Mastering Edexcel Core Pure Mathematics 2 | 掌握 Edexcel 核心纯数学2

📚 Mastering Edexcel Core Pure Mathematics 2 | 掌握 Edexcel 核心纯数学2

Core Pure Mathematics 2 (CP2) is a central component of Edexcel A level Further Mathematics. It extends Core Pure 1 ideas into advanced complex numbers, series, polar coordinates, hyperbolic functions, reduction formulae and differential equations. This article summarises the key techniques, common formulas and typical exam traps for CP2.

核心纯数学2（CP2）是 Edexcel A level 进阶数学的核心模块之一。它在核心纯数学1的基础上，进一步拓展复数、级数、极坐标、双曲函数、递推公式和微分方程等内容。本文总结 CP2 的主要解题方法、常用公式和常见考试易错点。

1. Complex Numbers in Exponential Form | 复数的指数形式

Any non-zero complex number can be written in exponential form z = re^iθ, where r = |z| is the modulus and θ = arg z is the argument. Euler's formula e^iθ = cos θ + i sin θ connects exponential notation with the modulus-argument form.

任何非零复数都可以写成指数形式 z = re^iθ，其中 r = |z| 是模，θ = arg z 是辐角。欧拉公式 e^iθ = cos θ + i sin θ 将指数记法与模-辐角形式联系起来。

e^iθ = cos θ + i sin θ,    z = re^iθ

Multiplication and division become very simple in exponential form: z₁z₂ = r₁r₂ e^{i(θ₁+θ₂)} and z₁ ÷ z₂ = r₁/r₂ e^{i(θ₁−θ₂)}. Powers also follow zⁿ = rⁿ e^inθ.

指数形式下乘法和除法变得非常简单：z₁z₂ = r₁r₂ e^{i(θ₁+θ₂)}，z₁ ÷ z₂ = r₁/r₂ e^{i(θ₁−θ₂)}。幂运算也遵循 zⁿ = rⁿ e^inθ。

For example, if z = 2e^iπ/6, then z³ = 2³e^iπ/2 = 8i. This is usually faster than expanding a trigonometric binomial.

例如，若 z = 2e^iπ/6，则 z³ = 2³e^iπ/2 = 8i。这通常比展开三角二项式更快。

2. De Moivre's Theorem and Trigonometric Identities | 棣莫弗定理与三角恒等式

De Moivre' also works for negative and rational powers, but fractional powers introduce multiple values because roots are not unique.

棣莫弗定理指出，对于任意整数 n，(cos θ + i sin θ)ⁿ = cos nθ + i sin nθ。该定理也适用于负整数和有理数幂，但分数次幂会引入多值性，因为根不是唯一的。

(cos θ + i sin θ)ⁿ = cos nθ + i sin nθ

ding (cos θ + i sin θ)³ gives cos 3θ = 4 cos³ θ − 3 cos θ and sin 3θ = 3 sin θ − 4 sin³ θ.

推导三角恒等式时，可以用二项式定理展开左边，然后分别令实部和虚部相等。例如，展开 (cos θ + i sin θ)³ 可得 cos 3θ = 4 cos³ θ − 3 cos θ 以及 sin 3θ = 3 sin θ − 4 sin³ θ。

ing powers of sin θ or cos θ as multiple angles, such as cos⁴ θ = ⅜ + ½ cos 2θ + ⅛ cos 4θ.

该方法在将 sin θ 或 cos θ 的幂表示为倍角时特别有用，例如 cos⁴ θ = ⅜ + ½ cos 2θ + ⅛ cos 4θ。

3. Roots of Unity and Polynomial Roots | 单位根与多项式根

The equation zⁿ = 1 has exactly n distinct roots, given by zₖ = e^2πik/n , 1, 2, ..., n−1. These roots lie on the unit circle and are equally spaced by angle 2π/n.

方程 zⁿ = 1 有 n 个不同的根，由 zₖ = e^2πik/n 给出，其中 k = 0, 1, 2, ..., n−1。这些根位于单位圆上，并且相邻两根之间的夹角为 2π/n。

zₖ = e^2πik/n,    k = 0, 1, ..., n−1

risation zⁿ − 1 = (z−1)(zⁿ⁻¹ + zⁿ⁻² + ... + z + 1), since the coefficient of zⁿ⁻¹ is zero.

所有 n 次单位根的和为零。这可以通过因式分解 zⁿ − 1 = (z−1)(zⁿ⁻¹ + zⁿ⁻² + ... + z + 1) 证明，因为 zⁿ⁻¹ 的系数为零。

nth root of unity, then the roots can be written as 1, ω, ω², ..., ωⁿ⁻¹. For n = 3, ω = e^2πi/3 and the roots are 1, ω, ω², with 1 + ω + ω² = 0 and ω³ = 1.

如果 ω 是一个 n 次本原单位根，则所有根可以写成 1, ω, ω², ..., ωⁿ⁻¹。当 n = 3 时，ω = e^2πi/3，三个根为 1, ω, ω²，且满足 1 + ω + ω² = 0 和 ω³ = 1。

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4. Complex Loci and Transformations | 复数轨迹与变换

in the complex plane include circles, perpendicular bisectors and half-lines. The equation |z − z₀| = r describes a circle with centre z₀ and radius r. |z − z₁| = |z − z₂| is the perpendicular bisector of the line segment joining z₁ and z₂.

复平面上的常见轨迹包括圆、垂直平分线和射线。方程 |z − z₀| = r 表示以 z₀ 为圆心、半径为 r 的圆。|z − z₁| = |z − z₂| 表示连接 z₁ 和 z₂ 的线段的垂直平分线。

a half-line starting at z₀ and making angle θ with the positive real direction. You should always check whether the endpoint z₀ is included or excluded.

方程 arg(z − z₀) = θ 表示从 z₀ 出发且与正实轴方向夹角为 θ 的射线。解题时必须注意端点 z₀ 是否包含在轨迹内。

lines. It is often easiest to substitute z = x + iy and separate real and imaginary parts.

变换 w = z + a 表示按向量 a 平移。w = iz 将复平面逆时针旋转 90°，而 w = 1/z 将圆和直线映射为圆或直线。通常最简便的方法是代入 z = x + iy，然后分离实部和虚部。

5. Series: Method of Differences and Standard Sums | 级数：差分法与标准和

CP2 expects fluency with the standard summation results. These are often combined with algebraic manipulation or partial fractions.

CP2 要求学生熟练掌握标准求和公式。这些公式通常会与代数变形或部分分式结合使用。

Σ r = n(n+1)/2,   Σ r² = n(n+1)(2n+1)/6,   Σ r³ = n²(n+1)²/4

. Summing from r = 1 to n makes most terms cancel, leaving only the first and last pieces.

差分法适用于一般项可以写成 uᵣ − uᵣ₊₁ 或类似差分形式的级数。对 r = 1 到 n

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