📚 WJEC Year 12 Further Maths: Formula & Theorems Quick Reference | WJEC 进阶数学公式定理速查手册
This quick reference guide brings together the essential formulas and theorems for the WJEC Year 12 Further Mathematics course. It is designed as a concise revision aid that covers pure mathematics topics such as complex numbers, matrices, polynomial roots, series, integration and differential equations. Use it to strengthen your grasp of key concepts and to prepare efficiently for assessments.
本速查手册汇集了 WJEC 12 年级进阶数学课程的核心公式和定理。它是一份精炼的复习工具，涵盖复数、矩阵、多项式根、级数、积分和微分方程等纯数学主题。使用它来巩固对关键概念的理解，高效备考。
1. Complex Numbers | 复数
A complex number can be written in Cartesian form as z = x + iy, where x is the real part and y is the imaginary part. The imaginary unit satisfies i² = −1.
复数可用笛卡尔形式表示为 z = x + iy，其中 x 为实部，y 为虚部。虚数单位满足 i² = −1。
i² = −1, z = x + iy
The modulus (or absolute value) of z is |z| = √(x² + y²). The argument arg(z) = θ is defined by tan θ = y/x, taking care to place θ in the correct quadrant using an Argand diagram.
z 的模（绝对值）为 |z| = √(x² + y²)。辐角 arg(z) = θ 由 tan θ = y/x 定义，需借助阿伽特图确定正确的象限。
|z| = √(x² + y²), θ = arg(z) with −π < θ ≤ π
The polar form uses the modulus and argument: z = r(cos θ + i sin θ), where r = |z|. Euler’s relation gives the compact notation z = r eiθ.
极坐标形式使用模和辐角：z = r(cos θ + i sin θ)，其中 r = |z|。欧拉公式给出紧凑记法 z = r eiθ。
z = r(cos θ + i sin θ) = r eiθ
De Moivre’s theorem: (cos θ + i sin θ)ⁿ = cos(nθ) + i sin(nθ). This is used to find powers and multiple-angle identities.
棣莫弗定理：(cos θ + i sin θ)ⁿ = cos(nθ) + i sin(nθ)。该定理用于求幂及多倍角恒等式。
The nth roots of unity satisfy zⁿ = 1 and are given by zₖ = e2πik/n for k = 0, 1, …, n−1. They lie equally spaced on the unit circle.
n 次单位根满足 zⁿ = 1，表达式为 zₖ = e2πik/n（k = 0, 1, …, n−1）。它们均匀分布在单位圆上。
Loci in the complex plane: |z − a| = r represents a circle of radius r centred at a; arg(z − a) = α represents a half-line from a making angle α with the positive real axis.
复平面上的轨迹：|z − a| = r 表示以 a 为圆心、半径为 r 的圆；arg(z − a) = α 表示从 a 出发，与正实轴夹角为 α 的半直线。
2. Matrices | 矩阵
For a 2×2 matrix A = [[a, b], [c, d]], the determinant is det(A) = ad − bc. If det(A) = 0 the matrix is singular and has no inverse.
对于 2×2 矩阵 A = [[a, b], [c, d]]，行列式为 det(A) = ad − bc。若 det(A) = 0，则矩阵为奇异矩阵，不可逆。
det(A) = |A| = ad − bc
The inverse of a non-singular 2×2 matrix is A⁻¹ = 1/(ad−bc) [[d, −b], [−c, a]]. For a 3×3 matrix, the inverse can be found using A⁻¹ = adj(A)/det(A), where adj(A) is the adjugate (transpose of the cofactor matrix).
非奇异 2×2 矩阵的逆为 A⁻¹ = 1/(ad−bc) [[d, −b], [−c, a]]。对于 3×3 矩阵，可利用 A⁻¹ = adj(A)/det(A) 求逆，其中 adj(A) 是伴随矩阵（余子式矩阵的转置）。
A⁻¹ = adj(A) / |A|
A system of linear equations AX = B has a unique solution X = A⁻¹B if A is invertible. Row operations can also be used with an augmented matrix [A | I] to find the inverse.
线性方程组 AX = B 若 A 可逆，则有唯一解 X = A⁻¹B。也可利用增广矩阵 [A | I] 通过行变换求逆。
Common linear transformations in the plane include rotations by θ: [[cos θ, −sin θ], [sin θ, cos θ]], reflections in the x-axis: [[1, 0], [0, −1]], and stretches by scale factor k: [[k, 0], [0, 1]].
平面中常见的线性变换包括旋转 θ 角：[[cos θ, −sin θ], [sin θ, cos θ]]；关于 x 轴反射：[[1, 0], [0, −1]]；以及以比例因子 k 的伸缩：[[k, 0], [0, 1]]。
3. Roots of Polynomials | 多项式根
For a quadratic equation ax² + bx + c = 0 with roots α and β, the sum and product are:
对于具有根 α 和 β 的二次方程 ax² + bx + c = 0，和与积为：
α + β = −b/a, αβ = c/a
For a cubic ax³ + bx² + cx + d = 0 with roots α, β, γ:
对于具有根 α, β, γ 的三次方程 ax³ + bx² + cx + d = 0：
Σα = α+β+γ = −b/a, Σαβ = αβ+βγ+γα = c/a, αβγ = −d/a
For a quartic ax⁴ + bx³ + cx² + dx + e = 0 with roots α, β, γ, δ:
对于具有根 α, β, γ, δ 的四次方程 ax⁴ + bx³ + cx² + dx + e = 0：
Σα = −b/a, Σαβ = c/a, Σαβγ = −d/a, αβγδ = e/a
Useful symmetrical sums include Σα² = (Σα)² − 2Σαβ and Σα²β = Σα·Σαβ − 3αβγ (for cubics). These allow evaluation of expressions without finding roots explicitly.
实用的对称求和包括 Σα² = (Σα)² − 2Σαβ 以及（对于三次方程）Σα²β = Σα·Σαβ − 3αβγ。此类恒等式无需显式求出根即可计算表达式。
To find a new polynomial whose roots are a simple transformation of the original roots (e.g. α + k, kα, α²), substitute x = (y−k) or x = y/k etc. into the original polynomial.
若要求一个根为原根简单变换（例如 α + k, kα, α²）的新多项式，可将 x = (y−k) 或 x =
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