AS AQA Further Maths Unit 1 June 2022 Paper Walkthrough | AS AQA 进阶数学 Unit 1 2022年6月真题解析

📚 AS AQA Further Maths Unit 1 June 2022 Paper Walkthrough | AS AQA 进阶数学 Unit 1 2022年6月真题解析

The AQA AS Further Maths Unit 1 paper is a core pure mathematics paper that tests fluency in complex numbers, matrices, roots of polynomials, series summation and proof. This walkthrough breaks down the main topic areas and key exam skills needed for the June 2022 paper series, with original worked examples designed to mirror the style and difficulty of the real questions.

AQA AS 进阶数学 Unit 1 是一份纯数学核心卷，重点考查复数、矩阵、多项式根、级数求和与证明等内容的熟练程度。本文将按主要考点拆分 2022 年 6 月试卷所需的核心技能，并通过原创示例来模拟真实考题的风格与难度。

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1. Paper Structure and Key Topics | 试卷结构与核心考点

The AQA AS Further Maths Unit 1 paper is typically 1 hour 30 minutes long and is worth 80 marks. It contains a mixture of short calculation questions and longer problem-solving questions, with the final part of each question often extending a basic idea into a more demanding algebraic challenge.

在 AQA AS 进阶数学 Unit 1 中，考试时长通常为 1 小时 30 分钟，满分 80 分。试卷由短计算题与较长问题解决题混合组成，每题的最后一部分常将基本概念延伸为更复杂的代数挑战。

Section / 板块	Approximate marks / 约分值	Main skills / 主要技能
Complex numbers / 复数	20	Modulus, argument, roots, loci
Matrices / 矩阵	20	Multiplication, inverses, transformations
Roots of polynomials / 多项式根	15	Cubic and quartic relationships
Summation and proof / 求和与证明	25	Standard sums, induction, contradiction

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2. Complex Numbers: Arithmetic and Argument | 复数：运算与辐角

For a complex number z = a + bi, the modulus is |z| = √(a² + b²), and the argument θ satisfies tan θ = b/a. The correct value of θ depends on which quadrant of the Argand diagram the point lies in, so you must always check the signs of a and b.

对于复数 z = a + bi，其模为 |z| = √(a² + b²)，辐角 θ 满足 tan θ = b/a。θ 的正确取值取决于点位于阿甘图上的哪个象限，因此必须先判断 a 和 b 的符号。

• If a > 0 and b > 0, then θ = arctan(b/a).

  若 a > 0 且 b > 0，则 θ = arctan(b/a)。

• If a < 0, add π to the acute angle obtained from arctan(b/a).

  若 a < 0，则在 arctan(b/a) 所得到的锐角基础上加 π。

• If b < 0, use a negative angle, or write θ in the range -π < θ ≤ π.

  若 b < 0，则使用负角，或将 θ 写成 -π < θ ≤ π 范围内的值。

Example: z = 2 + 3i. Then |z| = √(2² + 3²) = √13, and arg(z) = arctan(3/2) ≈ 0.9828 rad. Since a and b are both positive, this is in the first quadrant.

例：z = 2 + 3i，则 |z| = √(2² + 3²) = √13，arg(z) = arctan(3/2) ≈ 0.9828 弧度。由于 a 和 b 均为正，故位于第一象限。

|z| = √(a² + b²), and arg(z) = arctan(b/a) with the correct quadrant.

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3. Solving Equations with Complex Roots | 复数根方程的求解

When a quadratic equation with real coefficients has a non-real root, the other root is its complex conjugate. This result follows directly from the quadratic formula and is a frequent mark-rich question on the AS paper.

当实系数二次方程有一个非实数根时，另一个根必为其共轭复数。这一结论可由求根公式直接得到，也是 AS 试卷中常见的拿分考点。

Example: solve z² + 2z + 5 = 0. Using the quadratic formula, z = (-2 ± √(4 – 20)) / 2 = (-2 ± √(-16)) / 2 = -1 ± 2i.

例：解 z² + 2z + 5 = 0。由求根公式得 z = (-2 ± √(4 – 20)) / 2 = (-2 ± √(-16)) / 2 = -1 ± 2i。

If z = p + qi is a root of a real-coefficient equation, then z* = p – qi is also a root.

Notice that the sum of the roots is -2 and the product is 5, matching the coefficients of z² + 2z + 5 = 0. This can be used to check your solutions quickly.

注意根之和为 -2，根之积为 5，这与 z² + 2z + 5 = 0 的系数一致。可以利用这些关系快速检验答案。

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4. Matrices: Multiplication, Inverses and Transformations | 矩阵：乘法、逆矩阵与变换

In the AS Further Maths Unit 1 paper, questions on 2×2 matrices usually require multiplication, calculation of the determinant and inverse, and interpretation of transformations such as rotations, reflections or enlargements.

在 AS 进阶数学 Unit 1 试卷中，2×2 矩阵的题目通常要求进行乘法运算、计算行列式和逆矩阵，并理解旋转、反射或伸缩等变换的几何意义。

For a matrix M = [[a, b], [c, d]], the determinant is det(M) = ad – bc. The inverse is given by the following formula:

对于矩阵 M = [[a, b], [c, d]]，其行列式为 det(M) = ad – bc，逆矩阵由以下公式给出：

M⁻¹ = (1 / (ad – bc)) × [[d, -b], [-c, a]]

• Matrix multiplication is not commutative; AB is generally not equal to BA.

  矩阵乘法不满足交换律，AB 一般不等于 BA。

• The transformation matrix for a rotation by angle θ anticlockwise about the origin is [[cos θ, -sin θ], [sin θ, cos θ]].

  绕原点逆时针旋转 θ 角的变换矩阵为 [[cos θ, -sin θ], [sin θ, cos θ]]。

• A reflection in the x-axis has matrix [[1, 0], [0, -1]], and a reflection in the y-axis has matrix [[-1, 0], [0, 1]].

  关于 x 轴的反射矩阵为 [[1, 0], [0, -1]]，关于 y 轴的反射矩阵为 [[-1, 0], [0, 1]]。

Always state the order of transformations clearly. The matrix closest to the column vector is applied first when multiplying on the left by a combined matrix.

始终要清楚地说明变换的先后顺序。在左边的合成矩阵与列向量相乘时，靠近列向量的矩阵先作用。

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5. Using Matrices to Solve Simultaneous Equations | 用矩阵求解联立方程

A system of two linear simultaneous equations can be written in matrix form as Ax = b, where A is the 2×2 coefficient matrix, x is the column vector of unknowns, and b is the column vector of constants. If det(A) ≠ 0, then x = A⁻¹b gives the unique solution.

二元线性方程组可写成矩阵形式 Ax = b，其中 A 是 2×2 系数矩阵，x 是未知数列向量，b 是常数项列向量。若 det(A) ≠ 0，则 x = A⁻¹b 给出唯一解。

Example: solve 2x + 3y = 7 and x – 2y = -4. Write A = [[2, 3], [1, -2]]. Then det(A) = 2 × (-2) – 3 × 1 = -7, and

例：解 2x + 3y = 7 与 x – 2y = -4。将方程组写为 A = [[2, 3], [1, -2]]，则 det(A) = 2 × (-2) – 3 × 1 = -7，于是

A⁻¹ = (1 / -7) × [[-2, -3], [-1, 2]] = [[2/7, 3/7], [1/7, -2/7]]

Multiplying by b = [7, -4] gives x = 2/7 and y = 15/7. You should always substitute both values back into the original equations.

与 b = [7, -4] 相乘后得到 x = 2/7，y = 15/7。务必把两个值代回原方程验证。

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6. Roots of Polynomials | 多项式根的关系

For the cubic equation ax³ + bx² + cx + d = 0 with roots α, β and γ, the following relationships hold. These formulas are also extended to quartic equations in the Unit 1 specification.

对于三次方程 ax³ + bx² + cx + d = 0，设其根为 α、β、γ，则以下关系成立。Unit 1 考纲还要求将其推广到四次方程。

Relationship / 关系	Cubic / 三次方程	Quartic / 四次方程
Sum of roots / 根和	α + β + γ = -b/a	Sum = -b/a
Pairwise sum / 两两乘积和	αβ + αγ + βγ = c/a	Sum of products in pairs = c/a
Product / 根的乘积	αβγ = -d/a	Product = e/a

Example: for x³ – 6x² + 11x – 6 = 0, the sum of the roots is 6, the pairwise sum is 11, and the product is 6. In fact, the roots are 1, 2 and 3, which confirms these values.

例：对于 x³ – 6x² + 11x – 6 = 0，其根之和为 6，两两乘积和为 11，根的乘积为 6。事实上该方程的根为 1、2、3，验证了上述数值。

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7. Summation of Series | 级数求和

The standard results for the sum of the first n positive integers, their squares and their cubes are essential tools in the Unit 1 paper. You should be able to combine these results to find the sum of more complicated expressions.

前 n

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