Year 8 CAIE Further Mathematics: Quick Reference Formula and Theorem Handbook | 八年级CAIE进阶数学：公式定理速查手册

📚 Year 8 CAIE Further Mathematics: Quick Reference Formula and Theorem Handbook | 八年级CAIE进阶数学：公式定理速查手册

This quick reference handbook covers the essential formulas, theorems, and identities required for the Year 8 CAIE Further Mathematics syllabus. Use it as a revision tool to memorise key results and apply them confidently in problem-solving.

本速查手册涵盖了八年级CAIE进阶数学课程所必需的核心公式、定理和恒等式。将其用作复习工具，熟记关键结论，并在解题时自信运用。

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1. Algebraic Formulae and Identities | 代数公式与恒等式

The following algebraic identities are fundamental for simplifying expressions and solving equations.

以下代数恒等式是化简表达式和求解方程的基础。

Distributive Law: a(b + c) = ab + ac

乘法分配律：a(b + c) = ab + ac

Difference of Two Squares: a² − b² = (a + b)(a − b)

平方差公式：a² − b² = (a + b)(a − b)

Perfect Square Trinomial: (a + b)² = a² + 2ab + b² ; (a − b)² = a² − 2ab + b²

完全平方公式：(a + b)² = a² + 2ab + b² ; (a − b)² = a² − 2ab + b²

Sum of Cubes: a³ + b³ = (a + b)(a² − ab + b²)

立方和公式：a³ + b³ = (a + b)(a² − ab + b²)

Difference of Cubes: a³ − b³ = (a − b)(a² + ab + b²)

立方差公式：a³ − b³ = (a − b)(a² + ab + b²)

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2. Exponents and Radicals | 指数与根式

Mastering the laws of exponents and radicals is essential for handling powers and roots efficiently.

掌握指数律与根式的运算是高效处理乘方与开方的关键。

Product of Powers: aᵐ × aⁿ = aᵐ⁺ⁿ

同底数幂相乘：aᵐ × aⁿ = aᵐ⁺ⁿ

Power of a Power: (aᵐ)ⁿ = aᵐⁿ

幂的乘方：(aᵐ)ⁿ = aᵐⁿ

Power of a Product: (ab)ⁿ = aⁿ bⁿ

积的乘方：(ab)ⁿ = aⁿ bⁿ

Zero Exponent: a⁰ = 1 (a ≠ 0)

零指数：a⁰ = 1（a ≠ 0）

Negative Exponent: a⁻ⁿ = 1 / aⁿ (a ≠ 0)

负指数：a⁻ⁿ = 1 / aⁿ（a ≠ 0）

Product of Radicals: √a × √b = √ab for a, b ≥ 0

根式乘法：√a × √b = √ab（a, b ≥ 0）

Quotient of Radicals: √a / b = √a / √b for a ≥ 0, b > 0

根式除法：√a / b = √a / √b（a ≥ 0, b > 0）

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3. Linear Equations and Inequalities | 线性方程与不等式

Linear relationships can be expressed in various forms and solved using systematic algebraic steps.

线性关系可以用多种形式表示，并通过系统的代数步骤求解。

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

斜截式：y = mx + c，其中 m 是斜率，c 是 y 轴截距。

Point-slope form: y − y₁ = m(x − x₁)

点斜式：y − y₁ = m(x − x₁)

Slope between two points: m = (y₂ − y₁) / (x₂ − x₁)

两点间斜率：m = (y₂ − y₁) / (x₂ − x₁)

To solve a linear equation, isolate the variable by performing inverse operations on both sides.

解线性方程时，通过等式两边进行逆运算来隔离变量。

When solving linear inequalities, reverse the inequality symbol if multiplying or dividing by a negative number.

解线性不等式时，若乘以或除以负数，需要反转不等号的方向。

Compound inequality example: −3 < 2x + 1 ≤ 5 can be broken into two parts to solve.

复合不等式示例：−3 < 2x + 1 ≤ 5 可拆分成两个不等式来求解。

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4. Quadratic Equations | 二次方程

Quadratic equations play a central role in Further Mathematics; their solutions can be found by factoring, completing the square, or using the quadratic formula.

二次方程在进阶数学中占据核心地位；其解可通过因式分解、配方法或求根公式求得。

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

标准式：ax² + bx + c = 0，其中 a ≠ 0。

Quadratic Formula: x = [ −b ± √b² − 4ac ] / (2a)

求根公式：x = [ −b ± √b² − 4ac ] / (2a)

on has two distinct real roots; if Δ = 0, one repeated real root; if Δ < 0, no real roots.

判别式：Δ = b² − 4ac。若 Δ > 0，方程有两个不等实根；若 Δ = 0，有两个相等实根；若 Δ < 0，无实根。

Sum of roots: α + β = −b/a ; Product of roots: αβ = c/a

根与系数的关系：两根之和 α + β = −b/a ；两根之积 αβ = c/a

Factorised form when roots are α and β: a(x − α)(x − β) = 0

已知两根为 α 和 β 时的因式形式：a(x − α)(x − β) = 0

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5. Geometry: Angles and Triangles | 几何：角度与三角形

Key angle properties and triangle congruence conditions form the backbone of geometric reasoning.

重要的角度性质与三角形全等条件是几何推理的基石。

Angles on a straight line sum to 180°.

直线上的邻补角之和为 180°。

Angles around a point sum to 360°.

绕一点一周的角之和为 360°。

Vertically opposite angles are equal.

对顶角相等。

Sum of interior angles of a triangle is 180°.

三角形内角和为 180°。

Exterior angle of a triangle equals the sum of the two opposite interior angles.

三角形的一个外角等于与它不相邻的两个内角之和。

Conditions for triangle congruence: SSS, SAS, ASA, AAS, and RHS (right-angle, hypotenuse, side).

三角形全等条件：SSS（边边边）、SAS（边角边）、ASA（角边角）、AAS（角角边）以及 RHS（直角、斜边、边）。

Properties of isosceles triangles: base angles are equal and the perpendicular from the apex bisects the base.

等腰三角形性质：两底角相等，顶角平分线垂直平分底边。

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

Right-angled triangle relationships are fundamental in geometry and have wide applications.

直角三角形的边角关系是几何学的基础，应用广泛。

Pythagoras’ Theorem: a² + b² = c² (where c is the hypotenuse)

勾股定理：a² + b² = c²（c 为斜边）

Sine: sin θ = opposite / hypotenuse

正弦：sin θ = 对边 / 斜边

Cosine: cos θ = adjacent / hypotenuse

余弦：cos θ = 邻边 / 斜边

Tangent: tan θ = opposite / adjacent

正切：tan θ = 对边 / 邻边

Reciprocal identity: tan θ = sin θ / cos θ

倒数关系：tan θ = sin θ / cos θ

Exact values for special angles: sin 30° = 1/2, sin 45° = √2/2, sin 60° = √3/2; cos 30° = √3/2, cos 45° = √2/2, cos 60° = 1/2; tan 30° = 1/√3, tan 45° = 1, tan 60° = √3.

特殊角的精确值：sin 30° = 1/2，sin 45° = √2/2，sin 60° = √3/2；cos 30° = √3/2，cos 45° = √2/2，cos 60° = 1/2；tan 30° = 1/√3，tan 45° = 1，tan 60° = √3。

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7. Coordinate Geometry | 坐标几何

Key formulas for working with points, lines and segments in the Cartesian plane are summarised below.

下面总结了在笛卡尔平面中处理点、直线和线段的关键公式。

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

两点 (x₁, y₁) 和 (x₂, y₂) 间的距离：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)

直线的斜率：m = (y₂ − y₁) / (x₂ − x₁)

Equation of a horizontal line: y = k (slope 0).

水平线方程：y = k（斜率为 0）。

Equation of a vertical line: x = h (slope undefined).

垂直线方程：x = h（斜率不存在）。

Lines are parallel if their slopes are equal; lines are perpendicular if the product of their slopes is −1.

若两直线斜率相等则平行；若斜率之积为 −1 则互相垂直。

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8. Mensuration: Perimeter, Area and Volume | 测量：周长、面积与体积

These formulas are essential for calculating lengths, areas and volumes of common 2D and 3D shapes.

以下公式对于计算常见二维和三维图形的长度、面积和体积至关重要。

Perimeter of a rectangle: P = 2(l + w), where l is length and w is width.

矩形周长：P = 2(l + w)，l 为长，w 为宽。

Circumference of a circle: C = 2πr or C = πd

圆的周长：C = 2πr 或 C = πd

Area of a rectangle: A = lw

矩形面积：A = lw

Area of a triangle: A = ½ × b × h

三角形面积：A = ½ × b × h

Area of a parallelogram: A = b × h

平行四边形面积：A = b × h

Area of a trapezium: A = ½ (a + b) h, where a and b are the parallel sides.

梯形面积：A = ½ (a + b) h，其中 a 和 b 为平行边。

Area of a circle: A = πr²

圆的面积：A = πr²

Volume of a cuboid: V = l × w × h

长方体体积：V = l × w × h

Volume of a prism: V = area of cross-section × length

棱柱体积：V = 横截面积 × 长

Volume of a cylinder: V = πr²h

圆柱体积：V = πr²h

Surface area of a cuboid: SA = 2(lw + lh + wh)

长方体表面积：SA = 2(lw + lh + wh)

Surface area of a cylinder: SA = 2πr² + 2πrh (curved and top/bottom areas)

圆柱表面积：SA = 2πr² + 2πrh（包括两个底面和侧面）

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9. Number Theory and Sequences | 数论与数列

Understanding prime factorisation, LCM, HCF, and the properties of sequences aids in solving many numerical problems.

理解质因数分解、最小公倍数、最大公因数以及数列的性质有助于解决许多数值问题。

Prime numbers have exactly two distinct positive factors: 1 and the number itself.

质数恰好有两个不同的正因数：1 和它本身。

The LCM (Lowest Common Multiple) of two numbers is the smallest positive number that is a multiple of both.

两个数的最小公倍数（LCM）是它们的最小正公共倍数。

The HCF (Highest Common Factor) is the largest positive integer dividing both numbers without a remainder.

最大公因数（HCF）是能同时整除这两个数的最大正整数。

nth term is aₙ = a₁ + (n − 1)d, where a₁ is the first term and d is the common difference.

等差数列（AP）：第 n 项 aₙ = a₁ + (n − 1)d，其中 a₁ 为首项，d 为公差。

Sum of first n terms of an AP: Sₙ = n/2 (a₁ + aₙ) = n/2 [2a₁ + (n − 1)d]

等差数列前 n 项和：Sₙ = n/2 (a₁ + aₙ) = n/2 [2a₁ + (n − 1)d]

eometric sequence (GP): The nth term is aₙ = a₁ rⁿ⁻¹, where r is the common ratio (r ≠ 0).

等比数列（

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