Year 9 CCEA Further Maths Unit Test Mock Paper Walkthrough | 9年级CCEA进阶数学单元测试模拟卷解析

📚 Year 9 CCEA Further Maths Unit Test Mock Paper Walkthrough | 9年级CCEA进阶数学单元测试模拟卷解析

This walkthrough is designed to help Year 9 students prepare for a CCEA Further Maths unit test by working through a full mock paper. Each question is carefully explained step by step, covering topics from algebra and geometry to probability and graphs. Use this resource to boost your confidence and sharpen your problem‑solving skills.

本解析旨在帮助9年级学生通过一套完整的模拟试卷备考CCEA进阶数学单元测试。每道题均附有逐步解答，涵盖从代数、几何到概率和图像等主题。利用这一资源增强信心、提高解题能力。

1. Simplifying Algebraic Expressions | 化简代数表达式

Question: Simplify the expression 4a²b − 3ab² + 2a²b + 5ab².

问题：化简表达式 4a²b − 3ab² + 2a²b + 5ab²。

First, identify the like terms. The terms in a²b are 4a²b and 2a²b. The terms in ab² are −3ab² and 5ab².

首先，找出同类项。含 a²b 的项是 4a²b 和 2a²b。含 ab² 的项是 −3ab² 和 5ab²。

Combine the a²b terms: 4a²b + 2a²b = 6a²b.

合并 a²b 项：4a²b + 2a²b = 6a²b。

Combine the ab² terms: −3ab² + 5ab² = 2ab².

合并 ab² 项：−3ab² + 5ab² = 2ab²。

The simplified expression is 6a²b + 2ab².

化简后的表达式为 6a²b + 2ab²。

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2. Solving Linear Equations | 解一元一次方程

Question: Solve the equation 5x − 2 = 3x + 10.

问题：解方程 5x − 2 = 3x + 10。

Start by collecting the x‑terms on one side. Subtract 3x from both sides: 5x − 3x − 2 = 10, which gives 2x − 2 = 10.

首先将含 x 的项移到同侧。两边同时减去 3x：5x − 3x − 2 = 10，得到 2x − 2 = 10。

Add 2 to both sides: 2x = 12.

两边加 2：2x = 12。

Finally, divide both sides by 2 to find x = 6.

最后两边同除以 2，得到 x = 6。

Solution: x = 6

解：x = 6

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3. Inequalities on a Number Line | 数轴上的不等式

Question: Solve the inequality 2x + 5 < 11 and represent the solution on a number line.

问题：解不等式 2x + 5 < 11，并将解表示在数轴上。

Subtract 5 from both sides: 2x < 6.

两边减去 5：2x < 6。

Divide both sides by 2: x < 3.

两边除以 2：x < 3。

On a number line, draw an open circle at 3 and shade the line to the left, showing all values less than 3.

在数轴上，在 3 处画一个空心圆圈，并向左涂黑线条，表示所有小于 3 的值。

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4. Solving Simultaneous Equations | 解联立方程组

Question: Solve the simultaneous equations:
2x + y = 10
x − y = 2

问题：解联立方程组：
2x + y = 10
x − y = 2

Write the equations clearly:

2x + y = 10   (1)
x − y = 2     (2)

列出方程：

2x + y = 10   (1)
x − y = 2     (2)

Add equations (1) and (2) to eliminate y: (2x + x) + (y − y) = 10 + 2 → 3x = 12, so x = 4.

将方程 (1) 与 (2) 相加消去 y：(2x + x) + (y − y) = 10 + 2 → 3x = 12，因此 x = 4。

Substitute x = 4 into equation (2): 4 − y = 2 → y = 2.

将 x = 4 代入方程 (2)：4 − y = 2 → y = 2。

The solution is x = 4, y = 2.

解为 x = 4, y = 2。

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5. Finding the nth Term of a Linear Sequence | 求线性数列的第n项

Question: Find the nth term of the sequence: 5, 9, 13, 17, …

问题：求数列 5, 9, 13, 17, … 的第 n 项。

out the common difference: 9 − 5 = 4. The difference is 4, so the nth term begins with 4n.

计算公差：9 − 5 = 4。公差为 4，因此第 n 项以 4n 开始。

n = 1, 4n = 4, but the first term is 5, so you need to add 1. Thus the nth term is 4n + 1.

当 n = 1 时，4n = 4，但首项为 5，因此需要加 1。故第 n 项为 4n + 1。

Check with n = 2: 4(2) + 1 = 9, correct. n = 3: 4(3) + 1 = 13, correct.

检验 n = 2：4(2) + 1 = 9，正确。 n = 3：4(3) + 1 = 13，正确。

nth term = 4n + 1

第 n 项 = 4n + 1

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6. Angles in Regular Polygons | 正多边形内角

Question: Find the size of each interior angle of a regular octagon.

问题：求正八边形的每个内角大小。

sing the formula (n − 2) × 180°. For an octagon, n = 8: (8 − 2) × 180° = 6 × 180° = 1080°.

首先，利用公式 (n − 2) × 180° 计算内角和。对于八边形，n = 8：(8 − 2) × 180° = 6 × 180° = 1080°。

has all angles equal, so divide the sum by 8 to find one interior angle: 1080° ÷ 8 = 135°.

正八边形的每个内角相等，因此将总和除以 8：1080° ÷ 8 = 135°。

Each interior angle of a regular octagon is 135°.

正八边形的每个内角为 135°。

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7. Applying Pythagoras' Theorem | 应用毕达哥拉斯定理

Question: A ladder 5 m long leans against a vertical wall. The foot of the ladder is 3 m from the wall. How high up the wall does the ladder reach?

问题：一架 5 米长的梯子斜靠在一面垂直的墙上。梯脚距墙 3 米。梯子顶端能达到多高？

Let the height be h. By Pythagoras' theorem: h² + 3² = 5².

设高度为 h。根据毕达哥拉斯定理：h² + 3² = 5²。

h² + 9 = 25

h² + 9 = 25

Subtract 9 from both sides: h² = 16.

两边减 9：h² = 16。

Take the square root: h = 4 (since height is positive).

取平方根：h = 4（高度为正数）。

The ladder reaches 4 m up the wall.

梯子能达到 4 米高。

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8. Volume of a Cylinder | 圆柱体积

Question: A cylinder has a radius of 7 cm and a height of 10 cm. Find the volume, leaving your answer in terms of π.

问题：一个圆柱体底面半径为 7 cm，高为 10 cm。求体积，答案用 π 表示。

The formula for the volume of a cylinder is V = πr²h.

圆柱体积公式为 V = πr²h。

Substitute r = 7 and h = 10: V = π × 7² × 10 = π × 49 × 10 = 490π.

代入 r = 7，h = 10：V = π × 7² × 10 = π × 49 × 10 = 490π。

The volume is 490π cm³.

体积为 490π cm³。

If an approximate numerical answer is required (using π ≈ 3.14), it would be about 1539.4 cm³, but the question asks for the answer in terms of π.

若需近似值（取 π ≈ 3.14），约为 1539.4 cm³，但题目要求用 π 表示。

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9. Calculating Probabilities | 计算概率

Question: A bag contains 3 red marbles, 5 blue marbles and 2 green marbles. One marble is chosen at random. What is the probability that it is blue?

问题：一个袋子里有 3 个红球、5 个蓝球和 2 个绿球。随机摸出一个球，摸到蓝球的概率是多少？

Total number of marbles = 3 + 5 + 2 = 10.

球的总数 = 3 + 5 + 2 = 10。

Number of blue marbles = 5. So the probability P(blue) = 5/10.

蓝球数量 = 5。因此概率 P(蓝) = 5/10。

Simplify the fraction: 5/10 = 1/2.

化简分数：5/10 = 1/2。

The probability of picking a blue marble is ½.

摸到蓝球的概率为 ½。

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10. Plotting a Quadratic Graph | 绘制二次函数图像

Question: Plot the graph of y = x² − 2x − 3 for x values from −2 to 4.

问题：绘制函数 y = x² − 2x − 3 在 x 从 −2 到 4 范围内的图像。

First, complete a table of values. Substitute each integer x value into the equation.

首先，完成数值表。将每个整数 x 值代入方程。

x	−2	−1	0	1	2	3	4
y	5	0	−3	−4	−3	0	5

example, when x = −2: y = (−2)² − 2(−2) − 3 = 4 + 4 − 3 = 5. Do the same for all x values.

例如，当 x = −2 时：y = (−2)² − 2(−2) − 3 = 4 + 4 − 3 = 5。对所有 x 值重复计算。

Plot the points on a coordinate grid: (−2,5), (−1,0), (0,−3), (1,−4), (2,−3), (3,0), (4,5).

在坐标网格上描点：(−2,5), (−1,0), (0,−3), (1,−4), (2,−3), (3,0), (4,5)。

Join the points with a smooth U‑shaped curve (a parabola). The graph crosses the x‑axis at x = −1 and x = 3, and its minimum point is at (1,−4).

用平滑的 U 形曲线（抛物线）连接各点。图像与 x 轴相交于 x = −1 和 x = 3，最低点为 (1,−4)。

This quadratic graph illustrates the key features students need to identify: the roots, the y‑intercept, and the turning point.

该二次函数图像展示了学生需要识别的关键特征：根、y 轴截距和顶点。

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