📚 Year 11 Edexcel Maths: Core Topics Summary | Year 11 Edexcel 数学：核心知识点梳理
Year 11 Edexcel Maths brings together all the fundamental topics you need to master for your exams. From surds to vectors, from quadratic graphs to cumulative frequency, a clear grasp of the core principles will boost both your calculator and non-calculator paper scores. This guide walks you through each key area with essential formulas, theorems, and worked examples.
Year 11 Edexcel 数学汇集了备考所需的所有基础课题。从根式到向量，从二次函数图像到累积频率图，清晰掌握核心原理将同时提升你在可用计算器与不可用计算器试卷上的得分。本指南将带你梳理每一个关键领域，涵盖必背公式、定理与典型示例。
1. Surds and Indices | 根式与指数
Surds are irrational numbers left in root form. You must be able to simplify them, combine them, and rationalise denominators.
根式是以根号形式留下的无理数。你需要会化简根式、合并根式以及对分母有理化。
Key surd rules: √a × √b = √ab and √a / √b = √a/b. For example, √18 = √9×2 = 3√2.
关键根式法则： √a × √b = √ab 以及 √a / √b = √a/b。例如 √18 = √9×2 = 3√2。
To rationalise a denominator like 1/√2, multiply numerator and denominator by √2 to get √2/2. When the denominator is a + b√c, use the conjugate a − b√c.
对形如 1/√2 的分母进行有理化时，分子分母同乘 √2 得到 √2/2。当分母为 a + b√c 时，使用共轭式 a − b√c 来处理。
Index laws are just as important: aᵐ × aⁿ = aᵐ⁺ⁿ, (aᵐ)ⁿ = aᵐⁿ, a⁰ = 1, a⁻ⁿ = 1/aⁿ, and a1/n = ⁿ√a.
指数法则同样重要：aᵐ × aⁿ = aᵐ⁺ⁿ，(aᵐ)ⁿ = aᵐⁿ，a⁰ = 1，a⁻ⁿ = 1/aⁿ，以及 a1/n = ⁿ√a。
(8x³)²/³ = 8²/³ × x² = (∛8)² × x² = 4x²
2. Quadratic Equations | 二次方程
. The three main solution methods are factorising, completing the square, and using the quadratic formula.
二次方程形如 ax² + bx + c = 0。三种主要求解方法为因式分解、配方法和求根公式法。
x = [ −b ± √b² − 4ac ] / 2a
can be written as (px + q)(rx + s) = 0. Always set each bracket to zero to find the roots.
当二次式能写成 (px + q)(rx + s) = 0 时可用因式分解。记住令每个括号等于零以求解根。
ves two distinct real roots, Δ = 0 gives one repeated root, and Δ < 0 gives no real roots.
判别式 Δ = b² − 4ac 揭示根的性质：Δ > 0 时有两个不等实根，Δ = 0 时有一个重根，Δ < 0 时没有实根。
c in the form a(x + p)² + q, revealing the vertex (−p, q) and the line of symmetry x = −p.
配方法将二次式写成 a(x + p)² + q 的形式，由此可直接读出顶点 (−p, q) 与对称轴 x = −p。
3. Algebraic Fractions | 代数分式
Algebraic fractions are simplified, added, subtracted, multiplied, or divided using the same principles as numerical fractions – but factorising takes centre stage.
代数分式的化简、加减和乘除遵循与数值分式相同的原则——但其中因式分解是核心步骤。
letely. For example, (x² − 9)/(x² − 3x) = [(x−3)(x+3)]/[x(x−3)] = (x+3)/x, provided x ≠ 3.
约分前务必对分子和分母进行彻底因式分解。例如 (x² − 9)/(x² − 3x) = [(x−3)(x+3)]/[x(x−3)] = (x+3)/x，其中 x ≠ 3。
To add or subtract, find the lowest common denominator. Multiply through by the missing factors, then combine and simplify the numerators.
加减运算时，先找出最简公分母，乘上缺失的因式，再合并并化简分子。
For division, multiply by the reciprocal of the second fraction and then factorise to cancel common terms.
做除法时，乘以第二个分式的倒数，然后再通过因式分解约去公因式。
4. Linear and Simultaneous Equations | 线性与联立方程
Linear equations involving one variable are solved by isolating the unknown. With two unknowns, you need a pair of simultaneous equations.
含一个未知量的线性方程通过移项解出未知数。当有两个未知量时，需要一对联立方程。
The elimination method encourages adding or subtracting equations to cancel one variable. If coefficients differ, multiply one or both equations to match them.
消元法通过相加或相减方程以消去一个变量。若系数不同，则将一个或两个方程乘以某个数使系数一致。
The substitution method involves rearranging one equation to express one variable in terms of the other, then replacing it in the second equation.
代入法则是将其中一个方程变形，将一个变量用另一个变量表示，再将其代入第二个方程。
When one equation is quadratic and the other linear, substitution will yield a quadratic in one variable. Solve it, then back-substitute to find the corresponding values.
当一方程为二次、另一方程为一次时，通过代入可得到一个关于单一变量的二次方程。求解后再回代求出对应的另一变量值。
5. Inequalities | 不等式
Inequalities express a range of values. Solving them resembles solving equations, but multiplying or dividing by a negative number reverses the sign.
不等式表示一个取值范围。求解不等式与解方程类似，但当乘或除以负数时必须反转不等号。
cles for ≤ or ≥. For example, −2 < x ≤ 3 means x is greater than −2 and up to 3 inclusive.
在数轴上表示解集时，< 或 > 用空心圆，≤ 或 ≥ 用实心圆。例如 −2 < x ≤ 3 表示 x 大于 −2 且不超过 3。
Quadratic inequalities, like x² − 5x + 6 > 0, are solved by sketching the graph of the related quadratic function and identifying where it lies above the x‑axis.
二次不等式如 x² − 5x + 6 > 0，需画出对应二次函数的草图，找出图像位于 x 轴上方所对应的区间。
Always write your final answer using set notation or inequality signs, and check boundary points carefully.
最终答案务必用集合符号或不等式表示，并仔细验证边界点。
6. Graphs and Functions | 图像与函数
ssential. The linear graph has equation y = mx + c, where m is the gradient and c is the y‑intercept.
透彻理解图像形状及其变换至关重要。一次函数图像方程为 y = mx + c，其中 m 为斜率，c 为 y 轴截距。
abolas. If a > 0 the parabola has a minimum point; if a < 0 it has a maximum. The turning point can be found by completing the square.
二次函数图像 y = ax² + bx + c 为抛物线。若 a > 0，抛物线有最低点；若 a < 0，则有最高点。可通过配方法求得顶点。
= k/x appear frequently. You must also recognise exponential growth and decay curves y = aˣ.
三次函数 y = ax³ + bx² + cx + d 与反比例函数 y = k/x 的图像也常出现。你还要能识别指数增长与衰减曲线 y = aˣ。
Graph transformations include translations, stretches, and reflections. Replacing x with x − a moves the graph a units to the right; multiplying the function by −1 reflects it in the x‑axis.
图像变换包括平移、伸缩和反射。将 x 替换为 x − a 使图像右移 a 个单位；将函数乘以 −1 则使图像关于 x 轴反射。
7. Trigonometry | 三角学
Right‑ ratios sin θ = opposite/hypotenuse, cos θ = adjacent/hypotenuse, tan θ = opposite/adjacent – summarised as SOHCAHTOA.
直角三角形中的三角比基于 sin θ = 对边/斜边，cos θ = 邻边/斜边，tan θ = 对边/邻边——可用 SOHCAHTOA 记忆。
values for 0°, 30°, 45°, 60° and 90°. For instance, sin 30° = ½, cos 60° = ½, tan 45° = 1.
必须熟记 0°、30°、45°、60° 和 90° 的精确三角值。例如 sin 30° = ½，cos 60° = ½，tan 45° = 1。
The sine rule and cosine rule extend trigonometry to non‑right‑ triangles. Sine rule: a/sin A = b/sin B = c/sin C. Cosine rule: a² = b² + c² − 2bc cos A.
正弦定理与余弦定理将三角学拓展到任意三角形。正弦定理：a/sin A = b/sin B = c/sin C；余弦定理：a² = b² + c² − 2bc cos A。
The area of any triangle can be calculated using ½ ab sin C, where a and b are two sides and C is the included angle.
任意三角形的面积可用 ½ ab sin C 计算，其中 a 与 b 为两边，C 为其夹角。
8. Geometry: Circle Theorems | 几何：圆定理
Circle theorems are a staple of the Edexcel exam. You must be able to state and apply them to find missing angles.
圆定理是 Edexcel 考试的常客。你必须能陈述并应用这些定理求缺失角度。
| Theorem | Description |
|---|---|
| Angle at centre | Angle at centre is twice angle at circumference. |
| Angles in same segment | Angles subtended by the same chord are equal. |
| Angle in a semicircle | The angle in a semicircle is a right angle (90°). |
| Tangent and radius | A tangent meets the radius at 90°. |
| Alternate segment theorem | Angle between tangent and chord equals angle in alternate segment. |
| Cyclic quadrilateral | Opposite angles sum to 180°. |
Typically, questions provide a diagram with a few known angles; you then apply a chain of theorems to find the unknown. Always give reasons alongside your calculations.
典型题目会提供一个带有若干已知角度的图，你需要串联应用多个定理求未知角。务必在计算的同时写明理由。
9. Vectors | 向量
Vectors describe both magnitude and direction. They are written in column form or as expressions with unit vectors i and j.
向量兼有大小与方向，可以写成列向量形式或用单位向量 i 与 j 表达。
Vector addition follows the triangle law: to add vectors a and b, place the tail of b at the head of a; the resultant goes from the tail of a to the head of b.
向量加法遵循三角形法则：将向量 b 的起点置于向量 a 的终点，则合成向量由 a 的起点指向 b 的终点。
To show that three points A, B, and C are collinear, demonstrate that vectors AB and BC are parallel (i.e. one is a scalar multiple of the other) and share a common point.
要证明三点 A、B、C 共线，须证明向量 AB 与 BC 平行（即其中一个为另一的标量倍数）且它们有公共点。
For geometric proofs, express all relevant vectors in terms of a few basic vectors, then use properties like midpoints or ratios to establish the required relationship.
进行几何证明时，将相关向量均用若干基向量表示，再利用中点或比例等性质建立所需关系。
10. Probability | 概率
Probability is measured on a scale from 0 to 1. Tree diagrams help visualise successive events and are essential for conditional probability questions.
概率的度量范围在 0 至 1 之间。树形图可直观展示连续事件，在条件概率问题中尤为重要。
events, P(A and B) = P(A) × P(B). For mutually exclusive events, P(A or B) = P(A) + P(B).
对于独立事件，P(A 且 B) = P(A) × P(B)；对于互斥事件，P(A 或 B) = P(A) + P(B)。
Conditional probability P(A| ts the chance of A occurring given that B has already happened. It is calculated using P(A|B) = P(A and B)/P(B).
条件概率 P(A|B) 表示在事件 B 已发生的条件下 A 发生的概率，计算公式为 P(A|B) = P(A 且 B
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