📚 Year 10 CIE Additional Mathematics: Core Concepts Review | Year 10 CIE 进阶数学：核心知识点梳理
The Cambridge IGCSE Additional Mathematics (0606) syllabus moves beyond basic Mathematics to introduce advanced topics that form the bridge to A‑level studies. In Year 10, students consolidate the essential building blocks: functions, equations, trigonometry, calculus and vectors, among others. A clear grasp of these core concepts not only secures strong exam results but also builds the analytical skills needed for further success.
剑桥 IGCSE 进阶数学（0606）课程在基础数学之上引入更高阶的内容，为 A Level 学习架起桥梁。10 年级的核心任务就是夯实函数、方程、三角函数、微积分和向量等关键模块。透彻理解这些知识点，既能锁定优异成绩，更能培养深造必需的逻辑分析能力。
1. Functions and Relations | 函数与关系
x from its domain to exactly one element f(x) in its range. Common notations include f : x ↦ f(x) or y = f(x). Functions can be one‑to‑one or many‑to‑one; one‑to‑many mappings are not functions.
函数 f 将定义域中的每个元素 x 唯一地映射到值域中的一个元素 f(x)。常用记号包括 f : x ↦ f(x) 或 y = f(x)。函数可以是一对一或多对一的映射；一对多的映射不是函数。
ct of f, so that f⁻¹(f(x)) = x for all x in the domain. An inverse exists only if f is one‑to‑one.
复合函数 gf(x) 表示先作用 f 再作用 g：gf(x) = g(f(x))，顺序至关重要。反函数 f⁻¹(x) 逆转 f 的作用，对所有定义域内的 x 满足 f⁻¹(f(x)) = x。只有一一对应的函数才存在反函数。
To find an inverse, replace f(x) with y, swap x and y, then solve for y. The domain of f⁻¹ is the range of f, and vice versa.
求反函数时，将 f(x) 换成 y，交换 x 与 y，再解出 y。f⁻¹ 的定义域是 f 的值域，反之亦然。
2. Quadratic Functions | 二次函数
y is x = −b/(2a) and the vertex can be found by completing the square to obtain f(x) = a(x − h)² + k, where (h, k) are the vertex coordinates.
二次函数的一般式为 f(x) = ax² + bx + c，其中 a ≠ 0。它的图像是一条抛物线，a > 0 时开口向上，a < 0 时开口向下。对称轴为 x = −b/(2a)，顶点可通过配方法得到 f(x) = a(x − h)² + k，此时 (h, k) 即为顶点坐标。
Δ = b² − 4ac
peated real root; Δ < 0 gives no real roots. The roots are given by the quadratic formula:
判别式 Δ 决定根的性质：Δ > 0 有两个不等实根；Δ = 0 有一个重实根；Δ < 0 无实根。求根公式为：
x = [−b ± √b² − 4ac] / (2a)
β = −b/a and the product αβ = c/a provide powerful shortcuts for solving related problems.
最大值或最小值可从顶点式求得。根的和 α + β = −b/a，根的积 αβ = c/a，这些关系为解题提供了强大捷径。
3. Equations and Inequalities | 方程与不等式
Solving quadratic equations by factorisation, completing the square, or the quadratic formula is fundamental. Simultaneous equations often pair a linear equation with a quadratic; substitute the linear expression into the quadratic to obtain a single‑variable equation.
解二次方程的基本方法有因式分解、配方法和公式法。联立方程组常由一个线性方程和一个二次方程构成；将线性表达式代入二次方程，化为一元方程求解。
Quadratic inequalities such as ax² + bx + c > 0 are solved by finding critical values (roots) and testing intervals. A sketch of the parabola helps to identify where the inequality holds.
二次不等式如 ax² + bx + c > 0 的解法是先求出临界值（即根），再检验各区间。画一条抛物线草图即可直观确定不等式成立的区间。
Absolute value equations |ax + b| = c lead to two cases: ax + b = c or ax + b = −c. Absolute value inequalities, e.g. |ax + b| < c, translate to −c < ax + b < c.
绝对值方程 |ax + b| = c 可拆为两种情况：ax + b = c 或 ax + b = −c。绝对值不等式如 |ax + b| < c 可转化为 −c < ax + b < c。
4. Factors of Polynomials | 多项式的因式分解
inder. The Remainder Theorem states: when p(x) is divided by (x − a), the remainder is p(a). The Factor Theorem follows: (x − a) is a factor if and only if p(a) = 0.
多项式除法与数的长除法类似，可以用线性除式 (x − a) 去除多项式 p(x) 得到商和余式。余数定理指出：p(x) 除以 (x − a) 的余式为 p(a)。由此得出因式定理：(x − a) 是因式的充要条件是 p(a) = 0。
These theorems are indispensable for factorising cubic (or higher‑degree) polynomials. Typically, one root is found by trial (testing factors of the constant term), then the polynomial is factorised into a linear factor and a quadratic, which can be further broken down.
这两个定理是分解三次或更高次多项式的核心工具。通常先用尝试法找出一个根（检验常数项的因数），然后将多项式分解为一个一次因式和一个二次因式，再继续分解。
)=0, so (x − 2) is a factor. Division yields (x − 2)(x² − 2x − 3) = (x − 2)(x + 1)(x − 3).
例如：p(x) = x³ − 4x² + x + 6，检验 x = 2 得 p(2)=0，故 (x − 2) 是因式。除法得 (x − 2)(x² − 2x − 3) = (x − 2)(x + 1)(x − 3)。
5. Exponential and Logarithmic Functions | 指数与对数函数
aᵐ × aⁿ = aᵐ⁺ⁿ, aᵐ ÷ aⁿ = aᵐ⁻ⁿ, (aᵐ)ⁿ = aᵐⁿ, a⁰ = 1, a⁻ⁿ = 1/aⁿ. For rational exponents, am/n = ⁿ√aᵐ.
指数定律是基础：aᵐ × aⁿ = aᵐ⁺ⁿ，aᵐ ÷ aⁿ = aᵐ⁻ⁿ，(aᵐ)ⁿ = aᵐⁿ，a⁰ = 1，a⁻ⁿ = 1/aⁿ。对于有理指数，am/n = ⁿ√aᵐ。
The logarithm is the inverse of exponentiation: if aˣ N; logₐ (M/N) = logₐ M − logₐ N; logₐ (Mⁿ) = n logₐ M. Change of base rule: logₐ b = (log_𝚌 b) / (log_𝚌 a).
对数是取指数的逆运算：若 aˣ = y，则 x = logₐ y。主要性质：logₐ (MN) = logₐ M + logₐ N；logₐ (M/N) = logₐ M − logₐ N；logₐ (Mⁿ) = n logₐ M。换底公式：logₐ b = (log_𝚌 b) / (log_𝚌 a)。
onential equations are often solved by taking logs of both sides; logarithmic equations are solved by exponentiating or using properties.
自然对数记作 ln，以 e（≈ 2.718）为底。ln e = 1，ln 1 = 0。指数方程常通过两边取对数求解；对数方程则通过两边取指数或利用性质化简求解。
Exponential growth/decay model: N(t) = N₀ ekt
The function y = eˣ has gradient equal to its value, a crucial property for differentiation.
y = eˣ 的导数等于它自身，这一性质对微分极为重要。
6. Trigonometry | 三角学
gives radians. Radian measure simplifies arc length (s = rθ) and sector area (A = ½ r²θ).
进阶数学中角度主要用弧度制，π 弧度 = 180°。换算：弧度 × 180/π 得度数；度数 × π/180 得弧度。弧度制大大简化了弧长 (s = rθ) 和扇形面积 (A = ½ r²θ) 的计算。
Core trigonometric identities include sin²θ + cos²θ ≡ 1, tan θ ≡ sin θ / cos θ, and extensions like 1 + tan²θ ≡ sec²θ (though sec/cosec/cot may appear later). Know exact values for 0, π/6, π/4, π/3, π/2.
核心三角恒等式包括 sin²θ + cos²θ ≡ 1，tan θ ≡ sin θ / cos θ，以及扩展式如 1 + tan²θ ≡ sec²θ。必须熟记 0, π/6, π/4, π/3, π/2 的精确三角函数值。
odic nature of trig functions. For example, sin θ = 0.5 in 0 ≤ θ < 2π gives θ = π/6, 5π/6.
求解给定区间内的三角方程常需运用象限法则（CAST 图）以及三角函数的周期性。例如在 0 ≤ θ < 2π 内解 sin θ = 0.5，可得 θ = π/6, 5π/6。
u must understand their amplitude, period, and transformations like y = a sin(bx + c) + d.
y = sin x, cos x, tan x 的图像必须掌握；要能理解振幅、周期以及 y = a sin(bx + c) + d 等变换形式。
7. Vectors in Two Dimensions | 二维向量
A vector quantity has both magnitude and direction. In 2D, a vector can be written as a column vector v = (x, y)ᵀ or using unit vectors i and j: v = xi + yj. The magnitude |v| = √x² + y². A unit vector in the direction of v is v/|v|.
向量既有大小又有方向。二维向量可表示为列向量 (x, y)ᵀ 或使用单位向量 i 和 j：v = xi + yj。模长 |v| = √x² + y²。v 方向上的单位向量为 v/|v|。
Vector addition and subtraction are done component‑wise. Scalar multiplication kv changes the magnitude without rotating the direction (if k > 0). The zero vector is 0 = 0i + 0j.
向量加减按分量进行。标量乘法 kv 只改变大小而不改变方向（k > 0 时）。零向量为 0 = 0i + 0j。
Position vector of point A relative to origin O is OA. The vector from A to B is AB = OB − OA. Two vectors are parallel if one is a scalar multiple of the other. Collinear points have parallel position differences.
点 A 相对于原点 O 的位置向量是 OA。从 A 到 B 的向量为 AB = OB − OA。若一向量是另一向量的标量倍数，则两者平行。共线点之间的位置差向量互相平行。
8. Differentiation | 微分
Differentiation finds the gradient of a curve at a point. The derivative f′(x) = limh→0 r power functions: if y = xⁿ, then dy/dx = n xⁿ⁻¹. The rule works for any real exponent n.
微分求的是曲线上某一点的斜率。导数 f′(x) = limh→0 [f(x+h) − f(x)] / h。对于幂函数：若 y = xⁿ，则 dy/dx = n xⁿ⁻¹。该法则对任意实数指数 n 均成立。
Derivatives of sums/differences: (f ± g)′ = f′ ± g′. For eˣ, d/dx eˣ = eˣ . Derivatives of sin x and cos x are cos x and −sin x respectively (when x is in radians).
和差求导法则：(f ± g)′ = f′ ± g′。eˣ 的导数为 eˣ，ln x 的导数为 1/x。sin x 与 cos x 的导数分别为 cos x 和 −sin x（x 以弧度为单位）。
Equation of tangent at (x₁, y₁): y − y₁ = f′(x₁)(x − x₁). Normal gradient is −1/f′(x₁).
曲线在 (x₁, y₁) 处的切线方程：y − y₁ = f′(x₁)(x − x₁)。法线斜率为 −1/f′(x₁)。
f′(x)=0. Use first or second derivative tests to classify maximum, minimum or inflection.
二阶导数 d²y/dx² 描述曲线的凹凸性。驻点出现在 f′(x)=0 处。用一阶或二阶导数检验即可判断极大值、极小值或拐点。
9. Integration | 积分
). For power functions: ∫ xⁿ dx = xⁿ⁺¹/(n+1) + C, n ≠ −1. The special case ∫ (1/x) dx = ln|x| + C.
积分是微分的逆运算。不定积分 ∫ f(x) dx = F(x) + C，其中 F′(x) = f(x)。对于幂函数：∫ xⁿ dx = xⁿ⁺¹/(n+1) + C，n ≠ −1。特殊情况 ∫ (1/x) dx = ln|x| + C。
Integrals of standard functions include ∫ eˣ dx = eˣ s x dx = sin x + C. The constant of integration C is essential and must always be written.
标准函数的积分包括 ∫ eˣ dx = eˣ + C，∫ sin x dx = −cos x + C，∫ cos x dx = sin x + C。积分常数 C 至关重要，必须始终写出。
The definite integral ∫ₐᵇ f(x) dx calculates the signed area between the curve and the x‑axis from x = a to x = b. Evaluate the antiderivative at the limits: F(b) − F(a).
定积分 ∫ₐᵇ f(x) dx 计算的是曲线与 x 轴之间从 x = a 到 x = b 的带符号面积。计算方法是将原函数在上下限处求值：F(b) − F(a)。
Area between a curve and a line or between two curves is found by integrating the difference of the functions and carefully handling any crossing points.
曲线与直线之间或两条曲线之间的面积，可通过积分两函数之差并正确处理交点来求得。
10. Permutations and Combinations | 排列与组合
n! (n factorial) is the product of all positive integers up to n: n! = n × (n−1) × … × 3 × 2 × 1. By convention, 0! = 1.
n!（n 的阶乘）是 1 到 n 所有正整数的乘积：n! = n × (n−1) × … × 3 × 2 × 1。规定 0! = 1。
ems from n distinct items where order matters is given by the permutation formula: ⁿPᵣ = n! / (n−r)!. When all n items are arranged, number of permutations = n!.
从 n 个不同物品中取出 r 个并按顺序排列的方法数由排列公式给出：ⁿPᵣ = n! / (n−r)!。若排列所有 n 个物品，则排列数为 n!。
tems from n distinct items where order does not matter is the combination formula: ⁿCᵣ = n! / [r! (n−r)!]. ⁿCᵣ is also written as (n choose r).
从 n 个不同物品中选出 r 个且不考虑顺序的方法数由组合公式给出：ⁿCᵣ = n! / [r! (n−r)!]。ⁿCᵣ 也常记作 C(n, r)。
Key distinction: permutations are used for arranging items (like race positions, passwords), combinations are for selections (like committee members, lottery numbers). Many problems require multiplying permutations or combinations for parts of a task.
核心区别：排列用于有序情况（如比赛名次、密码），组合用于无序情况（如委员会成员、彩票号码）。许多实际问题需要将各个步骤的排列或组合数相乘。
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