Year 9 CCEA Further Maths: Formula & Theorem Quick-Reference Guide | 九年级 CCEA 进阶数学：公式定理速查手册

📚 Year 9 CCEA Further Maths: Formula & Theorem Quick-Reference Guide | 九年级 CCEA 进阶数学：公式定理速查手册

This quick-reference guide compiles all the essential formulas and theorems needed for the Year 9 CCEA Further Mathematics curriculum. Each section provides concise English explanations followed by Chinese translations, ensuring clarity for bilingual learners. Use this handbook to reinforce your understanding and to locate key concepts rapidly during revision.

本速查手册汇集了九年级 CCEA 进阶数学课程所需的所有核心公式与定理。每个小节均以简洁的英文说明搭配中文翻译，方便双语学习者掌握要点。请利用本手册巩固知识，并在复习时快速查阅关键概念。

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1. Algebraic Manipulation | 代数运算

of a binomial follows the pattern (a + b)² = a² + 2ab + b², and (a − b)² = a² − 2ab + b².

二项式的平方遵循公式 (a + b)² = a² + 2ab + b²，以及 (a − b)² = a² − 2ab + b²。

The difference of two squares can be factorised as a² − b² = (a + b)(a − b).

平方差公式可因式分解为 a² − b² = (a + b)(a − b)。

When factorising by grouping, look for a common factor: ab + ac = a(b + c).

提取公因式时，找出公共因子：ab + ac = a(b + c)。

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2. Quadratic Equations | 二次方程

A quadratic equation is written in the standard form ax² + bx + c = 0, where a ≠ 0.

二次方程的标准形式为 ax² + bx + c = 0，其中 a ≠ 0。

x = [ −b ± √b² − 4ac ] / (2a)

求根公式：x = [ −b ± √b² − 4ac ] / (2a)

wo distinct real roots, Δ = 0 gives one repeated real root, and Δ < 0 gives no real roots.

判别式 Δ = b² − 4ac 决定根的性质：Δ > 0 有两个不等实根，Δ = 0 有一个重根，Δ < 0 无实根。

found without solving: for ax² + bx + c = 0, sum of roots = −b/a, product of roots = c/a.

根与系数的关系：对于 ax² + bx + c = 0，两根之和为 −b/a，两根之积为 c/a。

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3. Inequalities | 不等式

Inequalities are solved much like equations, but remember to reverse the inequality sign when multiplying or dividing by a negative number.

解不等式的方法与解方程类似，但当乘以或除以一个负数时，必须反转不等号的方向。

ct inequality (< or >), while a closed circle represents a non-strict inequality (≤ or ≥).

在数轴上，空心圆表示严格不等式（< 或 >），实心圆表示非严格不等式（≤ 或 ≥）。

To solve a quadratic inequality such as x² − 5x + 6 > 0, first factorise to (x − 2)(x − 3) > 0, then determine the intervals where the product is positive.

解二次不等式（如 x² − 5x + 6 > 0）时，先因式分解为 (x − 2)(x − 3) > 0，再判断使乘积为正的区间。

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4. Indices and Surds | 指数与根式

The laws of indices: aᵐ × aⁿ = aᵐ⁺ⁿ, aᵐ ÷ aⁿ = aᵐ⁻ⁿ, (aᵐ)ⁿ = aᵐⁿ, a⁰ = 1, and a⁻ⁿ = 1/aⁿ.

指数律：aᵐ × aⁿ = aᵐ⁺ⁿ，aᵐ ÷ aⁿ = aᵐ⁻ⁿ，(aᵐ)ⁿ = aᵐⁿ，a⁰ = 1，a⁻ⁿ = 1/aⁿ。

For surds, √ab = √a × √b and √a/b = √a / √b b ≥ 0). Rationalising the denominator means removing a surd from the denominator, e.g. 1/√a = √a / a.

对于根式，√ab = √a × √b，√a/b = √a / √b（假设 a, b ≥ 0）。分母有理化是指将分母中的根号去掉，例如 1/√a = √a / a。

When simplifying expressions like √a²b, remember √a²b = a√b for a ≥ 0.

化简形如 √a²b 的式子时，记住 √a²b = a√b（其中 a ≥ 0）。

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5. Straight Line Graphs | 直线图像

ten in slope-intercept form: y = mx + c, where m is the gradient and c is the y-intercept.

直线的方程可以写成斜截式：y = mx + c，其中 m 是斜率，c 是 y 轴截距。

m = (y₂ − y₁) / (x₂ − x₁)

斜率公式：m = (y₂ − y₁) / (x₂ − x₁)

s. Two lines are perpendicular if the product of their gradients is −1, i.e. m₁ × m₂ = −1.

平行直线的斜率相等。若两条直线垂直，则它们的斜率乘积为 −1，即 m₁ × m₂ = −1。

₂, y₂) is ((x₁+x₂)/2, (y₁+y₂)/2), and the distance between them is √[(x₂−x₁)² + (y₂−y₁)²].

连接 (x₁, y₁) 和 (x₂, y₂) 的线段中点坐标为 ((x₁+x₂)/2, (y₁+y₂)/2)，两点间距离为 √[(x₂−x₁)² + (y₂−y₁)²]。

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6. Pythagoras and Trigonometry | 勾股定理与三角学

In a right-angled triangle, Pythagoras' theorem states a² + b² = c², where c is the hypotenuse.

在直角三角形中，勾股定理为 a² + b² = c²，其中 c 为斜边。

e sin θ = opposite/hypotenuse, cos θ = adjacent/hypotenuse, and tan θ = opposite/adjacent.

三个基本三角比：sin θ = 对边/斜边，cos θ = 邻边/斜边，tan θ = 对边/邻边。

θ (degrees)	sin θ	cos θ	tan θ
0°	0	1	0
30°	1/2	√3/2	1/√3
45°	1/√2	1/√2	1
60°	√3/2	1/2	√3
90°	1	0	undefined

used when two angles and a side are known or two sides and a non-included angle are known.

对于任意三角形，正弦定理为 a / sin A = b / sin B = c / sin C。这一般用于已知两角一边或两边及其一边对角的情况。

. It helps to find a side when two sides and the included angle are given, or to find an angle when all three sides are known.

余弦定理为 a² = b² + c² − 2bc cos A。用于已知两边及其夹角求第三边，或已知三边求角。

The area of a triangle can be calculated as ½ ab sin C, where a and b are two sides and C is the included angle.

三角形面积可用公式 ½ ab sin C 计算，其中 a 和 b 为两边，C 为它们的夹角。

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7. Circle Theorems | 圆定理

The angle at the centre of a circle is twice the angle at the circumference subtended by the same arc.

圆心角等于同弧所对的圆周角的两倍。

Angles in the same segment of a circle are equal; i.e. angles subtended by the same chord (or arc) on the same side are equal.

位于同一弓形内的圆周角相等；即同弧（或同弦）在同侧所对的圆周角相等。

The angle in a semicircle is always a right angle (90°).

半圆上的圆周角是直角（90°）。

The opposite angles of a cyclic quadrilateral add up to 180° (they are supplementary).

圆内接四边形的对角互补，和为 180°。

A tangent to a circle is perpendicular to the radius drawn to the point of contact.

圆的切线垂直于过切点的半径。

The alternate segment theorem: the angle between a tangent and a chord through the point of contact is equal to the angle in the alternate segment.

弦切角定理：切线与过切点的弦所夹的角等于弦另一侧圆内的圆周角。

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8. Sequences | 数列

e nth term is uₙ = a + (n − 1)d, where a is the first term and d is the common difference.

在等差数列中，第 n 项的通项公式为 uₙ = a + (n − 1)d，其中 a 为首项，d 为公差。

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 项和公式为 Sₙ = n/2 [2a + (n − 1)d] 或 Sₙ = n/2 (a + l)，其中 l 为末项。

, the nth term is uₙ = arⁿ⁻¹, where a is the first term and r is the common ratio (r ≠ 0).

在等比数列中，第 n 项公式为 uₙ = arⁿ⁻¹，其中 a 为首项，r 为公比（r ≠ 0）。

The sum of the first n terms of a geometric series is Sₙ = a(1 − rⁿ) / (1 − r) for |r| ≠ 1.

等比数列前 n 项和公式为 Sₙ = a(1 − rⁿ) / (1 − r)，当 |r| ≠ 1 时。

Special summations: Σn = n(n+1)/2, Σn² = n(n+1)(2n+1)/6, Σn³ = [n(n+1)/2]².

特殊求和公式：Σn = n(n+1)/2，Σn² = n(n+1)(2n+1)/6，Σn³ = [n(n+1)/2]²。

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9. Probability | 概率

bability of an event A is P(A) = number of favourable outcomes / total number of outcomes.

事件 A 的概率为 P(A) = 有利结果数 / 总结果数。

For mutually exclusive events, P(A or B) = P(A) + P(B).

对于互斥事件，P(A 或 B) = P(A) + P(B)。

For independent events, P(A and B) = P(A) × P(B).

对于独立事件，P(A 且 B) = P(A) × P(B)。

Conditional probability is given by P(A|B) = P(A ∩ B) / P(B), where P(A|B) means the probability of A given that B has occurred.

条件概率公式为 P(A|B) = P(A ∩ B) / P(B)，其中 P(A|B) 表示在 B 发生的条件下 A 发生的概率。

Tree diagrams help model sequential events: multiply along branches for 'and' probabilities, add branch outcomes for 'or' probabilities.

树状图有助于模拟连续事件：沿分支相乘得到“且”概率，将不同分支的结果相加得到“或”概率。

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10. Area and Volume | 面积与体积

Area of a triangle = ½ × base × height, or using trigonometry: ½ ab sin C.

三角形面积 = ½ × 底 × 高，或使用三角公式：½ ab sin C。

pendicular height; area of a trapezium = ½ (a + b)h, where a and b are the parallel sides.

平行四边形面积 = 底 × 垂直高；梯形面积 = ½ (a + b)h，其中 a 和 b 为上底和下底。

s arc length = (θ/360) × 2πr and area = (θ/360) × πr². In radians, arc = rθ, area = ½ r²θ.

圆：周长 = 2πr，面积 = πr²。圆心角为 θ°

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