Year 12 Edexcel Further Mathematics: Formula & Theorem Quick Reference Handbook | 公式定理速查手册

📚 Year 12 Edexcel Further Mathematics: Formula & Theorem Quick Reference Handbook | 公式定理速查手册

This handbook brings together the key formulas and theorems from the Year 12 Edexcel Further Mathematics syllabus in one place. It is designed as a rapid reference tool, helping you consolidate the core pure topics during revision. Each section pairs concise English explanations with their Chinese equivalents, and all mathematical notation is presented clearly without any LaTeX—just Unicode symbols and standard HTML.

本手册将 Year 12 Edexcel 进阶数学课程中的核心公式与定理汇总在一起，旨在成为快速查阅工具，帮助你在复习时巩固纯数核心内容。每个小节都采用简洁的英文解释与对应中文配对的形式，所有数学符号均不使用 LaTeX，仅以 Unicode 字符和标准 HTML 呈现，清晰明了。

1. Complex Numbers | 复数

i, where a, b are real numbers, a is the real part Re(z) and b is the imaginary part Im(z).

虚数单位 i 满足 i² = −1。复数记作 z = a + bi，其中 a, b 为实数，a 为实部 Re(z)，b 为虚部 Im(z)。

The complex conjugate of z is z* = a − bi. The modulus of z is |z| = √a² + b² nt θ = arg(z) satisfies tan θ = b/a, with the quadrant determined by the signs of a and b.

z 的共轭复数为 z* = a − bi。z 的模为 |z| = √a² + b²，其辐角 θ = arg(z) 满足 tan θ = b/a，并根据 a、b 的符号确定象限。

ollow the obvious rules: (a+bi)+(c+di) = (a+c)+(b+d)i and (a+bi)(c+di) = (ac−bd)+(ad+bc)i.

加法和乘法遵循直观法则：(a+bi)+(c+di) = (a+c)+(b+d)i，(a+bi)(c+di) = (ac−bd)+(ad+bc)i。

De Moivre' s nθ + i sin nθ for integer n. It is extremely useful for finding powers and roots of complex numbers.

棣莫弗定理指出：对整数 n，有 (cos θ + i sin θ)ⁿ = cos nθ + i sin nθ。该定理在求复数的幂与方根时极为有用。

ots which occur in conjugate pairs. If the discriminant is negative, the roots are a ± bi.

具有实系数的二次方程可有复数根，且必以共轭对出现。若判别式为负，根为 a ± bi。

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2. Argand Diagrams | 阿尔冈图

An Argand diagram represents complex numbers as points on a plane, with the x‑axis as the real axis and the y‑axis as the imaginary axis. The point (a, b) corresponds to z = a + bi.

阿尔冈图将复数表示为平面上的点，x 轴为实轴，y 轴为虚轴。点 (a, b) 对应 z = a + bi。

The modulus |z| is the distance from the origin to the point, and the argument arg(z) is the angle measured anticlockwise from the positive real axis.

模 |z| 是原点到该点的距离，辐角 arg(z) 是从正实轴逆时针测量的角度。

You can add and subtract complex numbers using vector addition. Multiplying by i rotates a vector by 90° anticlockwise.

复数的加减可通过向量加法实现。乘以 i 会将向量逆时针旋转 90°。

Loci such as |z − a| = r (a circle) or |z − a| = |z − b| (a perpendicular bisector) can be interpreted and sketched directly on the Argand plane.

诸如 |z − a| = r（圆）或 |z − a| = |z − b|（中垂线）的轨迹可直接在阿尔冈平面上解释并作图。

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3. Matrices | 矩阵

umns. Two matrices can be added if they have the same dimensions; multiplication is defined when the number of columns of the first equals the number of rows of the second.

矩阵是一个矩形数字阵列。m × n 矩阵有 m 行 n 列。两个矩阵若维度相同则可相加；乘法仅在第一矩阵的列数等于第二矩阵的行数时定义。

determinant is det(A) = ad − bc. If det(A) = 0, the matrix is singular and has no inverse.

对于 2×2 矩阵 A = [a, b; c, d]，其行列式为 det(A) = ad − bc。若 det(A) = 0，矩阵是奇异矩阵，无逆矩阵。

The inverse of a non‑singular 2×2 matrix is A⁻¹ = (1/det(A)) [d, −b; −c, a].

非奇异 2×2 矩阵的逆矩阵为 A⁻¹ = (1/det(A)) [d, −b; −c, a]。

aneous linear equations: write the system as AX = B, then X = A⁻¹B (when A is invertible).

矩阵可用于求解联立线性方程：将方程组写为 AX = B，则 X = A⁻¹B（当 A 可逆时）。

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4. Matrix Transformations | 矩阵变换

A 2×2 matrix acts on a position vector (x, y) to produce an image point (x', y'). The transformation can be a rotation, reflection, enlargement or shear.

2×2 矩阵作用于位置向量 (x, y) 产生像点 (x', y')。变换可以是旋转、反射、放大或剪切。

Rotation through angle θ anticlockwise: R = [cos θ, −sin θ; sin θ, cos θ].

逆时针旋转 θ 角：R = [cos θ, −sin θ; sin θ, cos θ]。

Reflection in the x‑axis: [1, 0; 0, −1]; in the y‑axis: [−1, 0; 0, 1]; in y = x: [0, 1; 1, 0]; in y = −x: [0, −1; −1, 0].

关于 x 轴反射：[1, 0; 0, −1]；关于 y 轴：[−1, 0; 0, 1]；关于 y = x：[0, 1; 1, 0]；关于 y = −x：[0, −1; −1, 0]。

Enlargement with scale factor k (centre at origin): [k, 0; 0, k]. A shear parallel to the x‑axis: [1, k; 0, 1]; parallel to the y‑axis: [1, 0; k, 1].

以原点为中心缩放因子 k 的放大：[k, 0; 0, k]。平行于 x 轴的剪切：[1, k; 0, 1]；平行于 y 轴：[1, 0; k, 1]。

Combined transformations correspond to multiplying the matrices in the correct order (rightmost matrix acts first).

复合变换对应矩阵按正确顺序相乘（最右边的矩阵最先作用）。

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5. Series | 级数

The sums of the first n natural numbers, squares and cubes are standard results:
Σ(r=1 to n) r = n(n+1)/2
Σ(r=1 to n) r² = n(n+1)(2n+1)/6
Σ(r=1 to n) r³ = n²(n+1)²/4

前 n 个自然数、平方和立方和的标准结果：
Σ(r=1 to n) r = n(n+1)/2
Σ(r=1 to n) r² = n(n+1)(2n+1)/6
Σ(r=1 to n) r³ = n²(n+1)²/4

Using these, you can sum any polynomial function of r by expanding and summing term by term.

利用这些结果，可通过将 r 的多项式展开并逐项求和来计算任何多项式函数的和。

The method of differences is used when the general term of a series can be written as f(r) − f(r+1) or a similar telescoping form. Most terms cancel, leaving only the first and last few terms.

差分法用于级数的一般项可写作 f(r) − f(r+1) 或类似的伸缩形式的情形。大部分项相消，只留下最初和最后的几项。

Be able to manipulate sums with index shifts and to recognise when a fraction can be split into partial fractions to create a telescoping series.

要能通过指标平移处理求和，并能识别何时可将分式拆分为部分分式以构造伸缩级数。

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6. Further Calculus | 进阶微积分

Standard derivatives: d/dx(sin x) = cos x, d/dx(cos x) = −sin x, d/dx(tan x) = sec² x.
d/dx(eˣ) = eˣ, d/dx(ln x) = 1/x.

标准导数：d/dx(sin x) = cos x，d/dx(cos x) = −sin x，d/dx(tan x) = sec² x。
d/dx(eˣ) = eˣ，d/dx(ln x) = 1/x。

Product rule: if y = uv, then dy/dx = u dv/dx + v du/dx.
Quotient rule: if y = u/v, then dy/dx = (v du/dx − u dv/dx)/v².
Chain rule: if y = f(g(x)), then dy/dx = f'(g(x)) g'(x).

乘积法则：若 y = uv，则 dy/dx = u dv/dx + v du/dx。
商法则：若 y = u/v，则 dy/dx = (v du/dx − u dv/dx)/v²。
链式法则：若 y = f(g(x))，则 dy/dx = f'(g(x)) g'(x)。

n is the reverse of differentiation. Key integrals: ∫ xⁿ dx = xⁿ⁺¹/(n+1) + C (n ≠ −1), ∫ eˣ dx = eˣ + C, ∫ 1/x dx = ln|x| + C.

积分是微分的逆运算。基本积分：∫ xⁿ dx = xⁿ⁺¹/(n+1) + C (n ≠ −1)，∫ eˣ dx = eˣ + C，∫ 1/x dx = ln|x| + C。

djust dx to du, and change limits if definite. Integration by parts: ∫ u dv = uv − ∫ v du.

换元积分法：用新变量 u 替换被积函数的一部分，将 dx 改为 du，若为定积分则改变积分限。分部积分法：∫ u dv = uv − ∫ v du。

Implicit differentiation is used when y is not given explicitly as a function of x. Differentiate both sides of an equation with respect to x, using the chain rule for terms involving y.

当 y 未显式表达为 x 的函数时使用隐函数求导。对方程两边关于 x 求导，含有 y 的项运用链式法则。

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7. Further Vectors | 进阶向量

A vector in 3D is written as a = xi + yj + zk or as a column vector (x, y, z)ᵀ. The magnitude is |a| = √x² + y² + z².

三维向量写作 a = xi + yj + zk 或列向量 (x, y, z)ᵀ。其模为 |a| = √x² + y² + z²。

The scalar (dot) product of two vectors a and b is a ⋅ b = |a||b| cos θ, where θ is the angle between them. In component form, a ⋅ b = a₁b₁ + a₂b₂ + a₃b₃.

两向量 a 与 b 的标量积（点积）为 a ⋅ b = |a||b| cos θ，θ 为其夹角。分量形式为 a ⋅ b = a₁b₁ + a₂b₂ + a₃b₃。

Two non‑zero vectors are perpendicular if and only if a ⋅ b = 0.

两非零向量垂直当且仅当 a ⋅ b = 0。

ing through point A with direction vector d is r = a + t d, where t is a scalar parameter.

过点 A 且方向向量为 d 的直线向量方程为 r = a + t d，t 为标量参数。

To find the angle between two lines, use the direction vectors: cos θ = |d₁ ⋅ d₂|/(|d₁||d₂|).

求两直线间夹角，利用方向向量：cos θ = |d₁ ⋅ d₂|/(|d₁||d₂|)。

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8. Roots of Polynomials | 多项式的根

For a quadratic equation ax² + bx + c = 0 with roots α and β:
Sum of roots: α + β = −b/a
Product of roots: αβ = c/a.

对于有根 α 和 β 的二次方程 ax² + bx + c = 0：
根之和：α + β = −b/a
根之积：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.

对于有根 α, β, γ 的三次方程 ax³ + bx² + cx + d = 0：
α + β + γ = −b/a
αβ + αγ + βγ = c/a
αβγ = −d/a。

Similar relationships hold for quartic equations. You can use symmetric sums of roots to form new equations whose roots are related to the original ones (e.g., squares, reciprocals, or linear transformations).

类似的关系适用于四次方程。可以利用根的对称和形成新方程，其根与原方程的根存在某种关系（例如平方、倒数或线性变换）。

Knowing these relationships helps to solve problems without finding the individual roots, especially when forming an equation with given roots.

了解这些关系有助于在不必求出每个根的情况下解题，尤其在已知根的特征时构造方程。

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9. Proof by Induction | 数学归纳法

Proof by induction is used to prove statements that depend on a positive integer n. The structure has four clear steps.

数学归纳法用于证明依赖于正整数 n 的命题。其结构有四个清晰的步骤。

Step 1 (Basis): Verify the statement is true for the smallest value of n, usually n = 1.

步骤 1（奠基）：验证命题对最小的 n 值成立，通常 n = 1。

hesis): Assume the statement is true for n = k (where k is an arbitrary positive integer).

步骤 2（归纳假设）：假设命题对 n = k 成立（k 为任意正整数）。

+ 1. This usually involves algebraic manipulation of the k+1 case to separate the k case.

步骤 3（归纳递推）：在此假设下证明命题对 n = k + 1 成立。通常需要对 k+1 的情况进行代数变形，分离出 k 的情形。

Step 4 (Conclusion): By mathematical induction, the statement is true for all positive integers n.

步骤 4（结论）：由数学归纳法可知，命题对所有正整数 n 成立。

ivisibility statements (e.g., n³ − n is divisible by 3), and sequences defined recursively.

常见应用包括求和公式（如 Σ r = n(n+1)/2）、整除性命题（如 n³ − n 可被 3 整除）以及递归定义的序列。

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10. Further Trigonometry | 进阶三角学

Compound angle formulas:
sin(A ± B) = sin A cos B ± cos A sin B
cos(A ± B) = cos A cos B ∓ sin A sin B
tan(A ± B) = (tan A ± tan B)/(1 ∓ tan A tan B)

和角公式：
sin(A ± B) = sin A cos B ± cos A sin B
cos(A ± B) = cos A cos B ∓ sin A sin B
tan(A ± B) = (tan A ± tan B)/(1 ∓ tan A tan B)

A, cos 2A = cos² A − sin² A = 2 cos² A − 1 = 1 − 2 sin² A, tan 2A = 2 tan A/(1 − tan² A).

倍角公式：sin 2A = 2 sin A cos A，cos 2A = cos² A − sin² A = 2 cos² A − 1 = 1 − 2 sin² A，tan 2A = 2 tan A/(1 − tan² A)。

Factor formulas express sums of sines/cosines as products:
((P+Q)/2) cos((P−Q)/2) etc. These are useful for solving equations and proving identities.

和差化积公式将正弦/余弦的和表示为积：
sin P + sin Q = 2 sin((P

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