Year 12 CAIE Further Mathematics: Formula & Theorem Quick Reference | Year 12 CAIE 进阶数学：公式定理速查手册

📚 Year 12 CAIE Further Mathematics: Formula & Theorem Quick Reference | Year 12 CAIE 进阶数学：公式定理速查手册

This quick reference handbook distills the essential formulas, theorems, and standard results from the CAIE Year 12 Further Mathematics syllabus. It is designed for rapid revision, covering Further Pure Mathematics 1 topics such as complex numbers, roots of polynomials, summation of series, matrices, vectors, proof by induction, polar coordinates, and rational functions. Each entry pairs an English statement with its Chinese counterpart to support bilingual learners.

本速查手册提炼了CAIE Year 12 进阶数学的核心公式、定理与标准结论，旨在帮助同学们快速复习。内容覆盖Further Pure Mathematics 1 的复数、多项式求根、级数求和、矩阵、向量、归纳法证明、极坐标及有理函数等专题。每一条目均提供英文与中文对照释义，以支持双语学习。

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1. Complex Numbers | 复数

i² = −1. A complex number is written as z = x + iy, where x = Re(z) and y = Im(z).

i² = −1。复数记作 z = x + iy，其中 x = Re(z) 为实部，y = Im(z) 为虚部。

The complex conjugate is z* = x − iy. Modulus: |z| = √x² + y². Argument: arg z = θ, where tan θ = y/x, with quadrant adjustments.

共轭复数为 z* = x − iy。模：|z| = √x² + y²。辐角：arg z = θ，满足 tan θ = y/x，并按象限调整。

Euler's relation: e^iθ = cos θ + i sin θ.

欧拉关系式：e^iθ = cos θ + i sin θ。

De Moivre's theorem: (cos θ + i sin θ)ⁿ = cos nθ + i sin nθ, for integer n.

棣莫弗定理：对于整数 n，(cos θ + i sin θ)ⁿ = cos nθ + i sin nθ。

ex number, express z in polar form z = r(cos θ + i sin θ). Then the n roots are given by r^1/n [ cos( (θ + 2kπ)/n ) + i sin( (θ + 2kπ)/n ) ], k = 0, 1, ..., n−1.

求复数的 n 次方根时，先将 z 写成极坐标形式 z = r(cos θ + i sin θ)，则 n 个根为 r^1/n [ cos( (θ + 2kπ)/n ) + i sin( (θ + 2kπ)/n ) ]，其中 k = 0, 1, ..., n−1。

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2. Roots of Polynomial Equations | 多项式方程的根

For a quadratic 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，设根为 α, β，则根之和 α + β = −b/a，根之积 αβ = c/a。

For a cubic ax³ + bx² + cx + d = 0 with roots α, β, γ: Σα = −b/a, Σαβ = c/a, αβγ = −d/a.

对于三次方程 ax³ + bx² + cx + d = 0，根为 α, β, γ，则 Σα = −b/a，Σαβ = c/a，αβγ = −d/a。

³ + cx² + dx + e = 0 with roots α, β, γ, δ: Σα = −b/a, Σαβ = c/a, Σαβγ = −d/a, αβγδ = e/a.

对于四次方程 ax⁴ + bx³ + cx² + dx + e = 0，根为 α, β, γ, δ，则 Σα = −b/a，Σαβ = c/a，Σαβγ = −d/a，αβγδ = e/a。

Use symmetric functions of roots to evaluate expressions like Σα² = (Σα)² − 2Σαβ.

利用根的对称式可计算如 Σα² = (Σα)² − 2Σαβ 等表达式。

f a given polynomial, substitute into the symmetric sums or use a transformation y = f(x).

若须求新多项式使其根与原方程的根有某种关系，可代入对称和或用变换 y = f(x) 处理。

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3. Summation of Series | 级数求和

Standard results for r = 1 to n:

对 r 从 1 到 n 的标准求和公式：

∑_r=1ⁿ r = ½ n(n+1)

∑_r=1ⁿ r = ½ n(n+1)

∑_r=1ⁿ r² = ⅙ n(n+1)(2n+1)

∑_r=1ⁿ r² = ⅙ n(n+1)(2n+1)

∑_r=1ⁿ r³ = ¼ n²(n+1)² = [½ n(n+1)]²

∑_r=1ⁿ r³ = ¼ n²(n+1)² = [½ n(n+1)]²

Use method of differences to sum expressions like ∑ 1/[r(r+1)] = 1 − 1/(n+1).

利用差分法可求如 ∑ 1/[r(r+1)] = 1 − 1/(n+1) 之类的和。

For ∑ (u_r+1 − u_r), the sum telescopes to u_n+1 − u₁.

对于 ∑ (u_r+1 − u_r)，和可裂项相消得 u_n+1 − u₁。

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4. Matrices | 矩阵

r a 2×2 matrix A = [a b; c d], det A = ad − bc. If det A ≠ 0, A⁻¹ = (1/det A)[d −b; −c a].

对于 2×2 矩阵 A = [a b; c d]，行列式 det A = ad − bc。若 det A ≠ 0，则 A⁻¹ = (1/det A)[d −b; −c a]。

The inverse is A⁻¹ = (1/det A) adj A, where adj A is the transpose of the cofactor matrix.

对于 3×3 矩阵，行列式可按某行或某列展开求解。其逆矩阵为 A⁻¹ = (1/det A) adj A，其中 adj A 是余子式矩阵的转置。

f det A ≠ 0. If det A = 0, the system has either no solution or infinitely many solutions.

线性方程组 AX = B 当 det A ≠ 0 时有唯一解 X = A⁻¹B。若 det A = 0，则方程组无解或有无穷多解。

λ satisfy det(A − λI) = 0. For each eigenvalue, solve (A − λI)X = 0 to find eigenvectors.

特征值 λ 满足特征方程 det(A − λI) = 0。对每个特征值求解 (A − λI)X = 0 可得特征向量。

2×2 matrix, the sum of eigenvalues equals trace(A) = a + d, and the product equals det A.

对于 2×2 矩阵，特征值之和等于迹 a+d，特征值之积等于行列式 det A。

rly independent eigenvectors, then P⁻¹AP = D, where D is a diagonal matrix of eigenvalues.

对角化：若矩阵有完全组线性无关的特征向量，则 P⁻¹AP = D，其中 D 是由特征值构成的对角矩阵。

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5. Vectors in 3D | 三维向量

A vector is a = (a₁, a₂, a₃) = a₁i + a₂j + a₃k. Magnitude |a| = √a₁² + a₂² + a₃².

向量 a = (a₁, a₂, a₃) = a₁i + a₂j + a₃k。模 |a| = √a₁² + a₂² + a₃²。

Dot product: a · b = a₁b₁ + a₂b₂ + a₃b₃ = |a||b| cos θ, where θ is the angle between the vectors.

数量积（点积）：a · b = a₁b₁ + a₂b₂ + a₃b₃ = |a||b| cos θ，其中 θ 为两向量夹角。

(a₂b₃ − a₃b₂, a₃b₁ − a₁b₃, a₁b₂ − a₂b₁). The result is perpendicular to both a and b, and |a × b| = |a||b| sin θ.

向量积（叉积）：a × b = (a₂b₃ − a₃b₂, a₃b₁ − a₁b₃, a₁b₂ − a₂b₁)。结果垂直于 a 和 b，且 |a × b| = |a||b| sin θ。

d, where a is a point on the line, d is the direction vector, and t is a scalar parameter.

直线的向量方程：r = a + t d，其中 a 为直线上一点，d 为方向向量，t 为参数。

n = a · n = d, where n is a normal vector to the plane. Alternatively, r = a + λ u + μ v.

平面的向量方程：r · n = a · n = d，其中 n 为平面的法向量。也可写成参数形式 r = a + λ u + μ v。

Distance from a point to a plane: |(r − a)·n|/|n|.

点到平面的距离为 |(r − a)·n|/|n|。

Angle between two lines/planes uses the dot product of direction/normal vectors.

两直线/两平面的夹角可通过其方向向量/法向量的点积求得。

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6. Proof by Induction | 归纳法证明

uctive step: prove true for n = k+1 using the assumption. (4) Conclusion: by mathematical induction, the statement is true for all positive integers n.

标准步骤：(1) 奠基：证明 n=1（或最小取值）时命题成立。(2) 归纳假设：假设 n=k 时命题为真。(3) 归纳递推：利用假设证明 n=k+1 时命题亦真。(4) 结论：由数学归纳法，命题对所有正整数 n 成立。

Common induction applications: summation formulas, divisibility proofs, matrix power formulas, and inequalities.

归纳法常见应用：求和公式证明、整除性证明、矩阵乘方公式、不等式证明。

Example: prove ∑_r=1ⁿ : LHS=1, RHS=1. Assume for n=k, then add (k+1) to both sides and simplify to the k+1 form.

例：证明 ∑_r=1ⁿ r = ½ n(n+1)。检验 n=1 成立。假设 n=k 时成立，两边同加 (k+1) 整理得 n=k+1 的形式。

n be expressed as a multiple of d, often by writing f(k+1) = m·f(k) + something divisible.

对于整除性问题，证明若 f(k) 能被 d 整除，则 f(k+1) 可表达为 d 的倍数，常用技巧为写出 f(k+1) = m·f(k) + （可被 d 整除的部分）。

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7. Polar Coordinates | 极坐标

In polar coordinates, a point is given by (r, θ), where r is the distance from the origin and θ is the angle measured from the positive x‑axis.

在极坐标中，点表示为 (r, θ)，r 是原点到该点的距离，θ 是从正 x 轴量起的角度。

Cartesian conversion: x = r cos θ, y = r sin θ; r² = x² + y², tan θ = y/x.

直角坐标转换：x = r cos θ, y = r sin θ；r² = x² + y², tan θ = y/x。

dioid r = a(1 + cos θ), circle r = a, rose curves r = a cos nθ, lemniscate r² = a² cos 2θ.

常见极坐标曲线：心脏线 r = a(1+cos θ)，圆 r = a，玫瑰线 r = a cos nθ，双纽线 r² = a² cos 2θ。

Area enclosed by a polar curve: A = ½ ∫_θ=α^β r² dθ.

极坐标曲线所围面积：A = ½ ∫_θ=α^β r² dθ。

To find the area between two polar curves, integrate ½ (r_outer² − r_inner²).

求两曲线间面积时，对 ½ (r_outer² − r_inner²) 积分。

When sketching, identify symmetry, maximum r-values, and values of θ for which r=0 (pole).

画草图时注意对称性、r 的最大值以及使 r=0 的 θ 值（通过极点）。

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8. Rational Functions and Graphs | 有理函数与图像

P(x)/Q(x), where P and Q are polynomials. Simplify by partial fractions where appropriate.

有理函数：f(x) = P(x)/Q(x)，其中 P, Q 为多项式。必要时用部分分式化简。

ctors: 1/[(x+a)(x+b)] = A/(x+a) + B/(x+b). Repeated factor: 1/(x+a)² = A/(x+a) + B/(x+a)².

部分分式：不同线性因式：1/[(x+a)(x+b)] = A/(x+a) + B/(x+b)。重因式：1/(x+a)² = A/(x+a) + B/(x+a)²。

For an irreducible quadratic factor ax²+bx+c, the partial fraction includes a term (Bx+C)/(ax²+bx+c).

若含不可约二次因式 ax²+bx+c，部分分式中对应项为 (Bx+C)/(ax²+bx+c)。

ctor does not cancel. Horizontal/slant asymptotes are found by examining limits as x → ±∞.

当 Q(x)=0 且该因式未约去时产生竖直渐近线。水平/斜渐近线通过考察 x → ±∞ 时的极限求得。

Curve sketching steps: intercepts, asymptotes, stationary points, behaviour at infinity, and sign of f(x) in intervals between critical values.

描画图像步骤：截距、渐近线、驻点、无穷远处行为，以及关键值区分区间内 f(x) 的符号。

(ax+b)/(cx+d), the horizontal asymptote is y = a/c and the vertical asymptote is x = −d/c.

对于 y = (ax+b)/(cx+d) 型函数，水平渐近线 y = a/c，竖直渐近线 x = −d/c。

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