Pre-U CIE Further Mathematics: Quick Reference Formula Handbook

引言 | Introduction

中文：Pre-U CIE 进阶数学（Further Mathematics）是剑桥国际考试中最具挑战性的数学课程之一。本文将系统梳理进阶数学的核心公式与定理，涵盖复数、矩阵、双曲函数、微分方程、极坐标和向量六大模块，帮助同学们快速查阅和复习。配合详细的讲解和例题，让你在考前做到胸有成竹。

English: Pre-U CIE Further Mathematics is one of the most challenging mathematics qualifications offered by Cambridge International. This article systematically consolidates the core formulas and theorems of Further Mathematics, covering six major modules: complex numbers, matrices, hyperbolic functions, differential equations, polar coordinates, and vectors. With detailed explanations and worked examples, you will be thoroughly prepared for the examination.

一、复数 | 1. Complex Numbers

1.1 基本形式与运算 | Basic Forms and Operations

中文：复数形如 z = a + bi，其中 a 为实部（Re(z)），b 为虚部（Im(z)），i^2 = -1。复数的共轭记为 z* = a – bi。复数的模（modulus）|z| = sqrt(a^2 + b^2)，辐角（argument）arg(z) 满足 tan(arg(z)) = b/a，注意需要根据象限确定正确角度。

English: A complex number takes the form z = a + bi, where a is the real part (Re(z)), b is the imaginary part (Im(z)), and i^2 = -1. The complex conjugate is denoted as z* = a – bi. The modulus is |z| = sqrt(a^2 + b^2), and the argument arg(z) satisfies tan(arg(z)) = b/a, noting that the correct angle must be determined according to the quadrant.

1.2 极坐标形式与棣莫弗定理 | Polar Form and De Moivre’s Theorem

中文：复数的极坐标形式为 z = r(cosθ + i sinθ) = r e^(iθ)，其中 r = |z|，θ = arg(z)。棣莫弗定理（De Moivre’s Theorem）指出：

[r(cosθ + i sinθ)]^n = r^n (cos(nθ) + i sin(nθ))

这一定理在求解复数方程 z^n = w 时极为有用。对于任意复数 w，其 n 次方根均匀分布在复平面上以原点为圆心、|w|^(1/n) 为半径的圆上。

English: The polar form of a complex number is z = r(cosθ + i sinθ) = r e^(iθ), where r = |z| and θ = arg(z). De Moivre’s Theorem states that:

[r(cosθ + i sinθ)]^n = r^n (cos(nθ) + i sin(nθ))

This theorem is extremely useful for solving complex equations of the form z^n = w. For any complex number w, its n-th roots are evenly distributed on a circle in the complex plane centered at the origin with radius |w|^(1/n).

1.3 复数与三角恒等式 | Complex Numbers and Trigonometric Identities

中文：利用棣莫弗定理和二项式展开，可以推导出多种三角恒等式。例如，将 (cosθ + i sinθ)^n 展开，分别取实部和虚部，即可得到 cos(nθ) 和 sin(nθ) 用 cosθ 和 sinθ 表示的公式。常见结果包括：cos(3θ) = 4cos^3θ – 3cosθ，sin(3θ) = 3sinθ – 4sin^3θ。

English: Using De Moivre’s Theorem combined with binomial expansion, various trigonometric identities can be derived. For instance, expanding (cosθ + i sinθ)^n and equating real and imaginary parts yields formulae expressing cos(nθ) and sin(nθ) in terms of cosθ and sinθ. Common results include: cos(3θ) = 4cos^3θ – 3cosθ and sin(3θ) = 3sinθ – 4sin^3θ.

二、矩阵与线性代数 | 2. Matrices and Linear Algebra

2.1 矩阵运算 | Matrix Operations

中文：矩阵加法要求两个矩阵具有相同的行数和列数。矩阵乘法 A(m x n) x B(n x p) = C(m x p)，其中 C_ij = sum(A_ik x B_kj) 对 k 从 1 到 n 求和。矩阵乘法一般不满足交换律（AB 不一定等于 BA），但满足结合律和分配律。

English: Matrix addition requires two matrices to have the same number of rows and columns. Matrix multiplication A(m x n) x B(n x p) = C(m x p), where C_ij = sum(A_ik x B_kj) for k from 1 to n. Matrix multiplication is generally not commutative (AB does not necessarily equal BA), but it is associative and distributive.

2.2 行列式与逆矩阵 | Determinants and Inverse Matrices

中文：2×2 矩阵的行列式：det(A) = ad – bc，其中 A = [[a, b], [c, d]]。3×3 矩阵的行列式可通过余因子展开计算。矩阵可逆的充要条件是行列式不为零。2×2 矩阵的逆矩阵公式为：

A^(-1) = (1/det(A)) [[d, -b], [-c, a]]

对于 3×3 矩阵，逆矩阵 = (1/det(A)) adj(A)，其中 adj(A) 是伴随矩阵（余因子矩阵的转置）。

English: Determinant of a 2×2 matrix: det(A) = ad – bc, where A = [[a, b], [c, d]]. For a 3×3 matrix, the determinant can be computed via cofactor expansion. A matrix is invertible if and only if its determinant is non-zero. The inverse formula for a 2×2 matrix is:

A^(-1) = (1/det(A)) [[d, -b], [-c, a]]

For a 3×3 matrix, the inverse = (1/det(A)) adj(A), where adj(A) is the adjugate matrix (transpose of the cofactor matrix).

2.3 特征值与特征向量 | Eigenvalues and Eigenvectors

中文：对于方阵 A，若存在标量 λ 和非零向量 v 满足 Av = λv，则 λ 为特征值，v 为对应的特征向量。特征方程为 det(A – λI) = 0。对于 2×2 矩阵，这是一元二次方程；对于 3×3 矩阵，则是三次方程。特征向量可用于矩阵的对角化，以及求解耦合微分方程组。

English: For a square matrix A, if there exists a scalar λ and a non-zero vector v such that Av = λv, then λ is an eigenvalue and v is the corresponding eigenvector. The characteristic equation is det(A – λI) = 0. For a 2×2 matrix, this is a quadratic equation; for a 3×3 matrix, it is cubic. Eigenvectors are used for diagonalization of matrices and for solving systems of coupled differential equations.

三、双曲函数 | 3. Hyperbolic Functions

3.1 定义与基本恒等式 | Definitions and Basic Identities

中文：双曲函数是指数函数的组合，其定义如下：

sinh x = (e^x – e^(-x)) / 2

cosh x = (e^x + e^(-x)) / 2

tanh x = sinh x / cosh x = (e^x – e^(-x)) / (e^x + e^(-x))

基本恒等式：cosh^2 x – sinh^2 x = 1（与三角恒等式 cos^2θ + sin^2θ = 1 类似但符号不同）。其他重要恒等式包括：sinh(2x) = 2 sinh x cosh x，cosh(2x) = cosh^2 x + sinh^2 x = 2cosh^2 x – 1 = 1 + 2sinh^2 x。

English: Hyperbolic functions are combinations of exponential functions, defined as follows:

sinh x = (e^x – e^(-x)) / 2

cosh x = (e^x + e^(-x)) / 2

tanh x = sinh x / cosh x = (e^x – e^(-x)) / (e^x + e^(-x))

The fundamental identity: cosh^2 x – sinh^2 x = 1 (similar to the trigonometric identity cos^2θ + sin^2θ = 1 but with a sign difference). Other important identities include: sinh(2x) = 2 sinh x cosh x, cosh(2x) = cosh^2 x + sinh^2 x = 2cosh^2 x – 1 = 1 + 2sinh^2 x.

3.2 反双曲函数与微分 | Inverse Hyperbolic Functions and Differentiation

中文：反双曲函数的对数形式非常重要：

arsinh x = ln(x + sqrt(x^2 + 1))

arcosh x = ln(x + sqrt(x^2 – 1)), 其中 x >= 1

artanh x = (1/2) ln((1 + x)/(1 – x)), 其中 |x| < 1

导数公式：d/dx(sinh x) = cosh x，d/dx(cosh x) = sinh x，d/dx(tanh x) = sech^2 x，d/dx(arsinh x) = 1/sqrt(x^2 + 1)，d/dx(arcosh x) = 1/sqrt(x^2 – 1)，d/dx(artanh x) = 1/(1 – x^2)。

English: The logarithmic forms of inverse hyperbolic functions are very important:

arsinh x = ln(x + sqrt(x^2 + 1))

arcosh x = ln(x + sqrt(x^2 – 1)), where x >= 1

artanh x = (1/2) ln((1 + x)/(1 – x)), where |x| < 1

Differentiation formulae: d/dx(sinh x) = cosh x, d/dx(cosh x) = sinh x, d/dx(tanh x) = sech^2 x, d/dx(arsinh x) = 1/sqrt(x^2 + 1), d/dx(arcosh x) = 1/sqrt(x^2 – 1), d/dx(artanh x) = 1/(1 – x^2).

四、微分方程 | 4. Differential Equations

4.1 一阶微分方程 | First-Order Differential Equations

中文：一阶可分离变量方程：dy/dx = f(x)g(y) 可写为 integral (1/g(y)) dy = integral f(x) dx。一阶线性微分方程的标准形式为 dy/dx + P(x)y = Q(x)，其积分因子（integrating factor）为 μ(x) = e^(integral P(x) dx)。通解为 y μ(x) = integral μ(x)Q(x) dx + C。

English: First-order separable equation: dy/dx = f(x)g(y) can be written as integral (1/g(y)) dy = integral f(x) dx. The standard form of a first-order linear differential equation is dy/dx + P(x)y = Q(x), with integrating factor μ(x) = e^(integral P(x) dx). The general solution is y μ(x) = integral μ(x)Q(x) dx + C.

4.2 二阶线性齐次微分方程 | Second-Order Linear Homogeneous ODEs

中文：标准形式：a d^2y/dx^2 + b dy/dx + c y = 0。设特解形式 y = e^(mx)，代入得到特征方程（auxiliary equation）：a m^2 + b m + c = 0。根据判别式 Δ = b^2 – 4ac 的不同情况，有以下通解形式：

Δ > 0（两个不等实根 m1, m2）：y = A e^(m1 x) + B e^(m2 x)

Δ = 0（重根 m）：y = (A + Bx) e^(mx)

Δ < 0（共轭复根 α +- iβ）：y = e^(αx) (A cos(βx) + B sin(βx))

English: Standard form: a d^2y/dx^2 + b dy/dx + c y = 0. Assume a particular solution of the form y = e^(mx), substituting yields the auxiliary equation: a m^2 + b m + c = 0. Depending on the discriminant Δ = b^2 – 4ac, the general solution takes the following forms:

Δ > 0 (two distinct real roots m1, m2): y = A e^(m1 x) + B e^(m2 x)

Δ = 0 (repeated root m): y = (A + Bx) e^(mx)

Δ < 0 (complex conjugate roots α +- iβ): y = e^(αx) (A cos(βx) + B sin(βx))

4.3 非齐次二阶微分方程 | Non-Homogeneous Second-Order ODEs

中文：标准形式：a d^2y/dx^2 + b dy/dx + c y = f(x)。通解 = 余函数（Complementary Function, CF）+ 特积分（Particular Integral, PI）。CF 即对应齐次方程的通解。PI 的求解采用待定系数法，根据 f(x) 的形式设特解：

f(x) = 多项式：设 y_p = 同次多项式

f(x) = k e^(px)：设 y_p = λ e^(px)，若 p 是特征根则乘以 x

f(x) = k cos(ωx) 或 k sin(ωx)：设 y_p = P cos(ωx) + Q sin(ωx)

English: Standard form: a d^2y/dx^2 + b dy/dx + c y = f(x). General solution = Complementary Function (CF) + Particular Integral (PI). CF is the general solution of the corresponding homogeneous equation. PI is found using the method of undetermined coefficients, assuming a particular solution based on the form of f(x):

f(x) = polynomial: assume y_p = polynomial of the same degree

f(x) = k e^(px): assume y_p = λ e^(px), multiply by x if p is a root of the auxiliary equation

f(x) = k cos(ωx) or k sin(ωx): assume y_p = P cos(ωx) + Q sin(ωx)

五、极坐标 | 5. Polar Coordinates

5.1 坐标系转换 | Coordinate System Conversion

中文：极坐标 (r, θ) 与直角坐标 (x, y) 的转换关系为：x = r cosθ，y = r sinθ。反之，r = sqrt(x^2 + y^2)，θ = arctan(y/x)（注意根据象限调整）。典型的极坐标曲线包括：r = a（圆），r = aθ（阿基米德螺线），r = a(1 + cosθ)（心形线），r^2 = a^2 cos(2θ)（双纽线）。

English: The conversion between polar coordinates (r, θ) and Cartesian coordinates (x, y) is: x = r cosθ, y = r sinθ. Conversely, r = sqrt(x^2 + y^2), θ = arctan(y/x) (adjusted for the correct quadrant). Typical polar curves include: r = a (circle), r = aθ (Archimedean spiral), r = a(1 + cosθ) (cardioid), r^2 = a^2 cos(2θ) (lemniscate).

5.2 极坐标系中的面积 | Area in Polar Coordinates

中文：极曲线 r = f(θ) 在角度区间 [α, β] 内所围成的扇形面积公式为：A = (1/2) integral[α to β] f(θ)^2 dθ。对于两条极曲线 r = r1(θ) 和 r = r2(θ) 之间的区域面积，使用 A = (1/2) integral (r1^2 – r2^2) dθ。需要特别注意的是，必须先确定积分区域中哪条曲线位于外侧。

English: The area bounded by the polar curve r = f(θ) in the angular interval [α, β] is given by: A = (1/2) integral[α to β] f(θ)^2 dθ. For the area between two polar curves r = r1(θ) and r = r2(θ), use A = (1/2) integral (r1^2 – r2^2) dθ. It is particularly important to first determine which curve lies on the outside within the integration region.

六、向量 | 6. Vectors

6.1 向量积 | Vector Products

中文：标量积（点积）：a · b = |a||b| cosθ = a1 b1 + a2 b2 + a3 b3，结果是一个标量。若 a · b = 0，则两向量垂直。

向量积（叉积）：a x b 的结果是一个向量，其方向由右手定则确定，大小为 |a x b| = |a||b| sinθ。在直角坐标系中：a x b = (a2 b3 – a3 b2)i + (a3 b1 – a1 b3)j + (a1 b2 – a2 b1)k。向量积在计算平面法向量和力矩方面有广泛应用。

English: Scalar product (dot product): a · b = |a||b| cosθ = a1 b1 + a2 b2 + a3 b3, yielding a scalar. If a · b = 0, the two vectors are perpendicular.

Vector product (cross product): a x b yields a vector whose direction is given by the right-hand rule, with magnitude |a x b| = |a||b| sinθ. In Cartesian coordinates: a x b = (a2 b3 – a3 b2)i + (a3 b1 – a1 b3)j + (a1 b2 – a2 b1)k. The vector product has wide applications in computing plane normal vectors and moments.

6.2 直线与平面的方程 | Equations of Lines and Planes

中文：空间中直线的向量方程：r = a + λd，其中 a 是直线上一点的位置向量，d 是方向向量，λ 是参数。直线的参数方程为：x = a1 + λ d1, y = a2 + λ d2, z = a3 + λ d3。平面的向量方程为 r · n = d，其中 n 是平面法向量，d 是常数。平面的笛卡尔形式：n1 x + n2 y + n3 z = d。

English: Vector equation of a line in space: r = a + λd, where a is the position vector of a point on the line, d is the direction vector, and λ is a parameter. The parametric equations of the line are: x = a1 + λ d1, y = a2 + λ d2, z = a3 + λ d3. The vector equation of a plane is r · n = d, where n is the normal vector to the plane and d is a constant. The Cartesian form of a plane: n1 x + n2 y + n3 z = d.

6.3 点线面的距离 | Distances Between Points, Lines, and Planes

中文：点 P 到平面 r · n = d 的距离为 |p · n – d| / |n|，其中 p 是点 P 的位置向量。两条异面直线的最短距离可通过叉积和混合积（标量三重积）计算。特别地，两条异面直线 r = a + λd1 和 r = b + μd2 之间的最短距离为 |(b – a) · (d1 x d2)| / |d1 x d2|。

English: The distance from point P to the plane r · n = d is |p · n – d| / |n|, where p is the position vector of point P. The shortest distance between two skew lines can be calculated using cross and scalar triple products. Specifically, the shortest distance between skew lines r = a + λd1 and r = b + μd2 is |(b – a) · (d1 x d2)| / |d1 x d2|.

七、级数求和 | 7. Summation of Series

7.1 标准求和公式 | Standard Summation Formulae

中文：以下标准求和公式是进阶数学中的基础工具：

sum[r=1 to n] 1 = n

sum[r=1 to n] r = n(n+1)/2

sum[r=1 to n] r^2 = n(n+1)(2n+1)/6

sum[r=1 to n] r^3 = [n(n+1)/2]^2

这些公式可用于求和各类多项式级数。通过将表达式拆分为 r 的各次幂的线性组合，再利用上述标准公式即可求出一般情况下的通项和。

English: The following standard summation formulae are fundamental tools in Further Mathematics:

sum[r=1 to n] 1 = n

sum[r=1 to n] r = n(n+1)/2

sum[r=1 to n] r^2 = n(n+1)(2n+1)/6

sum[r=1 to n] r^3 = [n(n+1)/2]^2

These formulae can be used to sum various polynomial series. By expressing a given expression as a linear combination of powers of r and then applying the standard formulae above, the general term sum can be determined.

7.2 数学归纳法 | Proof by Mathematical Induction

中文：数学归纳法是证明级数求和公式和其他数列性质的重要方法。归纳证明包括三个步骤：

第一步（基础情况）：证明当 n = 1（或某个初始值）时命题成立。

第二步（归纳假设）：假设当 n = k 时命题成立。

第三步（归纳递推）：利用归纳假设证明当 n = k+1 时命题也成立。

结论：由数学归纳法原理，命题对所有正整数 n 成立。

English: Mathematical induction is an essential method for proving summation formulae and other properties of sequences. An inductive proof consists of three steps:

Step 1 (Base case): Prove the proposition holds for n = 1 (or some initial value).

Step 2 (Inductive hypothesis): Assume the proposition holds for n = k.

Step 3 (Inductive step): Using the inductive hypothesis, prove the proposition also holds for n = k+1.

Conclusion: By the principle of mathematical induction, the proposition holds for all positive integers n.

八、备考策略与技巧 | 8. Exam Strategies and Tips

中文：Pre-U CIE 进阶数学考试不仅考察公式记忆，更注重理解与应用。以下是高效备考的关键建议：

建立公式卡片：将本文中的核心公式制作为记忆卡片，每日复习。特别关注双曲函数与三角函数之间的相似性与差异。
理解推导过程：不要死记硬背公式。理解 Standard Integral 的推导过程、棣莫弗定理的证明，以及各类微分方程解法背后的原理。
综合题练习：Pre-U 考试的综合性非常强。一道题可能同时涉及复数、向量和微分方程。建议多做历年真题中的综合题。
关注边界条件：许多公式有定义域限制（如 artanh x 要求 |x| < 1，arcosh x 要求 x >= 1）。在考试中忽略这些条件会导致失分。
时间管理：Pre-U 进阶数学的题量较大，平时练习时就要注意时间分配，确保每道题都能在规定时间内完成。

English: The Pre-U CIE Further Mathematics examination tests not only formula memorization but also deeper understanding and application. Here are key recommendations for effective preparation:

Create formula flashcards: Turn the core formulas in this article into flashcards and review them daily. Pay particular attention to the similarities and differences between hyperbolic and trigonometric functions.
Understand the derivations: Do not simply memorize formulas by rote. Understand the derivations of standard integrals, the proof of De Moivre’s Theorem, and the principles behind the various methods for solving differential equations.
Practice integrated problems: Pre-U examinations are highly integrative. A single question may simultaneously involve complex numbers, vectors, and differential equations. It is recommended to practice integrated questions from past papers extensively.
Pay attention to boundary conditions: Many formulae have domain restrictions (e.g., artanh x requires |x| < 1, arcosh x requires x >= 1). Ignoring these conditions in the examination will result in lost marks.
Time management: Pre-U Further Mathematics papers contain a substantial number of questions. During practice, always manage your time allocation to ensure every question can be completed within the allotted time.

九、麦克劳林级数 | 9. Maclaurin Series

9.1 标准展开式 | Standard Expansions

中文：麦克劳林级数是泰勒级数在 x = 0 处的特例。以下标准展开式必须牢记：

e^x = 1 + x + x^2/2! + x^3/3! + … = sum[n=0 to inf] x^n / n!

sin x = x – x^3/3! + x^5/5! – … = sum[n=0 to inf] (-1)^n x^(2n+1) / (2n+1)!

cos x = 1 – x^2/2! + x^4/4! – … = sum[n=0 to inf] (-1)^n x^(2n) / (2n)!

ln(1 + x) = x – x^2/2 + x^3/3 – … = sum[n=1 to inf] (-1)^(n+1) x^n / n, 对于 -1 < x <= 1

(1 + x)^n = 1 + nx + [n(n-1)/2!]x^2 + [n(n-1)(n-2)/3!]x^3 + … （二项式展开）

English: The Maclaurin series is the special case of the Taylor series at x = 0. The following standard expansions must be memorized:

e^x = 1 + x + x^2/2! + x^3/3! + … = sum[n=0 to inf] x^n / n!

sin x = x – x^3/3! + x^5/5! – … = sum[n=0 to inf] (-1)^n x^(2n+1) / (2n+1)!

cos x = 1 – x^2/2! + x^4/4! – … = sum[n=0 to inf] (-1)^n x^(2n) / (2n)!

ln(1 + x) = x – x^2/2 + x^3/3 – … = sum[n=1 to inf] (-1)^(n+1) x^n / n, for -1 < x <= 1

(1 + x)^n = 1 + nx + [n(n-1)/2!]x^2 + [n(n-1)(n-2)/3!]x^3 + … (binomial expansion)

9.2 函数的级数展开 | Series Expansion of Functions

中文：求函数 f(x) 的麦克劳林级数可通过两种方式：(1) 直接利用公式 f(x) = f(0) + f'(0)x + f”(0)x^2/2! + …；(2) 利用标准展开式进行代换。例如，e^(x^2) 的展开可通过将 e^u 的展开式中的 u 替换为 x^2 得到。此外，两个级数的乘积、除法以及复合函数的展开也是常考题型。

English: There are two approaches to finding the Maclaurin series of a function f(x): (1) directly using the formula f(x) = f(0) + f'(0)x + f”(0)x^2/2! + …; (2) using standard expansions with substitution. For example, the expansion of e^(x^2) can be obtained by replacing u with x^2 in the expansion of e^u. Additionally, the product of two series, division, and the expansion of composite functions are also commonly examined.

十、进一步积分技巧 | 10. Further Integration Techniques

10.1 递推公式 | Reduction Formulae

中文：递推公式（reduction formula）用于求解形如 I_n = integral x^n e^(ax) dx 或 I_n = integral sin^n x dx 的积分。通过分部积分法（integration by parts）建立 I_n 与 I_(n-1)（或更小的下标）之间的关系，反复递推直至可解。例如：

I_n = integral[0 to pi/2] sin^n x dx

递推公式：I_n = (n-1)/n · I_(n-2)，边界条件：I_0 = pi/2, I_1 = 1

由此可得 Wallis 公式，用于计算正弦和余弦各次幂的积分。

English: Reduction formulae are used to evaluate integrals of the form I_n = integral x^n e^(ax) dx or I_n = integral sin^n x dx. By using integration by parts, a relationship is established between I_n and I_(n-1) (or lower indices), and the reduction is repeated until the integral can be solved. For example:

I_n = integral[0 to pi/2] sin^n x dx

Reduction formula: I_n = (n-1)/n · I_(n-2), with boundary conditions: I_0 = pi/2, I_1 = 1

From this, Wallis’s formula can be derived for computing integrals of powers of sine and cosine.

10.2 弧长与旋转体表面积 | Arc Length and Surface Area of Revolution

中文：直角坐标系中曲线 y = f(x) 在区间 [a, b] 上的弧长：s = integral[a to b] sqrt(1 + (dy/dx)^2) dx。参数方程 x = x(t), y = y(t) 在区间 [t1, t2] 上的弧长：s = integral[t1 to t2] sqrt((dx/dt)^2 + (dy/dt)^2) dt。曲线绕 x 轴旋转所得曲面的表面积：S = 2π integral[a to b] y sqrt(1 + (dy/dx)^2) dx。

English: Arc length of the curve y = f(x) on the interval [a, b] in Cartesian coordinates: s = integral[a to b] sqrt(1 + (dy/dx)^2) dx. Arc length for parametric equations x = x(t), y = y(t) on the interval [t1, t2]: s = integral[t1 to t2] sqrt((dx/dt)^2 + (dy/dt)^2) dt. Surface area of revolution when the curve is rotated about the x-axis: S = 2π integral[a to b] y sqrt(1 + (dy/dx)^2) dx.

十一、典型例题精讲 | 11. Worked Examples

例题 1：复数方程 | Example 1: Complex Equations

中文：求解方程 z^4 = 8(1 + i sqrt(3))，将答案表示为极坐标形式 r(cosθ + i sinθ)。

解题步骤：

1. 将右边写为极坐标形式：8(1 + i sqrt(3)) = 8 · 2(cos(π/3) + i sin(π/3)) = 16(cos(π/3) + i sin(π/3))

2. 由棣莫弗定理，z^4 = 16(cos(π/3) + i sin(π/3))，则 z = 16^(1/4) [cos((π/3 + 2kπ)/4) + i sin((π/3 + 2kπ)/4)]，k = 0, 1, 2, 3

3. r = 16^(1/4) = 2；四个解的角度分别为：π/12, 7π/12, 13π/12, 19π/12

4. 四个解为：z = 2(cos(π/12) + i sin(π/12)), z = 2(cos(7π/12) + i sin(7π/12)), z = 2(cos(13π/12) + i sin(13π/12)), z = 2(cos(19π/12) + i sin(19π/12))

English: Solve the equation z^4 = 8(1 + i sqrt(3)), giving answers in polar form r(cosθ + i sinθ).

Solution steps:

1. Write the right side in polar form: 8(1 + i sqrt(3)) = 8 · 2(cos(π/3) + i sin(π/3)) = 16(cos(π/3) + i sin(π/3))

2. By De Moivre’s Theorem, z^4 = 16(cos(π/3) + i sin(π/3)), so z = 16^(1/4) [cos((π/3 + 2kπ)/4) + i sin((π/3 + 2kπ)/4)], k = 0, 1, 2, 3

3. r = 16^(1/4) = 2; the four angles are: π/12, 7π/12, 13π/12, 19π/12

4. The four solutions: z = 2(cos(π/12) + i sin(π/12)), z = 2(cos(7π/12) + i sin(7π/12)), z = 2(cos(13π/12) + i sin(13π/12)), z = 2(cos(19π/12) + i sin(19π/12))

例题 2：微分方程 | Example 2: Differential Equation

中文：求解微分方程 d^2y/dx^2 – 3 dy/dx + 2y = 10 e^x，初始条件为 y(0) = 2, y'(0) = 1。

解题步骤：

1. 特征方程：m^2 – 3m + 2 = 0，解得 m = 1 或 m = 2。

2. 余函数：CF = A e^x + B e^(2x)。

3. 求特积分：由于 f(x) = 10 e^x，设 PI = Cx e^x（注意乘以 x，因为 m = 1 是特征根）。代入方程得 C = -10。因此 PI = -10x e^x。

4. 通解：y = A e^x + B e^(2x) – 10x e^x。

5. 代入初始条件：y(0) = A + B = 2；y'(0) = A + 2B – 10 = 1。解得 A = 5, B = -3。

6. 特解：y = 5 e^x – 3 e^(2x) – 10x e^x。

English: Solve the differential equation d^2y/dx^2 – 3 dy/dx + 2y = 10 e^x, with initial conditions y(0) = 2, y'(0) = 1.

Solution steps:

1. Auxiliary equation: m^2 – 3m + 2 = 0, giving m = 1 or m = 2.

2. Complementary function: CF = A e^x + B e^(2x).

3. Find particular integral: Since f(x) = 10 e^x, assume PI = Cx e^x (note: multiply by x because m = 1 is a root of the auxiliary equation). Substituting yields C = -10. Hence PI = -10x e^x.

4. General solution: y = A e^x + B e^(2x) – 10x e^x.

5. Apply initial conditions: y(0) = A + B = 2; y'(0) = A + 2B – 10 = 1. Solving gives A = 5, B = -3.

6. Particular solution: y = 5 e^x – 3 e^(2x) – 10x e^x.

总结 | Summary

中文：Pre-U CIE 进阶数学是通往顶尖大学数学、物理和工程专业的重要敲门砖。掌握本文中整理的复数、矩阵、双曲函数、微分方程、极坐标和向量六大模块的核心公式，理解其推导过程和适用条件，辅以充分的真题练习，你一定能在考试中游刃有余。希望这份公式定理速查手册成为你复习路上的得力助手。

English: Pre-U CIE Further Mathematics is an important stepping stone towards top university programmes in mathematics, physics, and engineering. By mastering the core formulas across the six major modules covered in this article – complex numbers, matrices, hyperbolic functions, differential equations, polar coordinates, and vectors – understanding their derivations and conditions of applicability, and supplementing with ample past paper practice, you will undoubtedly excel in the examination. We hope this Quick Reference Formula Handbook serves as a valuable companion on your revision journey.