Edexcel Further Maths Core Pure 2: Key Topics and Exam Strategies | Edexcel 进阶数学 Core Pure 2：核心考点与备考策略

📚 Edexcel Further Maths Core Pure 2: Key Topics and Exam Strategies | Edexcel 进阶数学 Core Pure 2：核心考点与备考策略

Edexcel Further Maths Core Pure 2 extends Core Pure 1 with deeper complex numbers, polar coordinates, hyperbolic functions, series and differential equations. This revision guide summarises the key techniques and common exam pitfalls.

Edexcel 进阶数学 Core Pure 2 在 Core Pure 1 的基础上深入拓展复数、极坐标、双曲函数、级数和微分方程等内容。本复习指南总结关键技巧与常见考试陷阱。

1. Complex Numbers and de Moivre’s Theorem | 复数与棣莫弗定理

In Core Pure 2, complex numbers are expressed in modulus-argument form z = r(cos θ + i sin θ) and exponential form z = re^(iθ). De Moivre’s theorem states that for any integer n, (cos θ + i sin θ)ⁿ = cos nθ + i sin nθ. This gives a fast way to raise complex numbers to powers.

在 Core Pure 2 中，复数用模-辐角形式 z = r(cos θ + i sin θ) 和指数形式 z = re^(iθ) 表示。棣莫弗定理指出，对任何整数 n 有 (cos θ + i sin θ)ⁿ = cos nθ + i sin nθ。它提供了快速计算复数幂的方法。

zⁿ = rⁿ(cos nθ + i sin nθ) = rⁿ e^(inθ)

You must also use de Moivre’s theorem to derive multiple-angle identities, such as cos 3θ = 4 cos³ θ − 3 cos θ, by expanding (cos θ + i sin θ)³ and equating real parts.

你还必须会用棣莫弗定理推导倍角恒等式，例如 cos 3θ = 4 cos³ θ − 3 cos θ，方法是展开 (cos θ + i sin θ)³ 并比较实部。

A common mistake is taking the wrong quadrant for θ = arg z. Always sketch the Argand diagram: z = 1 + i√3 has r = 2 and θ = π/3, so z⁶ = 64(cos 2π + i sin 2π) = 64.

常见错误包括对 θ = arg z 取错象限。一定要画 Argand 图：例如 z = 1 + i√3 有 r = 2 且 θ = π/3，所以 z⁶ = 64(cos 2π + i sin 2π) = 64。

2. Roots of Unity and Complex Roots | 单位根与复数根

To solve zⁿ = w, write w in exponential form and take the nth root. The n distinct roots are evenly spaced around a circle of radius r^(1/n), separated by angle 2π/n.

要求解 zⁿ = w，先把 w 写成指数形式并开 n 次方。n 个不同根均匀分布在半径为 r^(1/n) 的圆上，相邻根的角度差为 2π/n。

zₖ = r^(1/n) [ cos((θ + 2kπ)/n) + i sin((θ + 2kπ)/n) ], k = 0, 1, …, n−1

Roots of unity occur when w = 1. For example, the cube roots of unity are 1, ω = −½ + (√3/2)i and ω² = −½ − (√3/2)i, and they satisfy 1 + ω + ω² = 0.

当 w = 1 时得到的根称为单位根。例如三次单位根为 1、ω = −½ + (√3/2)i 和 ω² = −½ − (√3/2)i，并且满足 1 + ω + ω² = 0。

Exam questions often ask for the sum or product of roots, or for a polynomial with given complex roots. Use the symmetry of the circle to simplify the calculation.

考题经常要求根的和或积，或写出具有给定复数根的多项式。利用圆的对称性可以简化计算。

3. Loci in the Complex Plane | 复平面上的轨迹

Loci problems ask you to describe or sketch sets of points satisfying conditions such as |z − a| = r, |z − a| = |z − b|, or arg(z − a) = θ.

轨迹问题要求描述或画出满足条件的点集，例如 |z − a| = r、|z − a| = |z − b| 或 arg(z − a) = θ。

The equation |z − a| = r is a circle centre a and radius r; |z − a| = |z − b| is the perpendicular bisector of the segment joining a and b; arg(z − a) =

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