📚 AQA AS Further Maths Unit 1 January 2019 Paper: Full Walkthrough & Key Concepts | AQA AS 进阶数学第1单元 2019年1月试卷：精讲与核心考点
The January 2019 AQA AS Further Mathematics Unit 1 paper (7366/1) examines the Core Pure syllabus, covering complex numbers, matrices, roots of polynomials, summation of series and proof by induction. This walkthrough breaks down each question area, highlights the mathematical techniques required, and provides fully worked examples that mirror the style and demand of the exam.
2019年1月 AQA AS 进阶数学第1单元试卷（7366/1）考查 Core Pure 核心纯数大纲，涵盖复数、矩阵、多项式求根、级数求和与数学归纳法。本文将逐题剖析各考查板块，提炼必备的数学技巧，并提供与真题风格和难度高度一致的完整解答示例。
1. Paper Structure & Assessment Overview | 试卷结构与考查概览
The Unit 1 paper is 1 hour 30 minutes in duration and carries a total of 80 marks. It contributes 60% of the AS Further Mathematics qualification. A scientific calculator is permitted, although questions are designed to reward clear written method, not just numerical answers.
第1单元试卷时长1小时30分钟，满分80分，占 AS 进阶数学总成绩的60%。考试允许使用科学计算器，但题目设计旨在奖励清晰的书写作答过程，而非仅仅数值答案。
The paper is structured to assess two distinct skill sets: routine procedural fluency in topics such as complex number arithmetic and matrix inversion, and multi-step problem-solving in areas such as induction and geometric interpretation on the Argand diagram.
试卷旨在评估两种不同能力：一是在复数运算、矩阵求逆等主题中的常规程序性熟练度；二是在归纳法、阿冈图几何解释等多步骤问题中的综合解题能力。
Sensible time management is essential. Candidates should aim to spend roughly one minute per mark, leaving the final 10 minutes for checking algebraic signs, determinant calculations and the conclusion of induction proofs.
合理的时间管理至关重要。考生应按"每分钟1分"的节奏答题，并留出最后10分钟检查代数符号、行列式计算以及归纳法证明的结论部分。
2. Complex Numbers: Modulus & Argument | 复数：模与辐角
fundamental operations on complex numbers. For a complex number z = a + bi, the modulus is |z| = √a² + b² = b/a, where the quadrant of the point (a, b) determines the correct principal value of θ.
2019年1月试卷的第1题考查复数的基本运算。对于复数 z = a + bi，模长为 |z| = √a² + b²，辐角满足 tan θ = b/a，其中点 (a, b) 所在象限决定 θ 主值的正确取法。
Worked example: Given z = 2 - 3i, find |z| and arg(z).
例题：已知 z = 2 - 3i，求 |z| 和 arg(z)。
|z| = √(2² + (-3)²) = √13
arg(z) = -arctan(3/2) ≈ -0.983 rad
ery common error is to give arg(z) = arctan(-3/2) as a positive angle, which is incorrect.
由于实部为正、虚部为负，z 位于第四象限，因此辐角应为负值。一个非常常见的错误是将 arg(z) 写作正角 arctan(-3/2)，这是不正确的。
Candidates must also remember that the principal argument lies in the interval (-π, π]. If the question requires an exact answer in terms of π, the result must be expressed accordingly; otherwise, a decimal to 3 significant figures is usually acceptable.
考生必须牢记主辐角位于区间 (-π, π] 内。若题目要求以含 π 的精确值作答，则需按精确形式书写；若未作要求，通常保留3位有效数字的小数即可。
3. Matrix Operations & Inverse Matrices | 矩阵运算与逆矩阵
inant is ad - bc. The inverse exists only when ad - bc ≠ 0, and is given by the following formula.
本卷的矩阵题要求考生熟练进行 2 × 2 矩阵的运算。对于矩阵 A = [a b; c d]，行列式为 ad - bc。仅当 ad - bc ≠ 0 时逆矩阵才存在，其计算公式如下。
A⁻¹ = 1/(ad - bc) × [d -b; -c a]
we compute det(A) = 2 × 2 - 3 × 1 = 1. Since the determinant is non-zero, the inverse is:
例如，已知 A = [2 3; 1 2]，计算得 det(A) = 2 × 2 - 3 × 1 = 1。由于行列式不为零，逆矩阵为：
A⁻¹ = [2 -3; -1 2]
be written in matrix form as Ax = b where x = [x; y]. Multiplying both sides by A⁻¹ gives:
同题中典型的后续问法是用逆矩阵求解二元一次方程组。考虑方程组 2x + 3y = 5 和 x + 2y = 2，可写成矩阵形式 Ax = b，其中 x = [x; y]。两边左乘 A⁻¹ 得：
[x; y] = A⁻¹b = [2 -3; -1 2] [5; 2] = [4; -1]
quations: 2(4) + 3(-1) = 5 and 4 + 2(-1) = 2. Both are satisfied, confirming the solution.
因此 x = 4，y = -1。务必代回原方程检验：2(4) + 3(-1) = 5，且 4 + 2(-1) = 2，两个方程均成立，验证了解的正确性。
Candidates frequently lose method marks by omitting the determinant calculation. Even when the determinant is simple, writing it explicitly demonstrates understanding and secures credit in a mark scheme that rewards full working.
考生常因省略行列式计算而丢失方法分。即使行列式很简单，明确写出计算过程也能体现理解，并在按步骤给分的评分标准中确保得分。
4. Matrix Transformations on the Plane | 平面上的矩阵变换
Another area tested in Unit 1 is the connection between matrices and geometric transformations. The matrix [cos θ -sin θ; sin θ cos θ] represents an anticlockwise rotation about the origin by angle θ, while a reflection in the x-axis is given by [1 0; 0 -1].
第1单元还考查矩阵与几何变换之间的联系。矩阵 [cos θ -sin θ; sin θ cos θ] 表示绕原点逆时针旋转角度 θ，而关于 x 轴的反射变换由矩阵 [1 0; 0 -1] 表示。
For a transformation matrix M, the determinant det(M) gives the area scale factor of the transformation. If det(M) is negative, the transformation reverses orientation. For example, the reflection matrix [1 0; 0 -1] has determinant -1, indicating an area scale factor of 1 with a reversal of orientation.
对于变换矩阵 M，行列式 det(M) 给出变换的面积缩放因子。若 det(M) 为负，则变换反转方向。例如，反射矩阵 [1 0; 0 -1] 的行列式为 -1，表示面积缩放因子为1且方向发生反转。
A question may ask for the image of a point under a transformation. To find the image of (2, 5) under rotation by 90° anticlockwise, apply the rotation matrix [0 -1; 1 0] to the column vector [2; 5]:
题目可能要求求某点在变换下的像。要求点 (2, 5) 在逆时针旋转90°下的像，将旋转矩阵 [0 -1; 1 0] 作用于列向量 [2; 5]：
[0 -1; 1 0] [2; 5] = [-5; 2]
The image is therefore (-5, 2). This type of calculation often appears as the first part of a longer question, providing marks that are straightforward if the matrix-vector product is performed accurately.
因此，像点为 (-5, 2)。这类计算通常作为较长的综合题的第一小问出现，只要矩阵与向量的乘法计算准确，这些分数是很容易拿到的。
5. Roots of Polynomials: Symmetric Functions | 多项式求根：对称函数
e α + β = -b and αβ = c. For a cubic x³ + px² + qx + r = 0 with roots α, β and γ, we have:
多项式求根问题重点考查根的对偶函数（对称函数）。对于二次方程 x² + bx + c = 0，设其根为 α 和 β，关键关系为 α + β = -b 和 αβ = c。对于三次方程 x³ + px² + qx + r = 0，设其根为 α、β 和 γ，则有：
α + β + γ = -p,
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