📚 Year 13 CAIE Further Mathematics: Quick Reference Handbook of Formulas and Theorems | Year 13 CAIE 进阶数学：公式定理速查手册
This handbook provides a concise summary of the essential formulas, theorems and standard results required for the Year 13 CAIE Further Mathematics syllabus (9231). It covers Further Pure Mathematics 2, Further Mechanics and Further Statistics. Use it as a quick revision aid to recall key facts before your examination.
本手册简要总结了 Year 13 CAIE 进阶数学(9231)课程所需的核心公式、定理和标准结果，涵盖 Further Pure Mathematics 2、Further Mechanics 和 Further Statistics。请将其作为考前快速回顾要点的速查工具。
1. Complex Numbers & de Moivre’s Theorem | 复数与棣莫弗定理
Euler’s relation links exponential and trigonometric forms of a complex number:
欧拉公式将复数的指数形式与三角形式联系起来：
eⁱθ = cos θ + i sin θ
De Moivre’s theorem extends this to integer powers:
棣莫弗定理将其推广到整数次幂：
(cos θ + i sin θ)ⁿ = cos(nθ) + i sin(nθ)
To find the nth roots of a complex number z = r(cos θ + i sin θ):
求复数 z = r(cos θ + i sin θ) 的 n 次方根：
z^(1/n) = r^(1/n) [ cos( (θ + 2kπ)/n ) + i sin( (θ + 2kπ)/n ) ], k = 0,1,…,n−1
The n distinct nth roots of unity sum to zero and lie on the unit circle.
n 个不同的 n 次单位根之和为零，且均匀分布在单位圆上。
2. Polar Coordinates | 极坐标
In polar coordinates, a curve is defined as r = f(θ). The Cartesian conversion is x = r cos θ, y = r sin θ.
在极坐标中，曲线由 r = f(θ) 定义。转换为直角坐标：x = r cos θ，y = r sin θ。
The area enclosed by a polar curve from θ = α to θ = β is:
极坐标曲线从 θ = α 到 θ = β 所围成的面积为：
A = ½ ∫_α^β r² dθ
The arc length of a polar curve between two angles is:
极坐标曲线在两角之间的弧长为：
s = ∫_α^β √( r² + (dr/dθ)² ) dθ
For tangents parallel or perpendicular to the initial line, examine dy/dθ = 0 or dx/dθ = 0.
要求平行或垂直于极轴的切线，可分析 dy/dθ = 0 或 dx/dθ = 0。
3. Hyperbolic Functions | 双曲函数
The hyperbolic functions are defined using exponential functions:
双曲函数通过指数函数定义：
sinh x = (eˣ − e⁻ˣ)/2, cosh x = (eˣ + e⁻ˣ)/2
tanh x = sinh x / cosh x
Fundamental identity and derivatives:
基本恒等式和导数：
cosh² x − sinh² x = 1
d/dx (sinh x) = cosh x, d/dx (cosh x) = sinh x, d/dx (tanh x) = sech² x
Inverse hyperbolic functions and their logarithmic forms:
反双曲函数及其对数形式：
arsinh x = ln( x + √(x²+1) )
arcosh x = ln( x + √(x²−1) ), x ≥ 1
artanh x = ½ ln( (1+x)/(1−x) ), |x| < 1
4. Matrices and Linear Transformations | 矩阵与线性变换
An eigenvalue λ and eigenvector x of a square matrix A satisfy Ax = λx. The characteristic equation is:
方阵 A 的特征值 λ 和特征向量 x 满足 Ax = λx。特征方程为：
det(A − λI) = 0
Diagonalisation: If a 2×2 or 3×3 matrix has a full set of eigenvectors, then A = PDP⁻¹, where D is a diagonal matrix of eigenvalues and P is the matrix of corresponding eigenvectors.
对角化：如果 2×2 或 3×3 矩阵具有一组完备的特征向量，则 A = PDP⁻¹，其中 D 是特征值对角阵，P 是对应的特征向量矩阵。
Standard linear transformations in 2D are represented by matrices. For example, a rotation through angle θ anticlockwise is:
二维平面上的标准线性变换由矩阵表示。例如，逆时针旋转 θ 角的变换矩阵为：
R = [ [cos θ, −sin θ], [sin θ, cos θ] ]
Reflection in the line y = (tan θ)x has matrix:
关于直线 y = (tan θ)x 的反射矩阵为：
[ [cos 2θ, sin 2θ], [sin 2θ, −cos 2θ] ]
Invariant points and lines are found by solving Ax = x or (A − I)x = 0.
不变点和不变线可通过解 Ax = x 或 (A − I)x = 0 求得。
5. Vectors: Lines, Planes & Distances | 向量：直线、平面与距离
Vector equation of a straight line through point A with direction vector b:
过点 A 且方向向量为 b 的直线向量方程：
r = a + t b, t ∈ ℝ
Vector equation of a plane with normal vector n and containing point A:
以 n 为法向量且过点 A 的平面向量方程：
r · n = a · n
Cartesian form: n₁x + n₂y + n₃z = d.
直角坐标形式：n₁x + n₂y + n₃z = d。
Distance from a point (x₁, y₁, z₁) to the plane ax + by + cz = d is:
点 (x₁, y₁, z₁) 到平面 ax + by + cz = d 的距离为：
|ax₁ + by₁ + cz₁ − d| / √(a² + b² + c²)
The shortest distance between two skew lines r = a + λb and r = c + μd is:
两异面直线 r = a + λb 与 r = c + μd 之间的最短距离为：
| (c − a) · (b × d) | / |b × d|
6. Further Differential Equations | 进一步微分方程
Second-order linear ODEs with constant coefficients: a d²y/dx² + b dy/dx + c y = f(x). First solve the auxiliary equation am² + bm + c = 0.
常系数二阶线性常微分方程：a d²y/dx² + b dy/dx + c y = f(x)。首先解辅助方程 am² + bm + c = 0。
The complementary function y_c depends on the roots:
补函数 y_c 取决于根的情况：
-
Real distinct roots m₁, m₂: y_c = A e^(m₁x) + B e^(m₂x)
实不等根 m₁, m₂：y_c = A e^(m₁x) + B e^(m₂x)
-
Repeated root m: y_c = (A + Bx) e^(mx)
重根 m：y_c = (A + Bx) e^(mx)
-
Complex roots α ± iβ: y_c = e^(αx) (C cos βx + D sin βx)
复根 α ± iβ：y_c = e^(αx) (C cos βx + D sin βx)
Particular integral forms for common f(x):
常见 f(x) 的特解形式：
| f(x) | Trial y_p |
|---|---|
| polynomial p(x) of degree n | general polynomial q(x) of degree n |
| k e^(px) | Ce^(px) (unless p is a root, then multiply by x) |
| k sin ωx or k cos ωx | P cos ωx + Q sin ωx |
For Euler equations a x² d²y/dx² + b x dy/dx + c y = 0, use the substitution x = eᵗ.
对于欧拉方程 a x² d²y/dx² + b x dy/dx + c y = 0，令 x = eᵗ 进行变量代换。
7. Maclaurin and Taylor Series | 麦克劳林与泰勒级数
The Maclaurin series expansion of a function about x = 0 is:
函数在 x = 0 处的麦克劳林级数展开为：
f(x) = f(0) + f'(0)x + f”(0)x²/2! + f”'(0)x³/3! + …
Taylor series about x = a is f(x) = Σ [f⁽ⁿ⁾(a)/n!] (x−a)ⁿ.
泰勒级数在 x = a 处为 f(x) = Σ [f⁽ⁿ⁾(a)/n!] (x−a)ⁿ。
Standard Maclaurin series:
标准麦克劳林级数：
eˣ = 1 + x + x²/2! + x³/3! + … valid for all x
sin x = x − x³/3! + x⁵/5! − … valid for all x
cos x = 1 − x²/2! + x⁴/4! − … valid for all x
ln(1+x) = x − x²/2 + x³/3 − x⁴/4 + … valid for −1 < x ≤ 1
(1+x)ⁿ = 1 + nx + n(n−1)x²/2! + … |x| < 1 (binomial series)
8. Numerical Methods | 数值方法
The Newton-Raphson method approximates a root of f(x) = 0 using:
牛顿-拉夫逊方法通过以下迭代式逼近 f(x) = 0 的根：
x_{n+1} = x_n − f(x_n)/f'(x_n)
For a first-order differential equation dy/dx = g(x,y), Euler’s method with step size h gives:
对于一阶微分方程 dy/dx = g(x,y)，步长为 h 的欧拉方法为：
y_{n+1} = y_n + h g(x_n, y_n)
The improved Euler method (midpoint or Heun’s) refines this using an average of slopes.
改进欧拉法（中点法或休恩法）通过斜率平均值进行修正。
For numerical integration, Simpson’s rule with n strips (n even) is:
对于数值积分，辛普森法则（n 须为偶数）为：
∫_a^b y dx ≈ (h/3)[ y₀ + 4(y₁+y₃+…) + 2(y₂+y₄+…) + yₙ ]
9. Further Mechanics: Centres of Mass & Collisions | 进一步力学：质心与碰撞
Centre of mass of a system of particles: R = (Σ mᵢ rᵢ) / Σ mᵢ.
质点系的质心：R = (Σ mᵢ rᵢ) / Σ mᵢ。
For a uniform lamina, the centre of mass of standard shapes:
对于均匀薄板，标准形状的质心：
-
Triangle: intersection of medians, at distance 1/3 from each base along the median.
三角形：中线交点，沿中线距底边 1/3 处。
-
Sector of a circle radius r, angle 2α: distance from centre = (2r sin α)/(3α).
半径为 r、圆心角为 2α 的扇形：距圆心距离 = (2r sin α)/(3α)。
Newton’s experimental law of restitution for two colliding bodies:
两物体碰撞的牛顿恢复系数实验定律：
e = (v₂ − v₁) / (u₁ − u₂)
where u and v are velocities before and after impact along the line of centres.
其中 u 和 v 分别为碰撞前后沿连心线方向的速度。
Conservation of momentum always applies. For an oblique collision, resolve velocities along the line of centres and perpendicular to it; perpendicular components remain unchanged.
动量始终守恒。对于斜碰撞，将速度沿连心线方向和垂直于其方向分解；垂直分量保持不变。
10. Further Statistics: Continuous Distributions | 进一步统计：连续分布
A continuous random variable X has probability density function (PDF) f(x) satisfying f(x) ≥ 0 and ∫ f(x) dx = 1 over the range.
连续随机变量 X 的概率密度函数 (PDF) f(x) 满足 f(x) ≥ 0 且在取值范围内 ∫ f(x) dx = 1。
Cumulative distribution function: F(x) = P(X ≤ x) = ∫_(-∞)^x f(t) dt.
累积分布函数：F(x) = P(X ≤ x) = ∫_(-∞)^x f(t) dt。
Standard continuous distributions:
标准连续分布：
| Distribution | PDF f(x) | Mean / Variance |
|---|---|---|
| Uniform: X ~ U[a,b] | f(x) = 1/(b−a), a ≤ x ≤ b | (a+b)/2 ; (b−a)²/12 |
| Exponential: X ~ Exp(λ) | f(x) = λ e^(-λx), x ≥ 0 | 1/λ ; 1/λ² |
| Normal: X ~ N(μ,σ²) | f(x) = (1/(σ√(2π))) e^(-(x−μ)²/(2σ²)) | μ ; σ² |
For the normal distribution, standardise using Z = (X − μ)/σ ∼ N(0,1). The central limit theorem states that the mean of a large sample from any distribution is approximately normally distributed.
对于正态分布，用 Z = (X − μ)/σ ~ N(0,1) 标准化。中心极限定理指出，来自任意分布的大样本均值近似服从正态分布。
The Poisson and binomial distributions can be approximated by a normal distribution under certain conditions, using continuity correction.
在一定条件下，泊松分布和二项分布可用正态分布近似，并需进行连续性校正。
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