Answers to the Exercises in Support Pack 3 | 支持包3练习答案

📚 Answers to the Exercises in Support Pack 3 | 支持包3练习答案

This article provides fully worked answers and exam-style commentary for the exercises in AQA A-Level Mathematics Support Pack 3. The solutions are grouped by topic so that you can check your working, identify slips, and strengthen the skills most often assessed.

本文为 AQA A-Level 数学支持包3中的练习提供完整答案和考试风格点评。答案按主题分组，便于核对解题过程、发现失误，并强化最常考查的技能。

1. Algebra: Quadratic and Polynomial Equations | 代数：二次与多项式方程

Exercise 1.1 Solve 2x² – 5x – 3 = 0. Factorising gives (2x + 1)(x – 3) = 0, so x = -1/2 or x = 3.

练习1.1 解 2x² – 5x – 3 = 0。因式分解得 (2x + 1)(x – 3) = 0，因此 x = -1/2 或 x = 3。

Exercise 1.2 Solve x³ – 4x² + x + 6 = 0 given that x = 2 is a root. Synthetic division gives (x – 2)(x² – 2x – 3) = 0, so (x – 2)(x – 3)(x + 1) = 0. The roots are x = -1, x = 2, x = 3.

练习1.2 已知 x = 2 是一个根，解 x³ – 4x² + x + 6 = 0。综合除法得 (x – 2)(x² – 2x – 3) = 0，因此 (x – 2)(x – 3)(x + 1) = 0。根为 x = -1、x = 2、x = 3。

Exercise 1.3 Find the remainder when 3x³ – 2x + 1 is divided by x – 2. By the remainder theorem, substitute x = 2: f(2) = 3(8) – 4 + 1 = 21.

练习1.3 求 3x³ – 2x + 1 除以 x – 2 的余数。由余数定理，代入 x = 2：f(2) = 3(8) – 4 + 1 = 21。

Examiner tip: For cubic equations, always test the small integer factors of the constant term first, such as ±1, ±2, ±3, ±6.

考官建议：对于三次方程，首先检验常数项的小整数因子，例如 ±1、±2、±3、±6。

2. Coordinate Geometry: Lines and Circles | 坐标几何：直线与圆

Exercise 2.1 Find the equation of the line through (2, 5) and (-1, -4). The gradient is m = (5 – (-4)) / (2 – (-1)) = 9 / 3 = 3, so using y – 5 = 3(x – 2) gives y = 3x – 1.

练习2.1 求经过点 (2, 5) 和 (-1, -4) 的直线方程。斜率为 m = (5 – (-4)) / (2 – (-1)) = 9 / 3 = 3，

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