📚 Year 12 CAIE Further Mathematics: Formula & Theorem Quick Reference Handbook | Year 12 CAIE 进阶数学：公式定理速查手册
This handbook provides a concise, syllabus-based collection of essential formulas and theorems for the Year 12 CAIE Further Mathematics course (9231). It covers Further Pure Mathematics 1 (FP1) and key results from Further Mechanics 1, enabling rapid revision and exam preparation. Each entry is presented with a clear explanation and its Chinese translation.
本手册基于教学大纲，简洁收录了 Year 12 CAIE 进阶数学 (9231) 的核心公式与定理，涵盖 Further Pure Mathematics 1 (FP1) 以及 Further Mechanics 1 的重要结论，助力快速复习与备考。每一条内容均提供英文解释及对应的中文翻译。
1. Polynomial Roots and Coefficients | 多项式根与系数
ax³ + bx² + cx + d = 0 with roots α, β, γ, the sum of the roots is ∑α = α + β + γ = -b/a.
对于三次方程 ax³ + bx² + cx + d = 0，根为 α, β, γ，根的和为 ∑α = α + β + γ = -b/a。
The sum of the products of roots taken two at a time is ∑αβ = αβ + βγ + γα = c/a.
每两个根的乘积之和为 ∑αβ = αβ + βγ + γα = c/a。
The product of the roots is αβγ = -d/a.
根的积为 αβγ = -d/a。
² + dx + e = 0, similar relationships hold: ∑α = -b/a, ∑αβ = c/a, ∑αβγ = -d/a, αβγδ = e/a.
对于四次方程 ax⁴ + bx³ + cx² + dx + e = 0，类似关系成立：∑α = -b/a, ∑αβ = c/a, ∑αβγ = -d/a, αβγδ = e/a。
These symmetric functions are used to find unknown coefficients or to form new polynomials with transformed roots, such as α², 1/α, or α+k.
这些对称函数可用于求未知系数，或构造具有变换后根的新多项式，例如 α², 1/α 或 α+k。
2. Rational Functions and Asymptotes | 有理函数与渐近线
l function f(x) = P(x)/Q(x) may have vertical asymptotes where Q(x)=0 and numerator is non‑zero.
有理函数 f(x) = P(x)/Q(x) 在分母 Q(x)=0 且分子不为零处可能存在垂直渐近线。
If the degree of P(x) is less than the degree of Q(x), the horizontal asymptote is y = 0.
若分子的次数低于分母的次数，水平渐近线为 y = 0。
horizontal asymptote is y = leading coefficient of P divided by leading coefficient of Q.
若次数相等，水平渐近线为 y = 分子首项系数除以分母首项系数。
an oblique asymptote found by polynomial division: y = quotient (ignoring the remainder).
当分子次数比分母次数恰好高一次时，图像有一条斜渐近线，可通过多项式除法求得：y = 商式（忽略余式）。
The curve may cross a horizontal or oblique asymptote, but it will never cross a vertical asymptote.
曲线可能与水平或斜渐近线相交，但绝不会与垂直渐近线相交。
3. Summation of Series | 级数求和
Standard finite sums for integer r are used throughout FP1:
FP1 中普遍使用下列有限项标准求和公式：
∑ (r = 1 to n) r = ½ n(n+1)
∑ (r = 1 to n) r² = n(n+1)(2n+1)/6
∑ (r = 1 to n) r³ = ¼ n²(n+1)² = [½ n(n+1)]²
The method of differences splits a term into f(r) - f(r+1) or f(r+1) - f(r) so that most terms cancel, leaving only the first and last parts.
差分法将通项拆分为 f(r) - f(r+1) 或 f(r+1) - f(r) 的形式，使大部分项互相抵消，仅剩首尾几项。
artial fractions like 1/(r(r+1)) = 1/r - 1/(r+1) are used to find sums of rational series.
常见分式如 1/(r(r+1)) = 1/r - 1/(r+1) 可用于求有理级数的和。
4. Matrices – Determinants and Inverses | 矩阵 – 行列式与逆矩阵
For a 2×2 matrix A = [a b; c d], the determinant is det(A) = ad - bc.
对于 2×2 矩阵 A = [a b; c d]，行列式为 det(A) = ad - bc。
A is invertible if and only if det(A) ≠ 0. Its inverse is A⁻¹ = (1/det(A)) × [d -b; -c a].
A 可逆当且仅当 det(A) ≠ 0。其逆矩阵为 A⁻¹ = (1/det(A)) × [d -b; -c a]。
expanding along a row or column, using cofactors and the 2×2 determinants of submatrices.
对于 3×3 矩阵，行列式可沿某一行或某一列展开，利用余子式及子矩阵的 2×2 行列式求得。
The product rule holds: det(AB) = det(A) det(B). Also, det(A⁻¹) = 1/det(A).
乘积法则成立：det(AB) = det(A) det(B)。此外，det(A⁻¹) = 1/det(A)。
5. Matrix Transformations | 矩阵变换
A matrix M represents a linear transformation in the plane. The image of point (x,y) is given by M [x; y].
矩阵 M 表示平面上的一个线性变换。点 (x,y) 的像由 M [x; y] 给出。
Common transformations are:
常见变换有：
- Reflection in the x‑axis: [1 0; 0 -1]
- 关于 x 轴的反射： [1 0; 0 -1]
- Reflection in the line y = x: [0 1; 1 0]
- 关于直线 y = x 的反射： [0 1; 1 0]
- Rotation through angle θ anticlockwise: [cos θ -sin θ; sin θ cos θ]
- 逆时针旋转角度 θ： [cos θ -sin θ; sin θ cos θ]
- Enlargement scale factor k, centre origin: [k 0; 0 k]
- 以原点为中心、缩放因子 k 的放大： [k 0; 0 k]
The area scale factor of a transformation equals the absolute value of the determinant of its matrix.
变换的面积缩放因子等于其矩阵行列式的绝对值。
Combined transformations correspond to multiplying their matrices in reverse order: the transformation applied first is written on the right.
复合变换对应矩阵的逆序相乘：先作用的变换写在右侧。
6. Polar Coordinates | 极坐标
Cartesian to polar: x = r cos θ, y = r sin θ; r = √x² + y², θ = arctan(y/x) (with quadrant adjustment).
直角坐标转极坐标：x = r cos θ, y = r sin θ；r = √x² + y²，θ = arctan(y/x)（需根据象限调整）。
The area enclosed by a polar curve r = f(θ) for α ≤ θ ≤ β is A = ½ ∫ (θ=α to β) r² dθ.
极坐标曲线 r = f(θ) 在 α ≤ θ ≤ β 内围成的面积为 A = ½ ∫ (θ=α 到 β) r² dθ。
/dθ (r sin θ) = 0; for tangents perpendicular to the initial line, set d/dθ (r cos θ) = 0.
求平行于极轴（水平）的切线：令 d/dθ (r sin θ) = 0；垂直于极轴的切线：令 d/dθ (r cos θ) = 0。
a circle, r = a(1+cos θ) is a cardioid, r = a cos nθ or r = a sin nθ produce rose curves.
常见曲线：r = a 为圆，r = a(1+cos θ) 为心形线，r = a cos nθ 或 r = a sin nθ 产生玫瑰线。
7. Vector Equations of Lines and Planes | 直线与平面的向量方程
can be written as r = a + λ b, where a is a point on the line and b is a direction vector.
三维空间中的直线可写为 r = a + λ b，其中 a 为直线上一点，b 为方向向量。
as r = a + λ b + μ c, or in scalar product form r·n = a·n = d, where n is a normal vector.
平面可表示为 r = a + λ b + μ c，或点积形式 r·n = a·n = d，其中 n 为法向量。
The angle between two planes equals the angle between their normals: cos θ = |n₁·n₂| / (|n₁||n₂|).
两个平面的夹角等于其法向量的夹角：cos θ = |n₁·n₂| / (|n₁||n₂|)。
The shortest distance from a point with position vector p to the plane r·n = d is |p·n - d| / |n|.
位置向量为 p 的点到平面 r·n = d 的最短距离为 |p·n - d| / |n|。
t distance between two skew lines can be found using the scalar triple product: distance = |(a₂ - a₁)·(b₁×b₂)| / |b₁×b₂|.
对于异面直线，最短距离可用标量三重积求得：距离 = |(a₂ - a₁)·(b₁×b₂)| / |b₁×b₂|。
The cross product a×b yields a vector perpendicular to both a and b, with magnitude |a||b| sin θ.
向量叉积 a×b 给出一个垂直于 a 和 b 的向量，其大小为 |a||b| sin θ。
8. Proof by Induction | 数学归纳法证明
The principle of mathematical induction has three steps: base case, inductive hypothesis, and inductive step.
数学归纳法包含三个步骤：基础情形、归纳假设和归纳递推。
Show the statement is true for n = 1 (or a specified starting integer).
证明命题对 n = 1（或指定的起始整数）成立。
Assume the statement is true for n = k (inductive hypothesis).
假设命题对 n = k 成立（归纳假设）。
statement for n = k+1. Then conclude that the statement holds for all positive integers n.
利用该假设，证明命题对 n = k+1 成立。然后得出结论：命题对所有正整数 n 成立。
Induction is widely used to prove summation formulas, divisibility, matrix powers, and inequalities in Further Mathematics.
在进阶数学中，归纳法广泛用于证明求和公式、整除性、矩阵的幂次以及不等式。
9. Further Mechanics – Kinematics & Projectiles | 进阶力学 – 运动学与抛射体
n in two dimensions, the vector form of SUVAT applies: v = u + a t, r = r₀ + u t + ½ a t².
对于二维匀加速运动，可使用 SUVAT 的向量形式：v = u + a t，r = r₀ + u t + ½ a t²。
upward as positive: horizontal acceleration = 0, vertical acceleration = -g (or downward).
在仅受重力作用的抛射体运动中，以向上为正：水平加速度 = 0，竖直加速度 = -g（或向下）。
Initial velocity: uₓ = u cos θ, uy = u sin θ. Positions at time t: x = (u cos θ) t, y = (u sin θ) t - ½ g t².
初速度：uₓ = u cos θ，uy = u sin θ。t 时刻的位置：x = (u cos θ) t，y = (u sin θ) t - ½ g t²。
Time of flight, maximum height, and horizontal range can be derived from these parametric equations.
飞行时间、最大高度和水平射程均可由这些参数方程导出。
10. Further Mechanics – Work, Energy & Impulse | 进阶力学 – 功、能与冲量
pulse of a constant force F acting for time t is I = F t = change in momentum = m v - m u.
恒力 F 作用时间 t 的冲量为 I = F t = 动量变化量 = m v - m u。
ervation of momentum applies in vector form: total momentum before = total momentum after.
冲量是矢量；对于碰撞或爆炸，动量守恒定律以矢量形式成立：碰撞前总动量 = 碰撞后总动量。
Work done by a force F moving its point of application a displacement s is W = F·s = |F| |s| cos θ (constant force).
力 F 使其作用点产生位移 s 所做的功为 W = F·s = |F| |s| cos θ（恒力情形）。
ic energy: KE = ½ m v². Gravitational potential energy (near Earth): GPE = m g h. The work‑energy principle states: net work done = change in kinetic energy.
动能：KE = ½ m v²。重力势能（近地表）：GPE = m g h。功能原理：合外力做功等于动能的变化量。
Power is the rate of doing work: P = dW/dt; for constant force and velocity, P = F·v.
功率是做功的速率：P = dW/dt；当力与速度恒定时，P = F·v。
Conservation of mechanical energy in the absence of friction: KE + GPE = constant.
无摩擦时机械能守恒：KE + GPE = 常数。
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