Pre-U Edexcel Further Maths: Formula & Theorem Quick Reference Handbook | Edexcel 进阶数学公式定理速查手册

📚 Pre-U Edexcel Further Maths: Formula & Theorem Quick Reference Handbook | Edexcel 进阶数学公式定理速查手册

This quick reference handbook distils the essential formulae, theorems and identities from the Edexcel Pre-U Further Mathematics syllabus. Use it to consolidate your revision, reinforce problem-solving pattern recognition, and ensure you can recall every critical result under timed conditions.

本速查手册提炼了 Edexcel Pre-U 进阶数学大纲中的核心公式、定理与恒等式。用它来巩固复习、强化解题模式识别，并确保在限时条件下能准确回忆起每一个关键结果。

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1. Complex Numbers | 复数

A complex number is written z = x + iy, where i² = −1. Its complex conjugate is z̅ = x − iy and modulus |z| = √(x² + y²). The argument θ satisfies tan θ = y/x, with quadrant adjustment.

复数写作 z = x + iy，其中 i² = −1。其共轭为 z̅ = x − iy，模 |z| = √(x² + y²)。辐角 θ 满足 tan θ = y/x，并需调整象限。

Euler’s relation links exponentials and trigonometric functions:

欧拉关系式将指数函数与三角函数联系起来：

De Moivre’s theorem for integer powers:

(cos θ + i sin θ)ⁿ = cos nθ + i sin nθ

棣莫弗定理（整数次幂）：

(cos θ + i sin θ)ⁿ = cos nθ + i sin nθ

The nth roots of unity are z = e^(2πik/n) for k = 0, 1, …, n−1. They form a regular n-gon on the Argand diagram. The sum of all roots of unity is zero.

n 次单位根为 z = e^(2πik/n)，k = 0, 1, …, n−1。它们在阿尔冈图上构成正 n 边形。所有单位根之和为零。

If z = reⁱᶿ, then zⁿ = rⁿ eⁱⁿᶿ. Multiplication rotates by adding arguments and multiplies moduli.

若 z = reⁱᶿ，则 zⁿ = rⁿ eⁱⁿᶿ。乘法通过辐角相加、模相乘来实现旋转。

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2. Matrices | 矩阵

For a 2×2 matrix A =

a	b
c	d

,
the determinant is det(A) = ad − bc. The inverse, when it exists, is
A⁻¹ = (1/det A) ^*_*

d	−b
−c	a

(the displayed element order must be preserved).

对二阶矩阵 A =

a	b
c	d

，行列式为 det(A) = ad − bc。逆矩阵（若存在）为
A⁻¹ = (1/det A) 乘

d	−b
−c	a

(按此位置排列)。

Eigenvalues λ satisfy det(A − λI) = 0. For a 2×2 matrix this yields a quadratic characteristic equation. Corresponding eigenvectors x are non‑zero vectors such that (A − λI)x = 0.

特征值 λ 满足 det(A − λI) = 0。对二阶矩阵产生二次特征方程。相应的特征向量 x 为非零向量，满足 (A − λI)x = 0。

A transformation matrix represents a linear mapping. The columns are the images of the standard basis vectors (1,0)ᵀ and (0,1)ᵀ. Composite transformations correspond to matrix multiplication in the correct order.

变换矩阵表示线性映射。其列分别为标准基向量 (1,0)ᵀ 与 (0,1)ᵀ 的像。复合变换对应正确次序的矩阵乘法。

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3. Vectors | 向量

The scalar (dot) product of vectors a and b is a·b = |a||b| cos θ. In Cartesian form, if a = a₁i + a₂j + a₃k, then a·b = a₁b₁ + a₂b₂ + a₃b₃. The angle between vectors is given by cos θ = (a·b)/(|a||b|).

向量 a 与 b 的标量积（点积）为 a·b = |a||b| cos θ。在直角坐标下，若 a = a₁i + a₂j + a₃k，则 a·b = a₁b₁ + a₂b₂ + a₃b₃。向量夹角由 cos θ = (a·b)/(|a||b|) 确定。

The 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.

向量积（叉积）a × b 产生垂直于 a 和 b 的向量，大小为 |a||b| sin θ。其分量式为
a × b = (a₂b₃ − a₃b₂)i + (a₃b₁ − a₁b₃)j + (a₁b₂ − a₂b₁)k.

A straight line can be written in vector form r = a + λd, where a is a point on the line and d is the direction vector. A plane is given by r·n = p, where n is the normal vector, or r = a + λu + μv.

直线可用向量方程 r = a + λd 表示，其中 a 为直线上一点，d 为方向向量。平面可由 r·n = p（n 为法向量）或 r = a + λu + μv 给出。

The shortest distance from a point P with position vector p to the line r = a + λd is |(p − a) × d| / |d|. For a plane r·n = p, the perpendicular distance from point Q is |(q·n − p)| / |n|.

点 P（位矢 p）到直线 r =

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