📚 Core Knowledge Review for Year 11 WJEC Further Maths | Year 11 WJEC 进阶数学：核心知识点梳理
This comprehensive revision guide covers the core topics of the WJEC Level 2 Additional Mathematics specification, designed for Year 11 learners aiming to bridge the gap between GCSE and A-level. Mastering these fundamental areas will not only secure success in the examination but also lay a robust groundwork for advanced study in pure mathematics, mechanics, and statistics.
本篇全面的复习指南涵盖 WJEC Level 2 进阶数学（Additional Mathematics）的核心主题，专为 Year 11 学生设计，旨在衔接 IGCSE 与 A-level 课程。掌握这些基础知识，不仅能确保在考试中取得成功，更能为未来学习纯数学、力学和统计打下坚实的根基。
1. Algebra and Functions | 代数与函数
A function is a rule that maps each input (x) to exactly one output (f(x)). You must be confident in manipulating algebraic fractions, expanding brackets, and simplifying expressions containing surds and indices. Always be able to find the domain (possible input values) and range (possible output values) of a given function.
函数是一种将每一个输入（x）唯一对应到一个输出（f(x)）的规则。你需要熟练掌握代数分式的运算、括号展开以及包含无理数（根式）和指数的表达式简化。并且要能确定给定函数的定义域（可能的输入值）和值域（可能的输出值）。
al gf(x). You should also be able to find inverse functions f⁻¹(x) by rearranging y = f(x) to make x the subject, then swapping x and y.
函数的复合记作 fg(x) = f(g(x))，表示将一个函数的结果作为另一个函数的输入。需要注意 fg(x) 通常不等于 gf(x)。你还应掌握求逆函数 f⁻¹(x) 的方法，即通过重新整理 y = f(x) 使 x 成为主体，然后交换 x 与 y。
- Example: If f(x) = 2x + 3 and g(x) = x², then fg(x) = 2x² + 3.
- 示例：若 f(x) = 2x + 3，g(x) = x²，则 fg(x) = 2x² + 3。
2. Quadratic Functions | 二次函数
be solved by factorising, completing the square, or using the quadratic formula: x = [-b ± √b² - 4ac two distinct real roots (Δ > 0), one repeated real root (Δ = 0), or no real roots (Δ < 0).
二次表达式 ax² + bx + c 可以通过因式分解、配方法或使用二次公式求解：x = [-b ± √b² - 4ac] / (2a)。判别式 Δ = b² - 4ac 决定了根的性质：两个不等实根（Δ > 0），一个重根（Δ = 0），或无实根（Δ < 0）。
The completed square form a(x + p)² + q reveals the turning point (-p, q). If a > 0, the parabola has a minimum point; if a < 0, it has a maximum point. You must be able to sketch the graph showing intercepts and the vertex.
配方法得到的 a(x + p)² + q 形式可以直接给出图像的顶点 (-p, q)。若 a > 0，抛物线开口向上，有最小值点；若 a < 0，开口向下，有最大值点。你必须能绘制草图，标出截距和顶点。
y = 2(x - 3)² + 1 → vertex (3, 1), minimum value 1
3. Polynomials and the Factor Theorem | 多项式与因式定理
rem tells us that when p(x) is divided by (x - a), the remainder is p(a). These theorems allow you to factorise cubic and higher-order polynomials by testing small integer values.
对于多项式 p(x)，因式定理指出：(x - a) 是 p(x) 的因式，当且仅当 p(a) = 0。余数定理则告诉我们，p(x) 除以 (x - a) 的余数为 p(a)。这些定理可以帮助你通过代入小整数来分解三次及更高次的多项式。
Once one factor is found, you can use long division or the method of equating coefficients to reduce the polynomial to a quadratic, which can then be factorised further. Always check for a common factor first.
一旦找到一个因式，你就可以使用长除法或比较系数法将多项式降次为二次式，从而继续分解。请始终先检查是否存在公因式。
- = 0, so (x - 2) is a factor. Dividing gives (x - 2)(x² - 2x - 3) = (x - 2)(x - 3)(x + 1).
- 示例：p(x) = x³ - 4x² + x + 6，由于 p(2) = 0，所以 (x - 2) 为因式。做除法得 (x - 2)(x² - 2x - 3) = (x - 2)(x - 3)(x + 1)。
4. Inequalities | 不等式
ues. Then sketch the parabola to identify intervals where the inequality holds. Remember to flip the inequality sign when multiplying or dividing by a negative number.
解二次不等式如 ax² + bx + c > 0 时，先解方程 ax² + bx + c = 0 求出临界值，然后画出抛物线草图以确定不等式成立的区间。注意，当乘以或除以一个负数时，必须改变不等号的方向。
e, combine into a single fraction, and find critical values from both numerator and denominator. Use a sign table or test points, never cross-multiply without considering the sign of the denominator.
对于分式不等式如 (x - 1)/(x + 2) ≤ 0，应将所有项移到一边，合并为一个分式，并从分子和分母中找出临界值。使用符号表或代入测试点，切不可在不考虑分母符号的情况下直接交叉相乘。
5. Exponentials and Logarithms | 指数与对数
The function y = aˣ thms: logₐ (xy) = logₐ x + logₐ y, logₐ (x/y) = logₐ x - logₐ y, and logₐ (xⁿ) = n logₐ x.
函数 y = aˣ（a > 0）可用于模拟指数增长（a > 1）或衰减（0 < a < 1）。它的反函数是对数：logₐ y = x。你需熟记对数运算法则：logₐ (xy) = logₐ x + logₐ y，logₐ (x/y) = logₐ x - logₐ y，以及 logₐ (xⁿ) = n logₐ x。
The natural exponential eˣ ogₑ x) appear frequently in calculus and growth models. Be able to solve equations like e²ˣ by taking ln of both sides, giving 2x = ln 5. You will also need to recognise graphs of aˣ and logₐ x and their asymptotes.
自然指数 eˣ 与自然对数 ln x（logₑ x）在微积分和增长模型中频繁出现。要能解决如 e²ˣ = 5 的方程，两边取自然对数得 2x = ln 5。你还需要识别 aˣ 和 logₐ x 的图像及其渐近线。
6. Trigonometry | 三角学
c/(sin C) and the cosine rule a² = b² + c² - 2bc cos A are essential for solving non-right-angled triangles.
必须熟记 0°、30°、45°、60° 和 90° 这些特殊角的 sin、cos、tan 的精确值。在直角三角形之外，正弦定理 a/(sin A) = b/(sin B) = c/(sin C) 和余弦定理 a² = b² + c² - 2bc cos A 是求解任意三角形的关键工具。
unit circle. Remember to find all solutions within the given interval, including those generated by the periodic nature of trigonometric functions.
你应能利用 CAST 图或单位圆求解限定区间（如 0° ≤ x ≤ 360°）内的三角方程，例如 2 sin x = 1。切记要考虑三角函数的周期性，找到区间内的全部解。
sin 45° = √2/2, cos 60° = 1/2, tan 30° = 1/√3
7. Differentiation | 微分
x = 12x³ - 4x. The derivative tells us whether a function is increasing (dy/dx > 0) or decreasing (dy/dx < 0).
微分可求出曲线的斜率（梯度）。对于 y = xⁿ，dy/dx = nxⁿ⁻¹。这一法则可推广至多项式的和与常数倍：若 y = 3x⁴ - 2x² + 5，则 dy/dx = 12x³ - 4x。导数可以告诉我们函数是递增（dy/dx > 0）还是递减（dy/dx < 0）。
lar, with gradient -1/m. Stationary points occur where dy/dx = 0; use the second derivative d²y/dx² to classify them as maximum, minimum, or points of inflection.
切线与法线：在点 (x₁, y₁) 处，切线斜率 m = dy/dx 在 x₁ 处的值，方程为 y - y₁ = m(x - x₁)。法线与切线垂直，斜率为 -1/m。驻点出现在 dy/dx = 0 处；利用二阶导数 d²y/dx² 可以区分极大值、极小值或拐点。
8. Integration | 积分
C is the constant of integration. You must be able to find the equation of a curve given its derivative and a point on the curve by determining the constant C.
积分是微分的逆运算。不定积分 ∫ xⁿ dx = xⁿ⁺¹/(n+1) + C，其中 C 为积分常数。你需要能根据给定的导数和曲线上的一点，确定常数 C，从而求出原曲线的方程。
ive. You should be able to calculate the area between a curve and a line, often by subtracting one integral from another.
定积分 ∫ₐᵇ f(x) dx 表示曲线 y = f(x) 与 x 轴在 x = a 到 x = b 之间的有号面积。位于 x 轴下方的面积为负值。你需要掌握计算曲线与直线之间所围面积的方法，通常是通过两个积分的相减来实现。
∫₀² (3x² + 1) dx = [x³ + x]₀² = (8 + 2) - 0 = 10
9. Vectors | 向量
A vector is a quantity with both magnitude and direction, often written as a column vector v = (x, y) or xi + yj. The magnitude is |v| = √x² + y². Unit vectors have magnitude 1. To add vectors, add their components; multiplication by a scalar changes the magnitude but not the direction (unless negative).
向量是既有大小又有方向的量，通常表示为列向量 v = (x, y) 或 xi + yj。其模长为 |v| = √x² + y²。单位向量的模长为 1。向量相加即对应分量相加；乘以一个标量会改变向量的大小但不改变方向（除非标量为负）。
e to solve problems involving parallel vectors, collinearity, and the ratio theorem for dividing a line segment in a given ratio.
A 点相对于原点 O 的位置向量为 OA。从 A 到 B 的向量为 AB = OB - OA。你应能解决涉及平行向量、共线性，以及按比例分割线段的定比分点问题。
10. Matrix Transformations | 矩阵变换
resent a linear transformation of the plane. Multiplying M by a position vector maps the point to a new location. Common transformations include rotations, reflections, enlargements, and shears; you must recognise the standard matrices for these.
2×2 矩阵 M = [[a, b], [c, d]] 可以表示平面的线性变换。用 M 去乘一个位置向量，可将该点映射到一个新的位置。常见的变换包括旋转、反射、放大和剪切；你必须识别上述变换的标准矩阵。
= ad - bc determines the area scale factor of the transformation: an area is multiplied by |det M| hould be able to find the image of a shape and the inverse transformation where it exists.
行列式 det M = ad - bc 决定了变换的面积比例因子：任何图形的面积都将乘以 |det M|。若 det M = 0，则变换是奇异的，会将平面压缩。你需要能够求出一个图形的像，以及在变换可逆时求出其逆变换。
| Transformation | Matrix |
|---|---|
| Reflection in x-axis | [[1,0],[0,-1]] |
| Rotation 90° anticlockwise | [[0,-1],[1,0]] |
| Enlargement scale factor k | [[k,0],[0,k]] |
11. Calculus Applications: Optimisation and Kinematics | 微积分应用：优化与运动学
iable, differentiate, and set dy/dx = 0. Typical scenarios include maximising area or volume and minimising surface area or cost.
优化问题涉及求某个量的最大值或最小值，通常需要将该量表示为单一变量的函数，然后求导并令 dy/dx = 0。常见情境包括面积或体积的最大化，以及表面积或成本的最小化。
, v = ds/dt and a = dv/dt = d²s/dt². Conversely, given v, displacement is the integral of velocity: s = ∫ v dt. Remember initial conditions to find constants.
在直线运动的运动学中，位移 s、速度 v 和加速度 a 通过微积分建立联系。若已知 s 为时间 t 的函数，则 v = ds/dt，a = dv/dt = d²s/dt²。反之，若已知 v，位移 v 的积分：s = ∫ v dt。注意利用初始条件确定积分常数。
12. Sequences, Binomial Expansion, and Proof | 数列、二项式展开与证明
(n-1)d]. The binomial expansion for (a + b)ⁿ can be applied for positive integer n using Pascal's triangle or the nCr button.
你需要认识等差数列（公差恒定），并能使用通项公式 uₙ = a + (n-1)d 与求和公式 Sₙ = n/2[2a + (n-1)d]。对于正整数 n，(a + b)ⁿ 的二项式展开可利用帕斯卡三角形或组合数（nCr）来实现。
Logical proof techniques include direct proof, proof by exhaustion, and proof by contradiction. These may appear in the context of algebraic identities, irrationality of √2, or statements about odd/even numbers. Structure your arguments clearly, stating assumptions and conclusions formally.
逻辑证明的方法包括直接证明、穷举证法和反证法。这些证明可能出现在代数恒等式、√2 的无理性证明，或关于奇数/偶数的命题中。请清晰地组织你的推理过程，规范地陈述假设与结论。
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