📚 Year 12 Cambridge Further Mathematics: Formula and Theorem Quick Reference Handbook | Year 12 剑桥进阶数学：公式定理速查手册
This article serves as a compact quick-reference guide to the key formulas and theorems encountered in Year 12 (AS Level) Cambridge Further Mathematics, with a focus on the Further Pure Mathematics 1 (FP1) syllabus. Having these results at your fingertips will support efficient revision and confident problem-solving under timed conditions.
本文是一份紧凑的速查指南，涵盖了 Year 12（AS Level）剑桥进阶数学 Further Pure Mathematics 1 (FP1) 课程中的关键公式与定理。将这些结论熟记于心，有助于高效复习并在限时条件下自信地解决问题。
1. Roots of Polynomial Equations | 多项式方程的根
For a quadratic equation ax² + bx + c = 0 with roots α and β, Vieta’s formulas give α + β = −b/a and αβ = c/a. These symmetric sums are used to form new equations without solving for the original roots.
对于根为 α 和 β 的二次方程 ax² + bx + c = 0，韦达定理给出 α + β = −b/a 和 αβ = c/a。这些对称和可用于无需解出原根即可构造新方程。
For a cubic ax³ + bx² + cx + d = 0 with roots α, β, γ: α+β+γ = −b/a, αβ+βγ+γα = c/a, αβγ = −d/a. Useful derived identities include Σα² = (Σα)² − 2Σαβ and Σαβ² = αβ(α+β) + … by algebra.
对于根为 α, β, γ 的三次方程 ax³ + bx² + cx + d = 0：α+β+γ = −b/a, αβ+βγ+γα = c/a, αβγ = −d/a。常用的导出恒等式有 Σα² = (Σα)² − 2Σαβ 以及通过代数运算得到的类似形式。
2. Rational Functions and Graphs | 有理函数与图形
For a rational function f(x) = P(x)/Q(x), vertical asymptotes occur where Q(x) = 0 and P(x) ≠ 0. Horizontal asymptotes depend on degrees: if deg P ≤ deg Q, y = 0 or y = a/b (when degrees equal). Oblique asymptotes arise when deg P = deg Q + 1, found via polynomial division.
对于有理函数 f(x) = P(x)/Q(x)，垂直渐近线出现在 Q(x) = 0 且 P(x) ≠ 0 处。水平渐近线取决于次数：如果 deg P ≤ deg Q，则 y = 0 或 y = a/b（当次数相等时）。斜渐近线出现在 deg P = deg Q + 1 时，通过多项式除法求得。
Graph transformations of y = f(x) follow standard rules: y = f(x) + a is a vertical translation, y = f(x + a) is a horizontal translation of −a units. Stretches: y = af(x) scales vertically by factor a, while y = f(ax) scales horizontally by factor 1/a.
函数 y = f(x) 的图形变换遵循标准规则：y = f(x) + a 是垂直平移，y = f(x + a) 是水平平移 −a 个单位。伸缩变换：y = af(x) 为垂直方向缩放 a 倍，而 y = f(ax) 为水平方向缩放 1/a 倍。
3. Summation of Series | 级数求和
The three standard results for the sum of the first n positive integers and their powers are: ∑r=1n r = n(n+1)/2, ∑r=1n r² = n(n+1)(2n+1)/6, and ∑r=1n r³ = n²(n+1)²/4. These are essential building blocks for manipulating linear combinations such as ∑ (3r² − 2r + 1).
三个前 n 个正整数及其幂的和的标准结果：∑r=1n r = n(n+1)/2，∑r=1n r² = n(n+1)(2n+1)/6，以及 ∑r=1n r³ = n²(n+1)²/4。它们是处理诸如 ∑ (3r² − 2r + 1) 等线性组合的基础。
For expressions of the form ∑ (ar + b), split the sum: ∑ (ar + b) = a∑ r + ∑ b, using the above results and noting that ∑r=1n c = cn. The method of differences is also tested, where telescoping terms simplify a sum like ∑ (1/(r) − 1/(r+1)).
对于 ∑ (ar + b) 形式的表达式，可将求和拆分：∑ (ar + b) = a∑ r + ∑ b，利用上述结果并注意 ∑r=1n c = cn。差分法也是考查内容，即通过裂项相消简化诸如 ∑ (1/r − 1/(r+1)) 的和。
4. Matrices | 矩阵
For a 2×2 matrix M = [[a, b], [c, d]], the determinant is det(M) = ad − bc. If the determinant is non‑zero, the inverse is given by M⁻¹ = (1/det(M)) * [[d, −b], [−c, a]]. For a 3×3 matrix, the determinant can be expanded via the first row: det(A) = a₁₁(minor) − a₁₂(minor) + a₁₃(minor).
对于 2×2 矩阵 M = [[a, b], [c, d]]，行列式为 det(M) = ad − bc。若行列式非零，逆矩阵由 M⁻¹ = (1/det(M)) * [[d, −b], [−c, a]] 给出。对于 3×3 矩阵，可按第一行展开计算行列式：det(A) = a₁₁(余子式) − a₁₂(余子式) + a₁₃(余子式)。
A system of linear equations AX = B can be solved using inverses: X = A⁻¹B, provided A is square and invertible. The solution is thus obtained by multiplying the inverse matrix by the constant column vector. Rank and row operations are also used when the inverse does not exist.
线性方程组 AX = B 可利用逆矩阵求解：X = A⁻¹B，只要矩阵 A 是方阵且可逆。由此可通过逆矩阵乘以常数列向量得到解。当逆矩阵不存在时，也会用到秩和行变换的方法。
5. Polar Coordinates | 极坐标
The relationship between Cartesian coordinates (x, y) and polar coordinates (r, θ) is: x = r cos θ, y = r sin θ. Conversely, r = √(x² + y²) and θ = arctan(y/x) (adjusting the quadrant). The area bounded by a polar curve r = f(θ) between θ = α and θ = β is A = ½ ∫αβ r² dθ.
直角坐标 (x, y) 与极坐标 (r, θ) 之间的关系为：x = r cos θ，y = r sin θ。反之，r = √(x² + y²)，θ = arctan(y/x)（需调整象限）。由极坐标曲线 r = f(θ) 在 θ = α 到 θ = β 之间围成的区域面积为 A = ½ ∫αβ r² dθ。
Symmetry is often exploited: if r(θ) = r(−θ), the curve is symmetric about the initial line; if r(θ) = r(π − θ), it is symmetric about the vertical half‑line θ = π/2. Substituting these simplifications can halve the integral calculation.
对称性常被利用：若 r(θ) = r(−θ)，曲线关于极轴对称；若 r(θ) = r(π − θ)，曲线关于半直线 θ = π/2 对称。应用这些简化可将积分计算量减半。
6. Vectors | 向量
A vector in three dimensions is written as ai + bj + ck or as a column vector. The position vector of a point P(x,y,z) is p = xi + yj + zk. The magnitude is |p| = √(x² + y² + z²). The scalar (dot) product is a·b = |a||b| cos θ = a₁b₁ + a₂b₂ + a₃b₃.
三维空间中的向量写成 ai + bj + ck 或列向量形式。点 P(x,y,z) 的位置向量为 p = xi + yj + zk，其模为 |p| = √(x² + y² + z²)。标量积（点积）为 a·b = |a||b| cos θ = a₁b₁ + a₂b₂ + a₃b₃。
The vector (cross) product a × b is perpendicular to both a and b with magnitude |a||b| sin θ. For a = a₁i + a₂j + a₃k and b = b₁i + b₂j + b₃k, it is calculated using the determinant of a 3×3 matrix: a × b = (a₂b₃ − a₃b₂)i − (a₁b₃ − a₃b₁)j + (a₁b₂ − a₂b₁)k.
向量积（叉积）a × b 同时垂直于 a 和 b，其大小为 |a||b| sin θ。对于 a = a₁i + a₂j + a₃k 和 b = b₁i + b₂j + b₃k，可通过 3×3 行列式计算：a × b = (a₂b₃ − a₃b₂)i − (a₁b₃ − a₃b₁)j + (a₁b₂ − a₂b₁)k。
The vector equation of a straight line is r = a + t d, where a is a point on the line and d is the direction vector. The Cartesian form is obtained by equating (x-a₁)/d₁ = (y-a₂)/d₂ = (z-a₃)/d₃ (if no denominator is zero).
直线的向量方程为 r = a + t d，其中 a 是直线上一点，d 为方向向量。直角坐标形式通过令 (x-a
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