📚 9H Homework Answer Book Prelims: Key Solutions and Revision Notes | 9H 作业答案簿预考：核心解答与复习笔记
This revision guide supports students working through the 9H homework answer book in preparation for preliminary examinations. It focuses on the most frequently tested topics, shows how to check answers efficiently, and highlights common mistakes that can cost marks. Use this alongside your own marked homework to close gaps before the prelim paper.
本复习指南旨在帮助使用 9H 作业答案簿备考预考的学生。内容聚焦最常考查的主题，展示如何高效检查答案，并指出可能导致失分的常见错误。请结合自己已批改的作业使用本指南，在预考前及时补上知识漏洞。
1. Algebraic Expressions and Simplification | 代数表达式与化简
+ 12y. When expanding brackets, apply the distributive law accurately: a(b + c) = ab + ac.
化简时首先仔细合并同类项。例如在 3x + 5y - 2x + 7y 中，将含 x 的项与含 y 的项分别合并，得到 x + 12y。去括号时要准确运用分配律：a(b + c) = ab + ac。
2(3x - 4) - 3(x + 2) = 6x - 8 - 3x - 6 = 3x - 14
Watch for sign errors when a negative sign sits outside a bracket. Multiply every term inside the bracket by the sign outside before simplifying. For example, -(x - 5) becomes -x + 5, not -x - 5.
当括号外为负号时要注意符号错误。化简前应先将括号外符号乘入括号内每一项。例如 -(x - 5) 应变为 -x + 5，而不是 -x - 5。
- Group like terms before adding or subtracting.
- Apply the distributive law to every term inside brackets.
- Check the sign of each term after expansion.
- 先将同类项分组，再进行加减。
- 对括号内每一项都应用分配律。
- 去括号后逐项检查符号。
2. Solving Linear Equations | 解一元一次方程
ind x = 6. Always verify your solution by substituting it back into the original equation.
解一元一次方程时，通过逆运算将未知数单独留在等式一边。例如 2x + 5 = 17，两边先同时减 5 得到 2x = 12，再两边同时除以 2，得到 x = 6。解出后务必代回原方程检验。
3(x - 4) = 2x + 5 → 3x - 12 = 2x + 5 → x = 17
ominators. For example, x/3 + 2 = 5 can be multiplied by 3 to become x + 6 = 15, so x = 9.
当方程中含有分数时，可将每一项乘以最小公分母以消去分母。例如 x/3 + 2 = 5 两边乘以 3，变为 x + 6 = 15，因此 x = 9。
- Perform the same operation on both sides of the equation.
- Clear fractions by multiplying by the lowest common denominator.
- Always substitute your final value back to check.
- 等式两边必须同时进行相同运算。
- 乘以最小公分母以清除分数。
- 最终答案必须代回原方程检验。
3. Quadratic Expressions and Factorisation | 二次表达式与因式分解
7x + 10, the numbers 2 and 5 work because 2 × 5 = 10 and 2 + 5 = 7, giving (x + 2)(x + 5).
对形如 x² + bx + c 的二次式进行因式分解，需要找到两个数，它们相乘等于 c，相加等于 b。例如 x² + 7x + 10 中，2 和 5 满足 2 × 5 = 10 且 2 + 5 = 7，因此分解为 (x + 2)(x + 5)。
x² - 9 = (x + 3)(x - 3)
r a common factor before using other methods. For 2x² + 8x, factor out 2x first to get 2x(x + 4).
要能识别平方差公式：a² - b² = (a + b)(a - b)。在使用其他方法之前还应先寻找公因式。例如 2x² + 8x 先提取公因式 2x，得到 2x(x + 4)。
- Find two numbers with the correct product and sum.
- Check for a common factor before factorising fully.
- Use the difference of two squares when appropriate.
- 找到乘积与和均符合要求的两个数。
- 完全分解前先检查是否存在公因式。
- 遇到平方差形式时使用平方差公式。
4. Simultaneous Equations | 联立方程
uations cancels y: 2x = 14, so x = 7. Substitute x = 7 into either equation to find y = 3.
联立方程可以用消元法或代入法求解。对于方程组 x + y = 10 和 x - y = 4，两式相加可消去 y：2x = 14，因此 x = 7。将 x = 7 代入任一方程，可得 y = 3。
2x + 3y = 12 and 2x - y = 4 → subtract to get 4y = 8, so y = 2 and x = 3
When coefficients are unequal, multiply one or both equations so that one variable has matching coefficients. Always check your solution by substituting both values into both original equations.
当系数不相等时，可给一个或两个方程乘以适当倍数，使某一未知数的系数相同。解出后必须将两个值同时代入两个原方程进行检验。
- Eliminate one variable by adding or subtracting equations.
- Multiply equations to match coefficients when needed.
- Check the solution in both original equations.
- 通过方程相加或相减消去一个未知数。
- 必要时给方程乘以适当倍数以配平系数。
- 将解同时代入两个原方程检验。
5. Coordinate Geometry and Straight Lines | 坐标几何与直线
₁) ÷ (x₂ - x₁). A positive gradient slopes upward, while a negative gradient slopes downward.
过两点 (x₁, y₁) 和 (x₂, y₂) 的直线斜率用公式 m = (y₂ - y₁) ÷ (x₂ - x₁) 计算。斜率为正时直线向上倾斜，斜率为负时直线向下倾斜。
m = (y₂ - y₁) ÷ (x₂ - x₁)
adient and c is the y-intercept. Parallel lines have equal gradients, while perpendicular lines have gradients that multiply to -1.
直线方程通常写作 y = mx + c，其中 m 为斜率，c 为 y 轴截距。平行直线斜率相等，垂直直线斜率乘积为 -1。
- Use the gradient formula with two coordinates.
- Identify m and c in the equation y = mx + c.
- Parallel lines share the same gradient; perpendicular gradients multiply to -1.
- 使用两点坐标代入斜率公式。
- 在 y = mx + c 中识别 m 和 c。
- 平行线斜率相同；垂直线斜率乘积为 -1。
6. Trigonometry in Right-Angled Triangles | 直角三角形中的三角学
÷ adjacent. Label the sides relative to the given angle before choosing the correct ratio.
在直角三角形中，三个三角比分别为 sin θ = 对边 ÷ 斜边，cos θ = 邻边 ÷ 斜边，tan θ = 对边 ÷ 邻边。选择正确比值前应相对于已知角标出各边。
sin θ = opposite ÷ hypotenuse, cos θ = adjacent ÷ hypotenuse, tan θ = opposite ÷ adjacent
otenuse and need the side opposite a 30° angle, use sin 30° = x ÷ 10, so x = 10 × 0.5 = 5.
求未知边时，应使用包含已知边和所求边的三角比。例如已知斜边长为 10，求 30° 角对边，使用 sin 30° = x ÷ 10，解得 x = 10 × 0.5 = 5。
- Label opposite, adjacent, and hypotenuse before solving.
- Choose the ratio with one known side and one unknown side.
- Use inverse trig functions to find an angle when two sides are known.
- 解题前先标出对边、邻边和斜边。
- 选择包含一个已知边和一个未知边的比值。
- 已知两边求角时使用反三角函数。
7. Perimeter, Area and Volume | 周长、面积与体积
and the circumference of a circle is 2πr. For a cuboid, volume = length × width × height.
熟记二维和三维图形的关键公式。三角形面积为 ½ × 底 × 高，圆面积为 πr²，圆周长为 2πr。长方体体积 = 长 × 宽 × 高。
| Shape | Formula |
| Triangle area | ½ × base × height |
| Circle area | πr² |
| Circumference | 2πr |
| Cylinder volume | πr²h |
Always include units in your final answer and convert units consistently. If a question mixes cm and m, convert to the same unit before calculating area or volume.
最终答案必须写明单位，并保持单位一致。如果题目混用 cm 和 m，应先换算成相同单位再计算面积或体积。
- Write the correct formula before substituting values.
- Check whether the question asks for perimeter, area, or volume.
- Convert all lengths to the same unit first.
- 代入数值前先写出正确公式。
- 看清题目要求求周长、面积还是体积。
- 先将所有长度换算为相同单位。
8. Probability and Venn Diagrams | 概率与韦恩图
lities always lie between 0 and 1. For mutually exclusive events, P(A or B) = P(A) + P(B).
事件发生的概率为 P(A) = 有利结果数 ÷ 总结果数。概率始终介于 0 和 1 之间。对于互斥事件，P(A 或 B) = P(A) + P(B)。
P(A or B) = P(A) + P(B) - P(A and B)
Venn diagrams help organise overlapping sets. Place the intersection value first, then work outward to fill the remaining regions. Always check that all regions sum to the total number of items.
韦恩图有助于整理有重叠的集合。先填写交集区域，再向外填写其余区域。务必检查所有区域之和等于总元素数。
- Simplify fractions when giving probability answers.
- Use the addition rule for non-mutually exclusive events.
- In Venn diagrams, fill the intersection before other regions.
- 给出概率答案时应约分。
- 对于非互斥事件使用加法法则。
- 画韦恩图时先填交集区域。
9. Statistics: Averages and Charts | 统计：平均数与图表
The three measures of average are mean, median, and mode. The mean is the sum of values divided by the number of values. The median is the middle value when data are ordered, and the mode is the most frequent value. The range is the difference between the largest and smallest values.
三种平均数分别是平均数、中位数和众数。平均数等于所有数值之和除以数据个数。中位数是数据排序后位于中间的值，众数是出现次数最多的值。极差是最大值与最小值之差。
Mean = sum of values ÷ number of values
When interpreting charts, read scales carefully and check whether the axis starts at zero. A misleading chart can exaggerate differences. For grouped frequency tables, use the midpoint of each class interval to estimate the mean.
解读图表时要仔细读取刻度，并检查坐标轴是否从零开始。误导性图表会夸大差异。对于分组频数表，可用每个组区间的中点来估算平均数。
- Order data before finding the median.
- Use the midpoint of interval for grouped data.
- Check axis scales to avoid misreading charts.
- 求中位数前先对数据排序。
- 分组数据使用组区间中点。
- 检查坐标轴刻度以避免误读图表。
10. Exam Technique and Common Errors | 考试技巧与常见错误
Before submitting your prelim paper, check that every answer is in the required form. Some questions ask for answers to a given number of decimal places or significant figures, while others require fractions or surds in simplest form.
提交预考试卷前，检查每个答案是否符合题目要求的形式。有些题目要求答案保留指定的小数位数或有效数字，另一些题目则要求最简分数或根式形式。
Common errors include dropping negative signs, forgetting units, misreading the order of operations, and not showing working. In prelims, method marks are often awarded even if the final answer is incorrect, so always write clear steps.
常见错误包括遗漏负号、忘记单位、看错运算顺序以及不写解题过程。预考中即使最终答案错误，步骤分也经常会给，因此务必写出清晰步骤。
- Read the instruction line before starting each question.
- Show all working to gain method marks.
- Leave time to check answers by substitution or estimation.
- 开始每题前先阅读答题要求。
- 写出完整步骤以获得过程分。
- 留出时间通过代入或估算检查答案。
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