📚 AS Cambridge Further Mathematics: Formula & Theorem Quick Reference Guide | AS剑桥进阶数学：公式定理速查手册
This concise reference covers the essential formulas, key theorems and standard results required for the AS Cambridge Further Mathematics (9231) syllabus. Use it for quick revision and last-minute checks before your exam. Each section presents the material in paired English and Chinese paragraphs, ensuring clarity and bilingual understanding.
本速查手册涵盖了AS剑桥进阶数学(9231)大纲所要求的核心公式、关键定理和标准结果。可用于考前快速复习和临场查阅。每个小节均以中英对照的方式呈现，确保清晰易懂。
1. Roots of Polynomial Equations | 多项式方程的根
For a cubic equation ax³ + bx² + cx + d = 0 with roots α, β, γ, the relationships between roots and coefficients are given by elementary symmetric sums.
α + β + γ = -b/a, αβ + βγ + γα = c/a, αβγ = -d/a
对于三次方程 ax³ + bx² + cx + d = 0 且根为 α, β, γ，根与系数的关系由初等对称多项式给出：α + β + γ = -b/a，αβ + βγ + γα = c/a，αβγ = -d/a。
For a quartic equation ax⁴ + bx³ + cx² + dx + e = 0 with roots α, β, γ, δ, the sums extend to:
Σα = -b/a, Σαβ = c/a, Σαβγ = -d/a, αβγδ = e/a
对于四次方程 ax⁴ + bx³ + cx² + dx + e = 0 其根为 α, β, γ, δ，相应关系为：Σα = -b/a，Σαβ = c/a，Σαβγ = -d/a，αβγδ = e/a。
To find a polynomial given its roots, construct factors (x – α) and expand. Recurrence relations for powers of roots use Σαⁿ linked to coefficients.
若已知根，可通过因式 (x – α) 相乘构造多项式。根的幂次和 Σαⁿ 通过系数递推关系相互联系。
2. Rational Functions and Graphs | 有理函数与图像
A rational function f(x) = P(x)/Q(x) may have vertical asymptotes where Q(x)=0 and P(x)≠0, and horizontal or oblique asymptotes determined by the degrees of P and Q.
If deg(P) < deg(Q): y = 0; if deg(P) = deg(Q): y = leading coefficient ratio; if deg(P) = deg(Q)+1: oblique asymptote.
有理函数 f(x) = P(x)/Q(x) 的垂直渐近线出现在 Q(x)=0 且 P(x)≠0 处；水平或斜渐近线由分子分母次数决定：deg(P) < deg(Q) 时 y=0；deg(P) = deg(Q) 时 y = 首项系数比；deg(P) = deg(Q)+1 时存在斜渐近线。
Improper fractions are simplified by long division into a polynomial plus a proper fraction. Curve sketching examines intercepts, stationary points, and behaviour near asymptotes.
假分式需通过长除法化为多项式加真分式。绘制图像时要考察截距、驻点以及渐近线附近的行为。
3. Summation of Series | 级数求和
Standard results for sums of natural numbers, their squares and cubes form the building blocks for more complicated finite series.
∑ᵣ₌₁ⁿ r = n(n+1)/2, ∑ᵣ₌₁ⁿ r² = n(n+1)(2n+1)/6, ∑ᵣ₌₁ⁿ r³ = n²(n+1)²/4
自然数、平方和与立方和的标准求和公式是处理复杂级数的基础：∑ r = n(n+1)/2，∑ r² = n(n+1)(2n+1)/6，∑ r³ = n²(n+1)²/4。
Use the method of differences to sum series involving terms like 1/[r(r+1)] by expressing them as partial fractions and observing telescoping cancellation.
1/[r(r+1)] = 1/r – 1/(r+1) → sum = 1 – 1/(n+1)
使用差分法求和，如 1/[r(r+1)] 可拆为部分分式并利用裂项相消：1/[r(r+1)] = 1/r – 1/(r+1)，其和为 1 – 1/(n+1)。
Further series may be reduced using standard formulas plus algebraic manipulation. Be prepared to prove results by induction.
更复杂的级数可通过标准公式与代数变形求和，也需掌握用数学归纳法证明求和公式。
4. Matrices | 矩阵
The determinant of a 2×2 matrix M = [[a, b], [c, d]] is det(M) = ad – bc. The inverse exists only if det ≠ 0.
M⁻¹ = (1/det(M)) [[d, -b], [-c, a]]
2×2 矩阵 M 的行列式为 det(M) = ad – bc，仅当 det ≠ 0 时逆矩阵存在：M⁻¹ = (1/det(M)) [[d, -b], [-c, a]]。
For a 3×3 matrix, the determinant can be computed by expansion along any row or column. The inverse may be found using the adjugate matrix: M⁻¹ = adj(M)/det(M).
3×3 矩阵的行列式可沿任一行或列展开计算；其逆矩阵可通过伴随矩阵求得：M⁻¹ = adj(M)/det(M)。
Matrices represent linear transformations. Rotation through angle θ (anticlockwise) is given by [[cosθ, -sinθ], [sinθ, cosθ]]. Reflection in the line y = (tanθ)x has matrix [[cos2θ, sin2θ], [sin2θ, -cos2θ]].
矩阵表示线性变换。逆时针旋转 θ 角的矩阵为 [[cosθ, -sinθ], [sinθ, cosθ]]；关于直线 y = (tanθ)x 的反射矩阵为 [[cos2θ, sin2θ], [sin2θ, -cos2θ]]。
The product of two transformation matrices corresponds to the composition of transformations. An invariant line or point satisfies Mx = λx or is mapped to itself under the transformation.
两个变换矩阵的乘积对应变换的复合。不变直线或不动点满足 Mx = λx 或经变换后仍在原位置。
5. Complex Numbers | 复数
A complex number z = x + iy has real part x, imaginary part y. Its conjugate is z̄ = x – iy, and modulus |z| = √(x² + y²). Argument arg(z) = θ is measured from the positive real axis.
z z̄ = |z|², |z₁ z₂| = |z₁||z₂|, arg(z₁ z₂) = arg(z₁) + arg(z₂)
复数 z = x + iy 的共轭为 z̄ = x – iy，模 |z| = √(x² + y²)，幅角 arg(z) = θ 从正实轴度量。基本性质：z z̄ = |z|²，|z₁ z₂| = |z₁||z₂|，arg(z₁ z₂) = arg(z₁) + arg(z₂)。
De Moivre’s theorem: (cosθ + i sinθ)ⁿ = cos(nθ) + i sin(nθ) for integer n. This is used to find powers and roots of complex numbers.
棣莫弗定理：对于整数 n，(cosθ + i sinθ)ⁿ = cos(nθ) + i sin(nθ)，可用于求复数的乘幂与方根。
To find the n-th roots of a complex number, write z = r(cosθ + i sinθ), then the k-th root is r^(1/n)[cos((θ+2kπ)/n) + i sin((θ+2kπ)/n)], k = 0,1,…,n-1.
求复数的 n 次根时，先将 z 写为 r(cosθ + i sinθ)，第 k 个根为 r^(1/n)[cos((θ+2kπ)/n) + i sin((θ+2kπ)/n)]，k = 0,1,…,n-1。
Loci in the Argand diagram: |z – a| = r is a circle; |z – a| = |z – b| is the perpendicular bisector; arg(z – a) = θ is a ray.
复平面上的轨迹：|z – a| = r 表示圆；|z – a| = |z – b| 表示垂直平分线；arg(z – a) = θ 表示从 a 出发的射线。
6. Vectors | 向量
Vector operations: for a = a₁i + a₂j + a₃k, b = b₁i + b₂j + b₃k. Scalar (dot) product: a·b = a₁b₁ + a₂b₂ + a₃b₃ = |a||b|cosθ. Vector (cross) product: a×b is a vector perpendicular to both a and b, with magnitude |a||b|sinθ.
a·b > 0 ⇒ acute angle; a·b = 0 ⇒ perpendicular.
向量 a = a₁i + a₂j + a₃k 与 b 的点积为 a·b = a₁b₁ + a₂b₂ + a₃b₃ = |a||b|cosθ；叉积 a×b 是一个垂直于 a 和 b 的向量，大小为 |a||b|sinθ。a·b > 0 对应锐角，a·b = 0 对应垂直。
The equation of a straight line passing through point A with direction vector d is r = a + t d, where t is a scalar parameter. Two lines intersect if there exists t and s satisfying the vector equation; skew lines do not intersect and are not parallel.
过点 A 且方向向量为 d 的直线方程为 r = a + t d，t 为参数。两直线相交即存在 t, s 满足方程；异面直线既不平行也不相交。
Plane equation with normal vector n: r·n = a·n = p. The distance from a point to a plane and the angle between a line and a plane are commonly tested.
Distance = |(r₀·n – p)| / |n|
平面的法线式方程： r·n = p。点到平面的距离公式为 |(r₀·n – p)| / |n|。线面角也是常见考点。
7. Proof by Induction | 数学归纳法
Proof by induction verifies that a statement P(n) holds for all positive integers n. The structure: (i) Base case: show P(1) true. (ii) Inductive step: assume P(k) true, prove P(k+1) true. (iii) Conclusion: by mathematical induction, P(n) is true for all n ∈ ℤ⁺.
归纳法用于证明命题 P(n) 对所有正整数 n 成立。步骤：（1）奠基：验证 P(1) 为真；（2）归纳步：假设 P(k) 成立，证明 P(k+1) 成立；（3）结论：由数学归纳法，P(n) 对所有正整数 n 成立。
Common applications include summation formulas, divisibility, matrix powers, and recurrence relations. For divisibility, express the k+1 case in terms of the k case plus an extra term clearly divisible by the required integer.
常见应用包括求和公式、整除性、矩阵乘方以及递推关系。处理整除时，需将 k+1 表达式用 k 的假设式子加上一个明显可被整除的项表示。
8. Polar Coordinates | 极坐标
In polar coordinates (r, θ), r ≥ 0 is the distance from the origin, and θ is the angle measured from the initial line. Conversion: x = r cosθ, y = r sinθ, r = √(x² + y²).
极坐标 (r, θ) 中 r ≥ 0 表示到极点的距离，θ 为极角。与直角坐标的转换：x = r cosθ，y = r sinθ，r = √(x² + y²)。
The area enclosed by a polar curve r = f(θ) from θ = α to θ = β is given by the integral ½ ∫ r² dθ. Use it to find loop areas and regions between curves.
Area = ½ ∫_α^β r² dθ
极曲线 r = f(θ) 从 θ = α 到 β 围成的面积为 ½ ∫ r² dθ，用于求圆环、曲线圈及曲线间区域的面积。
To sketch polar curves, check symmetry (e.g., f(θ) = f(-θ) ⇒ symmetric about initial line) and find maximum r. Tangents at the pole occur when r = 0 but dθ/dr ≠ 0.
绘制极曲线时要利用对称性（如 f(θ) = f(-θ) 则关于极轴对称），并找出 r 的最大值。极点处的切线出现在 r=0 且 dθ/dr ≠ 0 时。
9. 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. Key identities mirror trigonometric ones:
cosh² x – sinh² x = 1, 1 – tanh² x = sech² x
定义：sinh x = (eˣ – e⁻ˣ)/2，cosh x = (eˣ + e⁻ˣ)/2，tanh x = sinh x / cosh x。基本恒等式联系：cosh² x – sinh² x = 1，1 – tanh² x = sech² x。
Derivatives: d/dx (sinh x) = cosh x, d/dx (cosh x) = sinh x, d/dx (tanh x) = sech² x. Inverse hyperbolic functions are defined in logarithmic forms, e.g., arsinh x = ln(x + √(x²+1)).
导数：d(sinh x)/dx = cosh x, d(cosh x)/dx = sinh x, d(tanh x)/dx = sech² x。反双曲函数常用对数形式表示，如 arsinh x = ln(x + √(x²+1))。
Osborn’s rule helps convert trigonometric identities to hyperbolic ones: replace cos with cosh, sin with i sinh, and adjust signs of products of two sinhs.
Osborn 法则可将三角恒等式转换为双曲恒等式：将 cos 换为 cosh，sin 换为 i·sinh，并调整含有两个 sinh 乘积的符号。
10. Differential Equations | 微分方程
First-order separable equations: dy/dx = g(x)h(y) can be solved by separating variables ∫ 1/h(y) dy = ∫ g(x) dx. Always write the general solution with an arbitrary constant.
一阶可分离变量方程 dy/dx = g(x)h(y) 可通过分离变量求解：∫ 1/h(y) dy = ∫ g(x) dx。通解须含任意常数。
For linear first-order equations dy/dx + P(x)y = Q(x), use the integrating factor I = e^(∫ P dx). Multiply through by I to give d/dx (I y) = I Q, then integrate.
I = e^(∫ P dx), I y = ∫ I Q dx
一阶线性微分方程 dy/dx + P(x)y = Q(x) 使用积分因子 I = e^(∫ P dx)，等式两边乘以 I 后变为 d/dx (I y) = I Q，积分即得解。
Applications include growth and decay, cooling, and simple electric circuits. Understand how to incorporate initial conditions to find a particular solution.
应用包括增长与衰减、冷却问题、简单电路等。需掌握如何由初始条件确定特解。
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