梯形法则近似定积分：∫ₐᵇ f(x) dx ≈ h/2 [y₀ + 2(y₁ + y₂ + ... + yₙ₋₁) + yₙ]，其中 h = (b−a)/n。

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8. Vectors in 2D and 3D | 二维与三维向量

A vector can be represented as ai + bj (2D) or ai + bj + ck (3D). Magnitude: |v| = √a² + b² or √a² + b² + c². The dot (scalar) product is a·b = |a||b| cos θ = a₁b₁ + a₂b₂ + a₃b₃.

向量可表示为 ai + bj（二维）或 ai + bj + ck（三维）。模长：|v| = √a² + b² 或 √a² + b² + c²。点积（标量积）为 a·b = |a||b| cos θ = a₁b₁ + a₂b₂ + a₃b₃。

s are perpendicular if a·b = 0. The angle between vectors is found using cos θ = (a·b) / (|a||b| tion of a line: r = a + t b, where a is a point on the line and b is the direction vector.

若 a·b = 0，则两向量垂直。向量夹角由 cos θ = (a·b) / (|a||b|) 求得。直线的向量方程为 r = a + t b，其中 a 为线上一点，b 为方向向量。

For mechanics, position, velocity, and acceleration are vectors. Differentiation of position gives velocity; integration of acceleration gives velocity.

在力学中，位置、速度和加速度均为向量。对位置求导得速度；对加速度积分得速度。

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9. Kinematics in One Dimension | 一维运动学

u² + 2as; s = vt − ½ at². Here u = initial velocity, v = final velocity, s = displacement.

用于恒加速度 a 和时间 t 的 SUVAT 方程：v = u + at；s = ut + ½ at²；s = ½ (u + v)t；v² = u² + 2as；s = vt − ½ at²。其中 u = 初速度，v = 末速度，s = 位移。

t, velocity, and acceleration are linked through calculus: v = ds/dt, a = dv/dt = d²s/dt².

这些方程仅在加速度恒定时使用。对于变加速度，位移、速度和加速度通过微积分关联：v = ds/dt，a = dv/dt = d²s/dt²。

An object moving under gravity near the Earth' wards. In projectile motion, horizontal and vertical components are treated independently.

靠近地球表面的物体受重力加速度 g = 9.8 m s⁻² 向下作用。在抛体运动中，水平与竖直分量独立处理。

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10. Forces and Newton's Laws | 力与牛顿定律

Newton' solving forces in perpendicular directions is essential for equilibrium: ΣFₓ = 0, ΣFᵧ = 0.

牛顿第二定律：F = ma，F 为合外力。重量为 mg。沿垂直方向分解力对平衡至关重要：ΣFₓ = 0，ΣFᵧ = 0。

imiting/motion, where μ is the coefficient of friction and R is the normal reaction. Inclined plane problems often use components parallel and perpendicular to the slope.

摩擦力：静摩擦 F ≤ μR；极限/运动摩擦 F = μR，μ 为摩擦系数，R 为法向反力。斜面问题常需考虑平行和垂直于斜面的分量。

separately or as a system. Momentum p = mv; impulse I = Ft = mv − mu (change in momentum).

连接体问题涉及绳中张力 T，可单独分析各物体或整体分析。动量 p = mv；冲量 I = Ft = mv − mu（动量变化）。

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11. Probability and Statistical Distributions | 概率与统计分布

Basic probability: P(A∪B) = P(A) + P(B) − P(A∩B). Mutually exclusive events have P(A∩B) = 0. Conditional probability: P(A|B) = P(A∩B) / P(B). Independent events: P(A∩B) = P(A)P(B).

基本概率：P(A∪B) = P(A) + P(B) − P(A∩B)。互斥事件有 P(A∩B) = 0。条件概率：P(A|B) = P(A∩B) / P(B)。独立事件：P(A∩B) = P(A)P(B)。

The binomial distribution X ~ B(n, p) models n independent trials with constant probability p. P(X = r) = nCᵣ pʳ (1−p)ⁿ⁻ʳ. Mean = np, variance = np(1−p).

二项分布 X ~ B(n, p) 模拟 n 次独立试验，每次成功概率为定值 p。P(X = r) = nCᵣ pʳ (1−p)ⁿ⁻ʳ。均值 = np，方差 = np(1−p)。

ariable, E(X) = Σ x P(X = x), Var(X) = E(X²) − [E(X)]². The standard normal distribution Z ~ N(0,1) is used with standardisation Z = (X − μ) / σ.

对于离散型随机变量，E(X) = Σ x P(X = x)，Var(X) = E(X²) − [E(X)]²。标准正态分布 Z ~ N(0,1) 需用标准化公式 Z = (X − μ) / σ。

Given data, the sample mean and variance are x̄ = Σx / n, s² = Σ(x − x̄)² / (n−1). For large n, central limit theorem allows normal approximations.

对给定数据，样本均值和方差为 x̄ = Σx / n，s² = Σ(x − x̄)² / (n−1)。n 较大时，中心极限定理允适用正态近似。

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12. Numerical Methods and Iteration | 数值方法与迭代

solved analytically, iterative methods like the Newton-Raphson formula xₙ₊₁ = xₙ − f(xₙ)/f'(xₙ) can be used to find approximate roots.

当方程 f(x) = 0 无法解析求解时，可运用迭代方法如牛顿-拉夫森公式 xₙ₊₁ = xₙ − f(xₙ)/f'(xₙ) 来寻找近似根。

Simple rearrangement into x = g(x) yields an iteration xₙ₊₁ = g(xₙ), converging if |g'(x)| < 1 near the root. The trapezium and Simpson's rules approximate areas under curves.

简单重排为 x = g(x) 得到迭代式 xₙ₊₁ = g(xₙ)，若在根附近 |g'(x)| < 1 则收敛。梯形法则和辛普森法则用于近似曲线下面积。

Solving differential equations numerically uses Euler' ), where dy/dx = f(x,y) and h is the step size. This provides a first-order approximation.

数值解微分方程可采用欧拉方法：yₙ₊₁ = yₙ + h f(xₙ, yₙ)，其中 dy/dx = f(x,y)，h 为步长。这提供一阶近似。

Always check the degree of accuracy required and justify the number of iterations; sign change in f(x) over an interval confirms the presence of a root.

始终检查要求的精度并说明迭代次数；在区间内 f(x) 变号即可确认根的存在。

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