📚 Year 11 Edexcel Maths: Formula & Theorem Quick Reference Handbook | Year 11 Edexcel 数学：公式定理速查手册
This handbook brings together every essential formula, theorem, and key method you will need for the Year 11 Edexcel Mathematics exam. Use it as a daily revision companion, a last‑minute warm‑up, or a confidence booster before tackling past papers. All entries are aligned with the Edexcel Higher Tier specification, with clear explanations and practical tips to help you recall and apply them under exam pressure.
本手册汇集了 Year 11 Edexcel 数学考试所需的全部核心公式、定理和关键方法。你可以把它当作日常复习伴侣、考前热身工具或攻克历年真题的信心加速器。所有条目均严格按照 Edexcel 高阶课程要求编排，并附有清晰说明和实用技巧，帮助你在考场压力下迅速回忆并准确应用。
1. Linear Equations & Straight‑Line Graphs | 直线方程与图形
m = (y₂ − y₁)/(x₂ − x₁). A line’s equation can be written as y = mx + c, where c is the y‑intercept. Alternatively, use y − y₁ = m(x − x₁) for a point‑pe form. Parallel lines share the same gradient; perpendicular lines satisfy m₁ × m₂ = −1.
过点 (x₁, y₁) 和 (x₂, y₂) 的直线斜率为 m = (y₂ − y₁)/(x₂ − x₁)。直线方程可写为 y = mx + c，其中 c 是 y 轴截距。也可使用点斜式 y − y₁ = m(x − x₁)。平行线斜率相等；垂直线满足 m₁ × m₂ = −1。
o points, first calculate the gradient, then substitute one point into y − y₁ = m(x − x₁).
两点中点为 ((x₁ + x₂)/2, (y₁ + y₂)/2)，两点间距离为 √[(x₂ − x₁)² + (y₂ − y₁)²]。已知两点求直线方程时，先计算斜率，再将其中一点代入 y − y₁ = m(x − x₁)。
2. Quadratic Expressions & the Quadratic Formula | 二次表达式与求根公式
A quadratic expression ax² + bx + c can be factorised when two numbers multiply to ac and add to b. If factorisation is difficult, use the quadratic formula, given in the exam:
二次表达式 ax² + bx + c 可通过寻找两个乘积为 ac、和为 b 的数进行因式分解。若因式分解困难，可使用考试中提供的求根公式：
x = [−b ± √b² − 4ac] / (2a)
repeated real root, and Δ < 0 gives no real roots. To complete the square, write ax² + bx + c as a(x + p)² + q; the turning point is (−p, q).
判别式 Δ = b² − 4ac 决定根的性质：Δ > 0 有两个不等实根，Δ = 0 有一个重根，Δ < 0 无实根。配方法可将 ax² + bx + c 写成 a(x + p)² + q 的形式，其顶点坐标为 (−p, q)。
3. Inequalities & Regions | 不等式与区域
an inequality graphically, shade the unwanted region and use a solid line for ≤ or ≥, and a dashed line for < or >.
解一次不等式的方法与方程类似，但在乘以或除以负数时需反转不等号方向。用图形表示不等式时，将不满足条件的区域涂上阴影，≤ 或 ≥ 用实线，< 或 > 用虚线。
For quadratic inequalities like ax² + bx + c > 0, first solve the corresponding equation to find critical values, then test intervals on a number line. The solution is written as a combination of intervals, e.g. x < p or x > q.
对于二次不等式如 ax² + bx + c > 0，先解对应方程求出临界值，再在数轴上检验区间。解集可用区间组合表示，例如 x < p 或 x > q。
4. Simultaneous Equations | 联立方程组
Two linear equations can be solved by elimination (adding or subtracting multiples to cancel one unknown) or by substitution (rearranging one equation for a variable and substituting into the other). Always check your solution by plugging the values back into both original equations.
两个一次方程可通过消元法（乘以倍数后相加或相减消去一个未知数）或代入法（将一个方程变形后代入另一个）求解。务必把求得的解代回原方程组进行检验。
For one linear and one quadratic equation, substitute the linear expression into the quadratic, then solve the resulting quadratic equation. There can be zero, one or two pairs of solutions. Graphical interpretation: the points of intersection between a straight line and a parabola.
对于一个一次和一个二次方程，将一次表达式代入二次中，然后求解所得二次方程。解的个数可能为 0、1 或 2 对。其几何意义为直线与抛物线的交点。
5. Graphs of Functions & Transformations | 函数图像与变换
mx + c), quadratic (y = x²), cubic (y = x³), reciprocal (y = 1/x), and exponential (y = kˣ he turning point of a quadratic is found by completing the square or by using x = −b/(2a).
常见函数类型：一次函数 (y = mx + c)、二次函数 (y = x²)、三次函数 (y = x³)、反比例函数 (y = 1/x) 和指数函数 (y = kˣ)。必须能迅速识别各图形的形状。二次函数的顶点可通过配方法或用 x = −b/(2a) 求得。
Transformations: f(x + a) shifts the graph left by a; f(x) + a shifts it up by a; −f(x) reflects in the x‑axis; f(−x) reflects in the y‑axis. For stretched graphs, af(x) stretches vertically by a factor of a, and f(ax) stretches horizontally by a factor of 1/a. Always apply transformations in the correct order when combining them.
图形变换：f(x + a) 使图像向左平移 a 个单位；f(x) + a 向上平移 a；−f(x) 沿 x 轴反射；f(−x) 沿 y 轴反射。拉伸变换中，af(x) 垂直拉伸为原来的 a 倍，f(ax) 水平拉伸为原来的 1/a 倍。组合变换时必须按正确顺序操作。
6. Trigonometry in Right‑Angled & Non‑Right‑Angled Triangles | 直角三角形与斜三角形三角学
In a right‑ n θ = opposite/adjacent. Remember exact trigonometric values for 0°, 30°, 45°, 60° and 90° — these are often tested without a calculator.
在直角三角形中：sin θ = 对边/斜边，cos θ = 邻边/斜边，tan θ = 对边/邻边。务必熟记 0°、30°、45°、60° 和 90° 的精确三角函数值——这些常在不允许使用计算器的题目中出现。
For non‑right‑ f any triangle is ½ ab sin C. The sine rule is used when you know two angles and one side, or two sides and a non‑d angle. The cosine rule is used for two sides and the included angle, or for three sides.
斜三角形中，使用正弦定理：a/sin A = b/sin B = c/sin C，或余弦定理：a² = b² + c² − 2bc cos A。任意三角形面积公式为 ½ ab sin C。已知两角一边或两边一角（非夹角）时用正弦定理；已知两边及其夹角或三边时用余弦定理。
7. Circles: Arcs, Sectors & Circle Theorems | 圆：弧、扇形与圆定理
r². Convert angles to radians only if the question explicitly requires it; GCSE uses degrees.
圆周长 C = 2πr，圆面积 A = πr²。圆心角为 θ（角度制）的扇形：弧长 = (θ/360) × 2πr，扇形面积 = (θ/360) × πr²。仅在题目明确要求时转换为弧度；GCSE 使用角度制。
Circle theorems: the angle in a semicircle is 90°; the perpendicular from the centre to a chord bisects the chord; tangents from an external point are equal in length; the angle between a tangent and a radius is 90°. Always cite the theorem by name or description when giving a reason.
圆定理：半圆上的圆周角为 90°；从圆心垂直于弦的线段平分该弦；从圆外一点引出的两条切线长度相等；切线与半径的夹角为 90°。在书写推理过程时，务必使用定理名称或描述说明理由。
le is perpendicular to the radius at the point of contact; use this to find the gradient of the tangent.
圆心在 (0,0)、半径为 r 的圆方程为 x² + y² = r²。圆心在 (a,b) 时：(x − a)² + (y − b)² = r²。圆的切线在接触点处垂直于半径，可利用这一性质求出切线的斜率。
8. Volume & Surface Area of 3D Shapes | 立体图形的体积与表面积
s are provided on the Edexcel Higher Tier formula sheet: Volume of a prism = area of cross‑ a = πrl, where l is the slant height. For a sphere: volume = ⁴⁄₃ πr³, surface area = 4πr².
以下公式在 Edexcel 高阶试卷的公式表中提供：棱柱体积 = 底面积 × 长。棱锥体积 = ⅓ × 底面积 × 高。圆锥体积 = ⅓ πr²h，侧面积 = πrl，其中 l 为斜高。球体积 = ⁴⁄₃ πr³，表面积 = 4πr²。
For a frustum of a cone (truncated cone), subtract the volume of the smaller cone from the larger one, using similar triangles to find missing lengths. For composite solids, break the shape into simpler components and sum their volumes or surface areas — but be careful not to double‑count internal faces when finding surface area.
计算圆锥台（截头圆锥）的体积时，用大圆锥体积减去小圆锥体积，并利用相似三角形求出缺失长度。对于组合体，将图形拆分为简单形状后分别求体积或表面积——但在求表面积时需注意不要重复计算内部面。
9. Vectors | 向量
orm, (x y) represents a translation of x units right and y units up. The magnitude of vector (a b) is √a² + b².
向量既有大小又有方向，用粗体字母 a 或箭头 → 表示。列向量 (x y) 表示向右移动 x 个单位、向上移动 y 个单位。向量 (a b) 的大小为 √a² + b²。
Vectors can be added or subtracted (tip‑to‑ ultiples of each other. In geometry proofs, show that two vectors are multiples to prove lines are parallel, or that one point lies on a line by expressing its position vector as a combination of others.
向量可以进行加减（三角形法则）和数乘。AB = b − a。平行向量互为标量倍数。在几何证明中，通过证明两个向量成倍数关系可得出两直线平行，或通过将一个点的位置向量表示为其他向量的线性组合来证明该点在某条直线上。
10. Statistics: Averages, Charts & Spread | 统计：平均数、图表与离散程度
. The modal class is the class with the highest frequency. The median is the middle value in ordered data; for grouped data, use linear interpolation if required by the question.
平均数 = 数据总和 ÷ 数据个数。对于分组数据，用组中点估算平均数：Σ(f × 组中点) ÷ Σf。众数组是频数最高的组。中位数是有序数据的中间值；对于分组数据，若题目要求，则需使用线性内插法。
the median marked inside; whiskers extend to the minimum and maximum within 1.5 × IQR of the quartiles. For cumulative frequency graphs, plot the upper class boundary against cumulative frequency; the median and quartiles are found by reading off at ½, ¼ and ¾ of total frequency.
极差 = 最大值 − 最小值。四分位距 (IQR) = 上四分位数 (Q₃) − 下四分位数 (Q₁)。在箱线图中，箱子从 Q₁ 延伸至 Q₃，内部标出中位数；须线延伸至四分位数 1.5 倍 IQR 范围内的最小值和最大值。在累积频数图中，以上组界为横坐标、累积频数为纵坐标描点；中位数和四分位数分别从总频数的 ½、¼、¾ 处读取。
frequency ÷ class width; the area of each bar is proportional to frequency. When comparing distributions, always comment on a measure of centre (mean or median) and a measure of spread (range or IQR), and state them in context.
直方图中，频数密度 = 频数 ÷ 组距；每个条形的面积与频数成比例。比较数据分布时，务必从集中量数（平均数或中位数）和离散程度（极差或四分位距）两方面进行评述，并结合实际情境解释。
11. Probability & Venn Diagrams | 概率与维恩图
o 1. For independent events, P(A and B) = P(A) × P(B). For mutually exclusive events, P(A or B) = P(A) + P(B).
等可能结果中，概率 = 有利结果数 ÷ 所有可能结果总数。所有可能结果的概率之和为 1。独立事件：P(A 且 B) = P(A) × P(B)。互斥事件：P(A 或 B) = P(A) + P(B)。
probability is the probability of an event given that another event has occurred. Use P(A| lp multiply along branches (for combined events) and add probabilities of different branches. With replacement, probabilities remain constant; without replacement, denominators change.
条件概率指在另一事件已发生的条件下某事件发生的概率。使用 P(A|B) = P(A 且 B)/P(B)。树状图有助于沿分支相乘（组合事件）并相加不同分支的概率。有放回时概率保持不变；无放回时分母发生变化。
Venn diagrams represent sets visually. n(A ∪ B) = n(A) + n(B) − n(A ∩ t with the intersection and work outwards. Use a rectangle to represent the universal set.
维恩图直观展示集合关系。n(A ∪ B) = n(A) + n(B) − n(A ∩ B)。填充维恩图时，先确定交集部分，再向外展开。用矩形表示全集。
12. Direct & Inverse Proportion | 正比与反比
t k is found by substituting a known pair of values. Graphs of direct proportion are straight lines through the origin; inverse proportion produces a hyperbola shape.
若 y 与 x 成正比，写为 y = kx。若 y 与 x² 成正比，则 y = kx²。对于反比关系，y = k/x 或 y = k/x²。代入一组已知数值即可求出常数 k。正比关系的图像是过原点的直线；反比关系则呈双曲线形状。
e to find unknowns in contextual problems, such as the time taken to complete a job (inverse to the number of workers). Always write a concluding statement with units.
求常数 k 的公式：正比时 k = y/x；反比时 k = xy。利用这些关系可解答情景题，如完成一项工作所需时间（与工人数量成反比）。最后务必写出带单位的结论句。
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