A-Level Further Mathematics: End-of-Term Revision Guide | A-Level 进阶数学：期末复习提纲

📚 A-Level Further Mathematics: End-of-Term Revision Guide | A-Level 进阶数学：期末复习提纲

This comprehensive revision guide covers the key topics in A-Level Further Mathematics, providing a structured overview to help you consolidate your understanding before the end-of-term exam. Each section presents essential definitions, formulas, methods, and common pitfalls, with content directly aligned to the Core Pure syllabus and selected optional modules. Use this outline alongside past papers and your class notes for effective revision.

这份全面的复习提纲涵盖了A-Level进阶数学的核心主题，提供了一个有结构的概览，帮助你在期末考前巩固理解。每个部分都给出了基本定义、公式、方法和常见易错点，内容严格对标Core Pure考纲以及部分选修模块。请结合历年真题和课堂笔记一同使用，以实现高效复习。

1. Complex Numbers | 复数

Complex numbers extend real numbers by including the imaginary unit i, where i² = -1. A complex number can be written in Cartesian form z = a + bi, with real part a and imaginary part b. The complex conjugate is z* = a – bi. The modulus is |z| = √(a² + b²) and the argument is arg(z) = θ, where tanθ = b/a (adjusted for the correct quadrant). The polar form z = r(cosθ + i sinθ) and the exponential form z = r e^iθ using Euler’s formula are essential for multiplication, division and powers. De Moivre’s theorem states (cosθ + i sinθ)ⁿ = cos(nθ) + i sin(nθ) for integer n, and is the key to finding roots of complex numbers.

复数通过引入虚数单位 i（i² = -1）扩展了实数。复数可以用笛卡尔形式 z = a + bi 表示，实部为 a，虚部为 b。共轭复数为 z* = a – bi。模长 |z| = √(a² + b²)，辐角 arg(z) = θ，满足 tanθ = b/a（注意正确象限）。极坐标形式 z = r(cosθ + i sinθ) 以及利用欧拉公式得到的指数形式 z = r e^iθ 是进行乘、除和乘方运算的核心。棣莫弗定理指出 (cosθ + i sinθ)ⁿ = cos(nθ) + i sin(nθ) 对整数 n 成立，而且是求复数方根的关键。

To find the n-th roots of a complex number, first express it in polar form; then the roots are z_k = r^(1/n)[cos((θ+2πk)/n) + i sin((θ+2πk)/n)], k = 0,1,…,n-1. Geometric representation on the Argand diagram helps visualise loci such as |z – a| = r (circle) or arg(z – a) = θ (ray).

求复数的 n 次方根时，先将它表示为极坐标形式，则根为 z_k = r^(1/n)[cos((θ+2πk)/n) + i sin((θ+2πk)/n)]，k = 0,1,…,n-1。在阿干特图上的几何表示有助于理解轨迹，例如 |z – a| = r 表示圆，arg(z – a) = θ 表示射线。

2. Matrices and Transformations | 矩阵与变换

Matrices represent linear transformations. Multiplying a column vector by a matrix transforms coordinates. Key transformations in 2D include reflections (in the x-axis, y-axis, y = x, y = -x), rotation about the origin by angle θ, enlargement by scale factor k, and shears. Composing two transformations corresponds to multiplying the associated matrices in reverse order.

矩阵表示线性变换。列向量左乘矩阵便实现了坐标变换。2D空间中重要的变换包括：反射（关于 x 轴、y 轴、y = x、y = -x）、绕原点旋转 θ 角、按比例 k 放大以及剪切。两个变换的复合对应于按逆序相乘对应的矩阵。

The determinant of a 2×2 matrix gives the area scale factor; a zero determinant means the matrix is singular and has no inverse. Eigenvalues λ satisfy det(A – λI) = 0, and eigenvectors v satisfy Av = λv. Diagonalisation can be used to compute high powers of matrices. Systems of linear equations AX = B can be solved using the inverse matrix or row reduction.

2×2矩阵的行列式表示面积缩放因子；行列式为零意味着矩阵是奇异的，没有逆矩阵。特征

Published by TutorHao | A-Level Mathematics Revision Series | aleveler.com

更多咨询请联系16621398022（同微信）