📚 Year 12 CCEA Further Mathematics: Formula & Theorem Quick Reference Guide | CCEA 进阶数学：公式定理速查手册
This quick reference guide collects the essential formulas and theorems required for the CCEA AS Further Pure Mathematics (FP1) units. It covers algebra, complex numbers, matrices, vectors, series, induction, hyperbolic functions, calculus techniques and Maclaurin series, presented in a concise bilingual format for rapid revision.
本速查手册汇集了 CCEA AS 进阶纯数（FP1）单元所需的核心公式与定理，涵盖代数、复数、矩阵、向量、级数、归纳法、双曲函数、微积分技巧与麦克劳林级数，以简洁的中英双语呈现，方便快速查阅复习。
1. Algebra and Functions | 代数与函数
For a polynomial f(x), the remainder when divided by (x − a) is f(a).
Remainder Theorem: f(a) = remainder
多项式 f(x) 除以 (x − a) 的余数为 f(a)。
If f(a) = 0, then (x − a) is a factor of f(x).
Factor Theorem: (x − a) is a factor ⇔ f(a) = 0
若 f(a) = 0，则 (x − a) 是 f(x) 的一个因式。
For a quadratic equation ax² + bx + c = 0 with roots α and β:
α + β = −b/a, αβ = c/a
对于一元二次方程 ax² + bx + c = 0，其根为 α, β，则 α + β = −b/a，αβ = c/a。
For a cubic equation ax³ + bx² + cx + d = 0 with roots α, β, γ:
α + β + γ = −b/a, αβ + αγ + βγ = c/a, αβγ = −d/a
对于一元三次方程 ax³ + bx² + cx + d = 0，其根为 α, β, γ，则有 α + β + γ = −b/a，αβ + αγ + βγ = c/a，αβγ = −d/a。
Partial fractions for a rational function with distinct linear factors in the denominator:
f(x)/((x+a)(x+b)) = A/(x+a) + B/(x+b)
分母为不同线性因式的有理函数可拆分为部分分式：f(x)/((x+a)(x+b)) = A/(x+a) + B/(x+b)。
2. Complex Numbers | 复数
A complex number has the form z = x + iy, where i² = −1. The real part is x, the imaginary part is y.
z = x + iy, i² = −1
复数形式为 z = x + iy，其中 i² = −1。实部为 x，虚部为 y。
The complex conjugate is z* = x − iy. The modulus |z| = √(x² + y²). The argument arg z = θ, where tan θ = y/x, taking the principal value in (−π, π].
|z| = √(x² + y²), arg z = θ, tan θ = y/x
共轭复数为 z* = x − iy。模 |z| = √(x² + y²)。辐角 arg z = θ，其中 tan θ = y/x，取主值范围 (−π, π]。
Polar form: z = r(cos θ + i sin θ), where r = |z|, θ = arg z.
z = r(cos θ + i sin θ)
极坐标形式：z = r(cos θ + i sin θ)，其中 r = |z|，θ = arg z。
de Moivre’s theorem: (cos θ + i sin θ)ⁿ = cos nθ + i sin nθ, for any integer n.
(cos θ + i sin θ)ⁿ = cos nθ + i sin nθ
棣莫弗定理：对于任何整数 n，(cos θ + i sin θ)ⁿ = cos nθ + i sin nθ。
The n distinct nth roots of a complex number z = r(cos θ + i sin θ) are given by:
z^(1/n) = r^(1/n) [ cos((θ+2kπ)/n) + i sin((θ+2kπ)/n) ], k = 0, 1, …, n−1
复数 z = r(cos θ + i sin θ) 的 n 个不同 n 次方根为：z^(1/n) = r^(1/n) [ cos((θ+2kπ)/n) + i sin((θ+2kπ)/n) ]，其中 k = 0, 1, …, n−1。
3. Matrices | 矩阵
For 2×2 matrices A and B, matrix multiplication is defined as well as addition and scalar multiplication. The transpose Aᵀ is obtained by swapping rows and columns.
对于 2×2 矩阵可进行乘法、加法和数乘。转置矩阵 Aᵀ 由行与列互换得到。
The determinant of a 2×2 matrix M = [[a, b], [c, d]] is det M = ad − bc.
det(M) = ad − bc
2×2 矩阵 M = [[a, b], [c, d]] 的行列式为 det M = ad − bc。
The inverse of a 2×2 matrix, provided det M ≠ 0, is M⁻¹ = (1/det M) [[d, −b], [−c, a]].
M⁻¹ = (1/(ad−bc)) [[d, −b], [−c, a]]
若 det M ≠ 0，则 2×2 矩阵的逆矩阵为 M⁻¹ = (1/(ad−bc)) [[d, −b], [−c, a]]。
For a 3×3 matrix, the determinant can be calculated by expansion along any row or column. The inverse can be found using the adjugate (adjoint) matrix: A⁻¹ = (1/det A) adj A.
A⁻¹ = (1/det A) adj A
对于 3×3 矩阵，行列式可按任一行（或列）展开计算。逆矩阵可用伴随矩阵求得：A⁻¹ = (1/det A) adj A。
A system of linear equations can be written in matrix form Ax = b. If A is invertible, the unique solution is x = A⁻¹b.
Ax = b ⇒ x = A⁻¹b (when A⁻¹ exists)
线性方程组可写成矩阵形式 Ax = b。若 A 可逆，则唯一解为 x = A⁻¹b。
4. Vectors | 向量
The scalar (dot) product of two vectors a and b is a · b = |a||b| cos θ. In component form, if a = a₁i + a₂j + a₃k and b = b₁i + b₂j + b₃k, then a · b = a₁b₁ + a₂b₂ + a₃b₃.
a · b = |a||b| cos θ = a₁b₁ + a₂b₂ + a₃b₃
向量的数量积（点积）为 a · b = |a||b| cos θ。若 a = a₁i + a₂j + a₃k，b = b₁i + b₂j + b₃k，则 a · b = a₁b₁ + a₂b₂ + a₃b₃。
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 − (a₁b₃ − a₃b₁)j + (a₁b₂ − a₂b₁)k
向量的向量积（叉积）a × b 是一个同时垂直于 a 和 b 的向量，其模为 |a||b| sin θ。分量形式如上。
The vector equation of a straight line passing through point A with position vector a and parallel to direction b is r = a + λ b, λ ∈ ℝ.
r = a + λ b
过定点 A（位置向量为 a）且平行于方向 b 的直线向量方程为 r = a + λ b，λ ∈ ℝ。
The vector equation of a plane with normal vector n and passing through point with position vector a is r · n = a · n, or r · n = d where d = a · n.
r · n = d
法向量为 n 且过位置向量为 a 的点的平面方程为 r · n = a · n，常写作 r · n = d。
5. Sequences and Series: Standard Summations | 数列与级数：标准求和公式
Sum of the first n natural numbers:
Σ_{r=1}^{n} r = ½ n(n+1)
前 n 个自然数的和：Σ_{r=1}^{n} r = ½ n(n+1)
Sum of the squares of the first n natural numbers:
Σ_{r=1}^{n} r² = ⅙ n(n+1)(2n+1)
前 n 个自然数的平方和：Σ_{r=1}^{n} r² = ⅙ n(n+1)(2n+1)
Sum of the cubes of the first n natural numbers:
Σ_{r=1}^{n} r³ = ¼ n²(n+1)²
前 n 个自然数的立方和：Σ_{r=1}^{n} r³ = ¼ n²(n+1)²
These standard results can be combined and manipulated to sum polynomials in r. For example, Σ (ar² + br + c) = a Σ r² + b Σ r + c n.
这些标准结果可组合处理 r 的多项式求和，如 Σ (ar² + br + c) = a Σ r² + b Σ r + c n。
6. Proof by Induction | 数学归纳法
The method of proof by induction consists of four steps: (1) Prove the statement is true for the base case (usually n = 1). (2) Assume the statement is true for n = k. (3) Show that the statement is then true for n = k + 1. (4) Conclude that the statement holds for all positive integers n by mathematical induction.
数学归纳法包含四个步骤：(1) 验证基础情形（通常 n = 1）命题成立。(2) 假设 n = k 时命题成立。(3) 证明 n = k + 1 时命题也成立。(4) 由归纳法断言命题对所有正整数 n 成立。
This technique is used to prove summation formulas, inequalities, divisibility results and sequences defined recursively.
此法常用于证明求和公式、不等式、整除性质以及递归定义的数列公式。
7. Summation by Method of Differences | 差分法求和
If a general term can be expressed as uᵣ = f(r) − f(r+1), then the sum telescopes:
Σ_{r=1}^{n} uᵣ = f(1) − f(n+1)
若通项可写为 uᵣ = f(r) − f(r+1)，则求和可裂项相消：Σ_{r=1}^{n} uᵣ = f(1) − f(n+1)。
A classic application is Σ_{r=1}^{n} 1/(r(r+1)). Using partial fractions, 1/(r(r+1)) = 1/r − 1/(r+1). Hence the sum simplifies to 1 − 1/(n+1).
Σ 1/(r(r+1)) = 1 − 1/(n+1)
典型应用为 Σ 1/(r(r+1))。利用部分分式 1/(r(r+1)) = 1/r − 1/(r+1)，求和即得 1 − 1/(n+1)。
The method can be extended to expressions like 1/((r+1)(r+2)) by adjusting the form to f(r) = 1/(r+1).
此方法可推广至形如 1/((r+1)(r+2)) 的式子，只需适当选取 f(r)。
8. Hyperbolic Functions | 双曲函数
Definitions:
sinh x = (eˣ − e⁻ˣ)/2, cosh x = (eˣ + e⁻ˣ)/2, tanh x = sinh x / cosh x
定义：sinh x = (eˣ − e⁻ˣ)/2，cosh x = (eˣ + e⁻ˣ)/2，tanh x = sinh x / cosh x。
Reciprocal functions: coth x = 1/tanh x, sech x = 1/cosh x, cosech x = 1/sinh x.
倒数函数：coth x = 1/tanh x，sech x = 1/cosh x，cosech x = 1/sinh x。
Key identity:
cosh² x − sinh² x = 1
基本恒等式：cosh² x − sinh² x = 1。
Osborn’s rule: to obtain a hyperbolic identity from a trigonometric one, replace cos with cosh, sin with i sinh, and change the sign of any product or implied product of two sines. In practice, this often means that any term containing sin² becomes −sinh².
奥斯本规则：从三角恒等式得到双曲恒等式时，将 cos 换为 cosh，sin 换为 i sinh，并改变任何含有两个正弦乘积（含隐含乘积）的项的符号。实用中常表现为包含 sin² 的项变为 −sinh²。
Other useful identities: sinh 2x = 2 sinh x cosh x, cosh 2x = cosh² x + sinh² x = 2 cosh² x − 1 = 1 + 2 sinh² x.
sinh 2x = 2 sinh x cosh x, cosh 2x = cosh² x + sinh² x
其他常用恒等式：sinh 2x = 2 sinh x cosh x，cosh 2x = cosh² x + sinh² x = 2 cosh² x − 1 = 1 + 2 sinh² x。
9. Differentiation | 微分
Derivatives of hyperbolic functions:
d/dx (sinh x) = cosh x, d/dx (cosh x) = sinh x, d/dx (tanh x) = sech² x
双曲函数的导数：d/dx (sinh x) = cosh x，d/dx (cosh x) = sinh x，d/dx (tanh x) = sech² x。
Derivatives of inverse hyperbolic functions:
d/dx (arsinh x) = 1/√(x²+1), d/dx (arcosh x) = 1/√(x²−1) (x>1), d/dx (artanh x) = 1/(1−x²) (|x|<1)
反双曲函数的导数：d/dx (arsinh x) = 1/√(x²+1)，d/dx (arcosh x) = 1/√(x²−1) (x>1)，d/dx (artanh x) = 1/(1−x²) (|x|<1)。
By the chain rule, for a constant a, d/dx (sinh ax) = a cosh ax, and similar for other hyperbolic and inverse hyperbolic functions.
根据链式法则，对于常数 a，d/dx (sinh ax) = a cosh ax，其他双曲与反双曲函数类似。
10. Integration | 积分
Standard integrals of hyperbolic functions:
∫ sinh x dx = cosh x + C, ∫ cosh x dx = sinh x + C, ∫ sech² x dx = tanh x + C
双曲函数的标准积分：∫ sinh x dx = cosh x + C，∫ cosh x dx = sinh x + C，∫ sech² x dx = tanh x + C。
Integrals yielding inverse hyperbolic functions (useful for completing the square in the integrand):
∫ 1/√(x²+a²) dx = arsinh(x/a) + C, ∫ 1/√(x²−a²) dx = arcosh(x/a) + C (x>a), ∫ 1/(a²−x²) dx = (1/a) artanh(x/a) + C
导出反双曲函数的积分（常用于配方后积分）：∫ 1/√(x²+a²) dx = arsinh(x/a
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