📚 Year 12 CCEA Mathematics: Formula and Theorem Quick Reference Guide | Year 12 CCEA 数学：公式定理速查手册
This bilingual quick reference guide brings together the essential formulas and theorems you need for the CCEA GCSE Mathematics specification. Each section pairs clear English explanations with Chinese translations, helping you revise efficiently and build confidence across all key topics.
本中英对照速查手册汇编了 CCEA GCSE 数学课程所要求的核心公式与定理。每个要点均配有英文说明与中文翻译，帮助你高效复习，全面掌握各重点专题。
1. Number Essentials | 数字基础
Percentage change = (difference ÷ original) × 100%.
百分比变化 = (差值 ÷ 原值) × 100%。
To increase or decrease by a percentage, use a decimal multiplier: e.g., increase by 15% corresponds to ×1.15, decrease by 15% corresponds to ×0.85.
按百分比增长或减少可使用小数乘数：例如增长 15% 对应 ×1.15，减少 15% 对应 ×0.85。
r is annual interest rate (as a decimal), and t is time in years. Total amount A = P + I.
单利：I = P × r × t，其中 P 为本金，r 为年利率（以小数表示），t 为时间（年）。总金额 A = P + I。
e n is the number of compounding periods. For depreciation, use a subtraction in the bracket.
复利：A = P × (1 + r/100)ⁿ，若 r 为百分数；或 A = P × (1 + r)ⁿ，若 r 为小数，其中 n 为计息周期数。计算折旧则括号内为减法。
× 10ᵏ, where 1 ≤ a < 10 and k is an integer. Useful for very large or very small numbers.
标准型：N = a × 10ᵏ，其中 1 ≤ a < 10，k 为整数。用于表示极大或极小的数。
time; Density = mass ÷ volume; Pressure = force ÷ area. Rearrange freely using triangles.
速度 = 距离 ÷ 时间；密度 = 质量 ÷ 体积；压强 = 力 ÷ 面积。可使用公式三角形灵活变形。
2. Algebraic Fundamentals | 代数基础
Index laws: aᵐ × aⁿ = aᵐ⁺ⁿ, aᵐ ÷ aⁿ = aᵐ⁻ⁿ, (aᵐ)ⁿ = aᵐⁿ, a⁰ = 1 (a ≠ 0), a⁻ⁿ = 1/aⁿ.
指数法则：aᵐ × aⁿ = aᵐ⁺ⁿ，aᵐ ÷ aⁿ = aᵐ⁻ⁿ，(aᵐ)ⁿ = aᵐⁿ，a⁰ = 1 (a ≠ 0)，a⁻ⁿ = 1/aⁿ。
Expanding brackets: a(b + c) = ab + ac; (a + b)(c + d) = ac + ad + bc + bd.
展开括号：a(b + c) = ab + ac；(a + b)(c + d) = ac + ad + bc + bd。
3x(2x + 3). For a quadratic x² + bx + c, find two numbers that multiply to c and add to b.
因式分解：提取最高公因式（HCF），例如 6x² + 9x = 3x(2x + 3)。对于二次式 x² + bx + c，寻找两数其积为 c 其和为 b。
Difference of two squares: a² – b² = (a + b)(a – b).
平方差公式：a² – b² = (a + b)(a – b)。
Solving linear equations: isolate the unknown by performing the same operation on both sides. When an equation contains fractions, clear denominators first.
解一元一次方程：通过在等式两边执行相同运算来分离未知数。含有分数时先消去分母。
Solving linear inequalities: treat like an equation, but reverse the inequality sign when multiplying or dividing by a negative number.
解一元一次不等式：类似解方程，但当乘或除以负数时需反转不等号方向。
3. Quadratic Equations & Functions | 二次方程与函数
The standard form of a quadratic equation is ax² + bx + c = 0, where a ≠ 0.
一元二次方程的标准形式为 ax² + bx + c = 0，且 a ≠ 0。
Quadratic formula:
x = [ –b ± √b² – 4ac ] / 2a
二次方程求根公式：
x = [ –b ± √b² – 4ac ] / 2a
The discriminant Δ = b² – es two distinct real roots; Δ = 0 gives one repeated real root; Δ < 0 gives no real roots.
判别式 Δ = b² – 4ac 决定根的性质：Δ > 0 有两个不等实根；Δ = 0 有一个重根；Δ < 0 无实根。
Completing the square: rewrite x² + bx as (x + b/2)² – (b/2)². For ax² + bx + c, first factor out a.
配方法：将 x² + bx 变形为 (x + b/2)² – (b/2)²。对于 ax² + bx + c，先提取 a。
Vertex of a quadratic function f(x) = ax² + bx + c: the x-coordinate is – titute to find the y-coordinate. The parabola opens upward if a > 0 and downward if a < 0.
二次函数 f(x) = ax² + bx + c 的顶点：x 坐标为 –b/(2a)；代入可得 y 坐标。当 a > 0 时抛物线开口向上，a < 0 时开口向下。
To solve a quadratic inequality (e.g., x² – 5x + 6 < 0), find critical values from the equation, sketch the graph, and identify the interval where the inequality holds.
解二次不等式（如 x² – 5x + 6 < 0），先求出对应方程的关键值，结合图像判断使不等式成立的区间。
4. Sequences & Series | 数列与级数
linear (arithmetic) sequence has a constant difference d between consecutive terms. The nᵗʰ term is given by uₙ = a + (n – 1)d, where a is the first term.
线性（等差）数列相邻项之差 d 为常数。第 n 项公式为 uₙ = a + (n – 1)d，其中 a 为首项。
For quadratic sequences, the second difference is constant. The nᵗʰ bn + c. Determine a = half of the second difference, then use known terms to find b and c.
对于二次数列，二阶差分为常数。第 n 项通式为 an² + bn + c。先令 a 等于二阶段差分的一半，再代入已知项求 b 与 c。
Geometric sequences (Higher Tier): each term is multiplied by a common ratio r. The nᵗʰ term is arⁿ⁻¹.
等比数列（Higher Tier）：每一项乘以公比 r，第 n 项公式为 arⁿ⁻¹。
rs n², and Fibonacci-type sequences where each term is the sum of the two preceding terms.
常见特殊数列：三角形数 Tₙ = n(n+1)/2、平方数 n²，以及斐波那契型数列（每一项为前两项之和）。
5. Geometry & Angle Rules | 几何与角度定理
Angles on a straight line add up to 180°. Angles around a point sum to 360°. Vertically opposite angles are equal.
平角和为 180°。周角和为 360°。对顶角相等。
In a triangle, interior angles sum to 180°. An exterior angle equals the sum of the two opposite interior angles.
三角形内角和为 180°。外角等于不相邻的两内角之和。
Sum of interior angles of an n-sided polygon = (n – 2) × 180°. Sum of exterior angles (one per vertex) is always 360°.
n 边形内角和 = (n – 2) × 180°。外角和（每个顶点取一个外角）恒为 360°。
Parallel lines: alternate angles are equal, corresponding angles are equal, and co-interior (allied) angles sum to 180°.
平行线：内错角相等，同位角相等，同旁内角互补（和为 180°）。
Circle theorems (Higher Tier): the angle at the centre is twice the angle at the circumference subtended by the same arc. Angles in the same segment are equal. The angle in a semicircle is 90°. Opposite angles in a cyclic quadrilateral sum to 180°. A tangent meets a radius at 90°. The angle between a tangent and a chord equals the angle in the alternate segment.
圆定理（Higher Tier）：同弧所对的圆心角是圆周角的两倍。同弓形内的圆周角相等。半圆所对的圆周角为 90°。圆内接四边形对角互补。切线与半径的夹角为 90°。弦切角等于其弦所对的圆周角。
6. Pythagoras’ Theorem & Trigonometry | 勾股定理与三角学
In a right‑ e, Pythagoras’ theorem states: a² + b² = c², where c is the hypotenuse (the longest side).
在直角三角形中，勾股定理表述为：a² + b² = c²，其中 c 为斜边（最长边）。
TOA): sin θ = opposite/hypotenuse, cos θ = adjacent/hypotenuse, tan θ = opposite/adjacent.
三角函数比（SOH CAH TOA）：sin θ = 对边/斜边，cos θ = 邻边/斜边，tan θ = 对边/邻边。
Exact values for key angles:
| θ | sin θ | cos θ | tan θ |
|---|---|---|---|
| 30° | ½ | √3/2 | 1/√3 |
| 45° | 1/√2 | 1/√2 | 1 |
| 60° | √3/2 | ½ | √3 |
特殊角的精确值：
| θ | sin θ | cos θ | tan θ |
|---|---|---|---|
| 30° | ½ | √3/2 | 1/√3 |
| 45° | 1/√2 | 1/√2 | 1 |
| 60° | √3/2 | ½ | √3 |
Sine rule (Higher, non‑right‑angled triangles): a/sin A = b/sin B = c/sin C, or sin A/a = sin B/b = sin C/c.
正弦定理（Higher，非直角三角形）：a/sin A = b/sin B = c/sin C，或 sin A/a = sin B/b = sin C/c。
Cosine rule (Higher): a² = b² + c² – two sides and the included angle, or for finding an angle when all three sides are known.
余弦定理（Higher）：a² = b² + c² – 2bc cos A。已知两边及其夹角求对边，或已知三边求角均可使用。
Area of a triangle using sine: Area = ½ab sin C. (Higher Tier)
用正弦求三角形面积：面积 = ½ab sin C。（Higher Tier）
7. Area, Perimeter & Volume | 面积、周长与体积
Rectangle: Area = length × width, Perimeter = 2(l + w).
矩形：面积 = 长 × 宽，周长 = 2(长 + 宽)。
Triangle: Area = ½ × base × perpendicular height.
三角形：面积 = ½ × 底 × 垂直高度。
ium: Area = ½(a + b)h, where a and b are parallel sides and h is the perpendicular height.
平行四边形：面积 = 底 × 高。梯形：面积 = ½(a + b)h，其中 a 和 b 为平行边，h 为垂直高度。
: Circumference = 2πr or πd, Area = πr², where r is the radius and d = 2r is the diameter.
圆：周长 = 2πr 或 πd，面积 = πr²，其中 r 为半径，d = 2r 为直径。
(θ/360) × 2πr, Sector area = (θ/360) × πr², where θ is the angle at the centre in degrees.
弧长 = (θ/360) × 2πr，扇形面积 = (θ/360) × πr²，其中 θ 为圆心角的度数。
Prism volume = area of cross‑ gth. Cylinder volume = πr²h, curved surface area = 2πrh, total surface area = 2πrh + 2πr².
柱体体积 = 横截面积 × 长度。圆柱体积 = πr²h，侧面积 = 2πrh，总表面积 = 2πrh + 2πr²。
al height. Cone volume = ⅓πr²h, curved surface area = πrl where l is the slant height (l = √r² + h²). Sphere volume = ⁴⁄₃πr³, surface area = 4πr². (Higher Tier)
棱锥体积 = ⅓ × 底面积 × 垂直高度。圆锥体积 = ⅓πr²h，侧面积 = πrl，其中 l 为斜高 (l = √r² + h²)。球体积 = ⁴⁄₃πr³，表面积 = 4πr²。（Higher Tier）
8. Coordinate Geometry & Graphs | 坐标几何与图像
The gradient of a straight line through (x₁, y₁) and (x₂, y₂) is m = (y₂ – y₁) / (x₂ – x₁).
过点 (x₁, y₁) 与 (x₂, y₂) 的直线斜率（梯度）为 m = (y₂ – y₁) / (x₂ – x₁)。
t of two points: M = ( (x₁ + x₂)/2 , (y₁ + y₂)/2 ). Distance between two points: d = √[(x₂ – x₁)² + (y₂ – y₁)²].
两点间中点：M = ( (x₁ + x₂)/2 , (y₁ + y₂)/2 )。两点间距离：d = √[(x₂ – x₁)² + (y₂ – y₁)²]。
Equation of a straight line: slope‑intercept form y = mx + c (m is gradient, c is y‑intercept). Point‑slope form: y – y₁ = m(x – x₁).
直线方程：斜截式 y = mx + c（m 为斜率，c 为 y 轴截距）。点斜式：y – y₁ = m(x – x₁)。
Parallel lines have the same gradient. Perpendicular lines have gradients whose product is –1 (m₁ × m₂ = –1).
平行直线的斜率相等。垂直直线的斜率乘积为 –1（m₁ × m₂ = –1）。
ratic graphs y = ax² + bx + c are parabolas; cubic graphs y = ax³ + bx² + cx + d have an S‑shape; reciprocal graphs y = a/x form hyperbolas. Exponential graphs y = a × bˣ show rapid growth or decay.
二次图像 y = ax² + bx + c 为抛物线；三次图像 y = ax³ + bx² + cx + d 呈 S 形；反比例图像 y = a/x 为双曲线；指数图像 y = a × bˣ 展示快速增长或衰减。
To solve equations graphically, locate the intersection of two graphs. The x‑coordinates of intersection points give the solutions.
图像法解方程即求两个图像交点的 x 坐标，交点横坐标即为方程的解。
9. Transformations & Vectors | 变换与向量
same vector (ᵤ/ᵥ) (column vector written as x on top, y below). Vector addition: (ᵃ/₆) + (ᶜ/₇) = (ᵃ⁺ᶜ/₆⁺₇).
平移：各点按相同向量 (ᵤ/ᵥ)（列向量，上方为 x 位移，下方为 y 位移）移动。向量加法：(ᵃ/₆) + (ᶜ/₇) = (ᵃ⁺ᶜ/₆⁺₇)。
Reflection: mirroring in a line (e.g., x‑axis, y‑axis, y = x, y = –x). Each point is mapped perpendicularly to the mirror line.
反射：沿直线（如 x 轴、y 轴、y = x、y = –x）镜像对称，各点与镜像的连线垂直于反射轴。
en angle (clockwise or anticlockwise). A rotation of 180° about the origin maps (x, y) → (–x, –y).
旋转：绕旋转中心按指定角度（顺时针或逆时针）转动。关于原点旋转 180° 将 (x, y) 映射为 (–x, –y)。
Enlargement: a transformation with a centre of enlargement and a scale factor k. If k > 1 the shape gets larger; 0 < k < 1 the shape gets smaller. Distances from the centre are multiplied by k. A negative scale factor also produces a rotation of 180°.
放大：以放大中心和比例因子 k 进行变换。k > 1 图形放大；0 < k < 1 图形缩小。各点到中心的距离均乘以 k。负比例因子还会产生 180° 旋转效果。
A vector describes a translation: its magnitude (length) is found via Pythagoras, e.g., |(ᵃ/₆)| = √a² + b². Scalar multiplication: k × (ᵃ/₆) = (ᵏᵃ/ᵏ₆).
向量用于描述平移：其模（长度）由毕达哥拉斯定理求得，例如 |(ᵃ/₆)| = √a² + b²。标量乘法：k × (ᵃ/₆) = (ᵏᵃ/ᵏ₆)。
10. Probability | 概率
comes) ÷ (total number of possible outcomes), assuming equally likely outcomes. 0 ≤ P ≤ 1.
事件概率 = (有利结果数) ÷ (等可能结果总数)，且 0 ≤ P ≤ 1。
events A and B, P(A or B) = P(A) + P(B). For independent events, P(A and B) = P(A) × P(B).
互斥事件 A 与 B：P(A 或 B) = P(A) + P(B)。独立事件：P(A 且 B) = P(A) × P(B)。
Tree diagrams: multiply probabilities along branches for “and”, add for “or”. The sum of probabilities at each branch point is 1.
树形图：沿分支相乘求“且”的概率，相加求“或”。每个分支点的概率之和为 1。
y × number of trials. Relative frequency provides an experimental estimate of probability.
期望频数 = 概率 × 试验次数。相对频率用于给出概率的实验估计。
Conditional probability (Higher): P(A|B) = P(A and B) / P(B), where P(A|B) is the probability of A given that B has occurred.
条件概率（Higher）：P(A|B) = P(A 且 B) / P(B)，其中 P(A|B) 表示在 B 已发生的条件下 A 发生的概率。
11. Statistics | 统计
. Median = middle value when data are ordered. Mode = most frequent value. Range = maximum – minimum.
集中趋势量数：平均数 = 数据总和 ÷ 数据个数。中位数 = 排序后居中值。众数 = 出现次数最多的值。极差 = 最大值 – 最小值。
ated mean = Σ(f × midpoint) ÷ Σf. The modal class is the class with the highest frequency.
对于分组数据，估计平均数 = Σ(频率 × 组中点) ÷ Σ频率。众数组为频率最高的组。
Quartiles: Q₁ is the lower quartile (25% of data), Q₂ the median (50%), Q₃ the upper quartile (75%). Interquartile range (IQR) = Q₃ – Q₁.
四分位数：下四分位数 Q₁（位于 25% 处），中位数 Q₂（50%），上四分位数 Q₃（75%）。四分位距 (IQR) = Q₃ – Q₁。
Cumulative frequency graphs enable estimation of quartiles and percentiles. A box plot (box‑and‑whisker) displays minimum, Q₁, median, Q₃ and maximum.
累积频数图可用于估计四分位数与百分位数。箱线图展示最小值、Q₁、中位数、Q₃ 及最大值。
quency ÷ class width. The area of each bar is proportional to frequency. Use frequency density to plot bars when class widths are unequal.
直方图：频率密度 = 频率 ÷ 组距。每个直条的面积与频率成正比。当组宽不等时，必须用频率密度来绘制直条高度。
Scatter graphs show the relationship between two variables. Correlation can be positive, negative or none. A line of best fit (by eye) passes through the mean point (x̄, ȳ) and can be used to make predictions.
散点图显示两变量间的关系。相关性可分为正相关、负相关或不相关。最佳拟合线（目测）过点 (x̄, ȳ) 并可用于预测。
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