📚 Year 10 CCEA Maths: Formula & Theorem Quick Reference Guide | CCEA Year 10 数学公式定理速查手册
Welcome to your comprehensive quick-reference guide for Year 10 CCEA Mathematics. This handbook brings together all the essential formulas, theorems and key facts you need to master the foundation and higher tier topics, organised so that you can revise a whole topic in a single sitting. Keep it beside you for homework, classwork and the final weeks before your GCSE modules.
欢迎使用 Year 10 CCEA 数学全面速查手册。本手册汇集了掌握 GCSE 基础与高阶主题所需的所有基本公式、定理和关键知识点，并按主题编排，让你一次复习即可覆盖完整板块。无论是家庭作业、课堂练习，还是 GCSE 模块考前的最后冲刺，都请把它放在手边。
1. Algebraic Fundamentals | 代数基础
Index laws: a ≠ 0. When the index is negative, a⁻ⁿ = 1 / aⁿ, and a fractional index means a root, so a1/2 = √a and a1/3 = ∛ d examples, because CCEA questions frequently combine two or three laws in one expression.
指数法则： aᵐ × aⁿ = aᵐ⁺ⁿ，(aᵐ)ⁿ = aᵐⁿ，aᵐ ÷ aⁿ = aᵐ⁻ⁿ，且对于任何 a ≠ 0 都有 a⁰ = 1。当指数为负时，a⁻ⁿ = 1 / aⁿ；分数指数表示开方，因此 a1/2 = √a，a1/3 = ∛a。请以一般形式而非孤立例题来记忆，因为 CCEA 题目常在一个表达式中同时考查两三条法则。
Expanding single brackets: = ab + ac + ad. Be careful with negative multipliers: −3(x − 4) = −3x + 12, not −3x − 12.
展开单项括号： a(b + c) = ab + ac，并可推广为 a(b + c + d) = ab + ac + ad。注意负系数：−3(x − 4) = −3x + 12，而不是 −3x − 12。
Expanding double brackets: eneral case (ax + b)(cx + d), use FOIL (First, Outer, Inner, Last) to obtain acx² + (ad + bc)x + bd. Always collect like terms at the end.
展开两项括号： (x + a)(x + b) = x² + (a + b)x + ab。一般情形 (ax + b)(cx + d) 用 FOIL 法（首、外、内、末）展开得 acx² + (ad + bc)x + bd，最后务必合并同类项。
Factorising by common factor: ghest common factor first, including the common factor in the coefficients and in the powers of the variables.
提取公因式分解： ab + ac = a(b + c)。首先寻找最大公因式，既要看系数的公因数，也要看字母的公共次幂。
Difference of two squares: he most useful factorisation patterns and also appears inside surd and fraction questions.
平方差公式： a² − b² = (a + b)(a − b)。这是最有用的因式分解模式之一，也常出现在根式与分式化简题中。
a² − b² = (a + b)(a − b) | x² + (a + b)x + ab = (x + a)(x + b)
2. Equations and Inequalities | 方程与不等式
Solving a linear equation: every step, and always substitute your answer back into the original equation to check it.
解一元一次方程： 对于 ax + b = c，两边先减去 b，再除以 a，得 x = (c − b) / a。要写出每一步，并把答案代回原方程验算。
Quadratic formula: if ax² + bx + c = 0 with a ≠ 0, then x = [−b ± √b² − 4ac] / (2a). Write the formula down before substituting, and keep the whole numerator over 2a rather than dividing term by term, which is the single most common slip in this topic.
二次公式： 若 ax² + bx + c = 0 且 a ≠ 0，则 x = [−b ± √b² − 4ac] / (2a)。代入前先把公式抄下来，并注意分母 2a 作用于整个分子，不要把分子拆开分别除，这是本主题最常见的失误。
The discriminant: roots, if Δ = 0 there is exactly one repeated root, and if Δ < 0 there are no real roots.
判别式： Δ = b² − 4ac 决定一元二次方程实根的个数。Δ > 0 有两个不同实根，Δ = 0 恰有一个重根，Δ < 0 则没有实根。
Inequalities: resent solutions on a number line using an open circle for < or > and a closed circle for ≤ or ≥.
不等式： 解法与方程相同，但谨记：两边同乘或同除一个负数时，不等号方向要反向。在数轴上表示解集时，< 或 > 用空心圆，≤ 或 ≥ 用实心圆。
Compound inequalities: ≤ 6, then divide by 2 to obtain 1 < x ≤ 3. The final answer must keep x in the middle.
复合不等式： 对于 3 < 2x + 1 ≤ 7 这类式子，要对三个部分同时做相同运算：整体减 1 得 2 < 2x ≤ 6，再整体除以 2 得 1 < x ≤ 3。最终答案必须把 x 放在中间。
x = [−b ± √b² − 4ac] / (2a) | Δ = b² − 4ac
3. Sequences and the nth Term | 数列与第 n 项
Linear (arithmetic) sequences: he nth term begins 3n; testing n = 1 gives 3 + b = 5, so b = 2 and the nth term is 3n + 2.
线性（等差）数列： 第 n 项形如 an + b，其中 a 是公差，b 可由代入 n = 1 求出。例如数列 5, 8, 11, 14 的公差为 3，故第 n 项形如 3n；代入 n = 1 得 3 + b = 5，所以 b = 2，第 n 项为 3n + 2。
Quadratic sequences: if the second difference is constant and equal to 2a, the nth term contains an². Subtract an² from each term to leave a linear sequence, then work out the remaining nth term in the usual way.
二次数列： 若二阶差分为常数且等于 2a，则第 n 项含 an²。用每一项减去 an²，剩下的就是线性数列，再按常规方法求出剩余部分即可。
Geometric sequences: each term is found by multiplying the previous term by a fixed ratio r, so the nth term is ar⁽ⁿ⁻¹⁾ where a is the first term. Recognising whether a sequence is arithmetic or geometric is often the key first step in a problem.
等比数列： 每一项由前一项乘以固定公比 r 得到，故第 n 项为 ar⁽ⁿ⁻¹⁾，其中 a 是首项。判断数列是等差还是等比，往往是解题的第一步。
Special sequences to recognise: square numbers 1, 4, 9, 16, 25; cube numbers 1, 8, 27, 64, 125; triangular numbers 1, 3, 6, 10, 15; and the Fibonacci sequence 1, 1, 2, 3, 5, 8, 13, where each term is the sum of the two before it.
需要牢记的特殊数列： 平方数 1, 4, 9, 16, 25；立方数 1, 8, 27, 64, 125；三角数 1, 3, 6, 10, 15；以及斐波那契数列 1, 1, 2, 3, 5, 8, 13，其中每一项等于前两项之和。
4. Straight-Line Graphs and Coordinate Geometry | 直线图像与坐标几何
Equation of a straight line: s the y-intercept. To find the equation from two points, first calculate the gradient and then substitute one point to find c.
直线方程： y = mx + c，其中 m 是斜率，c 是 y 轴截距。已知两点求直线方程时，先算斜率，再代入其中一点求出 c。
Gradient between two points: ₂ − y₁) / (x₂ − x₁). A positive gradient rises from left to right, a negative gradient falls, a zero gradient is horizontal and an undefined gradient is vertical.
两点间斜率： 对点 (x₁, y₁) 与 (x₂, y₂)，m = (y₂ − y₁) / (x₂ − x₁)。正斜率自左向右上升，负斜率下降，斜率为零是水平线，斜率不存在是竖直线。
Midpoint and distance: /2), and the distance between them is √[(x₂ − x₁)² + (y₂ − y₁)²], which is just Pythagoras' theorem applied to the coordinate grid.
中点与距离： (x₁, y₁) 与 (x₂, y₂) 的中点为 ((x₁ + x₂)/2, (y₁ + y₂)/2)，两点距离为 √[(x₂ − x₁)² + (y₂ − y₁)²]，本质上是勾股定理在坐标格上的应用。
Parallel and perpendicular lines: fy m₁ × m₂ = −1, so the perpendicular gradient is the negative reciprocal of the original.
平行与垂直： 平行直线斜率相等，m₁ = m₂；垂直直线满足 m₁ × m₂ = −1，因此垂直斜率是原斜率的负倒数。
y = mx + c | m = (y₂ − y₁) / (x₂ − x₁) | m₁ × m₂ = −1
5. Ratio, Proportion and Percentages | 比例、比值与百分比
Simplifying and sharing in a ratio: divide every part by the highest common factor to simplify. To share an amount in the ratio a : b, add the parts to get a + b, divide the amount by that total to find one part, then multiply by each share.
化简比值与按比分配： 各项同时除以最大公因数即可化简。按 a : b 分配总量时，先求出份数之和 a + b，用总量除以份数之和得到一份的值，再分别乘以各份的数量。
Direct and inverse proportion: xy is constant. Find k from the given pair of values, then use it to answer the question.
正比例与反比例： 正比例写作 y = kx，即 y/x 为常数；反比例写作 y = k / x，即 xy 为常数。先用已知的一对数值求出 k，再代入求解。
Percentage change: hange = (change ÷ original) × 100. For repeated change, use multipliers: an increase of 15% is × 1.15 and a decrease of 15% is × 0.85.
百分比变化： 百分比变化 = (变化量 ÷ 原值) × 100。遇到连续变化时使用乘数法：增加 15% 是 × 1.15，减少 15% 是 × 0.85。
Compound interest and depreciation: incipal, r the rate per period and n the number of periods. Depreciation uses the same formula with a minus sign.
复利与折旧： 乘数法给出终值 = P × (1 ± r/100)ⁿ，其中 P 是本金，r 是每期利率，n 是期数。折旧使用同一公式，符号取负。
Final amount = P × (1 ± r/100)ⁿ
6. Angles, Polygons and Circles | 角度、多边形与圆
Core angle facts: angles on a straight line add to 180°, angles around a point add to 360°, vertically opposite angles are equal, and angles in a triangle add to 180°. For parallel lines, corresponding and alternate angles are equal and co-interior angles add to 180°.
基本角度事实： 直线上的邻角之和为 180°，围绕一点的角度之和为 360°，对顶角相等，三角形内角和为 180°。平行线中，同位角与内错角相等，同旁内角之和为 180°。
Angles in polygons: lar polygon, each exterior angle is 360° / n and each interior angle is 180° − (360° / n).
多边形的角： n 边形内角和为 (n − 2) × 180°。正多边形中，每个外角为 360° / n，每个内角为 180° − (360° / n)。
Circle formulas: = πd, and area A = πr². Remember to use the radius, not the diameter, in the area formula.
圆的公式： 周长 C = 2πr = πd，面积 A = πr²。注意面积公式中使用的是半径而不是直径。
Arcs and sectors: θ is the angle at the centre in degrees. A sector is simply a fraction of the full circle.
弧长与扇形： 弧长 = (θ / 360) × 2πr，扇形面积 = (θ / 360) × πr²，其中 θ 为圆心角（以度为单位）。扇形本质上是整个圆的一个比例部分。
C = 2πr | A = πr² | Interior angle sum = (n − 2) × 180°
7. Area, Surface Area and Volume | 面积、表面积与体积
Areas of plane shapes: um A = ½(a + b)h. For a triangle, the height must be perpendicular to the base you choose.
平面图形的面积： 长方形 A = l × w；三角形 A = ½ × 底 × 垂直高；平行四边形 A = 底 × 垂直高；梯形 A = ½(a + b)h。求三角形面积时，高必须垂直于所选底边。
Prisms and cylinders: ound by adding the areas of every face; for a closed cylinder, surface area = 2πr² + 2πrh.
棱柱与圆柱： 任何棱柱的体积都是截面积 × 长度，因此圆柱体积 V = πr²h。表面积等于各个面的面积之和；封闭圆柱的表面积 = 2πr² + 2πrh。
Cones and spheres (higher tier): ound using Pythagoras. The volume of a sphere is V = 4/3 πr³ and its surface area is 4πr².
圆锥与球（高阶）： 圆锥体积 V = ⅓πr²h，侧面积 πrl，其中 l 是用勾股定理求出的母线长。球的体积 V = 4/3 πr³，表面积 4πr²。
Units for volume: 1000 cm³. Unit conversion errors cost more marks than almost any other slip in this topic.
体积单位： 记住 1 cm³ = 1 ml，1 m³ = 1 000 000 cm³，1 升 = 1000 cm³。单位换算错误在本主题中失分比任何其他失误都多。
8. Pythagoras' Theorem and Trigonometry | 勾股定理与三角学
Pythagoras' theorem: orter side, rearrange to a² = c² − b². Always identify the hypotenuse before substituting.
勾股定理： 在直角三角形中，a² + b² = c²，其中 c 为斜边。求直角边时变形为 a² = c² − b²。代入前务必先确认斜边。
Trigonometric ratios (SOHCAHTOA): e / adjacent. Label the three sides relative to the angle you are using, then choose the ratio that links the two sides involved.
三角函数（SOHCAHTOA）： sin θ = 对边 / 斜边，cos θ = 邻边 / 斜边，tan θ = 对边 / 邻边。先相对所用角标记三边，再选择连接这两条边的比值。
Exact values to memorise: sin 30° = ½, cos 60° = ½, tan 45° = 1, sin 45° = cos 45° = √2 / 2, and sin 60° = cos 30° = √3 / 2. These appear in non-calculator papers.
需要背的精确值： sin 30° = ½，cos 60° = ½，tan 45° = 1，sin 45° = cos 45° = √2 / 2，sin 60° = cos 30° = √3 / 2。这些会出现在不可使用计算器的试卷中。
Sine and cosine rules (higher tier): ² − 2bc cos A. The area of any triangle is ½ab sin C. Use the cosine rule when you know three sides or two sides and the included angle.
正弦定理与余弦定理（高阶）： a / sin A = b / sin B = c / sin C，以及 a² = b² + c² − 2bc cos A。任意三角形面积 = ½ab sin C。已知三边或两边及其夹角时使用余弦定理。
a² + b² = c² | Area = ½ab sin C | a² = b² + c² − 2bc cos A
9. Probability | 概率
The probability scale: ty lies between 0 and 1, so 0 ≤ P(A) ≤ 1. The probability of an event not happening is P(A') = 1 − P(A), which is often the quickest route to an answer.
概率的取值范围： 任何概率都在 0 与 1 之间，即 0 ≤ P(A) ≤ 1。事件不发生的概率为 P(A') = 1 − P(A)，这往往是解题最快的途径。
The addition rule: vents that can occur together, subtract the overlap: P(A or B) = P(A) + P(B) − P(A and B).
加法法则： 对互斥事件，P(A 或 B) = P(A) + P(B)。若两事件可能同时发生，则要减去重叠部分：P(A 或 B) = P(A) + P(B) − P(A 且 B)。
The multiplication rule: dent events, P(A and B) = P(A) × P(B). If the events are not independent because the first
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