Year 9 CAIE Further Mathematics: Quick Reference Handbook of Formulas and Theorems | Year 9 CAIE 进阶数学：公式定理速查手册

📚 Year 9 CAIE Further Mathematics: Quick Reference Handbook of Formulas and Theorems | Year 9 CAIE 进阶数学：公式定理速查手册

This quick reference handbook compiles every essential formula, theorem and definition required for the Year 9 CAIE Further Mathematics syllabus. It is designed as a portable revision companion, helping you memorise, recall and apply key concepts across algebra, geometry, trigonometry, probability and statistics. Each section is presented with dual-language explanations so you can cross-check understanding in both English and Chinese. Keep this guide handy when practising past papers or revising for tests.

本速查手册汇集了 Year 9 CAIE 进阶数学大纲所要求的全部重要公式、定理与定义。它是一本便携式复习伴侣，帮助你记忆、回忆并运用代数、几何、三角学、概率与统计中的关键概念。每一节均提供中英双语解释，方便你对照理解。练习历年真题或备考测验时，请随身携带这份指南。

1. Quadratic Equations | 二次方程

orm of a quadratic equation is ax² + bx + c = 0, where a, b and c are constants and a ≠ 0.

二次方程的标准形式为 ax² + bx + c = 0，其中 a、b、c 是常数且 a ≠ 0。

Quadratic formula: x = [-b ± √b² - 4ac] / 2a

求根公式：x = [-b ± √b² - 4ac] / 2a

4ac is the discriminant. It reveals the nature of the roots without solving the equation.

表达式 D = b² - 4ac 称为判别式，它能在不求解方程的情况下揭示根的性质。

• If D > 0: two distinct real roots
• If D = 0: one repeated real root
• If D < 0: no real roots (the parabola does not cross the x‑axis)

• 若 D > 0：两个不相等的实根
• 若 D = 0：一个重实根
• 若 D < 0：无实根（抛物线不与 x 轴相交）

= -b/a and the product αβ = c/a. These relationships help form equations from given roots.

已知根 α 和 β，则根的和 α + β = -b/a，根的积 αβ = c/a。这些关系可用于由已知根构造方程。

Completing the square rewrites ax² + bx + c as a(x + b/(2a))² + [c - b²/(4a)]. This form instantly gives the turning point (-b/(2a) , c - b²/(4a)).

配方法将 ax² + bx + c 写成 a(x + b/(2a))² + [c - b²/(4a)]，该形式直接给出顶点坐标 (-b/(2a) , c - b²/(4a))。

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2. Pythagoras' Theorem and Trigonometry | 勾股定理与三角比

In any right‑ the sum of the squares of the other two sides: c² = a² + b², where c is the longest side.

在任意直角三角形中，斜边的平方等于两直角边的平方之和：c² = a² + b²，其中 c 为最长边。

The three primary trigonometric ratios for a right‑angled triangle are defined as:

对于直角三角形，三个基本三角比定义为：

sin θ = opposite/hypotenuse, cos θ = adjacent/hypotenuse, tan θ = opposite/adjacent

sin θ = 对边/斜边，cos θ = 邻边/斜边，tan θ = 对边/邻边

The exact values for special angles must be memorised. They appear frequently without calculators.

特殊角的精确三角值必须牢记。它们经常在非计算器题中出现。

Angle / 角度	sin	cos	tan
30°	½	√3/2	1/√3
45°	√2/2	√2/2	1
60°	√3/2	½	√3

cos²θ = 1 holds for all angles and is useful for solving equations or proving statements.

恒等式 sin²θ + cos²θ = 1 对所有角度均成立，常用于解方程或证明。

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3. Laws of Indices | 指数定律

Indices (or exponents) follow a set of strict rules that allow us to simplify expressions and solve equations involving powers.

指数遵循一套严格的运算法则，使我们能够化简含幂的表达式并解指数方程。

aᵐ × aⁿ = aᵐ⁺ⁿ

aᵐ ÷ aⁿ = aᵐ⁻ⁿ

(aᵐ)ⁿ = aᵐⁿ

a⁰ = 1 (a ≠ 0)

a⁻ⁿ = 1/aⁿ

a¹/ⁿ = ⁿ√a

contains both numbers and variables, apply each law carefully. For example, (2x³)² = 4x⁶.

当表达式同时包含数字和变量时，请仔细运用每一条定律。例如 (2x³)² = 4x⁶。

Fractional indices connect roots and powers: aᵐ/ⁿ = (ⁿ√a)ᵐ = ⁿ√aᵐ. This dual view is essential for simplifying surds and solving exponential equations.

分数指数将根式与幂联系起来：aᵐ/ⁿ = (ⁿ√a)ᵐ = ⁿ√aᵐ。这种双重视角对于化简根式和解指数方程至关重要。

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4. Sequences and Series | 数列与级数

ommon difference d. Its nth term is given by uₙ = a + (n - 1)d, where a is the first term.

等差数列具有公差 d。其第 n 项公式为 uₙ = a + (n - 1)d，其中 a 为首项。

ithmetic series is Sₙ = n/2 [2a + (n - 1)d] or Sₙ = n/2 (a + l), where l is the last term.

等差数列前 n 项和为 Sₙ = n/2 [2a + (n - 1)d] 或 Sₙ = n/2 (a + l)，其中 l 为末项。

A geometric sequence has a common ratio r. The nth term is uₙ = arⁿ⁻¹.

等比数列具有公比 r。其第 n 项为 uₙ = arⁿ⁻¹。

geometric series (r ≠ 1) is Sₙ = a(1 - rⁿ) / (1 - r). For infinite geometric series where |r| < 1, the sum to infinity is S∞ = a / (1 - r).

等比数列前 n 项和 (r ≠ 1) 为 Sₙ = a(1 - rⁿ) / (1 - r)。当 |r| < 1 时，无限等比级数的和为 S∞ = a / (1 - r)。

Recognising whether a sequence is arithmetic or geometric is a common exam skill. Check by subtracting consecutive terms (arithmetic) or dividing them (geometric).

识别数列是等差还是等比是常见的考试技巧。可通过相邻项的差（等差）或商（等比）来判断。

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5. Inequalities | 不等式

Linear inequalities are solved just like linear equations, with one critical exception: reversing the inequality sign when multiplying or dividing by a negative number.

线性不等式的解法与线性方程类似，但有一个关键例外：当乘或除以一个负数时，不等号方向需要反转。

Example: -2x < 6 becomes x > -3 after dividing by -2 and flipping the sign.

示例：-2x < 6 在除以 -2 并反转不等号后变为 x > -3。

displayed on a number line. Use an open circle for < or > and a closed circle for ≤ or ≥.

解可以在数轴上表示。使用空心圆圈表示 < 或 >，实心圆圈表示 ≤ 或 ≥。

Quadratic inequalities (e.g. x² - 4x + 3 > 0) are solved by sketching the parabola, identifying the roots and selecting the regions above or below the x‑axis as required.

二次不等式（如 x² - 4x + 3 > 0）通过画出抛物线、找到根并根据要求选择 x 轴上方或下方的区域来求解。

re tackled by splitting into two inequalities or by solving simultaneously, keeping the variable in the middle.

复合不等式如 2 < 3x

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