📚 CIE Year 12 Further Maths Formula & Theorem Quick Reference | 进阶数学公式定理速查手册
This concise handbook collects the essential formulas, identities and theorems required for the CIE Year 12 Further Mathematics course. It covers complex numbers, matrices, roots of polynomials, rational functions, polar coordinates, hyperbolic functions, integration techniques, first‑order differential equations, vectors in three dimensions and proof by induction. Each entry is presented in English followed by its Chinese counterpart, making it ideal for revision and quick checks.
本手册汇集了 CIE 12 年级进阶数学课程中必备的公式、恒等式与定理，涵盖复数、矩阵、多项式的根、有理函数、极坐标、双曲函数、积分技巧、一阶微分方程、三维向量以及数学归纳法。每个条目均以英文和中文配对呈现，是复习与快速查阅的理想资料。
1. Complex Numbers | 复数
A complex number can be written in Cartesian form as z = x + iy, where x, y ∈ ℝ and i² = –1. The real part is Re(z) = x, the imaginary part is Im(z) = y.
复数可以写成笛卡儿形式 z = x + iy，其中 x, y ∈ ℝ，且 i² = –1。实部 Re(z) = x，虚部 Im(z) = y。
The modulus (or absolute value) is |z| = √x² + y². The complex conjugate is z̅ = x – iy or z* = x – iy. A key identity is z z̅ = |z|² = x² + y².
模（绝对值）为 |z| = √x² + y²。共轭复数为 z̅ = x – iy 或 z* = x – iy。一个重要恒等式是 z z̅ = |z|² = x² + y²。
In polar form, z = r(cosθ + i sinθ) = r eiθ, where r = |z| and θ = arg(z). Multiplication and division become simpler: z₁z₂ = r₁r₂ ei(θ₁+θ₂).
极坐标形式中，z = r(cosθ + i sinθ) = r eiθ，其中 r = |z|，θ = arg(z)。乘法与除法变得简单：z₁z₂ = r₁r₂ ei(θ₁+θ₂)。
De Moivre’s theorem states: (cosθ + i sinθ)ⁿ = cos(nθ) + i sin(nθ) for any rational n. It is used to find powers and roots of complex numbers.
棣莫弗定理指出：对于任意有理数 n，(cosθ + i sinθ)ⁿ = cos(nθ) + i sin(nθ)。该定理用于求复数的幂与根。
The nth roots of a complex number are given by z1/n = r1/n [cos((θ+2kπ)/n) + i sin((θ+2kπ)/n)], where k = 0, 1, 2, …, n–1. They lie on a circle in the Argand diagram.
复数的 n 次根由 z1/n = r1/n [cos((θ+2kπ)/n) + i sin((θ+2kπ)/n)] 给出，其中 k = 0, 1, 2, …, n–1。它们分布在阿尔冈图中的一个圆上。
2. Matrices and Determinants | 矩阵与行列式
For a 2×2 matrix A = [[a, b], [c, d]], the determinant is det(A) = ad – bc. If det(A) ≠ 0, the inverse is A⁻¹ = (1/det(A)) [[d, –b], [–c, a]].
对于 2×2 矩阵 A = [[a, b], [c, d]]，行列式为 det(A) = ad – bc。若 det(A) ≠ 0，其逆矩阵为 A⁻¹ = (1/det(A)) [[d, –b], [–c, a]]。
A 3×3 determinant can be expanded using the first row: |A| = a(ei − fh) – b(di − fg) + c(dh − eg). A matrix is singular when its determinant is zero.
3×3 行列式可按第一行展开：|A| = a(ei − fh) – b(di − fg) + c(dh − eg)。当行列式为零时，矩阵是奇异的。
Matrix multiplication is associative but not commutative. The identity matrix I satisfies AI = IA = A. Transformations are often represented by matrices.
矩阵乘法满足结合律但不交换。单位矩阵 I 满足 AI = IA = A。变换常由矩阵表示。
A rotation about the origin through angle θ anticlockwise is given by [[cosθ, –sinθ], [sinθ, cosθ]]. A reflection in the x‑axis is [[1, 0], [0, –1]].
绕原点逆时针旋转 θ 角的变换矩阵为 [[cosθ, –sinθ], [sinθ, cosθ]]。关于 x 轴的反射矩阵为 [[1, 0], [0, –1]]。
The image of a vector v under transformation A is v' = A v. Successive transformations follow right‑to‑left multiplication: B(A v) = (BA)v.
向量 v 在变换 A 下的像为 v' = A v。相继变换的矩阵按从右到左相乘：B(A v) = (BA)v。
3. Roots of Polynomial Equations | 多项式方程根
For a cubic ax³ + bx² + cx + d = 0 (a ≠ 0), let the roots be α, β, γ. Then: 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（a ≠ 0），设根为 α, β, γ。则有：Σα = α+β+γ = –b/a，Σαβ = αβ+βγ+γα = c/a，αβγ = –d/a。
For a quartic ax⁴ + bx³ + cx² + dx + e = 0: Σα = –b/a, Σαβ = c/a, Σαβγ = –d/a, αβγδ = e/a.
对于四次方程 ax⁴ + bx³ + cx² + dx + e = 0：Σα = –b/a，Σαβ = c/a，Σαβγ = –d/a，αβγδ = e/a。
These relationships can be used to find symmetric expressions like Σα² = (Σα)² – 2Σαβ without solving the equation directly.
这些关系可用来求对称表达式，如 Σα² = (Σα)² – 2Σαβ，而不必直接解方程。
Given roots related by a transformation (e.g., squares or reciprocals), a new polynomial can be constructed using sums and products of the transformed roots.
若根之间通过某种变换联系（如平方或倒数），可利用变换后根的和与积构造出新的多项式。
4. Rational Functions and Partial Fractions | 有理函数与部分分式
A rational function is of the form f(x) = P(x)/Q(x) where P and Q are polynomials. Its vertical asymptotes occur where Q(x)=0 and P(x)≠0.
有理函数形如 f(x) = P(x)/Q(x)，其中 P, Q 为多项式。垂直渐近线出现在 Q(x)=0 且 P(x)≠0 处。
egree of Q) can be rewritten as a polynomial plus a proper fraction by algebraic division.
假分式（分子次数 ≥ 分母次数）可通过代数除法改写为多项式加真分式。
Partial fractions decompositions for proper rational functions: Distinct linear factors: A/(x–p) + B/(x–q). Repeated linear factor: A/(x–p) + B/(x–p)². Irreducible quadratic: (Ax+B)/(x²+cx+d).
真分式的部分分式分解：不同线性因式：A/(x–p) + B/(x–q)。重复线性因式：A/(x–p) + B/(x–p)²。不可约二次因式：(Ax+B)/(x²+cx+d)。
Partial fractions are especially useful for integrating rational functions and for series expansions.
部分分式在有理函数积分和级数展开中特别有用。
5. Polar Coordinates | 极坐标
In the polar coordinate system, a point is given by (r, θ), where r is the distance from the origin and θ is the angle measured anticlockwise from the positive x‑axis.
在极坐标系中，一点表示为 (r, θ)，其中 r 是到原点的距离，θ 是从正 x 轴逆时针测量的角度。
Conversions to Cartesian coordinates: x = r cosθ, y = r sinθ. Conversely, r = √x² + y² and tanθ = y/x (paying attention to the quadrant).
与笛卡儿坐标的转换：x = r cosθ，y = r sinθ。反之，r = √x² + y²，tanθ = y/x（注意象限）。
The area enclosed by a polar curve r = f(θ) from θ = α to θ = β is A = ∫ (1/2) r² dθ, with limits α and β.
由极坐标曲线 r = f(θ) 从 θ = α 到 θ = β 围成的面积为 A = ∫ (1/2) r² dθ，积分限为 α 与 β。
The slope of the tangent to a polar curve is found using dy/dx = (dy/dθ)/(dx/dθ), with x = r cosθ and y = r sinθ.
极坐标曲线切线的斜率可用 dy/dx = (dy/dθ)/(dx/dθ) 求得，其中 x = r cosθ，y = r sinθ。
Common polar curves include the cardioid r = a(1 + cosθ), the circle r = a, and the rose curves r = a cos(nθ).
常见极坐标曲线包括心形线 r = a(1 + cosθ)、圆 r = a 和玫瑰线 r = a cos(nθ)。
6. Hyperbolic Functions | 双曲函数
The hyperbolic functions are defined in terms of exponentials: sinh x = (eˣ – e⁻ˣ
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