📚 Year 11 Eduqas Maths: Core Topic Overview | Year 11 Eduqas 数学：核心知识点梳理
Year 11 is the crucial stage where students consolidate and extend their GCSE Mathematics knowledge ahead of the final examinations. The Eduqas specification covers a wide range of topics, from manipulating surds and indices to solving quadratic equations, applying circle theorems, and interpreting cumulative frequency graphs. This revision guide brings together the core concepts you must master, presented with clear bilingual explanations and key formulas. Whether you are aiming for a grade 5 or targeting the top grades 8-9, a thorough understanding of these fundamentals will boost your confidence and performance.
Year 11 是学生巩固和拓展 GCSE 数学知识、冲刺大考的关键阶段。Eduqas 考纲涵盖广泛，从根式与指数运算到二次方程求解，从圆定理应用到累积频率图解读。这份复习指南汇聚了必须掌握的核心概念，以简洁的中英双语讲解和关键公式呈现。无论你的目标是 5 分还是冲击 8-9 分，扎实掌握这些基础都将提升你的信心与表现。
1. Number: Surds and Indices | 数：根式与指数
A surd is an irrational root such as √2 or ³√5. To simplify surds, use the rules √(ab) = √a × √b and √(a/b) = √a / √b.
根式是无理根，例如 √2 或 ³√5。简化根式时使用规则 √(ab) = √a × √b 和 √(a/b) = √a / √b。
When a surd appears in the denominator, we rationalise it. For 1/√a, multiply numerator and denominator by √a to obtain √a/a. For expressions like 1/(a+√b), multiply top and bottom by its conjugate a−√b, giving (a−√b)/(a²−b).
当分母含有根式时，需进行有理化。对于 1/√a，分子分母同乘 √a 得到 √a/a。对于 1/(a+√b) 这类式子，上下同乘其共轭式 a−√b，得到 (a−√b)/(a²−b)。
Laws of indices are fundamental: aᵐ × aⁿ = aᵐ⁺ⁿ, (aᵐ)ⁿ = aᵐⁿ, a⁰ = 1, and a⁻ⁿ = 1/aⁿ. Fractional indices link roots and powers: a¹⸍ⁿ = ⁿ√a, and aᵐ⸍ⁿ = (ⁿ√a)ᵐ.
指数法则是基础：aᵐ × aⁿ = aᵐ⁺ⁿ，(aᵐ)ⁿ = aᵐⁿ，a⁰ = 1，且 a⁻ⁿ = 1/aⁿ。分数指数则将方根与乘方联系在一起：a¹⸍ⁿ = ⁿ√a， aᵐ⸍ⁿ = (ⁿ√a)ᵐ。
a⁻ⁿ = 1/aⁿ, a^(¹/₂) = √a, a^(³/₂) = (√a)³
2. Algebra: Quadratic Equations and Functions | 代数：二次方程与函数
A quadratic equation has the form ax² + bx + c = 0. The three main methods of solution are factorising, completing the square, and using the quadratic formula.
二次方程的一般形式为 ax² + bx + c = 0。三种主要解法是：因式分解、配方法和使用求根公式。
If the quadratic expression can be written as (px + q)(rx + s) = 0, then the solutions are x = −q/p and x = −s/r.
若能写成 (px + q)(rx + s) = 0，则解为 x = −q/p 和 x = −s/r。
The quadratic formula, derived from completing the square, works for any quadratic: x = [−b ± √(b² − 4ac)] / 2a.
由配方法导出的求根公式适用于所有二次方程：x = [−b ± √(b² − 4ac)] / 2a。
x = [−b ± √(b² − 4ac)] / 2a
The discriminant Δ = b² − 4ac tells you about the roots: if Δ > 0 there are two distinct real roots; Δ = 0 gives one repeated root; Δ < 0 means no real roots.
判别式 Δ = b² − 4ac 揭示根的性质：Δ > 0 时有两个相异实根；Δ = 0 时有一个重根；Δ < 0 时无实根。
Graphing y = ax² + bx + c yields a parabola. The sign of a determines whether it opens upward (a > 0) or downward (a < 0). The vertex is at (−b/(2a), f(−b/(2a))), and the axis of symmetry is x = −b/(2a).
y = ax² + bx + c 的图像是一条抛物线。a 的正负决定了开口向上 (a > 0) 还是向下 (a < 0)。顶点坐标为 (−b/(2a), f(−b/(2a)))，对称轴为 x = −b/(2a)。
3. Algebra: Simultaneous Equations | 代数：联立方程
Linear simultaneous equations can be solved by eliminating a variable or by substitution. Multiply equations to align coefficients, then add or subtract to eliminate.
线性联立方程可通过消元法或代入法求解。将方程乘以适当倍数使系数对齐，然后相加或相减消去一个变量。
When one equation is linear and the other is quadratic, substitute the linear expression into the quadratic. This will create a quadratic in one variable that can be solved normally.
当一方程为线性、另一方程为二次时，将线性表达式代入二次方程。这将得到一个仅含一个未知数的二次方程，按常规方法求解即可。
| Method | When to use |
| Elimination | Both equations linear; coefficients easily matched. |
| Substitution | One variable is already isolated, or linear + quadratic. |
Elimination: match x- or y- coefficients and subtract to find the first variable. Substitution: rearrange one equation to make x or y the subject, then replace in the other.
消元法：配齐 x 或 y 的系数后相减求出一个变量。代入法：将一个方程变形，将 x 或 y 表示出来，再代入另一个方程。
4. Ratio: Direct and Inverse Proportion | 比例：正比与反比
Two quantities are in direct proportion if y ∝ x, which means y = kx for some constant k. The graph is a straight line through the origin.
若 y ∝ x，则两个量成正比，即存在常数 k 使得 y = kx。图像是一条过原点的直线。
Inverse proportion y ∝ 1/x gives y = k/x; the graph is a hyperbola. Other proportionalities include y ∝ x² (parabolic pattern) and y ∝ 1/x².
若 y ∝ 1/x，则为反比关系 y = k/x，图像为双曲线。其他比例关系有 y ∝ x²（抛物线型）和 y ∝ 1/x²。
To find k, substitute a known pair of values into the proportion equation. You can then use the formula to calculate unknown values or construct a table.
求 k 时，将一对已知值代入比例方程。随后即可用公式计算未知量或列值表。
Direct: y = kx, Inverse: y = k/x, y = kx²
5. Geometry: Circle Theorems | 几何：圆定理
Circle theorems are essential for angle calculations in cyclic quadrilaterals, tangents, and chords. Here are the key theorems:
圆定理对于圆内接四边形、切线和弦的角度计算至关重要。以下是关键定理：
| Theorem | Statement |
| Angle in a semicircle | The angle at the circumference subtended by a diameter is 90°. |
| Angles in the same segment | Angles subtended by the same chord at the circumference are equal. |
| Cyclic quadrilateral | Opposite angles sum to 180°. |
| Tangent and radius | A tangent is perpendicular to the radius at the point of contact. |
| Alternate segment | The angle between a tangent and a chord equals the angle in the alternate segment. |
When solving problems, mark all known angles on the diagram. Use the fact that radii are equal to identify isosceles triangles. Combine theorems to find missing angles step by step.
解题时，在图上标出所有已知角度。利用半径相等识别等腰三角形。逐步结合定理求出缺失的角度。
6. Geometry: Trigonometry | 几何：三角学
Right-angled triangle trigonometry begins with SOH CAH TOA: sin = opposite/hypotenuse, cos = adjacent/hypotenuse, tan = opposite/adjacent.
直角三角形三角学从 SOH CAH TOA 开始：sin = 对边/斜边，cos = 邻边/斜边，tan = 对边/邻边。
Exact trigonometric values for 0°, 30°, 45°, 60° and 90° must be memorised. Use the special triangles (equilateral halved, right isosceles) to derive them.
需牢记 0°、30°、45°、60° 和 90° 的精确三角比值。可利用特殊三角形（等边三角形的一半、等腰直角三角形）推导。
| θ | 0° | 30° | 45° | 60° | 90° |
| sin | 0 | ½ | 1/√2 | √3/2 | 1 |
| cos | 1 | √3/2 | 1/√2 | ½ | 0 |
| tan | 0 | 1/√3 | 1 | √3 | undefined |
For non‑right‑angled triangles, the sine rule is used: a/sin A = b/sin B = c/sin C. It works when you know two angles and a side, or two sides and a non‑included angle.
对于非直角三角形，使用正弦定理：a/sin A = b/sin B = c/sin C。适用于已知两角一边，或两边及一非夹角的情形。
The cosine rule is a² = b² + c² − 2bc cos A. Apply it when you know two sides and the included angle, or all three sides and want an angle.
余弦定理为 a² = b² + c² − 2bc cos A。当已知两边及其夹角，或已知三边求角时使用。
The area of any triangle can be found with (1/2)ab sin C, where C is the included angle between sides a and b.
任意三角形的面积可用 (1/2)ab sin C 计算，其中 C 为边 a 和 b 的夹角。
Sine rule: a/sin A = b/sin B = c/sin C
Cosine rule: a² = b² + c² − 2bc cos A
7. Geometry: Vectors | 几何：向量
Vectors describe both magnitude and direction. They are often written as column vectors like (x, y) or using bold letters a. Addition and subtraction follow component‑wise operations.
向量描述大小和方向。常写作列向量如 (x, y) 或用粗体字母 a。加减法按分量逐一进行。
Scalar multiplication scales a vector: k × a = (kx, ky). Two vectors are parallel if one is a scalar multiple of the other.
标量乘法缩放向量：k × a = (kx, ky)。若一个向量是另一个的标量倍，则二者平行。
Vectors can be used to prove geometric results. For instance, if AB = k × CD, then AB is parallel to CD. Points A, B and C are collinear if AB is a scalar multiple of BC.
向量可用于证明几何结论。例如，若 AB = k × CD，则 AB 平行于 CD。若 AB 是 BC 的标量倍，则 A、B、C 三点共线。
When solving problems, express the required vector as a route combining known vectors. Frequently use the fact that AB = b − a where a and b are position vectors.
解题时，将所求向量表示为已知向量的路径组合。常用关系：AB = b − a，其中 a 和 b 为位置向量。
8. Statistics: Cumulative Frequency and Box Plots | 统计：累积频率与箱线图
A cumulative frequency table adds up frequencies row by row. Plot cumulative frequency against the upper class boundary to create a smooth cumulative frequency curve.
累积频率表逐行累加频数。以累积频率为纵坐标、上限值为横坐标绘制光滑的累积频率曲线。
The median is found at the position where the cumulative frequency reaches half the total. The lower quartile (LQ) is at one‑quarter, and the upper quartile (UQ) at three‑quarters. The interquartile range (IQR) = UQ − LQ.
中位数位于累积频率达到总数一半处。下四分位数 (LQ) 在四分之一处，上四分位数 (UQ) 在四分之三处。四分位距 (IQR) = UQ − LQ。
A box plot displays these five‑number summaries: minimum, LQ, median, UQ, maximum. Box plots are ideal for comparing distributions and identifying skewness.
箱线图展示这五个统计量：最小值、下四分位数、中位数、上四分位数、最大值。箱线图非常适合比较数据分布及识别偏态。
When drawing a box plot, use a scale, mark the five values, draw the box from LQ to UQ with the median line inside, and add whiskers to the extremes. Outliers may be plotted individually.
绘制箱线图时，使用刻度，标记五个值，从 LQ 到 UQ 画箱体并在内部画中位数线，须线延伸至极值。异常点可单独标记。
9. Probability: Tree Diagrams and Conditional Probability | 概率：树图与条件概率
Tree diagrams show the outcomes of multiple events. Branches are labelled with probabilities; to find the probability of a combined outcome, multiply along the branches.
树图展示多步事件的结果。分支上标注概率；求组合结果概率时，将路径上的概率相乘。
When events are independent, P(A and B) = P(A) × P(B). When events are dependent, the second probability changes based on the first outcome.
若事件独立，则 P(A 且 B) = P(A) × P(B)。若事件不独立，第二步概率需根据第一步结果调整。
Conditional probability is written as P(A|B), meaning “the probability of A given B”. The formula is P(A|B) = P(A ∩ B) / P(B).
条件概率记作 P(A|B)，表示“在 B 已发生的条件下 A 发生的概率”。公式为 P(A|B) = P(A ∩ B) / P(B)。
For tree diagrams with dependent events, apply the updated probabilities on the second branches. Always check that the probabilities on each set of branches sum to 1.
对于含非独立事件的树图，在第二阶段分支上使用更新后的概率。每次都需检查每组分支的概率之和是否为 1。
P(A|B) = P(A ∩ B) / P(B)
10. Algebra: Functions and Graph Transformations | 代数：函数与图像变换
Function notation f(x) is used to specify a rule. f(2) means substitute x = 2 into the expression. Composite functions like fg(x) mean apply g first, then f.
函数记号 f(x) 用于表示对应法则。f(2) 表示将 x = 2 代入表达式。复合函数如 fg(x) 表示先作用 g，再作用 f。
The inverse function f⁻¹(x) reverses the effect of f. To find it, write y = f(x), swap x and y, then solve for y.
反函数 f⁻¹(x) 逆转 f 的作用。求法：令 y = f(x)，交换 x 与 y，再解出 y。
Graph transformations correspond to changes in the function equation: f(x) + a translates the graph vertically by a units; f(x + a) translates horizontally by −a; a f(x) stretches vertically by factor a; f(ax) stretches horizontally by factor 1/a. A minus sign inside or outside reflects the graph.
图像变换对应函数式的变化：f(x) + a 将图像垂直平移 a 个单位；f(x + a) 水平平移 −a 个单位；a f(x) 垂直缩放 a 倍；f(ax) 水平缩放 1/a 倍。内或外的负号均可引起对称翻折。
| Transformation | Effect on y = f(x) |
| Vertical translation +k | y = f(x) + k |
| Horizontal translation +h | y = f(x − h) |
| Vertical stretch ×a | y = a f(x) |
| Horizontal stretch ×1/b | y = f(bx) |
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