IGCSE Cambridge Further Mathematics: Formula and Theorem Quick Reference Handbook | IGCSE Cambridge 进阶数学：公式定理速查手册

📚 IGCSE Cambridge Further Mathematics: Formula and Theorem Quick Reference Handbook | IGCSE Cambridge 进阶数学：公式定理速查手册

This handbook condenses the formulas, theorems and standard results that carry the most marks in Cambridge IGCSE Additional Mathematics (0606) – the Cambridge qualification that schools and tutoring centres most often label as “IGCSE Further Mathematics”. Every result is written in the exact notation Cambridge examiners expect, grouped by topic, so you can revise a whole strand in ten minutes before a paper. Use it after you have learned the topic, never instead of learning it.

本手册汇总了 Cambridge IGCSE Additional Mathematics (0606) 中最常考、最容易直接拿分的公式与定理。在剑桥体系中，真正对应 “进阶数学” 的资格考试就是 0606（部分学校写作 IGCSE Further Mathematics）。所有结论均按考官认可的书写方式整理，并按专题分组，方便你在考前十分钟快速过完一个板块。请把它当作学完知识后的复习工具，而不是替代教材。

---

1. Sets and Functions | 集合与函数

Set questions are almost always counting questions. Write the universal set, the number of elements in each region of a Venn diagram, and use the inclusion-exclusion identity. Then check that the total of all regions equals n(ξ).

集合题本质上几乎都是计数题。先画出全集 ξ，把 Venn 图每个区域的元素个数写清楚，再使用容斥恒等式，最后检验所有区域之和是否等于 n(ξ)。

n(A ∪ B) = n(A) + n(B) − n(A ∩ B)

n(A ∪ B ∪ C) = n(A) + n(B) + n(C) − n(A ∩ B) − n(A ∩ C) − n(B ∩ C) + n(A ∩ B ∩ C)

For functions, remember that a function must map every element of its domain to exactly one value, and that f⁻¹ exists only when f is one-to-one. The composite gf means “apply f first, then g”, and the graph of f⁻¹ is the reflection of y = f(x) in the line y = x.

关于函数，要牢记：定义域中每个元素必须对应唯一的值；只有一一函数（one-to-one）才存在反函数 f⁻¹。复合函数 gf 表示 “先做 f，再做 g”；y = f⁻¹(x) 的图像是 y = f(x) 关于直线 y = x 的对称图形。

fg(x) means apply g first, then f

f f⁻¹(x) = f⁻¹f(x) = x

---

2. Quadratic Functions and Equations | 二次函数与方程

The discriminant decides everything about a quadratic. Learn the three cases as exam language: “two distinct real roots”, “a repeated root”, “no real roots”.

判别式决定了二次方程的一切。三种情形必须能直接用考试语言回答：”两个不相等实根”、”重根”、”无实根”。

Δ = b² − 4ac

x = (−b ± √(b² − 4ac)) / 2a

α + β = −b/a    αβ = c/a

Completing the square gives the turning point directly: the vertex of y = ax² + bx + c sits at x = −b/2a, and the completed form is a(x + b/2a)² + (c − b²/4a). If a is positive the curve opens upwards and the vertex is a minimum.

配方可以直接读出顶点：y = ax² + bx + c 的对称轴为 x = −b/2a，配方结果为 a(x + b/2a)² + (c − b²/4a)。a 为正时开口向上，顶点为最小值点。

Δ 的取值	根的情况	图像与 x 轴
Δ > 0	两个不相等实根	相交于两点
Δ = 0	一个重根	相切于一点
Δ < 0	无实根	不相交

---

3. Polynomials, Remainder Theorem and Simultaneous Equations | 多项式、余式定理与联立方程

The remainder theorem converts division into substitution. If a polynomial f(x) is divided by (x − a), the remainder is f(a). The factor theorem is the special case where that remainder is zero.

余式定理把除法变成代入。多项式 f(x) 除以 (x − a) 的余数就是 f(a)；因式定理是余数为零的特例。

f(x) ÷ (x − a) → remainder f(a)

(x − a) is a factor of f(x) ⇔ f(a) = 0

f(x) ÷ (ax − b) → remainder f(b/a)

When a linear equation meets a quadratic, substitute and then use the discriminant on the resulting quadratic to decide how many intersection points exist. For equations with absolute value, split into cases: |x − a| = b gives x = a + b or x = a − b, and always substitute back to reject invalid solutions.

直线与二次曲线联立时，先代入消元，再用判别式判断交点个数。含绝对值的方程要分情况：|x − a| = b 给出 x = a + b 或 x = a − b；解完必须代回原方程检验，舍去不合理解。

---

4. Indices, Surds, Exponentials and Logarithms | 指数、根式、指数函数与对数

Index laws and logarithm laws are the same law written two ways. Learn them as a matched pair, because exam questions switch between the two forms freely.

指数律与对数律本质上是同一条规律的两种写法。要把它们成对记忆，因为考题经常在两种形式间自由切换。

aᵐ × aⁿ = aᵐ⁺ⁿ    aᵐ ÷ aⁿ = aᵐ⁻ⁿ    (aᵐ)ⁿ = aᵐⁿ

Find Cambridge IGCSE Further Maths Textbooks on eBay UK

New, used and second-hand copies of textbooks and revision guides are often much cheaper than retail — check current listings and prices before you buy.

Browse on eBay UK →