CCEA Year 10 Further Mathematics: Formula & Theorem Quick Reference Guide | CCEA 10年级进阶数学：公式定理速查手册

📚 CCEA Year 10 Further Mathematics: Formula & Theorem Quick Reference Guide | CCEA 10年级进阶数学：公式定理速查手册

This quick-reference revision guide covers essential formulas, theorems, and identities for the CCEA Year 10 Further Mathematics course. It is designed to help you revise efficiently and master key concepts tested in examinations.

本速查手册涵盖了CCEA 10年级进阶数学课程中的重要公式、定理和恒等式，帮助你高效复习并掌握考试中的核心概念。

1. Algebraic Expressions & Operations | 代数表达式与运算

Perfect square expansion: (a + b)² = a² + 2ab + b².

完全平方展开：(a + b)² = a² + 2ab + b²。

Square of a difference: (a – b)² = a² – 2ab + b².

差平方公式：(a – b)² = a² – 2ab + b²。

Difference of two squares: a² – b² = (a + b)(a – b).

平方差公式：a² – b² = (a + b)(a – b)。

Cube of a binomial: (a + b)³ = a³ + 3a²b + 3ab² + b³; and (a – b)³ = a³ – 3a²b + 3ab² – b³.

二项式立方：(a + b)³ = a³ + 3a²b + 3ab² + b³；以及 (a – b)³ = a³ – 3a²b + 3ab² – b³。

Sum and difference of cubes: a³ + b³ = (a + b)(a² – ab + b²); a³ – b³ = (a – b)(a² + ab + b²).

立方和与立方差：a³ + b³ = (a + b)(a² – ab + b²)；a³ – b³ = (a – b)(a² + ab + b²)。

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2. Indices & Surds | 指数与根式

Multiplication rule: a^m × aⁿ = a^m+n.

乘法法则：a^m × aⁿ = a^m+n。

Power of a power: (a^m)ⁿ = a^mn.

幂的幂：(a^m)ⁿ = a^mn。

Zero exponent: a⁰ = 1 (a ≠ 0).

零指数：a⁰ = 1（a ≠ 0）。

Negative exponent: a^–n = 1 / aⁿ.

负指数：a^–n = 1 / aⁿ。

Fractional exponent: a^1/n = ⁿ√a, and a^m/n = (ⁿ√a)^m = ⁿ√(a^m).

分数指数：a^1/n = ⁿ√a，且 a^m/n = (ⁿ√a)^m = ⁿ√(a^m)。

Simplifying surds: √(a × b) = √a × √b, and √(a/b) = √a / √b.

简化根式：√(a × b) = √a × √b，且 √(a/b) = √a / √b。

Rationalising denominator: 1/√a = √a / a; for a ± √b, multiply by conjugate.

分母有理化：1/√a = √a / a；对于 a ± √b，乘以共轭根式。

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3. Quadratic Equations & Functions | 二次方程与函数

Standard form: ax² + bx + c = 0, where a ≠ 0.

标准形式：ax² + bx + c = 0，a ≠ 0。

Quadratic formula: x = [–b ± √(b² – 4ac)] / (2a).

求根公式：x = [–b ± √(b² – 4ac)] / (2a)。

Discriminant: Δ = b² – 4ac. If Δ > 0, two distinct real roots; Δ = 0, one repeated root; Δ < 0, no real roots.

判别式：Δ = b² – 4ac。若 Δ > 0，有两个不等实根；Δ = 0，有一个重根；Δ < 0，无实根。

Sum and product of roots (Vieta): if roots are α and β, then α + β = –b/a, αβ = c/a.

韦达定理：若根为 α 和 β，则 α + β = –b/a，αβ = c/a。

Completing the square: ax² + bx + c = a(x + b/(2a))² + (c – b²/(4a)). Vertex form: a(x – h)² + k with vertex (h, k).

配方法：ax² + bx + c = a(x + b/(2a))² + (c – b²/(4a))。顶点式：a(x – h)² + k，顶点为 (h, k)。

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4. Polynomials & Factor Theorem | 多项式与因式定理

A polynomial in x is an expression of the form a_nxⁿ + a_n–1x^n–1 + … + a₀.

多项式形如 a_nxⁿ + a_n–1x^n–1 + … + a₀。

Remainder theorem: When f(x) is divided by (x – a), the remainder is f(a).

余数定理：f(x) 除以 (x – a) 的余数为 f(a)。

Factor theorem: (x – a) is a factor of f(x) if and only if f(a) = 0.

因式定理：(x – a) 是 f(x) 的因式当且仅当 f(a) = 0。

Polynomial division can be performed by long division or synthetic division to factorise cubic or higher-degree polynomials.

多项式除法可用长除法或综合除法分解三次及更高次多项式。

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5. Coordinate Geometry & Straight Lines | 坐标几何与直线

Distance between two points (x₁, y₁) and (x₂, y₂): d = √[(x₂ – x₁)² + (y₂ – y₁)²].

两点间距离：d = √[(x₂ – x₁)² + (y₂ – y₁)²]。

Midpoint: M = ( (x₁+x₂)/2 , (y₁+y₂)/2 ).

中点：M = ( (x₁+x₂)/2 , (y₁+y₂)/2 )。

Gradient (slope) of a line: m = (y₂ – y₁) / (x₂ – x₁).

直线斜率：m = (y₂ – y₁) / (x₂ – x₁)。

Equation of a straight line: y – y₁ = m(x – x₁) or y = mx + c, where c is the y-intercept.

直线方程：y – y₁ = m(x – x₁) 或 y = mx + c，c 为 y 截距。

Parallel lines: m₁ = m₂. Perpendicular lines: m₁ × m₂ = –1.

平行线：m₁ = m₂。垂直线：m₁ × m₂ = –1。

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6. Trigonometry: Basic Ratios & Identities | 三角函数：基本比与恒等式

In a right-angled triangle: sin θ = opposite/hypotenuse, cos θ = adjacent/hypotenuse, tan θ = opposite/adjacent.

在直角三角形中：sin θ = 对边/斜边，cos θ = 邻边/斜边，tan θ = 对边/邻边。

Exact values for special angles (in degrees):

特殊角精确值（角度制）：

θ	sin θ	cos θ	tan θ
30°	1/2	√3/2	1/√3
45°	1/√2	1/√2	1
60°	√3/2	1/2	√3

Pythagorean identity: sin² θ + cos² θ = 1.

毕达哥拉斯恒等式：sin² θ + cos² θ = 1。

Tangent identity: tan θ = sin θ / cos θ.

正切关系：tan θ = sin θ / cos θ。

Sine rule: a/sin A = b/sin B = c/sin C (for any triangle).

正弦定理：a/sin A = b/sin B = c/sin C（适用于任意三角形）。

Cosine rule: a² = b² + c² – 2bc cos A.

余弦定理：a² = b² + c² – 2bc cos A。

Area of a triangle: Area = ½ ab sin C.

三角形面积：面积 = ½ ab sin C。

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7. Radians, Arcs & Sectors | 弧度、弧与扇形

Definition: π radians = 180°, so 1 radian ≈ 57.3°.

定义：π 弧度 = 180°，所以 1 弧度 ≈ 57.3°。

To convert degrees to radians: multiply by π/180.

角度转弧度：乘以 π/180。

Arc length: s = rθ, where θ is in radians.

弧长：s = rθ，θ 以弧度为单位。

Sector area: A = ½ r² θ.

扇形面积：A = ½ r² θ。

Segment area = area of sector – area of triangle, or ½ r²(θ – sin θ) for small angles.

弓形面积 = 扇形面积 – 三角形面积，或 ½ r²(θ – sin θ)。

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8. Sequences & Series | 数列与级数

Arithmetic sequence: nth term u_n = a + (n – 1)d, where a is first term and d is common difference.

等差数列：第 n 项 u_n = a + (n – 1)d，a 为首项，d 为公差。

Sum of first n terms of AP: S_n = n/2 [2a + (n – 1)d] = n/2 (a + l), with l as last term.

等差数列前 n 项和：S_n = n/2 [2a + (n – 1)d] = n/2 (a + l)，l 为末项。

Geometric sequence: nth term u_n = ar^n–1, where r is common ratio.

等比数列：第 n 项 u_n = ar^n–1，r 为公比。

Sum of first n terms of GP (r ≠ 1): S_n = a(1 – rⁿ) / (1 – r).

等比数列前 n 项和（r ≠ 1）：S_n = a(1 – rⁿ) / (1 – r)。

Sum to infinity (|r| < 1): S_∞ = a / (1 – r).

无穷递缩等比数列求和（|r| < 1）：S_∞ = a / (1 – r)。

Sigma notation: Σ a_k from k=1 to n.

西格玛表示：∑_{k=1}^{n} a_k。

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9. Introduction to Differentiation | 微分入门

The derivative of f(x) is defined as f'(x) = lim_h→0 [f(x+h) – f(x)] / h.

导数定义：f'(x) = lim

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