📚 Year 12 Edexcel Further Maths: Quick Reference Formula & Theorem Handbook | 12年级爱德思进阶数学：公式定理速查手册
This handbook distils the essential formulas and theorems covered in Year 12 Edexcel Further Mathematics (Core Pure 1). Use it as a rapid revision tool and a checklist to ensure no key result is forgotten before the exam.
本手册浓缩了12年级爱德思进阶数学（核心纯数1）中的核心公式与定理。可将其作为快速复习工具和检查清单，确保考前没有遗漏任何关键结论。
1. Complex Numbers: Forms and Operations | 复数：形式与运算
Cartesian form: A complex number is written as z = x + iy, where x = Re(z) and y = Im(z) are real numbers, and i² = −1.
笛卡尔形式：复数写作 z = x + iy，其中 x = Re(z) 和 y = Im(z) 为实数，且 i² = −1。
Complex conjugate: z* = x − iy. Notice that z z* = x² + y² = |z|², a real number.
共轭复数：z* = x − iy。注意 z z* = x² + y² = |z|²，是一个实数。
Modulus: The distance from the origin is |z| = √(x² + y²).
模：到原点的距离为 |z| = √(x² + y²)。
Argument: arg(z) = θ, measured from the positive real axis, such that tan θ = y/x. The quadrant must be considered to determine the correct angle.
辐角：arg(z) = θ，从正实轴量起，满足 tan θ = y/x。必须考虑象限以确定正确的角度。
Modulus-argument form: z = r(cosθ + i sinθ), where r = |z| and θ = arg(z). This is also called polar form.
模-辐角形式：z = r(cosθ + i sinθ)，其中 r = |z| 且 θ = arg(z)。这又称为极坐标形式。
Multiplication and division in polar form:
z₁z₂ = r₁r₂ [cos(θ₁+θ₂) + i sin(θ₁+θ₂)]
极坐标形式下的乘法与除法：
z₁z₂ = r₁r₂ [cos(θ₁+θ₂) + i sin(θ₁+θ₂)]
Division: z₁/z₂ = (r₁/r₂)[cos(θ₁−θ₂) + i sin(θ₁−θ₂)], provided z₂ ≠ 0.
除法：z₁/z₂ = (r₁/r₂)[cos(θ₁−θ₂) + i sin(θ₁−θ₂)]，其中 z₂ ≠ 0。
Euler’s relation: eiθ = cosθ + i sinθ, thus we can write z = r eiθ.
欧拉公式：eiθ = cosθ + i sinθ，因此可记作 z = r eiθ。
2. De Moivre’s Theorem and Trigonometric Applications | 棣莫弗定理与三角应用
De Moivre’s theorem: For any integer n, (cosθ + i sinθ)ⁿ = cos(nθ) + i sin(nθ).
棣莫弗定理：对于任意整数 n，(cosθ + i sinθ)ⁿ = cos(nθ) + i sin(nθ)。
This result can also be extended to rational powers when finding roots, but care must be taken with multiple values.
该结果可在求根时推广到有理指数，但需注意多值性。
Deriving trigonometric identities: Expand (cosθ + i sinθ)ⁿ using the binomial theorem and equate real and imaginary parts to obtain expressions for cos(nθ) and sin(nθ) in terms of powers of cosθ and sinθ.
推导三角恒等式：用二项式定理展开 (cosθ + i sinθ)ⁿ，并令实部与虚部分别相等，即可得到用 cosθ 和 sinθ 的幂表示的 cos(nθ) 与 sin(nθ) 表达式。
Finding n-th roots: The n distinct n-th roots of a complex number z = r(cosθ + i sinθ) are given by
zk = r1/n [cos((θ+2πk)/n) + i sin((θ+2πk)/n)], for k = 0, 1, …, n−1.
求 n 次方根：复数 z = r(cosθ + i sinθ) 的 n 个互异 n 次方根为
zk = r1/n [cos((θ+2πk)/n) + i sin((θ+2πk)/n)], k = 0, 1, …, n−1.
The roots lie on a circle of radius r1/n and are equally spaced by an angle 2π/n.
这些根分布在半径为 r1/n 的圆上，且以角度 2π/n 等间隔排列。
3. Roots of Polynomials | 多项式的根
Quadratic equation ax² + bx + c = 0 with roots α, β:
α+β = −b/a, αβ = c/a.
二次方程 ax² + bx + c = 0 的根为 α, β：
α+β = −b/a，αβ = c/a。
Cubic equation ax³ + bx² + cx + d = 0 with roots α, β, γ:
Σα = α+β+γ = −b/a,
Σαβ = αβ+βγ+γα = c/a,
αβγ = −d/a.
三次方程 ax³ + bx² + cx + d = 0 的三个根为 α, β, γ：
Σα = α+β+γ = −b/a，
Σαβ = αβ+βγ+γα = c/a，
αβγ = −d/a。
Quartic equation ax⁴ + bx³ + cx² + dx + e = 0 with roots α, β, γ, δ:
Σα = −b/a, Σαβ = c/a, Σαβγ = −d/a, αβγδ = e/a.
四次方程 ax⁴ + bx³ + cx² + dx + e = 0 的四个根为 α, β, γ, δ：
Σα = −b/a， Σαβ = c/a， Σαβγ = −d/a， αβγδ = e/a。
Forming new equations: To find an equation whose roots are related to the original roots (e.g. squares, reciprocals), use substitution y = f(x) or express the required symmetric sums in terms of the original sums.
构造新方程：若要寻找根与原方程的根有特定关系的新方程（如平方、倒数），可采用代换 y = f(x) 或用原方程的对称和表达所需对称和。
4. Matrices – Determinants and Inverses | 矩阵：行列式与逆矩阵
2 × 2 determinant: For A = [a b; c d], det(A) = ad − bc.
2 × 2 行列式：对于 A = [a b; c d]，det(A) = ad − bc。
2 × 2 inverse: If det(A) ≠ 0, then A⁻¹ = (1/det A)[d −b; −c a].
2 × 2 逆矩阵：若 det(A) ≠ 0，则 A⁻¹ = (1/det A)[d −b; −
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