📚 Year 11 Eduqas Maths: Quick Reference Formula & Theorem Handbook | Year 11 Eduqas 数学：公式定理速查手册
This quick reference handbook compiles all the essential formulas and theorems needed for the Year 11 Eduqas GCSE Mathematics specification. Use it for rapid review, last-minute revision, and to reinforce your understanding before the exams. Every topic is presented with a concise English explanation immediately followed by a Chinese translation, making it ideal for bilingual learners or those seeking clarity through dual-language reinforcement.
本速查手册汇编了 Year 11 Eduqas GCSE 数学考纲所需的所有基本公式和定理，可用于快速回顾、考前冲刺，帮助巩固理解。每个主题先提供简洁的英文说明，紧接着给出中文对照，非常适合双语学习者或希望通过双语言强化来厘清概念的同学。
1. Solving Linear Equations | 解线性方程
To solve an equation, perform the same inverse operation on both sides to isolate the unknown. Always aim to get the variable alone on one side of the equals sign.
解方程时，对等号两边同时执行相同的逆运算以分离未知数。始终力求让变量单独位于等号的一侧。
Example: Solve 2x + 5 = 13. Subtract 5 from both sides → 2x = 8. Then divide both sides by 2 → x = 4.
示例：解方程 2x + 5 = 13。两边减 5 → 2x = 8。然后两边除以 2 → x = 4。
For equations with brackets, expand first. For those with unknowns on both sides, collect like terms onto one side before isolating.
对于带括号的方程，首先展开括号。若未知数出现在等号两边，先将同类项移到一边，再进行分离。
ax + b = c ⇒ x = (c – b) / a
2. Quadratic Equations & Completing the Square | 二次方程与配方法
A quadratic equation has the general form ax² + bx + c = 0, where a ≠ 0. Solutions can be found by factorising, using the quadratic formula, or by completing the square.
二次方程的一般形式为 ax² + bx + c = 0，其中 a ≠ 0。可通过因式分解、求根公式或配方法求解。
The quadratic formula gives the roots directly. The discriminant Δ = b² – 4ac determines the nature of the roots: two distinct real roots if Δ > 0, one repeated root if Δ = 0, and no real roots if Δ < 0.
求根公式直接给出根。判别式 Δ = b² – 4ac 决定根的性质：Δ > 0 有两个相异实根；Δ = 0 有两个相等实根；Δ < 0 无实根。
x = (-b ± √(b² – 4ac)) / (2a)
Completing the square rewrites the expression as a(x + p)² + q, revealing the turning point (–p, q) of the parabola.
配方法将表达式改写为 a(x + p)² + q 的形式，可揭示抛物线的顶点 (–p, q)。
Example: x² + 6x + 5 → (x + 3)² – 9 + 5 = (x + 3)² – 4.
示例：x² + 6x + 5 → (x + 3)² – 9 + 5 = (x + 3)² – 4。
3. Straight Line Graphs | 直线图
The equation of a straight line is y = mx + c, where m is the gradient and c is the y-intercept. The gradient m is calculated as the change in y divided by the change in x between two points.
直线方程为 y = mx + c，其中 m 为斜率，c 为 y 轴截距。斜率 m 通过两点间 y 的变化量除以 x 的变化量计算。
m = (y₂ – y₁) / (x₂ – x₁)
Parallel lines have equal gradients: m₁ = m₂. Perpendicular lines have gradients that are negative reciprocals: m₁ × m₂ = –1.
平行线斜率相等：m₁ = m₂。垂直线的斜率互为负倒数：m₁ × m₂ = –1。
To find the equation of a line given a point (x₁, y₁) and gradient m, use y – y₁ = m(x – x₁). The midpoint of a segment is ((x₁+x₂)/2, (y₁+y₂)/2).
已知一点 (x₁, y₁) 和斜率 m，可用点斜式 y – y₁ = m(x – x₁)。线段中点坐标为 ((x₁+x₂)/2, (y₁+y₂)/2)。
4. Area & Volume Formulae | 面积与体积公式
Memorise these core 2D and 3D measurement formulas. For compound shapes, split into simpler parts. Always include units squared for area and cubed for volume.
牢记以下核心的平面和立体测量公式。对于组合图形，拆分为简单图形分别计算。面积单位带平方，体积单位带立方。
| Shape | Formula (English) | 形状 | 公式 (中文) |
| Triangle | ½ × base × height | 三角形 | ½ × 底 × 高 |
| Parallelogram | base × perpendicular height | 平行四边形 | 底 × 高 |
| Trapezium | ½ (a + b) × h | 梯形 | ½ (上底 + 下底) × 高 |
| Circle | πr² (area); 2πr (circumference) | 圆 | πr² (面积); 2πr (周长) |
| Sector | (θ/360) × πr² (area); (θ/360) × 2πr (arc length) | 扇形 | (θ/360) × πr² (面积); (θ/360) × 2πr (弧长) |
| Prism | area of cross-section × length | 棱柱 | 横截面积 × 长度 |
| Cylinder | πr²h (volume); 2πrh + 2πr² (surface area) | 圆柱 | πr²h (体积); 2πrh + 2πr² (表面积) |
| Pyramid | ⅓ × base area × height | 棱锥 | ⅓ × 底面积 × 高 |
| Sphere | ⁴⁄₃ πr³ (volume); 4πr² (surface area) | 球 | ⁴⁄₃ πr³ (体积); 4πr² (表面积) |
5. Pythagoras & Basic Trigonometry | 勾股定理与基本三角学
In a right-angled triangle, Pythagoras’ theorem relates the sides: the square of the hypotenuse equals the sum of the squares of the other two sides.
在直角三角形中，勾股定理描述了边之间的关系：斜边的平方等于两直角边的平方和。
a² + b² = c² (c is the hypotenuse)
The primary trigonometric ratios connect angles and side lengths: sine, cosine and tangent. Recall SOH CAH TOA.
基本三角函数将角度与边长联系起来：正弦、余弦和正切。记住 SOH CAH TOA。
sin θ = opposite / hypotenuse, cos θ = adjacent / hypotenuse, tan θ = opposite / adjacent
Exact values for 0°, 30°, 45°, 60° and 90° must be memorised. For example, sin 30° = ½, cos 45° = √2/2, tan 60° = √3.
必须牢记 0°、30°、45°、60° 和 90° 的精确值。例如 sin 30° = ½, cos 45° = √2/2, tan 60° = √3。
6. Sine & Cosine Rules | 正弦与余弦定理
For non-right-angled triangles, the sine rule is used when given either two angles and a side, or two sides and a non-included angle.
对于非直角三角形，已知两角一边或两边及一个非夹角时，可使用正弦定理。
a / sin A = b / sin B = c / sin C
The cosine rule is applied when given three sides or two sides and the included angle. It generalises Pythagoras’ theorem.
已知三边，或两边及其夹角时，使用余弦定理。它是勾股定理的推广。
a² = b² + c² – 2bc cos A
The area of any triangle can be found using two sides and the included angle, which is especially useful when the perpendicular height is not known.
可用两边及其夹角求任意三角形面积，当高未知时此公式尤为实用。
Area = ½ ab sin C
7. Circle Theorems | 圆定理
Circle theorems describe angle relationships within a circle. They are essential for geometric proof questions covering Eduqas higher tier.
圆定理描述圆内的角度关系，是 Eduqas 高阶几何证明题的核心内容。
The angle at the centre is twice the angle at the circumference subtended by the same arc: ∠AOB = 2∠APB.
圆心角等于同弧所对圆周角的两倍：∠AOB = 2∠APB。
The angle in a semicircle is a right angle (90°). Angles in the same segment are equal.
半圆内的圆周角为直角 (90°)。同弧上的圆周角相等。
Opposite angles in a cyclic quadrilateral sum to 180°. The angle between a tangent and a chord equals the angle in the alternate segment.
圆内接四边形的对角互补，和为 180°。弦切角等于同弧所对的圆周角（弦切角定理）。
Tangents from an external point are equal in length. A radius drawn to the point of tangency is perpendicular to the tangent.
从同一点出发的两条切线长度相等。连接切点的半径垂直于该切线。
8. Probability Rules | 概率法则
The probability of an event is a number between 0 and 1. The sum of probabilities of all mutually exclusive outcomes is 1.
事件的概率是一个 0 到 1 之间的数。所有互斥结果的概率之和为 1。
For independent events, the probability of both occurring is the product of their individual probabilities: P(A and B) = P(A) × P(B).
若两事件独立，两者同时发生的概率为各自概率的乘积：P(A and B) = P(A) × P(B)。
For mutually exclusive events, the addition rule is used: P(A or B) = P(A) + P(B). If not mutually exclusive, subtract the intersection: P(A or B) = P(A) + P(B) – P(A and B).
若两事件互斥，使用加法法则：P(A or B) = P(A) + P(B)。若非互斥，则需减去交集部分：P(A or B) = P(A) + P(B) – P(A and B)。
Conditional probability ties to tree diagrams: P(A|B) = P(A and B) / P(B). Always update probabilities after replacing or not replacing items.
条件概率与树图相关：P(A|B) = P(A and B) / P(B)。务必将放回与不放回的概率更新考虑在内。
9. Statistical Measures & Charts | 统计量度与图表
Central tendency measures summarise a data set: mean = Σx / n, median is the middle value, mode is the most frequent value.
集中趋势量度概括数据集：平均数 = Σx / n，中位数为居中值，众数为出现频率最高的值。
Range = highest – lowest gives spread. The interquartile range (IQR = UQ – LQ) measures the spread of the middle 50%. Box plots display these five numbers neatly.
极差 = 最大值 – 最小值，描述整体离散程度。四分位距 (IQR = 上四分位数 – 下四分位数) 反映中间 50% 数据的离散程度。箱线图可直观展示这五个统计量。
Cumulative frequency graphs help find medians and quartiles. Frequency density is used for histograms with unequal class widths: frequency density = frequency ÷ class width.
累积频数图用于查找中位数与四分位数。直方图若组距不等，需用频数密度：频数密度 = 频数 ÷ 组距。
10. Indices, Surds & Standard Form | 指数、根号与标准形式
Index laws simplify expressions involving powers. The key rules are essential for algebraic manipulation.
指数法则用于化简幂运算表达式。以下关键法则在代数变形中不可或缺。
Multiplication: aᵐ × aⁿ = aᵐ⁺ⁿ. Division: aᵐ ÷ aⁿ = aᵐ⁻ⁿ. Power of a power: (aᵐ)ⁿ = aᵐⁿ. Zero exponent: a⁰ = 1. Negative exponent: a⁻ⁿ = 1/aⁿ.
乘法：aᵐ × aⁿ = aᵐ⁺ⁿ。除法：aᵐ ÷ aⁿ = aᵐ⁻ⁿ。幂的乘方：(aᵐ)ⁿ = aᵐⁿ。零指数：a⁰ = 1。负指数：a⁻ⁿ = 1/aⁿ。
Surds are irrational roots. Simplify by finding the largest square factor: √48 = √(16×3) = 4√3. Rationalise the denominator by multiplying numerator and denominator by the conjugate if needed.
根式是无理数的根号形式。通过提取最大平方因子进行化简：√48 = √(16×3) = 4√3。分母有理化时，分子分母同乘相应的共轭根式。
Standard form writes large or small numbers as a × 10ⁿ where 1 ≤ a < 10. e.g., 6500 = 6.5 × 10³, 0.00032 = 3.2 × 10⁻⁴.
标准形式将极大或极小的数表示为 a × 10ⁿ，其中 1 ≤ a < 10。例如 6500 = 6.5 × 10³, 0.00032 = 3.2 × 10⁻⁴。
11. Vectors | 向量
A vector has both magnitude and direction. Column vectors are written as a 2×1 matrix. Vector addition is done by adding the corresponding components.
向量既有大小又有方向。列向量写为 2×1 矩阵。向量加法通过将对应分量相加实现。
a = (x₁, y₁) expressed as column: [x₁; y₁]
Scalar multiplication scales the vector: ka = (kx₁, ky₁). Parallel vectors are scalar multiples of each other.
标量乘法对向量进行缩放：ka = (kx₁, ky₁)。若两向量互为标量倍数，则它们平行。
To prove points are collinear, show that the vectors connecting them are parallel and share a common point. Vectors are also used to solve geometric ratio problems.
要证明三点共线，可证明连接它们的向量互相平行，且有一个公共点。向量也常用于解决几何比例问题。
12. Direct & Inverse Proportion | 正比与反比
Two quantities are in direct proportion if their ratio is constant: y ∝ x ⇒ y = kx. The graph is a straight line through the origin.
若两个量的比值恒定，则它们成正比例关系：y ∝ x ⇒ y = kx。其图像是一条过原点的直线。
Inverse proportion means their product is constant: y ∝ 1/x ⇒ y = k/x. The graph is a hyperbola, never touching the axes.
反比例关系意味着它们的乘积恒定：y ∝ 1/x ⇒ y = k/x。其图像为双曲线，不与坐标轴相交。
For proportion problems, find the constant k using given paired values, then apply it to find unknown values. Algebraic methods often involve setting up proportional equations.
解决比例问题时，先用已知成对值求出常数 k，再代入求解未知量。代数方法常涉及建立比例方程。
Recognise proportion in context: converting currencies, scaling recipes, speed–time relationships, and work rates all rely on these relationships.
在情境中识别比例关系：货币兑换、食谱缩放、速度-时间关系及工作效率等问题都基于正比或反比关系。
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