Formula & Theorem Quick-Reference Handbook for CIE Year 10 Further Pure Maths | CIE 10年级进阶数学公式定理速查手册

📚 Formula & Theorem Quick-Reference Handbook for CIE Year 10 Further Pure Maths | CIE 10年级进阶数学公式定理速查手册

This article provides a carefully compiled quick-reference guide to all the essential formulae, theorems, and identities required for the CIE Year 10 Further Pure Mathematics course. It is designed for efficient revision and to support problem-solving in algebra, functions, trigonometry, calculus, and vector geometry. Each section pairs a concise English explanation with its Chinese equivalent, ensuring clarity for bilingual learners. Keep this handbook at your side while tackling past papers or consolidating your knowledge.

本文为 CIE 10年级进阶纯数课程精心汇编了所有核心公式、定理和恒等式的速查手册，旨在帮助高效复习并支持代数、函数、三角、微积分和向量几何中的解题。每个部分都将简洁的英文解释与中文配对，确保双语学习者能清晰理解。在练习往年真题或巩固知识时，请将本手册放在手边。

---

1. Algebraic Identities & Factorisation | 代数恒等式与因式分解

The three fundamental algebraic identities form the basis for expanding and factorising quadratic and cubic expressions. They must be applied rapidly and accurately in both directions.

三个基本代数恒等式是展开和因式分解二次与三次表达式的基础，必须能迅速且准确地进行双向应用。

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

完全平方：(a + b)² = a² + 2ab + b² 以及 (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)。

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)。

2)² − (b/2)². This technique is essential for deriving the quadratic formula and for sketching parabolas.

配方法：x² + bx = (x + b/2)² − (b/2)²。该技巧在推导二次公式和绘制抛物线草图时至关重要。

---

2. Quadratic Functions & the Discriminant | 二次函数与判别式

discriminant Δ (Delta) is defined as Δ = b² − 4ac. It determines the nature of the roots.

对于二次函数 f(x) = ax² + bx + c (a ≠ 0)，判别式 Δ 定义为 Δ = b² − 4ac。它决定了根的性质。

s one repeated real root. If Δ < 0, there are no real roots (two complex conjugate roots).

若 Δ > 0，有两个相异实根；若 Δ = 0，有一个重实根；若 Δ < 0，无实根（两个共轭复根）。

The quadratic formula gives the roots: x = [−b ± √b² − 4ac] / (2a).

二次求根公式为：x = [−b ± √b² − 4ac] / (2a)。

= c/a. These relationships with the coefficients are invaluable for forming new equations.

根的和：α + β = −b/a；根的积：αβ = c/a。这些与系数的关系对构造新方程非常有用。

rm: f(x) = a(x − h)² + k, where vertex is (h, k). The axis of symmetry is x = h = −b/(2a).

顶点式：f(x) = a(x − h)² + k，其中顶点为 (h, k)。对称轴为 x = h = −b/(2a)。

---

3. Equations & Inequalities | 方程与不等式

Solving equations often involves rearranging into standard forms and checking for extraneous solutions when dealing with surds or rational expressions.

解方程常需将其化为标准形式，并在处理根式或有理式时检验增根。

Surd equation technique: isolate the surd term, square both sides, then solve. Always verify solutions by substituting back into the original equation.

根式方程的解法：孤立根号项，两边平方后求解。务必代回原方程检验解的有效性。

Modulus equations: |f(x)| = k ⟹ f(x) = k or f(x) = −k, provided k ≥ 0. For inequalities such as |f(x)| < k, rewrite as −k < f(x) < k; for |f(x)| > k, use f(x) < −k or f(x) > k.

绝对值方程：|f(x)| = k 等价于 f(x) = k 或 f(x) = −k，前提是 k ≥ 0。对于不等式 |f(x)| < k，改写为 −k < f(x) < k；对于 |f(x)| > k，使用 f(x) < −k 或 f(x) > k。

etermine sign intervals using a sketch graph or a sign table. Pay attention to whether the inequality is strict or inclusive.

二次不等式：ax² + bx + c > 0，先解 ax² + bx + c = 0，然后利用图像草图或符号表确定符号区间。注意区分严格不等式与包含等号的情形。

---

4. Functions & Their Transformations | 函数及其变换

re each input has exactly one output. The notation f: x → f(x) or y = f(x) is widely used.

函数是定义域到值域的映射，每个输入只有一个输出。常用记号 f: x → f(x) 或 y = f(x)。

Composite function: (f ∘ g)(x) = f(g(x)). The order matters: apply g first, then f.

复合函数：(f ∘ g)(x) = f(g(x))。注意顺序：先作用 g，再作用 f。

g x and y, then solving for y. The graphs of f and f⁻¹ are symmetric about the line y = x.

反函数：f⁻¹(x) 仅在 f 为一一映射时存在。求法：设 y = f(x)，交换 x 与 y，然后解出 y。f 与 f⁻¹ 的图像关于直线 y = x 对称。

tor a; y = f(ax) is a horizontal stretch by factor 1/a; y = −f(x) is a reflection in the x‑axis; y = f(−x) is a reflection in the y‑axis.

图像变换：y = f(x) + a 是竖直平移 a 个单位；y = f(x + a) 是水平平移 −a 个单位；y = a f(x) 是竖直伸缩 a 倍；y = f(ax) 是水平伸缩 1/a 倍；y = −f(x) 是关于 x 轴的反射；y = f(−x) 是关于 y 轴的反射。

---

5. Indices, Surds & Exponentials | 指数、根式与指数函数

> 0 rational exponents: aᵐ × aⁿ = aᵐ⁺ⁿ; aᵐ ÷ aⁿ = aᵐ⁻ⁿ; (aᵐ)ⁿ = aᵐⁿ; a⁰ = 1; a⁻ⁿ = 1/aⁿ; a^1/n = ⁿ√a; a^m/n = ⁿ√aᵐ.

指数定律（a > 0，有理指数）：aᵐ × aⁿ = aᵐ⁺ⁿ；aᵐ ÷ aⁿ = aᵐ⁻ⁿ；(aᵐ)ⁿ = aᵐⁿ；a⁰ = 1；a⁻ⁿ = 1/aⁿ；a^1/n = ⁿ√a；a^m/n = ⁿ√aᵐ。

Surd simplification: √ab = √a × √b; √a/b = √a / √b. Rationalising the denominator is required: for 1/√a multiply top and bottom by √a; for 1/(a + √b) multiply by (a − √b).

根式化简：√ab = √a × √b；√a/b = √a / √b。需要分母有理化：对于 1/√a，上下同乘 √a；对于 1/(a + √b)，乘上共轭 (a − √b)。

The exponential function y = aˣ with a > 0, a ≠ 1 has domain ℝ h always passes through (0,1) and exhibits rapid growth for a > 1, or decay for 0 < a < 1.

指数函数 y = aˣ，其中 a > 0, a ≠ 1，定义域为 ℝ，值域为 (0, ∞)。其图像总经过点 (0,1)，当 a > 1 时快速增长，0 < a < 1 时为衰减。

---

6. Logarithms & Their Properties | 对数及其性质

If aˣ rithm is the inverse operation of exponentiation. Common log is log₁₀, natural log is ln (base e).

若 aˣ = N (a > 0, a ≠ 1)，则 x = logₐ N。对数是指数的逆运算。常用对数为 log₁₀，自然对数为 ln（底 e）。

Log laws: logₐ (MN) = logₐ M + logₐ N; logₐ (M/N) = logₐ M − logₐ N; logₐ (Mᵏ) = k logₐ M.

对数法则：logₐ (MN) = logₐ M + logₐ N；logₐ (M/N) = logₐ M − logₐ N；logₐ (Mᵏ) = k logₐ M。

Change of base: logₐ b = (log꜀ b) / (log꜀ a) for any valid base c. Typically c = 10 or e is used.

换底公式：logₐ b = (log꜀ b) / (log꜀ a)，适用于任何合法的底 c，通常取 c = 10 或 e。

Special values: logₐ 1 = 0; logₐ a = 1; ln e = 1; ln 1 = 0. The equation aˣ = b can be solved by taking logs: x = log b / log a.

特殊值：logₐ 1 = 0；logₐ a = 1；ln e = 1；ln 1 = 0。方程 aˣ = b 可通过对数求解：x = log b / log a。

---

广告 / Sponsored

7. Trigonometry: Ratios & Identities | 三角学：比值与恒等式

cos θ = adjacent/hypotenuse, tan θ = opposite/adjacent. SOH CAH TOA is a useful mnemonic.

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。SOH CAH TOA 是有效的助记口诀。

Exact values for key angles (θ in degrees):

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

dentities: sin²θ + cos²θ = 1; 1 + tan²θ = sec²θ; 1 + cot²θ = csc²θ. tan θ = sin θ / cos θ.

基本恒等式：sin²θ + cos²θ = 1；1 + tan²θ = sec²θ；1 + cot²θ = csc²θ。tan θ = sin θ / cos θ。

, c = AB: a/sin A = b/sin B = c/sin C (sine rule); a² = b² + c² − 2bc cos A (cosine rule).

对任意三角形 ABC，边长 a = BC, b = AC, c = AB：a/sin A = b/sin B = c/sin C（正弦定理）；a² = b² + c² − 2bc cos A（余弦定理）。

---

8. 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 (x₁, y₁) 与 (x₂, y₂) 的距离：d = √[(x₂ − x₁)² + (y₂ − y₁)²]。

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

中点坐标：M = ((x₁ + x₂)/2, (y₁ + y₂)/2)。

h gradients m₁ and m₂, they are parallel if m₁ = m₂; they are perpendicular if m₁·m₂ = −1.

直线的斜率（梯度）：m = (y₂ − y₁) / (x₂ − x₁)。两直线斜率分别为 m₁ 和 m₂，若 m₁ = m₂ 则平行；若 m₁·m₂ = −1 则垂直。

x₁) (point-slope form); y = mx + c (slope-intercept form); ax + by + c = 0 (general form).

直线方程：y − y₁ = m(x − x₁)（点斜式）；y = mx + c（斜截式）；ax + by + c = 0（一般式）。

---

9. Differentiation Rules | 微分法则

The derivative of f(x) is f' mₕ→₀ [f(x+h) − f(x)] / h. The power rule is the most frequently used: if f(x) = xⁿ, then f'(x) = n xⁿ⁻¹.

函数 f(x) 的导数为 f'(x) = limₕ→₀ [f(x+h) − f(x)] / h。最常用的是幂法则：若 f(x) = xⁿ，则 f'(x) = n xⁿ⁻¹。

Constant multiple rule: d/dx [k·f(x)] = k·f'(x). Sum/difference rule: d/dx [f(x) ± g(x)] = f'(x) ± g'(x).

常数倍法则：d/dx [k·f(x)] = k·f'(x)。和差法则：d/dx [f(x) ± g(x)] = f'(x) ± g'(x)。

The chain rule: if y = f(g(x)), then dy/dx = f'(g(x)) · g'(x). It is essential for differentiating composite functions such as (ax + b)ⁿ.

链式法则：若 y = f(g(x))，则 dy/dx = f'(g(x)) · g'(x)。它对复合函数如 (ax + b)ⁿ 的微分至关重要。

Product rule: if y = u(x)v(x), then dy/dx = u'v + uv'.

乘法法则：若 y = u(x)v(x)，则 dy/dx = u'v + uv'。

Quotient rule: if y = u/v, then dy/dx = (u'v − uv') / v².

除法法则：若 y = u/v，则 dy/dx = (u'v − uv') / v²。

ard functions: d/dx (sin x) = cos x; d/dx (cos x) = −sin x; d/dx (tan x) = sec² x; d/dx (eˣ) = eˣ; d/dx (ln x) = 1/x.

基本函数的导数：d/dx (sin x) = cos x；d/dx (cos x) = −sin x；d/dx (tan x) = sec² x；d/dx (eˣ) = eˣ；d/dx (ln x) = 1/x。

---

10. Integration: The Reverse of Differentiation | 积分：微分的逆运算

Indefinite integral (antiderivative): ∫ f(x) dx = F(x) + C, where F'(x) = f(x). The constant of integration C must be included.

不定积分（原函数）：∫ f(x) dx = F(x) + C，其中 F'(x) = f(x)。必须包含积分常数 C。

rule for integration: ∫ xⁿ dx = xⁿ⁺¹/(n+1) + C, for n ≠ −1. The special case ∫ x⁻¹ dx = ln |x| + C.

积分幂法则：∫ xⁿ dx = xⁿ⁺¹/(n+1) + C，n ≠ −1。特殊情况 ∫ x⁻¹ dx = ln |x| + C。

ite integral: ∫ₐᵇ f(x) dx = F(b) − F(a). It gives the net area between the curve and the x‑axis from x = a to x = b.

定积分：∫ₐᵇ f(x) dx = F(b) − F(a)。它表示曲线与 x 轴之间在 x = a 到 x = b 区间的净面积。

Area between a curve and the x‑ rea = ∫ₐᵇ f(x) dx. If the curve crosses the axis, split the interval where f changes sign.

曲线与 x 轴之间的面积：若在 [a, b] 上 f(x) ≥ 0，面积 = ∫ₐᵇ f(x) dx。若曲线穿越坐标轴，需在函数变号处分拆积分区间。

ion: if velocity v(t) = ds/dt, then displacement s(t) = ∫ v dt; acceleration a(t) = dv/dt.

运动学应用：若速度 v(t) = ds/dt，则位移 s(t) = ∫ v dt；加速度 a(t) = dv/dt。

---

11. Sequences & Series | 数列与级数

−1)d. Sum of first n terms Sₙ = n/2 [2a + (n−1)d] = n/2 (a + l), where l is the last term.

等差数列（AP）：第 n 项 uₙ = a + (n−1)d。前 n 项和 Sₙ = n/2 [2a + (n−1)d] = n/2 (a + l)，其中 l 为末项。

ic progression (GP): nth term uₙ = a rⁿ⁻¹. Sum of first n terms Sₙ = a(1 − rⁿ)/(1 − r) for |r| ≠ 1. Sum to infinity S∞ = a/(1 − r) provided |r| < 1.

等比数列（GP）：第 n 项 uₙ = a rⁿ⁻¹。前 n 项和 Sₙ = a(1 − rⁿ)/(1 − r)，|r| ≠ 1。无穷项和 S∞ = a/(1 − r)，要求 |r| < 1。

used to compactly represent a series. Properties: Σ(k uᵢ) = k Σuᵢ; Σ(uᵢ ± vᵢ) = Σuᵢ ± Σvᵢ.

求和符号 Σ 用于紧凑表示级数。性质：Σ(k uᵢ) = k Σuᵢ；Σ(uᵢ ± vᵢ) = Σuᵢ ± Σvᵢ。

Σi = n(n+1)/2; Σi² = n(n+1)(2n+1)/6. These can be combined to sum polynomial-type series.

标准结果：Σ1 = n；Σi = n(n+1)/2；Σi² = n(n+1)(2n+1)/6。这些可组合用于求和多项式型级数。

---

12. Vectors in Two Dimensions | 平面向量

. It can be expressed in column form as (x, y) or using unit vectors i and j: v = xi + yj.

向量具有大小和方向。可用列向量 (x, y) 或单位向量 i、j 表示：v = xi + yj。

Magnitude (length): |v| = √x² + y². A unit vector in the direction of v is v̂ = v / |v|.

模（长度）：|v| = √x² + y²。沿 v 方向的单位向量为 v̂ = v / |v|。

ion: (x₁, y₁) + (x₂, y₂) = (x₁ + x₂, y₁ + y₂). Scalar multiplication: k (x, y) = (kx, ky).

向量加法：(x₁, y₁) + (x₂, y₂) = (x₁ + x₂, y₁ + y₂)。数乘：k (x, y) = (kx, ky)。

ector AB = OB − OA = (x₂ − x₁, y₂ − y₁). Midpoint M of AB has position vector (OA + OB)/2.

点 A 的位置向量为 OA。向量 AB = OB − OA = (x₂ − x₁, y₂ − y₁)。AB 中点 M 的位置向量为 (OA + OB)/2。

scalar k. Collinear points: points A, B, C are collinear if AB is parallel to BC (or AC).

平行向量：若 v = k w（k 为标量），则 v 与 w 平行。共线点：若 AB 平行于 BC（或 AC），则 A、B、C 共线。

Dot (scalar) product: v · w = x₁x₂ + y₁y₂ = |v||w| s used to find the angle between two vectors and to test for perpendicularity (v · w = 0).

点积（标量积）：v · w = x₁x₂ + y₁y₂ = |v||w| cos θ。用于求向量间夹角和检验垂直（v · w = 0）。

---

Published by TutorHao | Further Pure Maths Revision Series | aleveler.com

更多咨询请联系16621398022（同微信）