KS3 CCEA Further Mathematics: Mock Unit Test Walkthrough | KS3 CCEA 进阶数学：单元测试模拟卷解析

📚 KS3 CCEA Further Mathematics: Mock Unit Test Walkthrough | KS3 CCEA 进阶数学：单元测试模拟卷解析

This walkthrough breaks down a full KS3 CCEA Further Mathematics unit test mock paper, covering each question with clear step-by-step solutions and examiner-style insights. By working through the paper alongside these notes, you can sharpen your problem-solving skills and build confidence for the real assessment.

这份解析逐题拆解了一套完整的KS3 CCEA进阶数学单元模拟卷，用清晰的步骤和阅卷视角给出了每一道题的详细解答。跟着笔记完整走一遍试卷，你可以强化解题思路，为正式考试积累信心。

1. Mock Test Overview | 模拟卷概览

The mock test is designed for 60 minutes and carries a total of 50 marks. It covers number, algebra, shape and space, data handling, and functional applications, reflecting the CCEA Further Mathematics emphasis on reasoning and problem-solving. Work through it as if in an exam, then use this walkthrough to check and deepen your understanding.

模拟卷时长为 60 分钟，满分 50 分，涵盖数、代数、图形与空间、数据处理以及实际应用题，贴合 CCEA 进阶数学对推理与问题解决的侧重。像正式考试一样完成试卷后，再用这篇解读来核对思路、加深理解。

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2. Solving Linear Equations | 解线性方程

Question: Solve 2(3x - 4) = 10.

题目：解方程 2(3x - 4) = 10。

Step 1: Expand the left side. Multiply 2 by each term inside the bracket to get 6x - 8.

步骤1：展开左边。用 2 乘以括号里的每一项，得到 6x - 8。

6x - 8 = 10

6x - 8 = 10

Step 2: Add 8 to both sides to isolate the term with x. This gives 6x = 18.

步骤2：两边同时加上 8，将含 x 的项单独留在左边，得到 6x = 18。

Step 3: Divide both sides by 6. The solution is x = 3.

步骤3：两边同时除以 6。解得 x = 3。

by substituting back: 2(3×3 - 4) = 2(9 - 4) = 2×5 = 10, which matches the right-hand side.

务必回代检验：2(3×3 - 4) = 2(9 - 4) = 2×5 = 10，与等式右边一致。

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3. Expanding Brackets | 多项式展开

Question: Expand and simplify (x + 3)(x - 5).

题目：展开并化简 (x + 3)(x - 5)。

Step 1: Use the distributive method (FOIL): First, Outer, Inner, Last.

步骤1：使用乘法分配律（首、外、内、尾）。

First: x × x = x². Outer: x × (-5) = -5x. Inner: 3 × x = 3x. Last: 3 × (-5) = -15.

首项：x × x = x²。外项：x × (-5) = -5x。内项：3 × x = 3x。尾项：3 × (-5) = -15。

Step 2: Write the expanded form as x² - 5x + 3x - 15.

步骤2：写出展开式 x² - 5x + 3x - 15。

Step 3: Combine like terms: -5x + 3x = -2x. Final answer: x² - 2x - 15.

步骤3：合并同类项：-5x + 3x = -2x。最终答案：x² - 2x - 15。

Watch out for signs; a common mistake is writing +3x -5x and losing a negative.

注意符号；常见的错误是把 +3x 和 -5x 的位置写混，导致丢失负号。

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4. Direct Proportion | 正比例问题

Question: 5 identical pens cost £3.50. Work out the cost of 8 pens.

题目：5 支相同的笔售价 £3.50，求 8 支笔的总价。

Step 1: Find the cost of one pen by unitary method: £3.50 ÷ 5 = £0.70.

步骤1：用归一法求单价：£3.50 ÷ 5 = £0.70。

Step 2: Multiply the unit cost by the required number: 8 × £0.70 = £5.60.

步骤2：用单价乘以所求数量：8 × £0.70 = £5.60。

ernatively, set up a proportion: 5/3.50 = 8/x, cross-multiply to get 5x = 28, so x = 5.60.

也可以列比例式：5/3.50 = 8/x，交叉相乘得 5x = 28，即 x = 5.60。

Both methods give the same answer, but the unitary approach is often quicker for simple ratios.

两种方法结果相同，不过在简单比例里归一法通常更快。

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5. Area of a Triangle | 三角形面积

Question: A triangle has base 12 cm and perpendicular height 7 cm. Calculate its area.

题目：一个三角形底边长 12 cm，垂直高 7 cm，计算其面积。

Recall the formula: Area = ½ × base × height.

回忆公式：面积 = ½ × 底 × 高。

A = ½ b h

A = ½ b h

Substitute the values: A = ½ × 12 × 7 = 6 × 7 = 42 cm².

代入数值：A = ½ × 12 × 7 = 6 × 7 = 42 cm²。

Always include the correct units; here the area is expressed in square centimetres. Forgetting units loses a mark.

别忘了写对单位；这里的面积单位是平方厘米。遗漏单位会丢分。

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6. Number Patterns | 数字规律

Question: Consider the sequence 2, 6, 10, 14, ... Find the next term and the nth term formula.

题目：观察数列 2, 6, 10, 14, ... 写出下一项和第 n 项公式。

Step 1: Identify the common difference: 6 - 2 = 4, 10 - 6 = 4, so d = 4.

步骤1：找出公差：6 - 2 = 4，10 - 6 = 4，因此公差 d = 4。

Step 2: The next term is 14 + 4 = 18.

步骤2：下一项为 14 + 4 = 18。

Step 3: The nth term of an arithmetic sequence is a + (n - 1)d. Here a = 2, d = 4.

步骤3：等差数列的第 n 项公式为 a + (n - 1)d。这里首项 a = 2，d = 4。

nth term = 2 + (n - 1)×4 = 4n - 2

第 n 项 = 2 + (n - 1)×4 = 4n - 2

Check: when n=1, 4(1) - 2 = 2; n=2, 4(2) - 2 = 6. It works.

检验：n=1 得 2，n=2 得 6，公式正确。

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7. Statistics: Mean, Median, Mode | 统计：平均数、中位数、众数

Question: The scores in a quiz are 5, 7, 3, 7, 8, 9. Calculate the mean, median, and mode.

题目：测验得分为 5, 7, 3, 7, 8, 9。计算平均数、中位数和众数。

Step 1: Mean: sum the values (5+7+3+7+8+9 = 39) and divide by 6. Mean = 39 ÷ 6 = 6.5.

步骤1：平均数：求和 (5+7+3+7+8+9 = 39)，除以 6，得 39 ÷ 6 = 6.5。

5, 7, 7, 8, 9. With 6 numbers, the median is the average of the 3rd and 4th: (7+7)÷2 = 7.

步骤2：中位数：排序得 3, 5, 7, 7, 8, 9。有 6 个数，中位数是第 3 和第 4 个数的平均数：(7+7)÷2 = 7。

Step 3: Mode is the most frequent value; here 7 appears twice, so mode = 7.

步骤3：众数是出现最频繁的值；这里 7 出现两次，所以众数 = 7。

Always order for median and remember to average the two middle values when the count is even.

求中位数时一定先排序，并在数据个数为偶数时记得取中间两个值的平均数。

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8. Basic Probability | 基础概率

Question: A bag contains 4 red, 3 blue, and 1 green counters. One counter is taken at random. Find P(red).

题目：袋中有 4 个红球、3 个蓝球和 1 个绿球。随机取一球，求取到红球的概率。

Total number of equally likely outcomes = 4 + 3 + 1 = 8.

等可能结果总数 = 4 + 3 + 1 = 8。

Number of successful outcomes (red) = 4.

有利结果数（红球）= 4。

P(red) = 4/8 = 1/2

P(红) = 4/8 = 1/2

Answers should always be simplified and can be given as a fraction, decimal, or percentage (½, 0.5, 50%).

答案必须化简，可以用分数、小数或百分数表示（½、0.5、50%）。

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9. Coordinates and Midpoint | 坐标与中点

Question: Find the midpoint of the line segment joining A(2, 5) and B(6, 9).

题目：求连接 A(2, 5) 和 B(6, 9) 的线段中点。

The midpoint formula: M = ((x₁ + x₂)/2 , (y₁ + y₂)/2).

中点公式：M = ((x₁ + x₂)/2 , (y₁ + y₂)/2)。

Substitute: x-coordinate = (2 + 6) ÷ 2 = 4, y-coordinate = (5 + 9) ÷ 2 = 7.

代入：横坐标 = (2 + 6) ÷ 2 = 4，纵坐标 = (5 + 9) ÷ 2 = 7。

The midpoint is (4, 7).

中点为 (4, 7)。

A common error is subtracting coordinates instead of adding; recall that the midpoint is an average.

常见错误是把坐标相减而不是相加；记住中点是取平均值。

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10. Application: Speed, Distance, Time | 应用题：速度、距离、时间

Question: A car travels at an average speed of 60 km/h for 2½ hours. How far does it travel?

题目：一辆汽车以平均 60 km/h 的速度行驶 2½ 小时，求它行驶的距离。

Use the formula: Distance = Speed × Time.

使用公式：距离 = 速度 × 时间。

Convert 2½ to an improper fraction or decimal: 2.5 or 5/2.

将 2½ 化成假分数或小数：2.5 或 5/2。

Distance = 60 × 2.5 = 150 km

距离 = 60 × 2.5 = 150 km

Always write the units; the answer is in kilometres. Check that the time unit matches the speed unit.

必须写单位；答案是千米。注意时间单位要与速度单位匹配。

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11. Inequalities | 不等式

Question: Solve 3x - 7 < 5 and represent the solution on a number line.

题目：解不等式 3x - 7 < 5，并将解表示在数轴上。

Step 1: Add 7 to both sides: 3x < 12.

步骤1：两边加 7 得 3x < 12。

Step 2: Divide by 3 (positive, so the inequality sign stays the same): x < 4.

步骤2：两边除以 3（正数，不等号方向不变）：x < 4。

For the number line, draw an open circle at 4 (since 4 is not included) and shade to the left, indicating all values less than 4.

在数轴上，在 4 处画空心圆圈（因为 4 不包含在内），并向左画阴影线，表示所有小于 4 的数。

Remember: if you multiply or divide by a negative number, reverse the inequality sign. That wasn't needed here.

记住：乘或除以负数时，不等号方向要反转。本题不需要反转。

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12. Common Mistakes and Exam Tips | 常见错误与备考建议

Check your expansion signs carefully; a missed negative can change the whole term. When simplifying fractions, always look for the highest common factor. In probability, the total number of outcomes must be correct. For sequences, verify the nth term with the first three terms. In statistics, reorder the dataset before finding the median. Always write units for area, volume, and speed.

仔细检查展开时的符号；漏掉一个负号会改变整个项。化简分数时，一定要找出最大公因数。概率问题里，所有可能结果的总数必须正确。对于数列，要用前三项验证第 n 项公式。统计中，求中位数前先把数据重新排序。面积、体积和速度题目都要写单位。

Finally, manage your time: spend one mark per minute and leave 5–10 minutes to review. Show all working, because method marks can be awarded even if the final answer is wrong.

最后，合理分配时间：一分钟拿一分，留出 5–10 分钟检查。务必展示解题过程，因为就算最终答案错误，过程分同样可以拿到。

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