📚 Year 12 WJEC Mathematics: Formula and Theorem Quick Reference Handbook | WJEC 12年级数学：公式定理速查手册
This quick reference handbook collects all the essential formulas, identities, and theorems you need for the Year 12 WJEC Mathematics specification. Whether you are revising pure mathematics, mechanics, or statistics, keeping these results at your fingertips will save time and boost exam confidence. Each section pairs English and Chinese explanations so that bilingual learners can cross-reference effortlessly.
这本速查手册汇集了WJEC 12年级数学考试所需的所有核心公式、恒等式和定理。无论你正在复习纯数学、力学还是统计，将这些结论牢记于心能节省时间、提升应试信心。每个小节都配有中英文对照解释，方便双语学习者随时查阅。
1. Algebraic Foundations | 代数基础
The index laws allow you to manipulate powers efficiently. For any real numbers a, b and integers m, n: aᵐ × aⁿ = aᵐ⁺ⁿ, aᵐ ÷ aⁿ = aᵐ⁻ⁿ, (aᵐ)ⁿ = aᵐⁿ, (ab)ⁿ = aⁿbⁿ, a⁰ = 1 (a ≠ 0), a⁻ⁿ = 1/aⁿ.
指数法则让你能够高效地处理幂运算。对于任意实数 a、b 和整数 m、n：aᵐ × aⁿ = aᵐ⁺ⁿ，aᵐ ÷ aⁿ = aᵐ⁻ⁿ，(aᵐ)ⁿ = aᵐⁿ，(ab)ⁿ = aⁿbⁿ，a⁰ = 1 (a ≠ 0)，a⁻ⁿ = 1/aⁿ。
Completing the square rewrites ax² + bx + c as a(x + p)² + q, where p = b/(2a) and q = c – b²/(4a). This form reveals the vertex of a parabola directly.
配方法将 ax² + bx + c 写为 a(x + p)² + q，其中 p = b/(2a)，q = c – b²/(4a)。该形式可以直接揭示抛物线的顶点。
The discriminant Δ = b² – 4ac determines the nature of the roots of ax² + bx + c = 0: two distinct real roots if Δ > 0, one repeated real root if Δ = 0, and no real roots if Δ < 0.
判别式 Δ = b² – 4ac 决定了方程 ax² + bx + c = 0 根的性质：Δ > 0 时有两个不等实根，Δ = 0 时有一个重根，Δ < 0 时无实根。
2. Quadratic Functions and Equations | 二次函数与方程
The quadratic formula gives the solutions of ax² + bx + c = 0:
二次公式给出方程 ax² + bx + c = 0 的解：
x = [−b ± √(b² − 4ac)] / (2a)
The sum of roots α + β = −b/a and the product αβ = c/a for the equation ax² + bx + c = 0 with roots α, β.
对于根为 α、β 的方程 ax² + bx + c = 0，根的和 α + β = −b/a，根的积 αβ = c/a。
To form a quadratic with given roots, use x² − (sum)x + product = 0.
要构造给定根的二次方程，使用 x² − (和)x + 积 = 0。
The graph of y = a(x − h)² + k is a parabola with vertex (h, k). If a > 0 the parabola opens upwards; if a < 0 it opens downwards.
y = a(x − h)² + k 的图像是以 (h, k) 为顶点的抛物线。若 a > 0，抛物线开口向上；若 a < 0，开口向下。
3. Coordinate Geometry | 坐标几何
The gradient m of a straight line passing through (x₁, y₁) and (x₂, y₂) is m = (y₂ − y₁) / (x₂ − x₁).
过点 (x₁, y₁) 和 (x₂, y₂) 的直线梯度 m = (y₂ − y₁) / (x₂ − x₁)。
The equation of a line with gradient m passing through (x₁, y₁) is y − y₁ = m(x − x₁). The general form is y = mx + c where c is the y-intercept.
梯度为 m 且过点 (x₁, y₁) 的直线方程为 y − y₁ = m(x − x₁)。一般式为 y = mx + c，其中 c 为 y 轴截距。
For parallel lines, gradients are equal: m₁ = m₂. For perpendicular lines, gradients satisfy m₁ × m₂ = −1.
平行线的梯度相等：m₁ = m₂。垂直线的梯度满足 m₁ × m₂ = −1。
The distance between two points (x₁, y₁) and (x₂, y₂) is √[(x₂ − x₁)² + (y₂ − y₁)²]. The midpoint is ((x₁ + x₂)/2, (y₁ + y₂)/2).
两点 (x₁, y₁) 和 (x₂, y₂) 之间的距离为 √[(x₂ − x₁)² + (y₂ − y₁)²]。中点为 ((x₁ + x₂)/2, (y₁ + y₂)/2)。
4. Sequences and Series | 数列与级数
An arithmetic sequence has a common difference d. The n-th term is uₙ = a + (n − 1)d. The sum of the first n terms is Sₙ = n/2 [2a + (n − 1)d] = n/2 (a + l), where l is the last term.
等差数列具有公差 d。第 n 项为 uₙ = a + (n − 1)d。前 n 项和为 Sₙ = n/2 [2a + (n − 1)d] = n/2 (a + l)，其中 l 为末项。
A geometric sequence has a common ratio r. The n-th term is uₙ = arⁿ⁻¹. The sum of the first n terms (r ≠ 1) is Sₙ = a(1 − rⁿ)/(1 − r). For |r| < 1, the sum to infinity is S∞ = a/(1 − r).
等比数列具有公比 r。第 n 项为 uₙ = arⁿ⁻¹。前 n 项和 (r ≠ 1) 为 Sₙ = a(1 − rⁿ)/(1 − r)。当 |r| < 1 时，无穷项和为 S∞ = a/(1 − r)。
Sigma notation Σ is used to write series concisely. Σₙ₌₁ᴺ uₙ = u₁ + u₂ + … + uₙ.
西格玛符号 Σ 用于简洁地表示级数。Σₙ₌₁ᴺ uₙ = u₁ + u₂ + … + uₙ。
5. Trigonometry | 三角学
In a right-angled triangle, sin θ = opposite/hypotenuse, cos θ = adjacent/hypotenuse, tan θ = opposite/adjacent.
在直角三角形中，sin θ = 对边/斜边，cos θ = 邻边/斜边，tan θ = 对边/邻边。
Two fundamental identities: tan θ = sin θ / cos θ, and sin²θ + cos²θ = 1.
两个基本恒等式：tan θ = sin θ / cos θ，以及 sin²θ + cos²θ = 1。
The sine rule: a/sin A = b/sin B = c/sin C. The cosine rule: a² = b² + c² − 2bc cos A. Use the sine rule for non-right triangles when you know an angle and its opposite side; use the cosine rule when you know two sides and the included angle or all three sides.
正弦定理：a/sin A = b/sin B = c/sin C。余弦定理：a² = b² + c² − 2bc cos A。当已知一角及其对边时，对非直角三角形使用正弦定理；当已知两边及其夹角或三边时，使用余弦定理。
Area of a triangle: ½ ab sin C. Radian measure: π rad = 180°. Arc length s = rθ, sector area A = ½ r²θ, where θ is in radians.
三角形面积：½ ab sin C。弧度制：π rad = 180°。弧长 s = rθ，扇形面积 A = ½ r²θ，其中 θ 以弧度为单位。
6. Exponentials and Logarithms | 指数与对数
The function y = aˣ has an inverse called the logarithm: x = logₐ y. In particular, if a = e (Euler’s number), the natural logarithm is ln y = logₑ y, and eˣ and ln x are inverses.
函数 y = aˣ 的反函数称为对数：x = logₐ y。特别地，当 a = e（欧拉数）时，自然对数为 ln y = logₑ y，且 eˣ 与 ln x 互为反函数。
Log laws: logₐ(xy) = logₐ x + logₐ y; logₐ(x/y) = logₐ x − logₐ y; logₐ(xⁿ) = n logₐ x. The change-of-base formula: logₐ b = logₓ b / logₓ a.
对数法则：logₐ(xy) = logₐ x + logₐ y；logₐ(x/y) = logₐ x − logₐ y；logₐ(xⁿ) = n logₐ x。换底公式：logₐ b = logₓ b / logₓ a。
To solve exponential equations, take logs of both sides. For example, aˣ = b becomes x ln a = ln b.
求解指数方程时，两边同时取对数。例如，aˣ = b 变为 x ln a = ln b。
7. Differentiation | 微分
The derivative of f(x) with respect to x is f'(x) or dy/dx. For a power function, d/dx (xⁿ) = nxⁿ⁻¹.
f(x) 关于 x 的导数为 f'(x) 或 dy/dx。对于幂函数，d/dx (xⁿ) = nxⁿ⁻¹。
The derivative of a sum is the sum of the derivatives. The gradient of a curve at a point can be found by evaluating dy/dx at that x-coordinate.
和的导数等于导数的和。曲线在某点的梯度可通过在该点 x 坐标处计算 dy/dx 求得。
Equation of a tangent at (x₁, y₁): y − y₁ = m(x − x₁), where m = f'(x₁). The normal has gradient −1/m.
过 (x₁, y₁) 的切线方程：y − y₁ = m(x − x₁)，其中 m = f'(x₁)。法线的梯度为 −1/m。
Stationary points occur where f'(x) = 0. Use the second derivative f”(x) to classify: if f”(x) > 0 it is a minimum; if f”(x) < 0 it is a maximum; if f''(x) = 0 the test is inconclusive.
驻点出现在 f'(x) = 0 处。用二阶导数 f”(x) 判别：若 f”(x) > 0 则为极小值点；若 f”(x) < 0 则为极大值点；若 f''(x) = 0 则此判别法失效。
8. Integration | 积分
Integration is the reverse process of differentiation. The indefinite integral of xⁿ (n ≠ −1) is ∫ xⁿ dx = xⁿ⁺¹/(n+1) + c.
积分是微分的逆过程。xⁿ (n ≠ −1) 的不定积分为 ∫ xⁿ dx = xⁿ⁺¹/(n+1) + c。
The definite integral ∫ₐᵇ f(x) dx gives the area under the curve y = f(x) between x = a and x = b, provided f(x) ≥ 0. If the curve lies below the x-axis, the integral yields a negative value, so separate areas must be treated with absolute values.
定积分 ∫ₐᵇ f(x) dx 给出曲线 y = f(x) 下方介于 x = a 与 x = b 之间的面积，前提是 f(x) ≥ 0。若曲线在 x 轴下方，积分值为负，因此必须用绝对值分别处理各部分面积。
The area between two curves y = f(x) and y = g(x) from a to b is ∫ₐᵇ [f(x) − g(x)] dx, assuming f(x) ≥ g(x) on [a, b].
两条曲线 y = f(x) 和 y = g(x) 在 a 到 b 之间的面积为 ∫ₐᵇ [f(x) − g(x)] dx，假设在 [a, b] 上 f(x) ≥ g(x)。
9. Vectors | 向量
A vector has both magnitude and direction. It can be written in component form as (x, y) or xi + yj. The magnitude of vector v = xi + yj is |v| = √(x² + y²).
向量既有大小又有方向。可以写成分量形式 (x, y) 或 xi + yj。向量 v = xi + yj 的大小为 |v| = √(x² + y²)。
The sum of vectors a = (x₁, y₁) and b = (x₂, y₂) is a + b = (x₁ + x₂, y₁ + y₂). Scalar multiplication: k a = (k x₁, k y₁).
向量 a = (x₁, y₁) 与 b = (x₂, y₂) 的和为 a + b = (x₁ + x₂, y₁ + y₂)。数乘：k a = (k x₁, k y₁)。
The dot product of a and b is a · b = x₁x₂ + y₁y₂ = |a||b| cos θ, where θ is the angle between them. If a · b = 0, the vectors are perpendicular.
a 与 b 的点积为 a · b = x₁x₂ + y₁y₂ = |a||b| cos θ，其中 θ 为二者的夹角。若 a · b = 0，则向量垂直。
Position vector of the point dividing AB in the ratio m:n is (n a + m b)/(m + n). The vector AB = b − a.
按比例 m:n 分割 AB 的点的位置向量为 (n a + m b)/(m + n)。向量 AB = b − a。
10. Mechanics – Kinematics | 力学 – 运动学
For constant acceleration a, initial velocity u, final velocity v, displacement s and time t, the SUVAT equations apply: v = u + at, s = ut + ½at², s = vt − ½at², v² = u² + 2as, s = ½(u + v)t.
对于恒定加速度 a、初速度 u、末速度 v、位移 s 和时间 t，适用 SUVAT 方程：v = u + at，s = ut + ½at²，s = vt − ½at²，v² = u² + 2as，s = ½(u + v)t。
On a velocity–time graph, the gradient gives acceleration and the area under the graph gives displacement.
在速度–时间图上，梯度给出加速度，图下面积给出位移。
Newton’s second law: F = ma, where F is the resultant force, m is mass and a is acceleration. Weight = mg, where g = 9.8 m s⁻².
牛顿第二定律：F = ma，其中 F 为合力，m 为质量，a 为加速度。重力 = mg，其中 g = 9.8 m s⁻²。
For a particle on a smooth inclined plane at angle θ, the component of weight down the plane is mg sin θ.
对于位于光滑斜面上、倾角为 θ 的质点，重力沿斜面向下的分量为 mg sin θ。
11. Statistics – Probability and Distributions | 统计 – 概率与分布
For mutually exclusive events A and B, P(A ∪ B) = P(A) + P(B). For independent events, P(A ∩ B) = P(A) × P(B). Conditional probability: P(A|B) = P(A ∩ B)/P(B).
对于互斥事件 A 与 B，P(A ∪ B) = P(A) + P(B)。对于独立事件，P(A ∩ B) = P(A) × P(B)。条件概率：P(A|B) = P(A ∩ B)/P(B)。
The mean of a discrete random variable X is E(X) = Σ x·P(X = x). Variance: Var(X) = E(X²) − [E(X)]². Standard deviation is the square root of variance.
离散随机变量 X 的期望值为 E(X) = Σ x·P(X = x)。方差：Var(X) = E(X²) − [E(X)]²。标准差为方差的平方根。
For the binomial distribution X ~ B(n, p), the probability of exactly r successes is P(X = r) = ⁿCᵣ pʳ (1−p)ⁿ⁻ʳ, where ⁿCᵣ = n!/(r!(n−r)!). Mean = np, variance = np(1−p).
对于二项分布 X ~ B(n, p)，恰好有 r 次成功的概率为 P(X = r) = ⁿCᵣ pʳ (1−p)ⁿ⁻ʳ，其中 ⁿCᵣ = n!/(r!(n−r)!)。期望值 = np，方差 = np(1−p)。
A box plot displays minimum, lower quartile Q₁, median Q₂, upper quartile Q₃ and maximum. Outliers are typically defined as values less than Q₁ − 1.5 × IQR or greater than Q₃ + 1.5 × IQR, where IQR = Q₃ − Q₁.
箱形图展示了最小值、下四分位数 Q₁、中位数 Q₂、上四分位数 Q₃ 和最大值。离群值通常定义为小于 Q₁ − 1.5 × IQR 或大于 Q₃ + 1.5 × IQR 的值，其中 IQR = Q₃ − Q₁。
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