📚 Year 10 Eduqas Maths: Formula & Theorem Quick Reference Handbook | 10年级Eduqas数学：公式定理速查手册
This quick reference handbook brings together the essential formulas, theorems and key facts you need to master for the Year 10 Eduqas GCSE Mathematics course. Use it as a daily study companion, a last‑minute revision sheet, or a confidence builder before topic tests. Every entry has been mapped closely to the Eduqas specification so you can focus on what really matters.
这本速查手册汇集了10年级Eduqas GCSE数学课程中你必须掌握的核心公式、定理和关键知识点。你可以把它当作日常学习伴侣、考前突击复习表，或者主题测验前的信心加油站。每一条内容都紧密贴合Eduqas考试大纲，帮助你抓住真正重要的内容。
1. Algebra: Key Formulae | 代数核心公式
The quadratic formula gives the solutions of ax² + bx + c = 0 (a ≠ 0) as:
二次方程求根公式给出方程 ax² + bx + c = 0（a ≠ 0）的解为：
x = [ –b ± √(b² – 4ac) ] / (2a)
The expression under the square root, b² – 4ac, is called the discriminant. It tells you about the nature of the roots: two distinct real roots if positive, one repeated root if zero, and no real roots if negative.
根号下的表达式 b² – 4ac 称为判别式。它告诉你根的性质：若为正，有两个不同的实数根；若为零，有两个相等的实数根；若为负，则没有实数根。
The equation of a straight line is usually written as y = mx + c, where m is the gradient and c is the y‑intercept. To find the equation of a line through (x₁, y₁) with gradient m, use y – y₁ = m(x – x₁).
直线方程通常写作 y = mx + c，其中 m 是斜率，c 是 y 轴截距。要求经过点 (x₁, y₁) 且斜率为 m 的直线方程，可使用 y – y₁ = m(x – x₁)。
When expanding brackets, remember the difference of two squares: a² – b² = (a + b)(a – b). For a perfect square: (a ± b)² = a² ± 2ab + b².
展开括号时，记住平方差公式：a² – b² = (a + b)(a – b)。完全平方公式：(a ± b)² = a² ± 2ab + b²。
Laws of indices are vital: aᵐ × aⁿ = aᵐ⁺ⁿ, aᵐ ÷ aⁿ = aᵐ⁻ⁿ, (aᵐ)ⁿ = aᵐⁿ, a⁰ = 1 (a ≠ 0), and a⁻ⁿ = 1/aⁿ.
指数定律至关重要：aᵐ × aⁿ = aᵐ⁺ⁿ，aᵐ ÷ aⁿ = aᵐ⁻ⁿ，(aᵐ)ⁿ = aᵐⁿ，a⁰ = 1（a ≠ 0），以及 a⁻ⁿ = 1/aⁿ。
Solving linear inequalities works much like solving equations, but remember to reverse the inequality sign when multiplying or dividing by a negative number.
解一元一次不等式与解方程类似，但要记住：当两边同乘或除以一个负数时，必须 反转不等号方向。
2. Geometry & Measures: Perimeter, Area and Volume | 几何与测量：周长、面积和体积
For a rectangle with length l and width w: Perimeter = 2l + 2w, Area = l × w. For a triangle with base b and perpendicular height h: Area = ½ × b × h.
对于长为 l、宽为 w 的长方形：周长 = 2l + 2w，面积 = l × w。对于底为 b、垂直高为 h 的三角形：面积 = ½ × b × h。
A parallelogram’s area is also base × perpendicular height. A trapezium with parallel sides a and b, height h: Area = ½(a + b)h.
平行四边形的面积同样是 底 × 垂直高。梯形两底边分别为 a 和 b，高为 h：面积 = ½(a + b)h。
For circles: Circumference = πd = 2πr, Area = πr². The radius r is half the diameter d. Use π ≈ 3.14 or the π button on your calculator.
圆：周长 = πd = 2πr，面积 = πr²。半径 r 是直径 d 的一半。使用 π ≈ 3.14 或计算器上的 π 键。
Volume of a cuboid: V = l × w × h. Volume of a prism: V = area of cross‑section × length. For a cylinder, the cross‑section is a circle, so V = πr²h.
长方体体积：V = l × w × h。棱柱体积：V = 横截面积 × 长度。对于圆柱体，横截面是圆，因此 V = πr²h。
You are expected to know Pythagoras’ theorem: in a right‑angled triangle with legs a and b, and hypotenuse c, a² + b² = c². Use it to find missing sides and to test whether a triangle is right‑angled.
你需要掌握毕达哥拉斯定理：在直角三角形中，设两直角边为 a、b，斜边为 c，则 a² + b² = c²。用它来求未知边长，也可检验三角形是否为直角三角形。
3. Trigonometry in Right‑Angled Triangles | 直角三角形的三角学
In any right‑angled triangle, the three trigonometric ratios are defined for an acute angle θ:
在任意直角三角形中，锐角 θ 的三个三角函数比定义为：
sin θ = opposite / hypotenuse
cos θ = adjacent / hypotenuse
tan θ = opposite / adjacent
Memorise this as “SOH CAH TOA”. Always label the sides relative to the angle you are using.
用 “SOH CAH TOA” 来记忆。始终要相对于所选的角来标注各边。
Exact trigonometric values for key angles should be known without a calculator:
你必须不借助计算器熟记关键角的精确三角值：
| 0° | 30° | 45° | 60° | 90° | |
|---|---|---|---|---|---|
| sin | 0 | ½ | 1/√2 or √2/2 | √3/2 | 1 |
| cos | 1 | √3/2 | 1/√2 or √2/2 | ½ | 0 |
| tan | 0 | 1/√3 | 1 | √3 | undefined |
4. Vectors | 向量
A vector describes a quantity with both magnitude and direction. It is often written as a bold letter, e.g. a, or as a column vector: (x, y)ᵀ (written vertically). On diagrams, an arrow from A to B is written as AB with an arrow above, or simply as a bold letter.
向量描述具有大小和方向的量。通常用粗体字母表示，如 a，或用列向量如 (x, y)ᵀ（竖着写）。在图形中，从 A 到 B 的向量可写为 AB（上面带箭头），或直接用粗体字母表示。
Vector addition: a + b is found by placing the tail of b at the head of a (or adding the corresponding components). Scalar multiplication: k a stretches the vector by factor k. If k is negative, the direction is reversed.
向量加法：a + b 通过将 b 的尾部接在 a 的头部来求得（或将对应分量相加）。标量乘法：k a 将向量按因子 k 伸缩。若 k 为负，则方向相反。
The magnitude (length) of a vector v = (x, y)ᵀ is |v| = √(x² + y²). A unit vector has magnitude 1.
向量 v = (x, y)ᵀ 的模（长度）为 |v| = √(x² + y²)。单位向量的模为 1。
5. Probability | 概率
The probability of an event A is given by P(A) = number of favourable outcomes / total number of possible outcomes. Probabilities always lie between 0 and 1 inclusive.
事件 A 的概率为 P(A) = 有利结果的个数 / 所有可能结果的个数。概率总是在 0 到 1 之间（含）。
For mutually exclusive events (they cannot happen at the same time): P(A or B) = P(A) + P(B).
对于互斥事件（它们不能同时发生）：P(A 或 B) = P(A) + P(B)。
For independent events (one does not affect the other): P(A and B) = P(A) × P(B).
对于独立事件（一个事件不影响另一个）：P(A 和 B) = P(A) × P(B)。
Tree diagrams help you list all possible outcomes for combined events. Multiply along branches for “and”, and add the relevant end probabilities for “or”.
树形图有助于列出组合事件的所有可能结果。沿着分支相乘对应“and”，将相关的末端概率相加对应“or”。
The complement rule: P(not A) = 1 – P(A).
补集规则：P(非 A) = 1 – P(A)。
6. Statistics: Averages and Measures of Spread | 统计：平均数与离散度量
Mean = (sum of all data values) ÷ (number of values). Median = middle value when ordered. Mode = most frequent value. Range = highest – lowest.
平均数 = （所有数据值的和）÷（数据的个数）。中位数 = 排序后位于中间的数值。众数 = 出现频率最高的值。极差 = 最大值 – 最小值。
For grouped data, the estimated mean uses midpoints: Estimated Mean = Σ(f × x) / Σf, where f is the frequency and x is the midpoint of each class interval.
对于分组数据，估算平均数使用组中点：估计平均数 = Σ(f × x) / Σf，其中 f 为频数，x 为每个区间的组中点。
A cumulative frequency diagram shows the running total of frequencies. The median, quartiles and interquartile range can be read from it.
累积频率图展示频率的累计情况。从图中可以读出中位数、四分位数以及四分位距。
Interquartile range (IQR) = upper quartile – lower quartile. It is a measure of spread that is not affected by extreme values.
四分位距 (IQR) = 上四分位数 – 下四分位数。它是一种不受极端值影响的离散度量。
7. Ratio, Proportion and Rates of Change | 比率、比例与变化率
Ratios can be simplified by dividing all parts by common factors. To share an amount in a given ratio, first find the total number of parts, then divide the amount accordingly.
通过将所有部分除以公因数可以简化比率。按比例分配一个量时，首先计算出总份数，然后相应对总量进行分配。
Two quantities are in direct proportion if y = kx (k constant). Their graph is a straight line through the origin. Inverse proportion means y = k / x, giving a rectangular hyperbola.
如果两个量满足 y = kx（k 为常数），则它们成正比例；图像是一条过原点的直线。反比例意味着 y = k / x，图像为一条双曲线。
Compound interest formula: A = P(1 + r/100)ⁿ, where A is the final amount, P the principal, r the interest rate per period, and n the number of periods. For depreciation, use a minus sign instead of plus.
复利公式：A = P(1 + r/100)ⁿ，其中 A 为最终金额，P 为本金，r 为每期利率，n 为期数。折旧则用减号代替加号。
Speed‑time graphs: the area under the graph equals the distance travelled; the gradient of the line gives acceleration. Constant speed is a horizontal line; uniform acceleration is a sloping straight line.
速度‑时间图：图线下方面积等于走过的距离；线的斜率给出加速度。匀速运动对应水平线，匀加速运动对应倾斜的直线。
8. Number: Standard Form and Percentages | 数字：标准形式和百分比
A number in standard form is written as a × 10ⁿ, where 1 ≤ a < 10 and n is an integer. This is essential for very large or very small numbers.
标准形式将数字写为 a × 10ⁿ，其中 1 ≤ a < 10，n 为整数。这对极大或极小的数字至关重要。
Percentage increase: New value = original × (1 + percentage/100). Percentage decrease: New value = original × (1 – percentage/100). To find the percentage change: % change = (change / original) × 100.
百分比增加：新值 = 原值 × (1 + 百分比/100)。百分比减少：新值 = 原值 × (1 – 百分比/100)。求百分比变化：% 变化 = (变化量 / 原值) × 100。
Reverse percentages: when given the final amount after a % change, divide by the multiplier to find the original. For example, after a 20% increase, the multiplier is 1.2, so original = final ÷ 1.2.
逆向百分比：当已知经过百分比变化后的量时，除以乘数就可以找到原值。例如，增长 20% 后的乘数为 1.2，因此原值 = 最终值 ÷ 1.2。
9. Sequences: The nth Term | 数列：第 n 项
For an arithmetic (linear) sequence, the term‑to‑term rule is a constant difference d. The nth term is given by uₙ = a + (n – 1)d, where a is the first term.
对于等差（线性）数列，相邻项的差为常数 d。第 n 项为 uₙ = a + (n – 1)d，其中 a 是首项。
A quadratic sequence has a second difference that is constant. Its nth term can be found by comparing with the generic form an² + bn + c. First use the second difference to find a, then solve for b and c.
二次数列的二阶差为常数。它的第 n 项可以通过与通用形式 an² + bn + c 对比求得。首先利用二阶差求出 a，再解出 b 和 c。
The Fibonacci sequence is a special pattern where each term is the sum of the two preceding ones: uₙ₊₂ = uₙ₊₁ + uₙ. You may be asked to generate terms or spot its occurrence in nature and geometry.
斐波那契数列是一种特殊的模式，每一项都是前两项之和：uₙ₊₂ = uₙ₊₁ + uₙ。你可能需要写出数列的项，或者识别它在自然界和几何中的出现。
10. Graphs of Functions | 函数图像
The graph of a quadratic function y = ax² + bx + c is a parabola. If a > 0 it is ∪‑shaped; if a < 0 it is ∩‑shaped. The turning point is a minimum or maximum respectively. The y‑intercept is at c; the x‑intercepts are found by solving ax² + bx + c = 0.
二次函数 y = ax² + bx + c 的图像是一条抛物线。若 a > 0 则为 ∪ 形；若 a < 0 则为 ∩ 形。其顶点分别对应最小值或最大值。y 轴截距在 c 处；x 轴截距通过解方程 ax² + bx + c = 0 求得。
Cubic graphs y = ax³ + … have a characteristic S‑shape or wave, while reciprocal graphs y = a/x have two separate branches with asymptotes along the axes.
三次函数图像 y = ax³ + … 呈现典型的 S 形或波浪形，而反比例函数图像 y = a/x 有两条分开的分支，并带有沿坐标轴的渐近线。
Exponential graphs y = kaˣ show rapid growth or decay. Recognise the basic shape and the intercept at (0, k).
指数函数图像 y = kaˣ 表现出快速增长或衰减。要熟悉其基本形状以及在 (0, k) 处的截距。
To interpret graphs: the gradient at a point tells you the rate of change; the area under a graph often has a real‑world meaning (e.g. distance travelled).
解读图像：某点的斜率告诉你变化率；图下的面积通常具有现实意义（例如走过的距离）。
11. Angles in Polygons | 多边形中的角
The sum of interior angles of an n‑sided polygon is (n – 2) × 180°. For a regular polygon, each interior angle = [(n – 2) × 180°] / n.
n 边形的内角和为 (n – 2) × 180°。对于正多边形，每个内角 = [(n – 2) × 180°] / n。
The sum of exterior angles of any convex polygon is 360°. For a regular polygon, each exterior angle = 360° / n.
任何凸多边形的外角和均为 360°。对于正多边形，每个外角 = 360° / n。
When parallel lines are crossed by a transversal, alternate angles are equal, corresponding angles are equal, and interior (co‑interior) angles add to 180°.
当一组平行线被一条截线所截时，内错角相等，同位角相等，同旁内角之和为 180°。
12. Circle Theorems – Basic Set | 基本圆定理
The angle in a semicircle is always a right angle (90°). The tangent to a circle is perpendicular to the radius at the point of contact.
半圆内的圆周角始终为直角（90°）。圆的切线在切点处垂直于半径。
Angles at the circumference subtended by the same arc are equal. The angle at the centre is twice the angle at the circumference when both are subtended by the same arc.
同一弧所对的圆周角相等。圆心角是同一弧所对圆周角的两倍。
Opposite angles in a cyclic quadrilateral sum to 180°. The perpendicular from the centre to a chord bisects the chord.
圆内接四边形的对角之和为 180°。从圆心到弦的垂线平分该弦。
Two tangents drawn from an external point to a circle are equal in length.
从圆外一点引出的两条切线长度相等。
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