📚 Year 11 CIE Additional Mathematics: Core Concepts Review | CIE 进阶数学：核心知识点梳理
This revision guide covers the essential topics for the CIE IGCSE Additional Mathematics (0606) syllabus, designed for Year 11 students. Mastering these core concepts will build a strong foundation for problem-solving and examination success.
本篇复习指南涵盖 CIE IGCSE 进阶数学（0606）课程的核心知识点，专为 Year 11 学生设计。扎实掌握这些内容将为解题和考试成功打下坚实基础。
1. Functions and Their Properties | 函数及其性质
A function f is a rule that maps each element x from a domain to exactly one element f(x) in the range. The vertical line test confirms whether a graph represents a function: any vertical line must intersect the graph at most once.
函数 f 是将定义域中的每个元素 x 映射到值域中唯一元素 f(x) 的规则。垂直线检验可以确认图形是否表示一个函数：任意垂直线最多与图形相交一次。
, and its graph is the reflection of y = f(x) in the line y = x. Composite functions such as fg(x) mean applying g first, then f.
函数要有反函数必须是一一映射，可通过水平线检验验证。反函数 f⁻¹ 逆转了这个映射，其图形是 y = f(x) 关于直线 y = x 的对称图形。复合函数如 fg(x) 表示先作用 g，再作用 f。
≥ 0 has range f(x) ≥ 2. Always consider the shape of the graph when determining the range.
一项重要技能是根据给定的定义域求函数的值域。例如，f(x) = x² + 2 且 x ≥ 0 时，值域为 f(x) ≥ 2。判断值域时总要结合图像的形状。
2. Quadratic Functions and Their Graphs | 二次函数及其图像
x - h)² + k, where the vertex is (h, k). If a > 0, the parabola has a minimum point; if a < 0, it has a maximum.
一般式为 y = ax² + bx + c。通过配方法可将其改写为 y = a(x - h)² + k，此时顶点为 (h, k)。若 a > 0，抛物线有最低点；若 a < 0，则存在最高点。
repeated real root, and Δ < 0 gives no real roots. This also helps solve quadratic inequalities by factorising and sketching the sign diagram.
判别式 Δ = b² - 4ac 决定了方程根的类型。Δ > 0 时有两个不等实根，Δ = 0 时有一个重根，Δ < 0 时无实根。这也有助于通过因式分解和符号图来求解二次不等式。
x = (-b ± √b² - 4ac) / 2a
例如，解 x² - 5x + 6 > 0，可因式分解为 (x - 2)(x - 3) > 0，画出符号图后得到解集 x < 2 或 x > 3。
3. Equations and Inequalities | 方程与不等式
Solving polynomial equations often involves factorising or applying the remainder theorem. Simultaneous equations may combine one linear and one quadratic, solved by substitution.
解多项式方程通常要用因式分解或运用余式定理。联立方程可能是一次与二次的组合，可通过代入法求解。
Modulus equations such as |ax + b| = c can be rewritten as ax + b = ±c. Inequalities like |ax + b| < c lead to -c < ax + b < c, whereas |ax + b| > c splits into two separate intervals.
含有绝对值（模）的方程如 |ax + b| = c 可转化为 ax + b = ±c。不等式 |ax + b| < c 可化为 -c < ax + b < c，而 |ax + b| > c 则要拆成两个独立区间来求解。
When solving rational inequalities, bring all terms to one side, combine into a single fraction, and then use a sign diagram. Always exclude values that make the denominator zero.
解分式不等式时，应把所有项移到一边、通分，再用符号图分析。务必排除使分母为零的值。
4. Indices, Surds, Exponentials and Logarithms | 指数、根式、指数函数与对数函数
Laws of indices: aᵐ × aⁿ = aᵐ⁺ⁿ, (aᵐ)ⁿ = aᵐⁿ, a⁰ = 1, a⁻ⁿ = 1/aⁿ. For surds, simplify √ab = √a × √b and rationalise denominators using the conjugate.
指数运算法则：aᵐ × aⁿ = aᵐ⁺ⁿ，(aᵐ)ⁿ = aᵐⁿ，a⁰ = 1，a⁻ⁿ = 1/aⁿ。对于根式，化简遵循 √ab = √a × √b，并用共轭根式有理化分母。
The exponential function y = eˣ mic laws: logₐ (xy) = logₐ x + logₐ y, logₐ (x/y) = logₐ x - logₐ y, logₐ (xⁿ) = n logₐ x.
指数函数 y = eˣ 与自然对数 y = ln x 互为反函数。重要的对数运算法则：logₐ (xy) = logₐ x + logₐ y，logₐ (x/y) = logₐ x - logₐ y，logₐ (xⁿ) = n logₐ x。
To solve exponential equations, take ln on both sides: e.g., 3ˣ = 5 → x ln 3 = ln 5 → x = ln 5 / ln 3. Graphs of y = aˣ e rapidly; graphs of y = logₐ x pass through (1,0) and have a vertical asymptote at x = 0.
解指数方程时两边取自然对数，如 3ˣ = 5 → x ln 3 = ln 5 → x = ln 5 / ln 3。y = aˣ（a > 1）的图像快速增长；y = logₐ x 的图像经过 (1,0) 且以 x = 0 为垂直渐近线。
5. Polynomials and the Remainder Theorem | 多项式与余式定理
ded by (x - a) is P(a). If P(a) = 0, then (x - a) is a factor. This is the factor theorem.
对于多项式 P(x)，除以 (x - a) 的余式为 P(a)。如果 P(a) = 0，则 (x - a) 是一个因式，这就是因式定理。
To factorise a cubic, test factors of the constant term. Once one factor is found, use long division or synthetic division to reduce to a quadratic, which can then be factorised further.
分解三次多项式时，尝试代入常数项的因数。找到一个因式后，用长除法或综合除法降次为一个二次式，再继续分解。
= 0, so (x - 2) is a factor. Division yields (x - 2)(x² - x - 6) = (x - 2)(x - 3)(x + 2).
例如：P(x) = x³ - 3x² - 4x + 12，P(2) = 0，故 (x - 2) 为因式。除法得到 (x - 2)(x² - x - 6) = (x - 2)(x - 3)(x + 2)。
6. Coordinate Geometry and Linearisation | 坐标几何与线性化
is useful. Parallel lines have equal gradients; perpendicular lines satisfy m₁ × m₂ = -1.
过两点 (x₁, y₁) 和 (x₂, y₂) 的直线斜率（梯度）为 m = (y₂ - y₁)/(x₂ - x₁)。常用方程形式为 y - y₁ = m(x - x₁)。平行线的斜率相等；相互垂直的直线满足 m₁ × m₂ = -1。
ln y = ln a + n ln x, so a plot of ln y against ln x gives gradient n and intercept ln a.
线性化可将非线性关系转化为直线，从而估算常数。对于 y = axⁿ，取自然对数得 ln y = ln a + n ln x，绘制 ln y 对 ln x 的图像，斜率即为 n，截距为 ln a。
For y = Abˣ gradient ln b and intercept ln A. Remember to reverse the transformation to find A and b.
对于 y = Abˣ，变为 ln y = ln A + x ln b，以 ln y 对 x 作图，斜率为 ln b，截距为 ln A。求出后需通过逆变换得到 A 和 b 的值。
7. Trigonometry | 三角学
Know the exact values of sin, cos, tan for 0°, 30°, 45°, 60°, 90° and their radian equivalents. The ASTC diagram helps determine the sign of trig ratios in each quadrant.
需牢记 0°、30°、45°、60°、90°以及对应的弧度制下 sin、cos、tan 的精确值。ASTC 示意图有助于判断各象限内三角比的符号。
lve equations and prove identities. When solving trig equations, remember to give all solutions within the specified domain, often using the periodic properties of the functions.
基本恒等式：sin²θ + cos²θ = 1，tanθ = sinθ / cosθ。利用这些恒等式解三角方程并证明恒等式。解三角方程时，记得根据函数的周期性给出指定区间内的所有解。
Graphs of y = a sin(bx) + c involve amplitude |a|, period 2π/|b|, and vertical shift c. Similar transformations apply to cos and tan graphs.
y = a sin(bx) + c 的图像涉及振幅 |a|、周期 2π/|b| 和垂直平移 c。余弦和正切图像也有类似的变换。
8. Circular Measure | 圆弧度量
, where θ is in radians. The area of a segment is found by subtracting the triangle area from the sector area.
弧度制至关重要：π 弧度 = 180°。弧长公式 s = rθ，扇形面积 A = ½ r²θ，其中 θ 以弧度为单位。弓形面积等于扇形面积减去相应的三角形面积。
is particularly useful when the triangle is isosceles with two sides equal to the radius.
对于弓形内的非直角三角形，面积可用 A = ½ r² sinθ 计算。当三角形为等腰三角形且两腰为半径时，这特别有用。
8 cm and angle 1.2 rad has arc length 8 × 1.2 = 9.6 cm, and area ½ × 64 × 1.2 = 38.4 cm².
例如：半径为 8 cm、圆心角为 1.2 rad 的扇形，弧长为 8 × 1.2 = 9.6 cm，面积为 ½ × 64 × 1.2 = 38.4 cm²。
9. Vectors in Two Dimensions | 平面向量
ritten as a column vector or using i, j notation. The magnitude of vector v = a i + b j is |v| = √a² + b².
向量具有大小和方向，可用列向量或 i、j 形式表示。向量 v = a i + b j 的模长为 |v| = √a² + b²。
Vector operations include addition (component-wise), scalar multiplication, and subtraction. The position vector of a point is its displacement from the origin O.
向量的运算包括加法（对应分量相加）、数乘和减法。点的位置向量是它相对于原点 O 的位移。
Two vectors are parallel if one is a scalar multiple of the other. Points A, B, and C are collinear if vectors AB and BC are parallel, which often involves solving for an unknown scalar.
若一个向量是另一个向量的数乘，则两者平行。若 AB 和 BC 平行，则 A、B、C 三点共线，通常需要解出某个未知标量。
10. Permutations and Combinations | 排列与组合
n! product of all positive integers up to n. Permutations count ordered arrangements: nPr = n! / (n - r)!. Combinations count selections where order does not matter: nCr = n! / (r! (n - r)!).
n!（n 的阶乘）是所有正整数从 1 到 n 的乘积。排列计算有序的安排：nPr = n! / (n - r)!。组合计算无需考虑顺序的选择：nCr = n! / (r! (n - r)!)。
In examination problems, identify whether order matters. Words like 'arrange' or 'line up' suggest permutations; 'select' or 'choose' suggests combinations. Be careful with repetitions and restrictions.
在考题中，要辨别顺序是否重要。“排列”“排序”等词提示使用排列；“选择”“选出”提示使用组合。对于重复和限制条件要特别留意。
Cr also appear in the expansion of (a + b)ⁿ. They satisfy nC0 = nCn = 1 and nCr = nC(n-r).
二项式系数 nCr 也出现在 (a + b)ⁿ 的展开式中，满足 nC0 = nCn = 1 及 nCr = nC(n-r)。
11. Sequences and Series | 数列与级数
a + (n-1)d]. For geometric progression (GP): uₙ = arⁿ⁻¹, Sₙ = a(1 - rⁿ)/(1 - r) for r ≠ 1.
等差数列：通项 uₙ = a + (n - 1)d，前 n 项和 Sₙ = n/2 [2a + (n - 1)d]。等比数列：通项 uₙ = arⁿ⁻¹，前 n 项和 Sₙ = a(1 - rⁿ)/(1 - r)（r ≠ 1）。
If |r| = a/(1 - r). This is useful in problems involving recurring decimals or convergent series.
若 |r| < 1，等比数列的无穷和存在：S∞ = a/(1 - r)。这在循环小数或收敛级数问题中十分有用。
The binomial theorem for a positive integer
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