Year 12 CCEA Further Mathematics: Quick Formula & Theorem Handbook | Year 12 CCEA 进阶数学：公式定理速查手册

📚 Year 12 CCEA Further Mathematics: Quick Formula & Theorem Handbook | Year 12 CCEA 进阶数学：公式定理速查手册

This handbook provides a concise, at-a-glance reference for the key formulas and theorems encountered in Year 12 CCEA Further Mathematics. It covers the Pure Mathematics topics essential for AS-level study, structured for rapid revision and easy retrieval. Each section presents the core identities, techniques, and standard results in clear format, with English followed by Chinese explanations.

本手册为 Year 12 CCEA 进阶数学的关键公式与定理提供简明速查参考，涵盖 AS 阶段必备的纯数主题，结构清晰，便于快速复习和查阅。每个小节都以英文说明紧跟中文阐释，突出重点。

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1. Complex Numbers | 复数

A complex number has the form z = a + bi, where a and b are real numbers and i is the imaginary unit with the property i² = −1. The real part is Re(z) = a, and the imaginary part is Im(z) = b.

复数具有形式 z = a + bi，其中 a 和 b 是实数，i 为虚数单位，满足 i² = −1。实部为 Re(z) = a，虚部为 Im(z) = b。

The complex conjugate of z is z* = a − bi. Key properties include: (z*)* = z, z + z* = 2a, z − z* = 2bi, and z z* = a² + b² = |z|².

共轭复数 记为 z* = a − bi。重要性质有：(z*)* = z, z + z* = 2a, z − z* = 2bi, z z* = a² + b² = |z|²。

The modulus of z is |z| = √(a² + b²). The argument arg(z) = θ is the angle satisfying tanθ = b/a, taking account of the quadrant in which z lies.

复数 z 的模为 |z| = √(a² + b²)。辐角 arg(z) = θ 是满足 tanθ = b/a 的角，需依据所在象限确定正确值。

|z| = √(a² + b²), θ = arg(z)

Polar form: z = r(cosθ + i sinθ), where r = |z| and θ = arg(z). Multiplication in polar form is straightforward: z₁z₂ = r₁r₂[cos(θ₁ + θ₂) + i sin(θ₁ + θ₂)].

极坐标形式：z = r(cosθ + i sinθ)，其中 r = |z|，θ = arg(z)。极坐标下的乘法简便：z₁z₂ = r₁r₂[cos(θ₁ + θ₂) + i sin(θ₁ + θ₂)]。

De Moivre’s Theorem: For any integer n, (cosθ + i sinθ)ⁿ = cos(nθ) + i sin(nθ). It can also be extended to rational exponents to find nth roots of a complex number.

棣莫弗定理：对任意整数 n，(cosθ + i sinθ)ⁿ = cos(nθ) + i sin(nθ)。该定理可推广到有理指数，用于求复数的 n 次方根。

(cosθ + i sinθ)ⁿ = cos(nθ) + i sin(nθ)

Euler’s formula: eⁱθ = cosθ + i sinθ, which gives the compact exponential form z = reⁱθ.

欧拉公式：eⁱθ = cosθ + i sinθ，由此可得简洁的指数形式 z = reⁱθ。

eⁱθ = cosθ + i sinθ, z = reⁱθ

nth roots of unity: The solutions to zⁿ = 1 are z = e^(2πki/n) = cos(2πk/n) + i sin(2πk/n), where k = 0, 1, …, n−1.

n 次单位根：方程 zⁿ = 1 的解为 z = e^(2πki/n) = cos(2πk/n) + i sin(2πk/n)，其中 k = 0, 1, …, n−1。

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2. Roots of Polynomial Equations | 多项式方程的根

For a quadratic equation ax² + bx + c = 0 with roots α and β, the sum of roots is α + β = −b/a and the product is αβ = c/a.

对于二次方程 ax² + bx + c = 0，设两根为 α 和 β，则根的和 α + β = −b/a，根的积 αβ = c/a。

α + β = −b/a, αβ = c/a

For a cubic equation ax³ + bx² + cx + d = 0 with roots α, β, γ:

Sum: α + β + γ = −b/a

Sum of products taken two at a time: αβ + αγ + βγ = c/a

Product: αβγ = −d/a.

对于三次方程 ax³ + bx² + cx + d = 0，设根为 α, β, γ：

根的和：α + β + γ = −b/a

两根乘积之和：αβ + αγ + βγ = c/a

根的积：αβγ = −d/a。

For a quartic ax⁴ + bx³ + cx² + dx + e = 0 with roots α, β, γ, δ, analogous relations hold: sum = −b/a, sum of pairwise products = c/a, sum of triple products = −d/a, product = e/a.

对于四次方程 ax⁴ + bx³ + cx² + dx + e = 0，根为 α, β, γ, δ，有类似关系：和 = −b/a，两两积之和 = c/a，三三积之和 = −d/a，四根之积 = e/a。

These symmetric sums are invaluable when forming new equations whose roots are related to those of a given polynomial, e.g. by a linear substitution or reciprocal transformation.

这些对称和在构造新方程时十分有用，例如通过线性代换或倒数变换，使新方程的根与原方程的根有特定关系。

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3. Matrices & Determinants | 矩阵与行列式

For a 2 × 2 matrix A =
( a b
c d ),
the determinant is det(A) = ad − bc. The matrix is singular if det(A) = 0.

对于 2 × 2 矩阵 A =
( a b
c d )，
行列式为 det(A) = ad − bc。若 det(A) = 0，矩阵为奇异矩阵。

det(A) = ad − bc

The inverse of a non-singular 2 × 2 matrix is given by

A⁻¹ = (1/(ad − bc))
( d −b
−c a ).

非奇异 2 × 2 矩阵的逆矩阵为

A⁻¹ = (1/(ad − bc))
( d −b
−c a )。

Matrix multiplication is associative but not commutative: in general, AB ≠ BA. The transpose satisfies (AB)ᵀ = Bᵀ Aᵀ, where ᵀ denotes transpose.

矩阵乘法满足结合律但不满足交换律：一般地，AB ≠ BA。转置满足 (AB)ᵀ = Bᵀ Aᵀ，其中 ᵀ 表示转置。

A system of linear equations can be solved using the inverse matrix

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