AS CIE Further Maths: Quick Reference Handbook of Formulas and Theorems | AS CIE 进阶数学：公式定理速查手册

📚 AS CIE Further Maths: Quick Reference Handbook of Formulas and Theorems | AS CIE 进阶数学：公式定理速查手册

This handbook compiles the essential formulas, theorems, and key points for the AS Level CIE Further Mathematics syllabus (9231). It covers Further Pure Mathematics 1, as well as the optional Further Mechanics and Further Statistics papers. Use this as a quick revision guide – each section presents concepts in both English and Chinese to reinforce understanding.

本手册汇编了 AS 阶段 CIE 进阶数学 (9231) 课程的核心公式、定理与要点，涵盖进阶纯数1以及选考的进阶力学与进阶统计。可用作快速复习指南 – 每一节均以中英双语呈现概念，帮助加深理解。

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1. Roots of Polynomial Equations | 多项式方程的根

+ c = 0, with roots α and β, the sum of roots is α+β = −b/a, and the product is αβ = c/a.

对于二次方程 ax² + bx + c = 0，若根为 α 和 β，则根之和 α+β = −b/a，根之积 αβ = c/a。

bx² + cx + d = 0, with roots α, β, γ: Σα = α+β+γ = −b/a, Σαβ = αβ+βγ+γα = c/a, αβγ = −d/a.

对于三次方程 ax³ + bx² + cx + d = 0，根为 α, β, γ： Σα = α+β+γ = −b/a， Σαβ = αβ+βγ+γα = c/a， αβγ = −d/a。

+ cx² + dx + e = 0, with roots α, β, γ, δ: Σα = −b/a, Σαβ = c/a, Σαβγ = −d/a, αβγδ = e/a.

对于四次方程 ax⁴ + bx³ + cx² + dx + e = 0，根为 α, β, γ, δ： Σα = −b/a， Σαβ = c/a， Σαβγ = −d/a， αβγδ = e/a。

., y = 3x + 1), substitute x = (y−1)/3 into the original polynomial equation and simplify.

若要寻找根为原根线性变换后的方程（例如 y = 3x + 1），可将 x = (y−1)/3 代入原多项式方程并化简。

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2. Rational Functions and Graphs | 有理函数与图像

grees of numerator and denominator; if degrees are equal, it is y = leading coefficient of numerator / leading coefficient of denominator.

垂直渐近线出现在分母为零且分子不为零之处。水平渐近线的求法是比较分子与分母的次数；若次数相等，则为 y = 分子首项系数 / 分母首项系数。

An oblique (slant) asymptote occurs when the degree of the numerator is exactly one more than the denominator. It can be found by polynomial division, writing the function as a linear expression plus a proper rational remainder.

斜渐近线出现在分子的次数比分母恰好高一次时。可通过多项式除法求得，将函数写成一个线性表达式加上一个真分式。

Sketch the graph by identifying intercepts, asymptotes, stationary points (via differentiation), and behaviour near asymptotes. Check symmetry where possible.

绘制图像时应确定截距、渐近线、驻点（通过求导），并考察渐近线附近的行为。可能时还应检查对称性。

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3. Summation of Series | 级数求和

summation formulas for integers are essential: Σ[r=1 to n] r = ½ n(n+1), Σ[r=1 to n] r² = ⅙ n(n+1)(2n+1), Σ[r=1 to n] r³ = ¼ n²(n+1)² = (½ n(n+1))².

整数次幂求和的标准公式必须掌握： Σ[r=1 to n] r = ½ n(n+1)， Σ[r=1 to n] r² = ⅙ n(n+1)(2n+1)， Σ[r=1 to n] r³ = ¼ n²(n+1)² = (½ n(n+1))²。

olynomial sequence can be split: Σ (ar² + br + c) = a Σ r² + b Σ r + Σ c, where Σ c = c n.

对于常数倍的求和， Σ kr = k Σ r。多项式序列的求和可以拆分为各项之和： Σ (ar² + br + c) = a Σ r² + b Σ r + Σ c，其中 Σ c = c n。

like 1×2 + 2×3 + ... + n(n+1) by expressing the general term in terms of r and expanding.

这些公式可用于给出诸如 1×2 + 2×3 + ... + n(n+1) 的封闭形式，方法是将通项用 r 表示并展开，再逐项求和。

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4. Matrices | 矩阵

ant of a 2×2 matrix A = [[a, b], [c, d]] is det(A) = ad − bc. A is singular if det(A) = 0.

2×2 矩阵 A = [[a, b], [c, d]] 的行列式为 det(A) = ad − bc。若 det(A) = 0，则 A 为奇异矩阵。

For the inverse of a 2×2 matrix: A⁻¹ = (1/det(A)) [[d, −b], [−c, a]], provided det(A) ≠ 0.

2×2 矩阵的逆矩阵为：A⁻¹ = (1/det(A)) [[d, −b], [−c, a]]，前提是 det(A) ≠ 0。

: if A is m×n and B is n×p, the product C = AB has (i,j) entry given by Σ[k=1 to n] aᵢₖ bₖⱼ. Multiplication is not generally commutative.

矩阵乘法：若 A 为 m×n 阵，B 为 n×p 阵，则乘积 C = AB 的 (i,j) 元为 Σ[k=1 to n] aᵢₖ bₖⱼ。乘法一般不满足交换律。

ons AX = B, if A is invertible, X = A⁻¹B. Equivalently, use row reduction to echelon form.

求解线性方程组 AX = B 时，若 A 可逆，则 X = A⁻¹B。也可用行变换化为阶梯形求解。

FP1, eigenvalues and eigenvectors are mainly used for transformations and powers of matrices.

特征值 λ 满足 det(A − λI) = 0。特征向量 x 满足 Ax = λx。FP1 中特征值与特征向量主要用于线性变换及矩阵的高次幂计算。

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5. Polar Coordinates | 极坐标

,θ) by: x = r cos θ, y = r sin θ, and r² = x² + y², tan θ = y/x (consider quadrant for θ).

直角坐标 (x,y) 与极坐标 (r,θ) 的关系为：x = r cos θ, y = r sin θ，且 r² = x² + y²，tan θ = y/x（需根据象限确定 θ）。

considering symmetry and plotting key points. Common curves include cardioids, limacons, and roses.

极坐标方程 r = f(θ) 的图像可通过考虑对称性并描出关键点来绘制。常见曲线包括心形线、蜗线、玫瑰线等。

The area enclosed by a polar curve r = f(θ) from θ = α to θ = β is A = ½ ∫[α to β] r² dθ.

由极坐标曲线 r = f(θ) 在 θ = α 到 θ = β 之间所围成的面积为 A = ½ ∫[α to β] r² dθ。

/dθ ≠ 0) and vertical when dx/dθ = 0 (provided dy/dθ ≠ 0), where x = r cos θ, y = r sin θ.

在极坐标曲线上，当 dy/dθ = 0 且 dx/dθ ≠ 0 时切线为水平；当 dx/dθ = 0 且 dy/dθ ≠ 0 时切线为垂直，其中 x = r cos θ, y = r sin θ。

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6. Vectors | 向量

The scalar (dot) product of vectors a and b is a · b = |a||b| component form, for a = a₁i + a₂j + a₃k, b = b₁i + b₂j + b₃k, a · b = a₁b₁ + a₂b₂ + a₃b₃.

向量 a 与 b 的数量积（点积）为 a · b = |a||b| cos θ，θ 为两向量夹角。用分量表示，若 a = a₁i + a₂j + a₃k, b = b₁i + b₂j + b₃k，则 a · b = a₁b₁ + a₂b₂ + a₃b₃。

vector (cross) product a × b yields a vector perpendicular to both a and b, with magnitude |a||b| sin θ. In components: a × b = (a₂b₃ − a₃b₂)i + (a₃b₁ − a₁b₃)j + (a₁b₂ − a₂b₁)k.

叉积 a × b 给出一个同时垂直于 a 和 b 的向量，大小为 |a||b| sin θ。分量形式：a × b = (a₂b₃ − a₃b₂)i + (a₃b₁ − a₁b₃)j + (a₁b₂ − a₂b₁)k。

ere a is a point on the line and d is the direction vector. In 3D, the distance between two skew lines can be found using cross product of direction vectors.

直线的向量方程：r = a + t d，其中 a 为直线上一点，d 为方向向量。在三维中，两条异面直线的距离可通过方向向量的叉积求得。

The area of a parallelogram spanned by vectors a and b is |a × b|; the area of the related triangle is ½|a × b|.

以向量 a 和 b 为邻边的平行四边形面积为 |a × b|；对应三角形的面积为 ½|a × b|。

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7. Proof by Induction | 数学归纳法

true. 2. Inductive hypothesis: assume P(k) is true for some arbitrary k ≥ 1. 3. Inductive step: prove that P(k+1) follows from P(k). Conclude P(n) holds for all n ∈ ℕ.

数学归纳法用于证明对所有正整数 n，命题 P(n) 成立。证明分三步：1. 基础情形：验证 P(1) 为真。2. 归纳假设：假设对某个任意 k ≥ 1，P(k) 成立。3. 归纳步骤：证明 P(k+1) 可由 P(k) 推出。由此断言 P(n) 对所有正整数 n 均成立。

Common induction proofs involve summation formulas, divisibility (e.g., 9ⁿ − 1 is divisible by 8), matrix powers, and sequences defined by recurrence relations.

常见的归纳证明包括求和公式、整除性（如 9ⁿ − 1 可被 8 整除）、矩阵的幂次以及由递推关系定义的数列。

For divisibility, show that f(k+1) − m·f(k) (for a suitable integer m) is divisible by the required number, relying on the inductive hypothesis for f(k).

对于整除性证明，可设法证明 f(k+1) − m·f(k)（m 为适当整数）能被所要求的数整除，并利用归纳假设对 f(k) 的结论。

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8. Further Mechanics | 进阶力学 (选考)

(u sin θ) t − ½ g t². Time of flight T = (2u sin θ)/g, range R = (u² sin 2θ)/g, maximum height H = (u² sin² θ)/(2g).

抛体运动：初速 u，仰角 θ，水平位移 x = (u cos θ) t，竖直位移 y = (u sin θ) t − ½ g t²。飞行时间 T = (2u sin θ)/g，射程 R = (u² sin 2θ)/g，最大高度 H = (u² sin² θ)/(2g)。

ween force and displacement. Kinetic energy = ½ mv², gravitational potential energy = mgh.

恒力做功：W = F d cos θ，其中 d 为位移，θ 为力与位移的夹角。动能 = ½ mv²，重力势能 = mgh。

Principle of conservation of mechanical energy: if no external work except gravity, total mechanical energy (kinetic + potential) remains constant.

机械能守恒原理：若除重力外无其他外力做功，则总机械能（动能 + 势能）保持不变。

mentum, I = mv − mu. In collisions, total momentum is conserved if no external force acts.

动量：p = mv。冲量-动量定理：冲量 = 动量变化，I = mv − mu。碰撞中若合外力为零，总动量守恒。

Newton' aration)/(speed of approach). For perfectly elastic collisions e = 1, for inelastic e < 1.

牛顿碰撞定律：恢复系数 e = (分离速度)/(接近速度)。完全弹性碰撞 e = 1，非弹性碰撞 e < 1。

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9. Further Statistics | 进阶统计 (选考)

(mean) is E(X) = Σ x p(x). The variance Var(X) = E(X²) − [E(X)]², where E(X²) = Σ x² p(x).

对于离散随机变量 X，期望值（均值）E(X) = Σ x p(x)。方差 Var(X) = E(X²) − [E(X)]²，其中 E(X²) = Σ x² p(x)。

eometric distribution models the number of trials up to and including the first success: X ~ Geo(p) with P(X = x) = (1 − p)^x−1 p, x = 1,2,3,... The mean is E(X) = 1/p, variance Var(X) = (1−p)/p².

几何分布描述为获得第一次成功所需的试验次数：X ~ Geo(p)，P(X = x) = (1 − p)^x−1 p，x = 1,2,3,...。均值 E(X) = 1/p，方差 Var(X) = (1−p)/p²。

(aX + bY) = aE(X) + bE(Y). If X and Y are independent, Var(aX + bY) = a²Var(X) + b²Var(Y).

独立随机变量的和：E(aX + bY) = aE(X) + bE(Y)。若 X 与 Y 独立，Var(aX + bY) = a²Var(X) + b²Var(Y)。

For a random sample, the distribution of the sample mean X̄ from a population with mean μ and variance σ² has E(X̄) = μ and Var(X̄) = σ²/n. The Central Limit Theorem (CLT) says X̄ is approximately normal for large n, regardless of the population distribution.

对于从均值为 μ 、方差为 σ² 的总体中抽取的随机样本，样本均值 X̄ 满足 E(X̄) = μ，Var(X̄) = σ²/n。中心极限定理 (CLT) 指出，当 n 较大时，不论总体分布如何，X̄ 近似服从正态分布。

Confidence intervals: For a population mean with known σ, a 95% confidence interval is x̄ ± 1.96 (σ/√n). For unknown σ and small n, use the t-distribution; for large n, the normal approximates it.

置信区间：已知总体标准差 σ 时，均值的 95% 置信区间为 x̄ ± 1.96 (σ/√n)。当 σ 未知且 n 较小时，使用 t 分布；n 较大时可用正态近似。

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