Year 10 Cambridge Additional Mathematics: Quick Formula & Theorems Reference | Year 10 剑桥进阶数学：公式定理速查手册

📚 Year 10 Cambridge Additional Mathematics: Quick Formula & Theorems Reference | Year 10 剑桥进阶数学：公式定理速查手册

This is a concise, exam-focused reference handbook for Year 10 Cambridge Additional Mathematics. It collects essential formulas, theorems and key reminders for quick revision. Use it to check your understanding before tackling past papers.

这是一份针对 Year 10 剑桥进阶数学的精炼考点速查手册。它汇总了必背公式、核心定理与重要提醒，方便你快速复习，在刷真题前查漏补缺。

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1. Algebraic Identities and Surds | 代数恒等式与根式

(a + b)² = a² + 2ab + b²
(a − b)² = a² − 2ab + b²
(a + b)(a − b) = a² − b²

(a + b)² = a² + 2ab + b²
(a − b)² = a² − 2ab + b²
(a + b)(a − b) = a² − b²

When simplifying surds, write √ab = √a × √b and √a/b = √a / √b. Always rationalise the denominator so that no surd appears at the bottom.

化简二次根式时，利用 √ab = √a × √b 以及 √a/b = √a / √b。分母有理化是必须的，确保分母不含根号。

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2. Quadratic Equations and Functions | 二次方程与二次函数

For ax² + bx + c = 0, the roots are given by the quadratic formula:

对于 ax² + bx + c = 0，求根公式为：

x = [−b ± √b² − 4ac] / (2a)

c. If Δ > 0, two distinct real roots; Δ = 0, one repeated real root; Δ < 0, no real roots.

判别式 Δ = b² − 4ac。Δ > 0 ⇄ 两个不等实根；Δ = 0 ⇄ 一个重根；Δ < 0 ⇄ 无实根。

)² + k, the vertex is (h, k). Completing the square turns y = ax² + bx + c into this form.

二次函数 y = a(x − h)² + k 的顶点为 (h, k)。通过配方法可将一般式化为顶点式。

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3. Exponents and Logarithms | 指数与对数

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

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

(xⁿ) = n logₐx. Also logₐa = 1, logₐ1 = 0. The change of base rule: logₐb = logₓb / logₓa.

对数运算法则：logₐ(xy) = logₐx + logₐy，logₐ(x/y) = logₐx − logₐy，logₐ(xⁿ) = n logₐx。特殊值：logₐa = 1，logₐ1 = 0。换底公式：logₐb = logₓb / logₓa。

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4. Polynomials and the Remainder Theorem | 多项式与余式定理

P(x) is divided by (x − a), the remainder is P(a). If P(a) = 0, then (x − a) is a factor.

多项式 P(x) 除以 (x − a) 的余数为 P(a)。若 P(a) = 0，则 (x − a) 是 P(x) 的因式。

ctorise cubics such as x³ + ax² + bx + c. Always test simple values like x = ±1, ±2 first.

因式定理常用于分解三次式，如 x³ + ax² + bx + c。可先代入 x = ±1, ±2 等简单值进行检验。

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5. Binomial Expansion | 二项式展开

For positive integer n: (1 + x)ⁿ = 1 + nx + [n(n−1)/2!]x² + [n(n−1)(n−2)/3!]x³ + ... + xⁿ.

对于正整数指数 n：(1 + x)ⁿ = 1 + nx + [n(n−1)/2!]x² + [n(n−1)(n−2)/3!]x³ + ... + xⁿ。

More generally, (a + b)ⁿ = Σ (nCr) aⁿ⁻ʳ bʳ, where nCr = n! / [r!(n−r)!]. The expansion is valid for all x when n is a positive integer.

一般地，(a + b)ⁿ = Σ (nCr) aⁿ⁻ʳ bʳ，其中 nCr = n! / [r!(n−r)!]。当 n 为正整数时，该展开对全体 x 有效。

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6. Sequences and Series | 数列与级数

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

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

irst n terms Sₙ = a(1 − rⁿ)/(1 − r) for r ≠ 1. Infinite sum S∞ = a/(1 − r) valid only when |r| < 1.

等比数列：第 n 项 uₙ = arⁿ⁻¹。前 n 项和 Sₙ = a(1 − rⁿ)/(1 − r)，r ≠ 1。无穷和 S∞ = a/(1 − r)，仅当 |r| < 1 时成立。

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7. Coordinate Geometry | 坐标系几何

n (x₁, y₁) and (x₂, y₂): d = √[(x₂ − x₁)² + (y₂ − y₁)²]. Midpoint: ((x₁+x₂)/2, (y₁+y₂)/2).

两点间距离：d = √[(x₂ − x₁)² + (y₂ − y₁)²]。中点坐标：((x₁+x₂)/2, (y₁+y₂)/2)。

₁ = m(x − x₁). Parallel lines have equal gradients; perpendicular lines satisfy m₁m₂ = −1.

直线的斜率：m = (y₂ − y₁)/(x₂ − x₁)。直线方程：y − y₁ = m(x − x₁)。平行线斜率相等；垂直线斜率满足 m₁m₂ = −1。

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8. Trigonometry | 三角学

/hypotenuse, cos θ = adjacent/hypotenuse, tan θ = opposite/adjacent. Remember SOH CAH TOA.

直角三角形中：sin θ = 对边/斜边，cos θ = 邻边/斜边，tan θ = 对边/邻边。牢记 SOH CAH TOA。

Exact values for special angles:

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	√3/3	1	√3	undefined

Sine and cosine rules: a/sin A = b/sin B = c/sin C; a² = b² + c² − 2bc cos A. Use for non‑right‑angled triangles. Area = ½ ab sin C.

正弦定理与余弦定理：a/sin A = b/sin B = c/sin C；a² = b² + c² − 2bc cos A。用于非直角三角形。面积 = ½ ab sin C。

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9. Vectors | 向量

A vector can be written as a = (x, y) or a = xi + yj. Magnitude |a| = √x² + y². A unit vector has magnitude 1: a/|a|.

向量可表示为 a = (x, y) 或 a = xi + yj。模长 |a| = √x² + y²。单位向量模长为1，公式为 a/|a|。

If points A and B have position vectors a and b, then vector AB = b − a. The midpoint of AB has position vector (a + b)/2.

若点 A、B 的位置向量分别为 a 和 b，则向量 AB = b − a。线段 AB 中点的位置向量为 (a + b)/2。

Collinear vectors satisfy c = ka for some scalar k. Parallel vectors are scalar multiples.

共线向量满足 c = ka（k 为标量）。平行向量彼此成标量倍数关系。

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10. Introduction to Calculus | 微积分初步

rivative of xⁿ: d/dx (xⁿ) = nxⁿ⁻¹. For a constant k, d/dx (k) = 0. The gradient function f'(x) gives the slope of the tangent to y = f(x).

xⁿ 的导数：d/dx (xⁿ) = nxⁿ⁻¹。常数的导数为0。梯度函数 f'(x) 给出曲线 y = f(x) 在某点切线的斜率。

Stationary points occur when f'(x) = 0. Use the second derivative f''(x) to determine nature: f''(x) > 0 gives a minimum, f''(x) < 0 gives a maximum.

驻点发生在 f'(x) = 0 处。用二阶导数 f''(x) 判断性质：f''(x) > 0 为极小值点，f''(x) < 0 为极大值点。

≠ −1. Definite integral ∫ₐᵇ f(x) dx gives the area under y = f(x) between x = a and x = b.

不定积分：∫ xⁿ dx = xⁿ⁺¹/(n+1) + c，n ≠ −1。定积分 ∫ₐᵇ f(x) dx 表示曲线 y = f(x) 在区间 [a, b] 上与 x 轴围成的面积。

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