9665 FM01 International AS Further Mathematics Mark Scheme 2017 v2 – Exam Technique Breakdown | 9665 FM01 国际AS进阶数学2017年评分方案题型精析

📚 9665 FM01 International AS Further Mathematics Mark Scheme 2017 v2 – Exam Technique Breakdown | 9665 FM01 国际AS进阶数学2017年评分方案题型精析

The 9665 FM01 mark scheme from 2017 (version 2) provides a clear blueprint for how examiners award marks in International AS Further Mathematics. By dissecting the mark allocation across typical question types, students can learn to present solutions that hit every method (M) and accuracy (A) point, minimizing careless errors. This article breaks down eight common topics from the paper, explaining the scoring logic and offering model approaches.

2017年第二版的9665 FM01评分方案清晰展示了国际AS进阶数学的阅卷标准。通过拆解各个典型题型的分值分配，学生能够学会如何呈现解答以抓住每一个方法分(M)和准确分(A)，从而减少粗心失分。本文解析试卷中的八大常见主题，阐释评分逻辑并提供范例思路。

1. Complex Numbers & Polynomial Roots | 复数与多项式根

A recurring question type asks for the remaining roots of a cubic or quartic equation given one complex root. The 2017 mark scheme emphasizes the use of the complex conjugate root theorem (M1). For a real-coefficient equation, if z = a + bi is a root, its conjugate z* = a – bi must also be a root. To find the unknown real root, students often multiply factors or use relationships among roots (sum and product). Full marks are awarded for correctly identifying the conjugate (A1), forming a quadratic factor (M1), and deriving the real root (A1).

试卷中常出现给定一个复数根求解三次或四次方程其余根的题型。2017年评分方案强调运用共轭复根定理（方法分M1）。对于实系数方程，若z = a + bi是一个根，其共轭z* = a – bi必为另一根。为求未知实根，学生可利用根与系数的关系（和与积）或相乘因式。满分步骤包括正确写出共轭根（A1）、构造二次因式（M1）并求出实根（A1）。

Example: “The cubic equation z³ – 4z² + 6z – 4 = 0 has a root z = 1 + i. Find the other two roots.” Using the mark scheme logic, state that 1 – i is also a root (M1 A1). The corresponding quadratic factor is (z – (1+i))(z – (1-i)) = z² – 2z + 2 (M1). Since the sum of all three roots equals 4, the real root is 4 – (1+i) – (1-i) = 2 (A1). Alternatively, divide the cubic by the quadratic to obtain z = 2. The solution set is {

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