📚 Year 10 Cambridge Additional Mathematics: Mock Test Paper Solutions | 剑桥 Year 10 进阶数学：模拟测试卷解析
Mock tests are an essential part of your Cambridge IGCSE Additional Mathematics preparation. This walkthrough analyses a full unit test covering functions, algebra, coordinate geometry, trigonometry and introductory calculus. By working through each question you will identify key concepts, common pitfalls and effective problem-solving techniques, and you will see exactly how marks are awarded in a structured paper.
模拟测试是准备剑桥 IGCSE 进阶数学的重要环节。本文详细解析一份涵盖函数、代数、坐标几何、三角学和微积分初阶的完整单元测试卷。通过逐一攻克每道题目，你将掌握核心概念、常见失分点以及高效的解题方法，并清楚看到一份结构化试卷中分数究竟是如何给出的。
1. Paper Overview and Key Topics | 试卷概览与重点考点
This mock test is designed to mimic a 45-minute unit assessment covering the first half of the Cambridge IGCSE Additional Mathematics syllabus. Topics include algebra, functions, equations and inequalities, coordinate geometry, trigonometry and introductory calculus. The paper contains 10 compulsory questions with a mix of short-answer and structured problems, and the total mark is 60.
本模拟试卷模拟一份 45 分钟的单元测试，覆盖剑桥 IGCSE 进阶数学课程前半部分。主题包括代数、函数、方程与不等式、坐标几何、三角学和微积分基础。试卷包含 10 道必答题，题型涵盖简答题和结构化问题，满分 60 分。
Key skills tested: polynomial factorisation, solving quadratic inequalities, handling exponential and logarithmic equations, coordinate geometry of lines and circles, composite and inverse functions, trigonometric identities, differentiation for tangents and normals, and definite integration for area. Time management and careful checking for domain/validity are critical.
重点考查技能：多项式因式分解、解二次不等式、处理指数与对数方程、直线和圆的坐标几何、复合函数与反函数、三角恒等式、微分求切线与法线，以及定积分求面积。时间管理和仔细检验定义域/有效性至关重要。
The table below shows how the marks are distributed. Notice that calculus alone carries roughly a quarter of the paper, so it deserves dedicated revision time rather than last-minute cramming.
下表显示各题分值分布。可以看到仅微积分部分就占了约四分之一的分值，因此它值得专门安排复习时间，而不是考前临时抱佛脚。
| Question | Topic | Main Skill | Marks |
|---|---|---|---|
| 1 | Polynomials | Factor theorem, division | 5 |
| 2 | Inequalities | Quadratic inequality, Δ | 5 |
| 3–4 | Indices and logs | Same-base comparison, laws of logs | 10 |
| 5–6 | Coordinate geometry | Lines, circles, tangents | 12 |
| 7 | Functions | Composites, inverses | 6 |
| 8 | Trigonometry | Identities and equations | 8 |
| 9–10 | Calculus | Tangents, stationary points, area | 14 |
A sensible strategy is to spend no more than 4 minutes on any 5-mark question during your first pass. Mark anything you cannot finish, move on, and return with the remaining 10 minutes. Never leave a structured question completely blank: method marks for stating a correct formula or substituting correctly are usually available even when the final answer is wrong.
合理的策略是第一遍作答时，任何 5 分的题目都不要超过 4 分钟。做不完的题先标记，继续往下做，最后用剩余 10 分钟回头处理。结构化问题千万不要完全留空白：即使最终答案错误，写出正确公式或正确代入通常仍能拿到方法分。
2. Q1: Polynomial Division and Factor Theorem | 多项式除法与因式定理
² − 3x + 2 and knowing that x = −1 is a root, find all the roots of the equation f(x) = 0.
已知 f(x) = 2x³ − 3x² − 3x + 2，且已知 x = −1 是方程的一个根，求方程 f(x) = 0 的全部根。
Step 1 – ) is indeed a factor. Always show this substitution line; it is worth one mark on its own.
第一步——用因式定理验证：f(−1) = 2(−1)³ − 3(−1)² − 3(−1) + 2 = −2 − 3 + 3 + 2 = 0，所以 (x + 1) 确实是因式。这一行代入务必写出，它本身就值 1 分。
Step 2 – divide. Synthetic division with the coefficients 2, −3, −3, 2 and the root −1 gives quotient coefficients 2, −5, 2 and remainder 0:
第二步——做除法。用系数 2、−3、−3、2 与根 −1 进行综合除法，得到商式系数 2、−5、2，余数为 0：
2x³ − 3x² − 3x + 2 = (x + 1)(2x² − 5x + 2)
Step 3 – factorise the quadratic: 2x² − 5x + 2 = (2x − 1)(x − 2), because 2 × 2 = 4
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