📚 AS AQA Further Maths Paper 1 (Jan 2020) | AS AQA 进阶数学试卷一（2020年1月）
Welcome to this comprehensive revision guide for the AS AQA Further Mathematics Paper 1, January 2020 sitting. This paper tests your command of the compulsory core topics that underpin all further maths work, from complex numbers and matrices to proof and vectors. We will break down each skill area, show you exactly what the examiners look for, and give you strong model answers.
欢迎阅读本复习指南，其内容针对 2020 年 1 月 AS AQA 进阶数学试卷一。该试卷考查你对进阶数学核心必修主题的掌握，包括复数、矩阵、证明方法与向量。我们将逐项拆解每一考点，向您展示考官评分的关键，并提供高质量的标准答案。
1. Complex Number Arithmetic | 复数运算
ex number, multiply numerator and denominator by the complex conjugate of the denominator.
2020 年 1 月试卷以复数基本运算开篇。核心结论包括对形如 z = a + bi 的复数进行加减乘除，其中 i² = −1。复数相除时，需将分子与分母同时乘上分母的共轭复数。
z = a + bi, z̄ = a − bi, z × z̄ = a² + b²
Example: simplify (3 + 4i) ÷ (1 − i). Multiply top and bottom by (1 + i):
示例：化简 (3 + 4i) ÷ (1 − i)。将分子分母同乘 (1 + i)：
)(1 + i) ÷ ((1 − i)(1 + i)) = (3 + 3i + 4i + 4i²) ÷ (1 − i²) = (−1 + 7i) ÷ 2 = −0.5 + 3.5i
3 − 4 = −1. This type of calculation frequently appears as part (a) of a question, worth 2– ct real and imaginary terms; a common error is leaving i² instead of replacing it with −1.
注意符号：4i × i = 4i² = −4，因此实数部分为 3 − 4 = −1。此类计算常作为题目第 (a) 小题出现，占 2–3 分。两个复数直接相乘时，使用分配律并分别合并实部与虚部；常见错误是保留 i² 而未将其替换为 −1。
2. Argand Diagrams and Modulus–Argument Form | 阿甘图与模–辐角形式
er z = x + yi is plotted at the point (x, y). The modulus is the distance from the origin:
在阿甘图中，x 轴为实轴，y 轴为虚轴。复数 z = x + yi 对应坐标点 (x, y)。模长是到原点的距离：
|z| = √x² + y², arg(z) = θ, tan θ = y ÷ x
s, measured in radians and normally taken in the range (−π, π]. When converting to modulus–argument form use z = r(cos θ + i sin θ).
辐角是自正实轴逆时针转过的角度，以弧度为单位，通常取 (−π, π]。转换为模–辐角形式时使用 z = r(cos θ + i sin θ)。
estions often ask you to locate a point on the Argand diagram or to shade a region such as |z − a − bi| the Cartesian form; then use x = r cos θ and y = r sin θ, taking care with quadrant signs.
考试题目常要求你在阿甘图中标出复数点，或标出满足条件的区域，例如 |z − a − bi| ≤ r 表示以 (a, b) 为圆心、半径为 r 的圆盘。有的题目给出模与辐角，要求写出直角坐标形式；此时使用 x = r cos θ、y = r sin θ，务必注意象限符号。
3. Roots of Polynomials | 多项式的根
hip between roots and coefficients. For a cubic ax³ + bx² + cx + d = 0 with roots α, β, γ:
试卷中包含这样的问题：给出三次或四次方程的一个根，要求利用根与系数的关系求出其余各根。对于三次方程 ax³ + bx² + cx + d = 0，设根为 α、β、γ：
α + β + γ = −b ÷ a, αβ + βγ + γα = c ÷ a, αβγ = −d ÷ a
are real, any non-real roots occur in conjugate pairs. So if z = p + qi is a root, then z̄ nts, the conjugate-pair rule applies twice, and the sum of roots condition becomes a quick
Published by TutorHao | AS Mathematics Revision Series | aleveler.com
更多咨询请联系16621398022（同微信）
或截图后用「扫一扫 → 相册」选择本图