📚 CAIE Further Mathematics: Quick Reference to Key Formulae & Theorems | CAIE 进阶数学：公式定理速查手册
This concise handbook covers the essential formulae and theorems for the CAIE A-Level Further Mathematics (9231) syllabus. Each section presents the most frequently required results, making it an ideal revision aid for both Pure and Mechanics/Statistics components. Keep this guide close when practising past papers to reinforce your speed and accuracy.
这本简明手册涵盖了 CAIE A-Level 进阶数学 (9231) 大纲中最核心的公式与定理。每一节都列出了最高频的考试要点，无论复习纯数还是力学/统计部分，它都能帮助你快速查阅。在做历年真题时随身参考这本手册，可以有效提升解题速度和准确度。
1. Complex Numbers | 复数
The modulus-argument form and de Moivre’s theorem are fundamental. Powers and roots of complex numbers are easily handled in polar form.
模－辐角形式和棣莫弗定理是基础。使用极坐标形式可以轻松处理复数的乘幂和开方。
z = x + iy = r(cos θ + i sin θ) = r eiθ
Rectangular, polar and exponential forms of a complex number.
复数的代数、三角和指数三种表示形式。
de Moivre: (cos θ + i sin θ)n = cos(nθ) + i sin(nθ)
De Moivre’s theorem holds for any integer n.
棣莫弗定理对任意整数 n 成立。
Roots of unity: The n roots of zn = 1 are given by zk = e2πk i/n, k = 0,1,…,n−1. They sum to zero and lie evenly around the unit circle.
单位根：方程 zn = 1 的 n 个根为 zk = e2πk i/n，k = 0,1,…,n−1。它们的和为零，均匀分布在单位圆上。
Complex loci: |z − a| = r is a circle; |z − a| = |z − b| is the perpendicular bisector; arg(z − a) = θ is a ray from a.
复数轨迹：|z − a| = r 表示圆；|z − a| = |z − b| 表示中垂线；arg(z − a) = θ 表示从 a 出发的射线。
2. Matrices and Linear Transformations | 矩阵与线性变换
The determinant and inverse of a 2×2 and 3×3 matrix must be computed fluently. Transformations in the plane are represented by 2×2 matrices.
必须熟练计算二阶和三阶矩阵的行列式与逆矩阵。平面上的线性变换可以用二阶矩阵表示。
det A = |A|; A−1 = (1/|A|) adj(A)
Inverse of a matrix using the adjugate and determinant.
利用伴随矩阵和行列式求逆矩阵。
Reflection in y = x: [0 1; 1 0]
Reflection matrix for the line y = x.
关于直线 y = x 的反射矩阵。
| Transformation | Matrix (2×2) |
| Rotation by θ anticlockwise | [cosθ −sinθ; sinθ cosθ] |
| Stretch, factor k along x-axis | [k 0; 0 1] |
| Shear, x-direction, factor k | [1 k; 0 1] |
Common 2×2 transformation matrices; rotation, stretch and shear.
常见的二阶变换矩阵：旋转、拉伸和剪切。
Eigenvalues and eigenvectors: Solve det(A − λI) = 0. An eigenvector x satisfies Ax = λx. Diagonalisation: P−1AP = D, where columns of P are eigenvectors.
特征值与特征向量：解 det(A − λI) = 0。特征向量 x 满足 Ax = λx。对角化：P−1AP = D，P 的列为特征向量。
3. Vectors | 向量
Vector equations of lines and planes, plus scalar and vector products, are central to 3‑D geometry questions.
直线的向量方程、平面的向量方程以及标量积和向量积是三维几何问题的核心。
Line: r = a + t d
Vector equation of a line through point a, direction d.
经过点 a、方向向量为 d 的直线方程。
Plane: r·n = a·n or r = a + λ d₁ + μ d₂
Plane in scalar product form and in parametric form.
平面的点法式方程和参数式方程。
a·b = |a||b|cos θ, a×b = |a||b|sin θ n̂
Scalar (dot) product and vector (cross) product.
标量积（点乘）与向量积（叉乘）的定义。
Distance from point B to line r = a + t d: d = |(b − a) × d|/|d|.
点 B 到直线 r = a + t d 的距离：d = |(b − a) × d|/|d|。
Distance between skew lines: find common perpendicular vector n = d₁ × d₂, then distance = |(a₂ − a₁)·n|/|n|.
异面直线间的距离：求公垂向量 n = d₁ × d₂，距离 = |(a₂ − a₁)·n|/|n|。
4. Hyperbolic Functions | 双曲函数
Definitions, identities and calculus of hyperbolic functions mirror trigonometric functions but with sign differences.
双曲函数的定义、恒等式及其微积分与三角函数类似，仅符号有所不同。
sinh x = (ex − e−x)/2, cosh x = (ex + e−x)/2
Expressions for sinh and cosh in terms of exponentials.
用指数函数表示双曲正弦和双曲余弦。
cosh²x − sinh²x = 1
Fundamental hyperbolic identity, analogous to cos²θ + sin²θ = 1.
最基本的双曲恒等式，类似于三角恒等式 cos²θ + sin²θ = 1。
Inverse hyperbolic functions: arsinh x = ln(x + √(x²+1)), arcosh x = ln(x + √(x²−1)) for x ≥ 1, artanh x = ½ ln((1+x)/(1−x)) for |x| < 1.
反双曲函数：arsinh x = ln(x + √(x²+1))，arcosh x = ln(x + √(x²−1))（x ≥ 1），artanh x = ½ ln((1+x)/(1−x))（|x| < 1）。
d/dx (sinh x) = cosh x, d/dx (cosh x) = sinh x
Derivatives of basic hyperbolic functions.
基本双曲函数的导数。
5. Differential Equations | 微分方程
First-order linear ODEs, second-order homogeneous and non-homogeneous ODEs with constant coefficients are examined regularly.
一阶线性常微分方程、二阶常系数齐次和非齐次常微分方程是常考题型。
Integrating factor: dy/dx + P(x)y = Q(x) ⇒ I(x) = e∫P dx
Solve first-order linear ODEs using the integrating factor method.
使用积分因子法求解一阶线性常微分方程。
Second-order: a y” + b y’ + c y = 0
Auxiliary equation: a m² + b m + c = 0. Real distinct roots: y = A em₁x + B em₂x; repeated root m: y = (A + Bx)emx; complex roots α ± iβ: y = eαx(A cos βx + B sin βx).
辅助方程：a m² + b m + c = 0。实不等根：y = A em₁x + B em₂x；重根 m：y = (A + Bx)emx；复根 α ± iβ：y = eαx(A cos βx + B sin βx)。
Non-homogeneous case: Particular integral (PI) guessed from the form of RHS (polynomial, ekx, sin/cos, or products). General solution = complementary function + PI.
非齐次情形：特解 (PI) 根据右端项形式设定（多项式、ekx、正余弦或乘积）。通解 = 补函数 + 特解。
Substitutions: Often use u = y/x, v = x + y, etc., to reduce to a separable equation. Chain rule is essential.
代换法：常设 u = y/x、v = x + y 等，将方程化为可分离变量的形式，必须熟练运用链式法则。
6. Polar Coordinates | 极坐标
Curves are defined as r = f(θ). Areas, tangent slopes and arc lengths are key applications.
极坐标曲线由 r = f(θ) 定义。求面积、切线斜率和弧长是主要应用。
Area = ½ ∫θ₁θ₂ r² dθ
Area enclosed by a polar curve between two rays.
极坐标曲线在两条射线之间围成的面积公式。
Slope of tangent: dy/dx = (r’ sin θ + r cos θ)/(r’ cos θ − r sin θ), where r’ = dr/dθ. Parallel/perpendicular tests are common.
切线斜率：dy/dx = (r’ sin θ + r cos θ)/(r’ cos θ − r sin θ)，其中 r’ = dr/dθ。常考与极轴平行或垂直的切线。
Arc length: s = ∫ √(r² + (dr/dθ)²) dθ. Used for cardioids, circles and other standard curves.
弧长：s = ∫ √(r² + (dr/dθ)²) dθ。常用于心形线、圆等标准曲线。
Common curves: Circle r = 2a cos θ (through pole), cardioid r = a(1 + cos θ), rose curves r = a sin(nθ).
常见曲线：圆 r = 2a cos θ（过极点），心形线 r = a(1 + cos θ)，玫瑰线 r = a sin(nθ)。
7. Roots of Polynomial Equations | 多项式方程的根
Relations between roots and coefficients for quadratic, cubic and quartic equations. Transformations of roots are used to find new equations.
二次、三次和四次方程的根与系数之间的关系，以及通过根的变换来求新方程。
For x² + px + q = 0: Σα = α+β = −p, Σαβ = αβ = q
Quadratic: sum of roots, product of roots.
二次方程：根之和，根之积。
For x³ + px² + qx + r = 0: Σα = −p, Σαβ = q, αβγ = −r
Cubic: sums of roots taken one, two and three at a time.
三次方程：单根和、两两积之和、三根积。
Quartic: Σα, Σαβ, Σαβγ, αβγδ with alternating signs.
四次方程：Σα、Σαβ、Σαβγ、αβγδ，符号交替。
Transformation of roots: For equation with roots f(α), let y = f(x), express x in terms of y and substitute. For example, to find equation whose roots are squares, let y = x².
根的变换：求根为 f(α) 的方程，设 y = f(x)，解出 x 代回原方程。例如求根为平方的新方程，令 y = x²。
Useful identities: Σα² = (Σα)² − 2Σαβ. This can avoid solving the cubic.
常用恒等式：Σα² = (Σα)² − 2Σαβ，可避免直接解出三次方程的根。
8. Summation of Series | 级数求和
Standard results for sums of integers, squares and cubes, plus the method of differences and Maclaurin series expansions.
自然数、平方和、立方和的标准和公式，外加差分法和麦克劳林级数展开。
Σr = n(n+1)/2, Σr² = n(n+1)(2n+1)/6, Σr³ = n²(n+1)²/4
Sums of the first n natural numbers, squares and cubes.
前 n 个自然数、平方和与立方和的求和公式。
Method of differences: Express the general term as f(r) − f(r+1) or f(r+1) − f(r) so that summation telescopes. Common with rational functions.
差分法：将通项写成 f(r) − f(r+1) 或 f(r+1) − f(r) 的形式，使求和时前后抵消。常用于有理分式。
Maclaurin series: f(x) = f(0) + x f'(0) + (x²/2!) f”(0) + (x³/3!) f”'(0) + … . Recognise standard series: ex, sin x, cos x, ln(1+x), (1+x)n.
麦克劳林级数：f(x) = f(0) + x f'(0) + (x²/2!) f”(0) + (x³/3!) f”'(0) + … 。熟记 ex、sin x、cos x、ln(1+x)、(1+x)n 的标准展开。
9. Proof by Induction | 归纳证明
Induction is used for divisibility, summation formulae, matrix powers and inequalities. The structure must be rigorous.
数学归纳法常用来证明整除性、求和公式、矩阵幂和不等式，结构必须严谨。
Standard steps: (1) Base case: show true for n = 1 (or smallest value). (2) Inductive hypothesis: assume true for n = k. (3) Inductive step: prove true for n = k + 1 using the hypothesis. (4) Conclusion: by PMI, true for all n ≥ 1.
标准步骤：（1）基础步骤：验证 n = 1（或最小取值）时成立。（2）归纳假设：假设 n = k 时成立。（3）归纳递推：利用假设证明 n = k + 1 时成立。（4）结论：由数学归纳法，对所有 n ≥ 1 成立。
Divisibility example: To prove 8n − 3n is divisible by 5, write 8k+1 − 3k+1 = 8·8k − 3·3k = 5·8k + 3(8k − 3k). Both terms are multiples of 5.
整除性示例：证明 8n − 3n 能被 5 整除，可写 8k+1 − 3k+1 = 8·8k − 3·3k = 5·8k + 3(8k − 3k)，两项均为 5 的倍数。
Matrix powers: Prove An has a given form by induction, using Ak+1 = Ak A.
矩阵幂：通过归纳法证明 An 具有给定形式，相乘时利用 Ak+1 = Ak A。
10. Numerical Methods | 数值方法
Iterative methods for solving f(x) = 0, numerical integration and approximations for differential equations.
求解方程 f(x) = 0 的迭代法、数值积分以及微分方程的近似解法。
Newton-Raphson: xn+1 = xn − f(xn)/f'(xn)
Fast convergence near a root, but requires derivative and a good initial guess.
在根附近收敛很快，但需要计算导数以及合适的初值。
Mid-ordinate rule: ∫ab f(x) dx ≈ h Σ f(a + (k − ½)h), with h = (b−a)/n.
中点法则：∫ab f(x) dx ≈ h Σ f(a + (k − ½)h)，其中 h = (b−a)/n。
Simpson’s rule: (h/3)[y₀ + 4(y₁+y₃+…) + 2(y₂+y₄+…) + yn], n even.
辛普森法则：(h/3)[y₀ + 4(y₁+y₃+…) + 2(y₂+y₄+…) + yn]，要求 n 为偶数。
Euler method: yr+1 = yr + h f(xr, yr), step size h. Improved Euler and other routines may appear.
欧拉方法：yr+1 = yr + h f(xr, yr)，步长为 h。也可能出现改进欧拉法等变形。
Fixed-point iteration: Rewrite f(x)=0 as x = g(x). The sequence converges if |g'(x)| < 1 near the root.
不动点迭代：将 f(x)=0 改写为 x = g(x)，若在根附近有 |g'(x)| < 1 则迭代收敛。
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