CCEA Year 13 Further Mathematics: Quick Reference Formula and Theorem Handbook | CCEA 进阶数学（13年级）：公式定理速查手册

📚 CCEA Year 13 Further Mathematics: Quick Reference Formula and Theorem Handbook | CCEA 进阶数学（13年级）：公式定理速查手册

This handbook provides a concise summary of essential formulas and theorems for the CCEA Year 13 Further Mathematics course. It is designed as a quick reference for revision, covering key topics from pure mathematics and mechanics. All results are stated in a form suitable for direct application in examinations, with careful attention to standard notation and conditions of validity.

本手册针对 CCEA 13 年级进阶数学课程，对核心公式与定理进行精要总结。它可作为复习速查工具，涵盖纯数学与力学中的重点内容。所有结论均以可直接用于考试的标准化形式呈现，同时特别注明了适用条件与记号规范。

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1. Complex Numbers | 复数

A complex number is expressed in Cartesian form as z = a + bi, where a, b ∈ ℝ and i² = -1. The real part is Re(z) = a, and the imaginary part is Im(z) = b.

复数在笛卡尔形式下表示为 z = a + bi，其中 a, b ∈ ℝ 且 i² = -1。其实部 Re(z) = a，虚部 Im(z) = b。

The complex conjugate of z is z* = a - bi. The modulus is |z| = √a² + b², and the argument θ = arg(z) satisfies tanθ = b/a, with due regard to the quadrant.

z 的共轭复数为 z* = a - bi。模为 |z| = √a² + b²，辐角 θ = arg(z) 满足 tanθ = b/a，并需根据象限确定角度。

Euler's formula states e^iθ = cosθ + i sinθ. De Moivre's theorem: (cosθ + i sinθ)ⁿ = cos(nθ) + i sin(nθ) for integer n.

欧拉公式：e^iθ = cosθ + i sinθ。棣莫弗定理：对整数 n，(cosθ + i sinθ)ⁿ = cos(nθ) + i sin(nθ)。

The n distinct nth roots of a complex number are given by z_k = r^1/n [cos((θ+2kπ)/n) + i sin((θ+2kπ)/n)], k = 0,1,…,n-1.

复数的 n 个不同 n 次方根为 z_k = r^1/n [cos((θ+2kπ)/n) + i sin((θ+2kπ)/n)]，k = 0,1,…,n-1。

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2. Matrices and Determinants | 矩阵与行列式

) = ad - bc. The inverse A⁻¹ exists if det(A) ≠ 0 and is given by (1/det(A)) [d -b; -c a].

对于 2×2 矩阵 A = [a b; c d]，行列式为 det(A) = ad - bc。若 det(A) ≠ 0，则逆矩阵存在，且 A⁻¹ = (1/det(A)) [d -b; -c a]。

The inverse can be found using the adjugate matrix: A⁻¹ = (1/det A) adj A, where adj A = C^T (cofactor matrix transposed).

对于 3×3 矩阵，行列式可通过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，其中 adj A = C^T（余子式矩阵的转置）。

s associative but not commutative in general. The identity matrix I satisfies AI = IA = A.

矩阵乘法满足结合律，但一般不满足交换律。单位矩阵 I 满足 AI = IA = A。

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3. Roots of Polynomials | 多项式方程根的关系

a quadratic equation ax² + bx + c = 0 with roots α, β: sum α + β = -b/a, product αβ = c/a.

对于二次方程 ax² + bx + c = 0，根为 α, β：根之和 α + β = -b/a，根之积 αβ = c/a。

For a cubic ax³ + bx² + cx + d = 0 with roots α, β, γ: ∑α = -b/a, ∑αβ = c/a, αβγ = -d/a.

对于三次方程 ax³ + bx² + cx + d = 0，根为 α, β, γ：∑α = -b/a，∑αβ = c/a，αβγ = -d/a。

³ + cx² + dx + e = 0 with roots α, β, γ, δ: ∑α = -b/a, ∑αβ = c/a, ∑αβγ = -d/a, αβγδ = e/a.

对于四次方程 ax⁴ + bx³ + cx² + dx + e = 0，根为 α, β, γ, δ：∑α = -b/a，∑αβ = c/a，∑αβγ = -d/a，αβγδ = e/a。

etric expressions involving the roots, such as ∑α² and ∑α²β, without solving the equation.

这些关系可用于求涉及根的对称表达式（如 ∑α²、∑α²β），而无需解方程。

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4. Summation of Series | 级数求和

Standard summation formulas for integers r from 1 to n:

r 从 1 到 n 的标准求和公式：

∑_r=1ⁿ r = n(n+1)/2

∑_r=1ⁿ r² = n(n+1)(2n+1)/6

∑_r=1ⁿ r³ = [n(n+1)/2]²

ese can be used to sum polynomial expressions by linearity, e.g. ∑(3r² - 2r) = 3∑r² - 2∑r.

利用线性性质可求多项式表达式的和，例如 ∑(3r² - 2r) = 3∑r² - 2∑r。

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5. Proof by Mathematical Induction | 数学归纳法证明

The method of proof by induction consists of three steps: (1) Base case – ify the statement is true for the smallest value (usually n = 1). (2) Inductive hypothesis – assume the statement holds for some arbitrary n = k. (3) Inductive step – 1, using the assumption. Conclude that the statement is true for all positive integers n.

数学归纳法证明包含三步：(1) 奠基——验证命题对最小的值（通常为 n = 1）成立。(2) 归纳假设——假设命题对某个任意 n = k 成立。(3) 归纳递推——利用该假设证明命题对 n = k + 1 也成立。从而断定命题对所有正整数 n 成立。

This technique is frequently used to verify summation formulas, divisibility results, and matrix powers.

该技巧常用于验证求和公式、整除性质以及矩阵的幂次。

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6. Vector Geometry | 向量几何

For two vectors a and b, the scalar (dot) product is a · b = |a||b| component form, if a = (a₁, a₂, a₃) and b = (b₁, b₂, b₃), then a · b = a₁b₁ + a₂b₂ + a₃b₃.

对于两个向量 a 和 b，数量积（点积）为 a · b = |a||b| cosθ，其中 θ 为两向量夹角。若 a = (a₁, a₂, a₃) 且 b = (b₁, b₂, b₃)，则 a · b = a₁b₁ + a₂b₂ + a₃b₃。

vector (cross) product a × b yields a vector perpendicular to both a and b, with magnitude |a × b| = |a||b| sinθ. In component form: a × b = (a₂b₃ - a₃b₂, a₃b₁ - a₁b₃, a₁b₂ - a₂b₁).

向量积（叉积） a × b 得到一个同时垂直于 a 和 b 的向量，其模为 |a × b| = |a||b| sinθ。坐标形式：a × b = (a₂b₃ - a₃b₂, a₃b₁ - a₁b₃, a₁b₂ - a₂b₁)。

d is r = a + λd. The equation of a plane through A with normal vector n is r · n = a · n.

过点 A（位置向量 a）且平行于方向向量 d 的直线方程为 r = a + λd。过点 A 且法向量为 n 的平面方程为 r · n = a · n。

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7. Further Calculus – Volumes of Revolution | 进阶微积分——旋转体体积

and x = b is rotated completely about the x-axis, the volume generated is given by V = π ∫_a^b y² dx.

由曲线 y = f(x)、x 轴、x = a 与 x = b 围成的区域绕 x 轴旋转一周所得的体积为 V = π ∫_a^b y² dx。

= x(t), y = y(t) for t from t₁ to t₂, the volume of revolution about the x-axis is V = π ∫_t₁^t₂ y² (dx/dt) dt.

若曲线由参数方程 x = x(t), y = y(t) 定义，t 从 t₁ 到 t₂，则绕 x 轴旋转的体积为 V = π ∫_t₁^t₂ y² (dx/dt) dt。

For rotation about the y-axis, analogous formulas apply with x² dy when x is expressed as a function of y.

绕 y 轴旋转时，当 x 可表示为 y 的函数，可使用类似公式 V = π ∫ x² dy。

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8. Maclaurin Series Expansions | 麦克劳林级数展开

The Maclaurin series for a function f(x) is f(x) = f(0) + f'(0)x + f''(0)x²/2! + … + f⁽ⁿ⁾(0)xⁿ/n! + …, valid for |x| within the radius of convergence.

函数 f(x) 的麦克劳林级数为 f(x) = f(0) + f'(0)x + f''(0)x²/2! + … + f⁽ⁿ⁾(0)xⁿ/n! + …，在收敛半径内的 x 成立。

Standard series to memorise (all valid for all x unless stated):

需记忆的标准展开式（除另行说明均对所有 x 有效）：

e^x = 1 + x + x²/2! + x³/3! + …

sin x = x - x³/3! + x⁵/5! - …

cos x = 1 - x²/2! + x⁴/4! - …

ln(1 + x) = x - x²/2 + x³/3 - x⁴/4 + … (|x| < 1, x ≠ -1)

(1 + x)ⁿ = 1 + nx + n(n-1)x²/2! + … (|x| < 1, n ∈ ℝ)

These expansions can be used to approximate functions and to evaluate limits.

这些展开式可用于函数的近似计算与极限求解。

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9. Trigonometric Identities and Equations | 三角恒等式与方程

Fundamental identities: sin²θ + cos²θ = 1; 1 + tan²θ = sec²θ; 1 + cot²θ = csc²θ.

基本恒等式：sin²θ + cos²θ = 1；1 + tan²θ = sec²θ；1 + cot²θ = csc²θ。

Compound angle formulas: sin(A ± B) = sinA cosB ± cosA sinB; cos(A ± B) = cosA cosB ∓ sinA sinB; tan(A ± B) = (tanA ± tanB) / (1 ∓ tanA tanB).

和角公式：sin(A ± B) = sinA cosB ± cosA sinB；cos(A ± B) = cosA cosB ∓ sinA sinB；tan(A ± B) = (tanA ± tanB) / (1 ∓ tanA tanB)。

2 sinθ cosθ; cos2θ = cos²θ - sin²θ = 2cos²θ - 1 = 1 - 2sin²θ; tan2θ = 2tanθ / (1 - tan²θ).

倍角公式：sin2θ = 2 sinθ cosθ；cos2θ = cos²θ - sin²θ = 2cos²θ - 1 = 1 - 2sin²θ；tan2θ = 2tanθ / (1 - tan²θ)。

= 2 sin((A+B)/2) cos((A-B)/2), etc. These are useful for solving trigonometric equations.

和差化积公式：sinA + sinB = 2 sin((A+B)/2) cos((A-B)/2) 等，这些在解三角方程时非常有用。

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10. Mechanics: Constant Acceleration (SUVAT) | 力学：匀加速直线运动

For motion in a straight line with constant acceleration a, initial velocity u, final velocity v, displacement s, and time t, the five SUVAT equations are:

对于恒定加速度 a 的直线运动，初速度 u、末速度 v、位移 s、时间 t，五个 SUVAT 方程为：

v = u + at

s = ut + ½ at²

s = ½ (u + v)t

v² = u² + 2as

s = vt - ½ at²

All quantities are vectors; signs must be assigned consistently for a chosen positive direction.

所有量均为矢量；需选定正方向并一致使用符号。

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11. Mechanics: Forces and Newton’s Laws | 力学：力与牛顿定律

Newton' t force on an object equals its mass times acceleration: F = ma, where F is the net force.

牛顿第二定律：物体所受合外力等于其质量与加速度的乘积：F = ma，其中 F 为合力。

ownwards. The normal reaction R is the force exerted by a surface perpendicular to itself.

重力为 W = mg，方向竖直向下。法向反力 R 是表面对物体垂直于接触面施加的力。

Maximum static friction (limiting friction) is given by F_max ction. Kinetic friction is often approximated as constant and equal to μR after motion begins.

最大静摩擦力（极限摩擦力）为 F_max = μR，其中 μ 为摩擦系数。滑动摩擦力通常近似为常数，运动开始后仍等于 μR。

e = change in momentum: I = Ft = mv - mu. Both momentum and impulse are vector quantities.

动量为 p = mv。冲量等于动量的变化：I = Ft = mv - mu。动量和冲量均为矢量。

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