AQA International AS FP1 Pure Maths: Core Methods and Exam Skills | AQA 国际 AS FP1 纯数学：核心方法与应试技巧

📚 AQA International AS FP1 Pure Maths: Core Methods and Exam Skills | AQA 国际 AS FP1 纯数学：核心方法与应试技巧

Welcome to the AQA International AS Further Pure 1 (FP1) unit. This paper develops pure mathematics beyond the standard AS course: complex numbers, roots of polynomials, numerical solutions, matrices, series, proof by induction and parametric coordinate systems. The focus is on precise algebraic technique, clear reasoning and efficient exam methods.

欢迎学习 AQA 国际 AS 进阶纯数学 1（FP1）单元。本卷在标准 AS 数学基础上拓展纯数学内容：复数、多项式根、数值求解、矩阵、级数、数学归纳法和参数坐标系。学习重点是精确的代数技巧、清晰的推理和高效的应试方法。

1. Complex Numbers: Basic Operations and Conjugates | 复数：基本运算与共轭

Define i = √(−1), so i² = −1. A complex number has the form z = a + bi, where a is the real part Re(z) and b is the imaginary part Im(z). Two complex numbers are equal only when their real parts and imaginary parts are both equal.

定义 i = √(−1)，因此 i² = −1。复数形如 z = a + bi，其中 a 是实部 Re(z)，b 是虚部 Im(z)。两个复数相等，当且仅当它们的实部和虚部分别相等。

For z = a + bi, the complex conjugate is z* = a − bi. Conjugates are used to simplify products and divisions because z z* = a² + b², a real number. This identity is central to many FP1 problems.

复数 z = a + bi 的共轭为 z* = a − bi。共轭可用于化简乘积与除法，因为 z z* = a² + b² 是一个实数。这个恒等式是许多 FP1 问题的核心。

(a + bi)(a − bi) = a² + b²

When solving a quadratic with no real roots, completing the square or using the quadratic formula leads to a pair of complex conjugates, such as z = −2 ± 3i.

解没有实根的二次方程时，配方法或求根公式会得到一对共轭复根，例如 z = −2 ± 3i。

2. Dividing Complex Numbers and Quadratic Roots | 复数除法与二次方程根

To divide by a complex number, multiply the numerator and denominator by the conjugate of the denominator. For z₁ = a + bi and z₂ = c + di, write z₁/z₂ = (a + bi)(c − di) / (c² + d²). This always produces a single complex number in the form p + qi.

除以复数时，将分子和分母同时乘以分母的共轭。若 z₁ = a + bi 且 z₂ = c + di，则 z₁/z₂ = (a + bi)(c − di) / (c² + d²)。这样总能整理成 p + qi 的形式。

Every quadratic equation with real coefficients has either two real roots or a complex conjugate pair. If one complex root is known, the other root is its conjugate. This fact saves time in exam questions.

每个实系数二次方程要么有两个实根，要么有一对共轭复根。如果已知一个复根，另一个根就是它的共轭。这一事实在考试中可以节省时间。

Be careful with signs when subtracting complex parts in the numerator. Expand each bracket fully before separating real and imaginary terms.

分子中复部相减时要注意符号。先将每个括号完全展开，再分离实部和虚部。

3. Roots of Polynomials: Symmetric Functions | 多项式根：对称函数

For a quadratic ax² + bx + c = 0 with roots α and β

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