📚 Year 10 AQA Maths: Formula & Theorem Quick Reference Handbook | Year 10 AQA 数学：公式定理速查手册
This article provides a compact revision guide covering all the essential formulas, theorems and key results you need for the Year 10 AQA Mathematics course. Each section presents the most important points in clear English, immediately followed by its Chinese equivalent, so you can study confidently in either language. Keep this handbook handy for homework, end-of-topic tests and mock exam preparation.
本文提供一个紧凑的复习指南，涵盖 Year 10 AQA 数学课程中所有重要的公式、定理和关键结论。每个部分先用清晰的英文呈现要点，紧接着给出对应的中文解释，让你用任一种语言都能自信地学习。把这份速查手册放在手边，无论是做作业、章节测验还是模拟考试备考都很有用。
1. Algebraic Foundations | 代数基础
the FOIL method (First, Outer, Inner, Last) to obtain (x + a)(x + b) = x² + (a + b)x + ab.
展开括号使用分配律：a(b + c) = ab + ac。两个二项式相乘时，应用 FOIL 方法（首、外、内、尾）得到 (x + a)(x + b) = x² + (a + b)x + ab。
Factorising is the reverse of expanding. Always look for the highest common factor (HCF) first. For a quadratic expression x² + bx + c, find two numbers that multiply to c and add to b, then write as (x + p)(x + q).
因式分解是展开的逆过程。总是优先寻找最大公因数 (HCF)。对于二次式 x² + bx + c，找到两个数，其乘积为 c 且和为 b，然后写成 (x + p)(x + q)。
The quadratic formula gives solutions to ax² + bx + c = 0:
二次公式给出 ax² + bx + c = 0 的解：
x = (-b ± √b² – 4ac) / (2a)
The discriminant Δ = b² – are two distinct real roots, if Δ = 0 one repeated real root, and if Δ < 0 no real roots.
判别式 Δ = b² – 4ac 告诉我们根的性质：若 Δ > 0，有两个相异实根；若 Δ = 0，有一个重根；若 Δ < 0，无实根。
2. Equations of Straight Lines | 直线方程
The gradient m of a line passing through points (x₁, y₁) and (x₂, y₂) is given by m = (y₂ – y₁) / (x₂ – x₁). A horizontal line has m = 0, a vertical line has undefined gradient.
过点 (x₁, y₁) 和 (x₂, y₂) 的直线斜率 m 由 m = (y₂ – y₁) / (x₂ – x₁) 给出。水平线斜率为 0，竖直线斜率无定义。
where m is the gradient and c is the y-intercept. To find the equation from a graph, read off the y-intercept and calculate the gradient using two clear points.
斜截式为 y = mx + c，其中 m 是斜率，c 是 y 轴截距。要从图像求方程，读出 y 轴截距，并利用两个清晰点计算斜率。
e same gradient: m₁ = m₂. Two lines are perpendicular if the product of their gradients is –1: m₁ × m₂ = –1. This is the key relationship for solving perpendicular line problems.
两直线平行当且仅当它们的斜率相等：m₁ = m₂。两直线垂直当且仅当它们的斜率乘积为 –1：m₁ × m₂ = –1。这是解决垂直线问题的关键关系。
3. Indices and Standard Form | 指数与标准形式
The index laws are essential for simplifying expressions with powers. For any non-zero base a:
指数定律是化简幂表达式的关键。对于任意非零底数 a：
aᵐ × aⁿ = aᵐ⁺ⁿ (add the indices when multiplying) | 同底数幂相乘，指数相加
aᵐ ÷ aⁿ = aᵐ⁻ⁿ (subtract the indices when dividing) | 同底数幂相除，指数相减
(aᵐ)ⁿ = aᵐⁿ (multiply the indices for a power of a power) | 幂的乘方，指数相乘
a⁰ = 1 (any non-zero number to the power zero equals 1) | 任何非零数的零次方等于 1
a⁻ⁿ = 1 / aⁿ (negative index gives the reciprocal) | 负指数表示倒数
a1/n = ⁿ√a (fractional index 1/n means the nth root) | 分数指数 1/n 表示 n 次方根
r large numbers n is positive; for small numbers (less than 1) n is negative. This is extremely useful in science and engineering.
标准形式把一个数写成 A × 10ⁿ，其中 1 ≤ A < 10，n 为整数。大数的 n 为正；小于 1 的小数 n 为负。在科学与工程中极为有用。
4. Ratio, Proportion and Percentage Change | 比率、比例与百分比变化
A ratio a : b can be simplified by dividing both parts by their HCF. To share a quantity in a given ratio, find the total number of parts, then divide the quantity by the total parts and multiply by each part of the ratio.
比 a : b 可通过将两项同时除以它们的最大公因数来化简。按给定比率分配数量时，先求出总份数，然后将总数除以总份数，再乘以比中的各部份。
). Graphs of direct proportion are straight lines through the origin; inverse proportion gives a hyperbola.
若 y = kx（k 为常数），则两量成正比（x 翻倍，y 也翻倍）。反比关系为 y = k/x（x 翻倍，y 减半）。正比图像是过原点的直线；反比图像是一条双曲线。
Percentage increase or decrease is calculated as:
百分比增减计算如下：
Percentage change = (change / original) × 100%
after n years = P(1 + r/100)ⁿ, where P is the principal and r is the annual interest rate.
复利用重复乘法计算：n 年后的总额 = P(1 + r/100)ⁿ，其中 P 为本金，r 为年利率。
5. Geometry: Angles and Polygons | 几何：角度与多边形
Basic angle rules: angles on a straight line sum to 180°; angles around a point sum to 360°; vertically opposite angles are equal. In parallel lines, alternate angles are equal, corresponding angles are equal, and interior (co-interior) angles sum to 180°.
基本角度规则：直线上的角之和为 180°；绕一点一周的角之和为 360°；对顶角相等。在平行线中，内错角相等，同位角相等，同旁内角之和为 180°。
e two opposite interior angles. The sum of the interior angles of an n-sided polygon is (n – 2) × 180°.
任意三角形内角和为 180°。三角形的一个外角等于与其不相邻的两个内角之和。n 边形的内角和为 (n – 2) × 180°。
In a regular polygon, all sides and angles are equal. Each interior angle = (n – 2) × 180° / n, and each exterior angle = 360° / n.
正多边形的所有边和角都相等。每个内角 = (n – 2) × 180° / n，每个外角 = 360° / n。
6. Pythagoras’ Theorem and Trigonometry | 勾股定理与三角函数
In a right-angled triangle with legs a, b and hypotenuse c, Pythagoras’ theorem states:
在直角三角形中，设两直角边为 a、b，斜边为 c，勾股定理指出：
a² + b² = c²
Use this to find an unknown side when the other two are known. Remember to identify the hypotenuse correctly (the longest side, opposite the right angle).
已知任意两边时，可用此公式求第三边。务必正确识别斜边（最长边，正对直角）。
The three trigonometric ratios for an acute angle θ in a right triangle are:
直角三角形中锐角 θ 的三个三角函数比：
sin θ = opposite / hypotenuse (对边 / 斜边)
cos θ = adjacent / hypotenuse (邻边 / 斜边)
tan θ = opposite / adjacent (对边 / 邻边)
These ratios allow us to find missing sides or angles. For angles, use the inverse functions sin⁻¹, cos⁻¹, tan⁻¹ on your calculator.
这些比让我们能够计算缺失的边或角。求角度时，在计算器上使用反函数 sin⁻¹、cos⁻¹、tan⁻¹。
7. Area and Volume Formulas | 面积与体积公式
Area formulas for common 2D shapes:
常见平面图形的面积公式：
Rectangle: Area = length × width 矩形：面积 = 长 × 宽 |
Triangle: Area = ½ × base × height 三角形：面积 = ½ × 底 × 高 |
Parallelogram: Area = base × perpendicular height 平行四边形：面积 = 底 × 垂直高 |
zium: Area = ½ (a + b) × h, where a and b are parallel sides and h is perpendicular height 梯形：面积 = ½ (a + b) × h，a 和 b 为平行边，h 为垂直高 |
For 3D shapes, volume measures space occupied:
对于立体图形，体积衡量所占空间：
Cuboid: Volume = length × width × height 长方体：体积 = 长 × 宽 × 高 |
Prism: Volume = area of cross-section × length 棱柱：体积 = 横截面积 × 长 |
Cylinder: Volume = πr²h 圆柱：体积 = πr²h |
Sphere: Volume = (4/3)πr³ 球：体积 = (4/3)πr³ |
Cone: Volume = (1/3)πr²h 圆锥：体积 = (1/3)πr²h |
8. Circles: Circumference, Area, Arcs and Sectors | 圆：周长与面积，弧长与扇形面积
For a circle with radius r and diameter d = 2r:
对于半径为 r、直径为 d = 2r 的圆：
Circumference (perimeter) C = 2πr = πd
周长 C = 2πr = πd
Area A = πr²
面积 A = πr²
When dealing with a sector (a slice of the circle) with central angle θ (in degrees), the arc length is a fraction of the circumference, and the sector area is the same fraction of the total area:
处理中心角为 θ（度）的扇形（圆的一片）时，弧长是周长的若干分之一，扇形面积也是整个圆面积的相同分数：
Arc length = (θ / 360) × 2πr
Sector area = (θ / 360) × πr²
If the angle is given in radians, simply replace 360° with 2π, but Year 10 work typically uses degrees.
若角度以弧度给出，只需将 360° 换成 2π 即可，但 Year 10 通常使用度。
9. Probability Fundamentals | 概率基础
number of favourable outcomes / total number of possible outcomes, provided all outcomes are equally likely. Probabilities always lie between 0 and 1.
事件 A 的概率 P(A) = 有利结果的数量 / 所有可能结果的总数，前提是每种结果等可能。概率总是在 0 到 1 之间。
The probability that A does not happen is P(not A) = 1 – same time; then P(A or B) = P(A) + P(B). For independent events, P(A and B) = P(A) × P(B).
A 不发生的概率为 P(非 A) = 1 – P(A)。若两事件不能同时发生，则它们互斥；此时 P(A 或 B) = P(A) + P(B)。对于独立事件，P(A 且 B) = P(A) × P(B)。
Tree diagrams help visualise successive events. Multiply probabilities along branches for combined outcomes, and add probabilities for different branches leading to the same final result. Always check that branch probabilities at each node sum to 1.
树图有助于可视化接连发生的事件。沿分枝相乘得到组合结果的概率，将导向同一最终结果的不同分枝的概率相加。务必检查每个节点处分枝概率之和为 1。
10. Statistics: Averages and Diagrams | 统计：平均数与图表
The three measures of central tendency (averages) are:
三种集中趋势（平均数）的度量是：
Mean: sum of all values divided by the number of values. For grouped data, use the midpoint of each class. | 平均数：所有数值之和除以数值个数。对于分组数据，用每组的组中值。
Median: the middle value when data are ordered. In a frequency table, find the (n+1)/2-th position. | 中位数：将数据排序后中间的那个值。在频数表中，找到第 (n+1)/2 个位置。
Mode: the value that appears most frequently. | 众数：出现频次最高的值。
Range is a measure of spread: maximum – minimum. Cumulative frequency diagrams are used to estimate the median, quartiles and interquartile range. A box plot (box-and-whisker) displays minimum, lower quartile (Q1), median (Q2), upper quartile (Q3) and maximum.
极差是一种离散程度的度量：最大值 – 最小值。累积频数图用于估计中位数、四分位数和四分位距。箱线图（箱须图）展示了最小值、下四分位数 (Q1)、中位数 (Q2)、上四分位数 (Q3) 和最大值。
Interquartile range (IQR) = Q3 – Q1. It is a better measure of spread than the range because it excludes outliers.
四分位距 (IQR) = Q3 – Q1。它是比极差更好的离散程度度量，因为它排除了异常值影响。
11. Inequalities | 不等式
al to). Solving linear inequalities follows similar rules to equations, with one vital exception: if you multiply or divide by a negative number, you must reverse the inequality sign.
不等式使用符号 >（大于）、<（小于）、≥（大于或等于）和 ≤（小于或等于）。解一元一次不等式的规则与方程类似，但有一个关键例外：若乘以或除以一个负数，必须反转不等号方向。
Solution sets can be expressed in set notation, e.g., {x : x > 3}, or on a number line using open circles for strict inequalities and closed circles for those including equality.
解集可用集合符号表示，如 {x : x > 3}，或在数轴上用空心圆表示严格不等、用实心圆表示包含等号的情况。
To solve a quadratic inequality like x² – 5x + 6 > 0, first solve the corresponding quadratic equation to find critical values, then test regions on a number line to determine which intervals satisfy the inequality.
解如 x² – 5x + 6 > 0 的二次不等式时，先解对应的二次方程找临界值，然后在数轴上测试各区间以确定哪些区间满足不等式。
12. Vectors | 向量
A vector is a quantity with both magnitude and direction, represented by a directed line segment. In component form, a vector from A to B is written as a column: the top number is the horizontal displacement (x-change), the bottom number is the vertical displacement (y-change).
向量是有大小和方向的量，用有向线段表示。在分量形式中，从 A 到 B 的向量写成一个列向量：上方数字为水平位移（x 变化量），下方数字为垂直位移（y 变化量）。
Vector addition: to add vectors, add their corresponding components. Geometrically, place the tail of the second vector at the head of the first; the resultant goes from tail of first to head of second.
向量加法：将对应分量相加。几何上，把第二个向量的起点接到第一个向量的终点；合向量从第一个起点指向第二个终点。
Scalar multiplication: multiplying a vector by a scalar k multiplies each component by k, changing the magnitude but not the direction (if k > 0). If k is negative, the direction reverses.
标量乘法：向量乘以标量 k 将每个分量乘以 k，从而改变大小但不改变方向（当 k > 0 时）。若 k 为负，则方向反转。
Two vectors are parallel if one is a scalar multiple of the other. Vectors are often used to prove geometric facts, such as the midpoint of a segment or that two sides of a quadrilateral are parallel and equal.
若一个向量是另一个的标量倍数，则两向量平行。向量常用于证明几何事实，例如线段的中点，或者四边形的两边平行且相等。
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