📚 Year 11 OCR Maths: Quick Reference Formula & Theorem Handbook | Year 11 OCR 数学：公式定理速查手册
This is a concise, exam-focused summary of all essential formulas and theorems for the OCR GCSE (9-1) Mathematics Higher Tier (Year 11). Use it to check your recall before topic tests, mocks, and the final exams. Every entry is paired with a brief explanation so you understand not just what the formula is, but when and why to use it.
这是一份简洁的、针对 OCR 普通中等教育证书数学高阶（Year 11）的必考公式与定理速查手册。适合在单元测验、模拟考和最终大考前用来检查记忆。每一条公式都配有简短说明，帮助你不仅知道公式是什么，还能理解何时使用、为何使用它。
1. Number Essentials | 数的要点
A strong number foundation makes algebra and ratio problems much easier. These are the must-know facts and multiplicative relationships.
打好数的基础能让代数和比例问题简单得多。以下是你必须掌握的基本事实和乘法关系。
- BIDMAS / BODMAS: Brackets, Indices (Order), Division/Multiplication, Addition/Subtraction – the order of operations.
运算顺序：括号，指数，乘除，加减。 - Product Rule for Counting there are m ways for one event and n ways for another, the total number of ways is m × n.
计数乘法原理：若一件事有 m 种方式，另一件事有 n 种方式，则总方式数为 m × n。 - Multiples and Factors: LCM (Lowest Common Multiple) and HCF (Highest Common Factor) can be found by writing numbers in prime factor form. HCF uses the smallest powers of common primes; LCM uses the largest powers.
倍数与因数：通过质因数分解求最小公倍数 (LCM) 与最大公因数 (HCF)。HCF 取各公共质数的最小指数；LCM 取最大指数。 - Standard Form written as a × 10ⁿ where 1 ≤ a < 10 and n is an integer. For example, 0.0032 = 3.2 × 10⁻³.
标准形式：一个数写作 a × 10ⁿ，其中 1 ≤ a < 10，n 为整数。例如 0.0032 = 3.2 × 10⁻³。
2. Algebra – Core Manipulation | 代数 — 核心变形
Algebraic fluency is essential across the entire syllabus. Below are the key skills for rearranging, expanding, and factorising.
代数运算的熟练度贯穿整个大纲。以下是移项、展开和因式分解的关键技能。
- Expanding brackets: a(b + c) = ab + ac. For double brackets: (x + a)(x + b) = x² + (a+b)x + ab.
展开括号：a(b + c) = ab + ac。双括号展开：(x + a)(x + b) = x² + (a+b)x + ab。 - Factorising quadratics (x + q) where p+q = b and pq = c. For ax²+bx+c, use splitting the middle term or grouping.
二次三项式因式分解：x² + bx + c = (x + p)(x + q)，其中 p+q = b，pq = c。形如 ax²+bx+c 时，使用十字相乘法或分组法。 - Laws of indices: aᵐ × aⁿ = aᵐ⁺ⁿ, aᵐ ÷ aⁿ = aᵐ⁻ⁿ, (aᵐ)ⁿ = aᵐⁿ, a⁰ = 1 (a≠0), a⁻ⁿ = 1/aⁿ, a¹/² = √a, aᵐ/ⁿ = ⁿ√aᵐ.
指数运算律：aᵐ × aⁿ = aᵐ⁺ⁿ；aᵐ ÷ aⁿ = aᵐ⁻ⁿ；(aᵐ)ⁿ = aᵐⁿ；a⁰ = 1 (a≠0)；a⁻ⁿ = 1/aⁿ；a¹/² = √a；aᵐ/ⁿ = ⁿ√aᵐ。 - Solving linear equations: Do the same to both sides. E.g. 3x − 7 = 5 → 3x = 12 → x = 4.
解一元一次方程：等式两边进行相同运算。如 3x − 7 = 5 → 3x = 12 → x = 4。
3. Sequences | 数列
Recognising patterns and using the nth term formula is a common non-calculator skill.
识别模式并使用通项公式是常见的非计算器题型技能。
- Arithmetic (linear) sequences rm = a + (n − 1)d where a is the first term, d is the common difference. E.g. 5, 9, 13, 17… gives 5 + 4(n − 1) = 4n + 1.
等差数列（线性数列）：通项 = a + (n − 1)d，其中 a 为首项，d 为公差。如 5, 9, 13, 17… 的通项为 5 + 4(n − 1) = 4n + 1。 - Quadratic sequences: The second difference is constant. The nth term has the form an² + bn + c. Find a from half the second difference, then solve for b and c.
二次数列：二阶差为常数。通项形式为 an² + bn + c。a 取二阶差的一半，再解 b 和 c。 - Geometric sequences: Each term is multiplied by a constant ratio r. nth term = arⁿ⁻¹.
等比数列：每一项乘以固定公比 r。通项 = arⁿ⁻¹。 - Fibonacci-type sequences: Each term is the sum of the two preceding terms. Tₙ = Tₙ₋₁ + Tₙ₋₂.
类斐波那契数列：每一项是前两项之和。Tₙ = Tₙ₋₁ + Tₙ₋₂。
4. Graphs & Coordinate Geometry | 图形与坐标几何
Understanding how algebraic equations connect to geometric shapes on a grid is central to many problem-solving questions.
理解代数方程与网格上几何图形的联系，是许多综合问题的核心。
- Straight line graphs ines have equal gradients; perpendicular lines have gradients with product −1 (m₁m₂ = −1).
直线图形：y = mx + c，m 为斜率（纵差/横差），c 为 y 截距。平行线斜率相等；垂直线斜率之积为 −1 (m₁m₂ = −1)。 - Midpoint of a line segment: Midpoint = ((x₁+x₂)/2, (y₁+y₂)/2).
线段中点：中点 = ((x₁+x₂)/2, (y₁+y₂)/2)。 - Distance between two points: d = √[(x₂ − x₁)² + (y₂ − y₁)²]. (Based on Pythagoras’ theorem.)
两点间距离：d = √[(x₂ − x₁)² + (y₂ − y₁)²]。（基于毕达哥拉斯定理。） - Quadratic graphs parabola. Roots are where y = 0. The turning point can be found by completing the square.
二次函数图形：y = ax² + bx + c 为抛物线。根为 y=0 时的 x 值。通过配方法可求得顶点（转折点）。 - Equation of a circle (centre origin): x² + y² = r². For centre (a, b): (x − a)² + (y − b)² = r².
圆的方程（圆心在原点）：x² + y² = r²。圆心 (a, b) 时：(x − a)² + (y − b)² = r²。
5. Quadratic Equations & the Formula | 二次方程与求根公式
Quadratic equations often have two solutions. You need to be equally confident factorising, using the formula, or completing the square.
二次方程常常有两个解。你需要熟练掌握因式分解、公式法和配方法。
- The Quadratic Formula: For ax² + bx + c = 0, x = [−b ± √b² − 4ac] / (2a). This is given on the formula sheet, but you must be able to substitute correctly.
二次求根公式：对于 ax² + bx + c = 0，x = [−b ± √b² − 4ac] / (2a)。公式表上会提供，但你必须会正确代入数值。 - Discriminant: D = b² − 4ac. D > 0 ⇒ two real roots; D = 0 ⇒ one repeated real root; D < 0 ⇒ no real roots.
判别式：D = b² − 4ac。D > 0 ⇒ 两个相异实根；D = 0 ⇒ 一个重实根；D < 0 ⇒ 无实根。 - Completing the square: x² + bx = (x + b/2)² − (b/2)². This form reveals the turning point of a quadratic graph.
配方法：x² + bx = (x + b/2)² − (b/2)²。这种形式可以揭示二次函数图像的顶点坐标。
6. Pythagoras & Trigonometry | 毕达哥拉斯与三角学
Right-angled triangle problems can be solved with Pythagoras and SOHCAHTOA; non-right-angled triangles need the sine and cosine rules.
直角三角形问题可以用毕达哥拉斯定理和 SOHCAHTOA 解决；非直角三角形则需要正弦定理和余弦定理。
| Formula / 公式 | Usage / 用法 |
|---|---|
| a² + b² = c² (c is the hypotenuse) | Pythagoras’ theorem – find a missing side in a right-angled triangle. 毕达哥拉斯定理 — 求直角三角形中的未知边长。 |
| sin θ = opp/hyp, cos θ = adj/hyp, tan θ = opp/adj | SOHCAHTOA – basic trig ratios for right-angled triangles. SOHCAHTOA — 直角三角形的基本三角比。 |
| Exact trig values: sin 30°=½, sin 45°=1/√2, sin 60°=√3/2 cos 30°=√3/2, cos 45°=1/√2, cos 60°=½ tan 30°=1/√3, tan 45°=1, tan 60°=√3 | Must be memorised for non-calculator papers. 非计算器考试必须熟记的精确三角函数值。 |
| a / sin A = b / sin B = c / sin C | Sine Rule – when you have two angles and any side, or two sides and a non-included angle. 正弦定理 — 已知两角一边，或两边及一对角时使用。 |
| a² = b² + c² − 2bc cos A | Cosine Rule – when you have two sides and the included angle, or three sides. 余弦定理 — 已知两边及其夹角，或已知三边时使用。 |
| Area = ½ ab sin C | Area of a triangle formula – when you know two sides and the included angle. 三角形面积公式 — 已知两边及夹角时使用。 |
7. Geometry – Angles, Polygons & Circle Theorems | 几何 — 角、多边形与圆定理
Geometric reasoning often appears in multi-step proof questions. Know your circle theorems and angle facts cold.
几何推理常出现在多步证明题中。必须熟记圆定理和角的基本性质。
- Angles on a straight line sum to 180°; angles around a point sum to 360°.
平角：一直线上的角之和为 180°；周角：绕一点的角之和为 360°。 - Vertically opposite angles are equal. Alternate angles, corresponding angles, and allied (co-interior) angles rules apply for parallel lines.
对顶角相等。平行线中，内错角相等、同位角相等、同旁内角互补。 - Sum of interior angles of an n-sided polygon: (n − 2) × 180°. Exterior angles sum to 360°.
多边形内角和：n 边形内角和为 (n − 2) × 180°。外角和恒为 360°。 - Circle Theorems (memorise all eight standard ones): Angle at centre is twice angle at circumference; angle in a semicircle is 90°; angles in the same segment are equal; opposite angles in a cyclic quadrilateral sum to 180°; tangent is perpendicular to radius; angle between tangent and chord equals angle in the alternate segment; perpendicular from centre to chord bisects the chord; equality of tangents from a point.
圆定理（记住全部八个）：圆心角等于圆周角的两倍；直径所对的圆周角是直角；同弧上的圆周角相等；圆内接四边形对角互补；切线与过切点的半径垂直；弦切角等于同弧上的圆周角（交错弦角定理）；圆心到弦的垂线平分弦；从圆外一点引的两条切线等长。
8. Perimeter, Area & Volume | 周长、面积与体积
You must know the standard shapes. Compound shapes can be split into rectangles, triangles, and sectors.
你必须掌握标准形状的公式。复杂图形可分割为矩形、三角形和扇形。
| Shape / 形状 | Perimeter / Area / Volume / 周长 / 面积 / 体积 |
|---|---|
| Rectangle | Area = l × w; Perimeter = 2(l + w) |
| Triangle | Area = ½ × base × height (also ½ ab sin C for non-right triangles) |
| Parallelogram | Area = base × perpendicular height |
| Trapezium | Area = ½ (a + b) × h, where a and b are parallel sides. |
| Circle | Circumference C = 2πr = πd; Area A = πr² |
| Sector of a circle | Arc length = (θ/360) × 2πr; Sector area = (θ/360) × πr² |
| Cylinder | Curved surface area = 2πrh; Total surface area = 2πrh + 2πr²; Volume = πr²h |
| Cone (given formulas) | Curved surface area = πrl; Total surface area = πrl + πr²; Volume = ⅓πr²h |
| Sphere (given formulas) | Surface area = 4πr²; Volume = ⁴⁄₃πr³ |
| Pyramid | Volume = ⅓ × base area × height |
| Prism | Volume = area of cross-section × length |
For cones and spheres, the formula is printed on the exam paper. However, you must be able to identify radius, height, or slant height correctly.
圆锥和球的公式在试卷上会提供。但你必须能正确地识别半径、高或斜高。
9. Ratio, Proportion & Rates of Change | 比率、比例与变化率
Multiplicative reasoning runs through direct and inverse proportion, percentages, and compound measures.
乘法推理贯穿正比例、反比例、百分数和复合单位的运算。
- Direct proportion: y ∝ x ⇒ y = kx. Graph is a straight line through the origin.
正比例：y ∝ x ⇒ y = kx。图像是过原点的直线。 - Inverse proportion: y ∝ 1/x ⇒ y = k/x. Graph is a hyperbola.
反比例：y ∝ 1/x ⇒ y = k/x。图像是双曲线。 - Percentage change: % change = (new − original) / original × 100%. Use decimal multipliers: increase by 15% → ×1.15; decrease by 15% → ×0.85.
百分数变化：变化率 = (新值 − 原值) / 原值 × 100%。使用小数乘数：增加15% 即 ×1.15；减少15% 即 ×0.85。 - Compound interest al = P × (1 + r/100)ⁿ, where P is principal, r is rate per period, n is number of periods.
复利：总额 = P × (1 + r/100)ⁿ，P 为本金，r 为每期利率，n 为期数。 - Speed, Density, Pressure e; Density = mass / volume; Pressure = force / area. (Formula triangles are helpful aids.)
速度、密度、压强：速度 = 路程 / 时间；密度 = 质量 / 体积；压强 = 压力 / 受力面积。（公式三角形是可用的辅助工具。）
10. Probability | 概率
Probability always lies between 0 (impossible) and 1 (certain). You may express it as a fraction, decimal, or percentage.
概率值总是在 0（不可能）和 1（必然）之间。你可以用分数、小数或百分比表示。
- Basic rule: P(event) = number of favourable outcomes / total number of outcomes.
基本法则：P(事件) = 有利结果数 / 所有可能结果总数。 - AND rule: For independent events, P(A and B) = P(A) × P(B).
“且”的法则：对于独立事件，P(A 且 B) = P(A) × P(B)。 - OR rule: For mutually exclusive events, P(A or B) = P(A) + P(B).
“或”的法则：对于互斥事件，P(A 或 B) = P(A) + P(B)。 - Conditional probability P(A given B) = P(A and B) / P(B). On tree diagrams, second-level branches are conditional.
条件概率：P(已知B发生后A发生) = P(A且B) / P(B)。在树状图中，第二层的分支就是条件概率。 - Expected frequency: Expected = probability × number of trials.
期望频数：期望值 = 概率 × 试验次数。
11. Statistics – Data Handling | 统计 — 数据处理
Interpretation of diagrams, averages, and spread is tested heavily. Know the difference between representing data and analysing it.
图表的解读、平均数与离散度的计算是考试重点。要分清数据呈现与数据分析的区别。
- Averages values. Median = middle value when ordered. Mode = most frequent value. Range = max − min.
平均数：平均数 = 数值总和 / 总个数。中位数 = 排序后中间的值。众数 = 出现频率最高的值。极差 = 最大值 − 最小值。 - Averages from grouped frequency tables: Find midpoints of classes, multiply by frequency, sum, then divide by total frequency. Modal class is the class with the highest frequency.
分组频数表求平均值：找出组中点，乘以频数，求和后除以总频数。众数组是频数最高的组。 - Histograms density. The area of each bar represents frequency. On an exam histogram, frequency density is given on the vertical axis.
直方图：频数 = 组距 × 频数密度。每一条块的面积代表频数。考试中直方图的纵轴通常标为频数密度。 - Cumulative frequency ₃), and interquartile range (IQR = Q₃ − Q₁) can be read from a cumulative frequency curve.
累积频数：从低往高将频数累加。从累积频数曲线图中可读出中位数、下四分位数 (Q₁)、上四分位数 (Q₃) 以及四分位距 (IQR = Q₃ − Q₁)。 - Box plots: Show minimum, Q₁, median, Q₃, maximum. They allow visual comparison of distributions.
箱线图：显示最小值、Q₁、中位数、Q₃、最大值。可用于直观地比较分布。 - Scatter graphs: Correlation does not imply causation. A line of best fit can be used to estimate values (interpolation is reliable; extrapolation is less reliable).
散点图：相关性不代表因果关系。最佳拟合线可用于估计数值（内插法较可靠，外推法可靠性较低）。
12. Vectors & Transformations | 向量与变换
Vectors describe movement in magnitude and direction. Transformations map objects to images on the Cartesian plane.
向量描述大小和方向上的运动。变换是在笛卡尔平面上将图形映射到像。
- Vector notation: Column vector [x, y] or xi + yj. Magnitude of vector (x, y) is √x² + y². Add vectors component-wise.
向量表示法：列向量 [x, y] 或 xi + yj。向量 (x, y) 的模长（大小）是 √x² + y²。向量的加法为对应分量相加。 - Parallel vectors: Vectors are parallel if one is a scalar multiple of the other.
平行向量：如果一个向量是另一个向量的标量倍，那么它们平行。 - Translations: Described by a column vector. Shape moves without rotation or reflection.
平移：用一个列向量描述。图形平移时不旋转、不反转。 - Reflections: Specified by a mirror line (e.g. x = a, y = b, y = x, y = −x).
反射：由对称轴（如 x = a, y = b, y = x, y = −x）确定。 - Rotations: Require centre, angle, and direction (e.g. 90° clockwise about (0,0)).
旋转：需要明确旋转中心、旋转角度和旋转方向（如绕点(0,0) 顺时针旋转 90°）。 - Enlargements: Specified by a centre and a positive scale factor k. Side lengths multiply by k; area multiplies by k². Negative scale factor produces a rotation-like enlargement.
放大：由放大中心和正的比例因子 k 确定。边长乘以 k；面积乘以 k²。负比例因子会产生类似旋转的放大。
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