📚 Prelims 9H Answers | 模拟考试 9H 答案解析
This revision guide covers typical answer patterns and mark-worthy steps for a mathematics Preliminary Paper 9H. Use the worked examples and answer bank to check your method, avoid common errors and improve accuracy under timed conditions.
本复习指南涵盖数学模拟考试 9H 试卷中典型答案模式与得分步骤。通过示例解答和答案库检查你的方法，避免常见错误，并在限时条件下提高准确性。
1. Understanding the 9H Mark Scheme | 理解 9H 评分标准
In a 9H preliminary paper, method marks are often awarded for correct substitution, factorisation, or use of a formula even if the final answer is wrong. Examiners look for clear reasoning, not just a final value.
在 9H 模拟试卷中，即使最终答案错误，正确代入、因式分解或使用公式通常也能获得方法分。阅卷人看重清晰的推理过程，而不仅仅是最终数值。
Always write each step on a new line. For example, if you use the quadratic formula, show the values of a, b and c before substituting them into the formula.
始终将每一步写在新的一行。例如，如果你使用二次公式，先把 a、b 和 c 的值写出来，再代入公式。
Final answers should be written on the answer line or in the box provided. If the question asks for an exact answer, leave surds or π instead of rounding.
最终答案应写在答题线或指定方框内。如果题目要求精确答案，请保留根号或 π，而不是四舍五入。
2. Algebraic Manipulation: Worked Answers | 代数运算：答案示例
erm: 3x×2x = 6x², 3x×5 = 15x, -2×2x = -4x, -2×5 = -10. Collect like terms: 6x² + 11x - 10.
示例 1：展开并化简 (3x - 2)(2x + 5)。逐项相乘：3x×2x = 6x²，3x×5 = 15x，-2×2x = -4x，-2×5 = -10。合并同类项：6x² + 11x - 10。
Factorising is the reverse process: for x² - 7x + 12, find two numbers that multiply to 12 and add to -7. The answer is (x - 3)(x - 4).
因式分解是逆过程：对于 x² - 7x + 12，找到两个数，乘积为 12，和为 -7。答案是 (x - 3)(x - 4)。
and -1. Split the middle term: 2x² + 6x - x - 3 = 2x(x + 3) - 1(x + 3) = (2x - 1)(x + 3).
示例 2：因式分解 2x² + 5x - 3。将二次项系数与常数项相乘：2 × (-3) = -6。找到两个数，乘积为 -6，和为 5：这两个数是 6 和 -1。拆分中间项：2x² + 6x - x - 3 = 2x(x + 3) - 1(x + 3) = (2x - 1)(x + 3)。
Completing the square is also tested: x² + 6x + 2 can be written as (x + 3)² - 7. Keep the constant term separate and check by expanding.
配方法也是考点：x² + 6x + 2 可写成 (x + 3)² - 7。将常数项分离出来，并通过展开检验。
3. Solving Equations and Inequalities | 解方程与不等式
x - 3 = 0, then factor by grouping: 2x(x - 3) + 1(x - 3) = 0, giving (2x + 1)(x - 3) = 0.
对于二次方程 2x² - 5x - 3 = 0，拆分中间项：2x² - 6x + x - 3 = 0，然后分组分解：2x(x - 3) + 1(x - 3) = 0，得到 (2x + 1)(x - 3) = 0。
are x = -1/2 or x = 3. Check each solution in the original equation to avoid sign errors.
解为 x = -1/2 或 x = 3。将每个解代入原方程检验，避免符号错误。
, substitute the first into the second: x + (2x + 1) = 7, so 3x = 6 and x = 2. Then y = 5.
对于联立方程 y = 2x + 1 和 x + y = 7，将第一个方程代入第二个：x + (2x + 1) = 7，因此 3x = 6，x = 2。然后 y = 5。
For inequalities such as 3x - 4 < 5x + 2, subtract 3x from both sides: -4 < 2x + 2, then -6 < 2x, so x > -3. Remember to reverse the symbol when multiplying or dividing by a negative number.
对于不等式 3x - 4 < 5x + 2，两边减去 3x：-4 < 2x + 2，然后 -6 < 2x，因此 x > -3。记住当乘以或除以负数时要反转不等号。
If you need to solve x² > 9, consider both branches: x > 3 or x < -3. Sketching a parabola helps visualise the solution set.
如果需要解 x² > 9，考虑两个分支：x > 3 或 x < -3。画抛物线草图有助于直观理解解集。
4. Coordinate Geometry and Graphs | 坐标几何与图像
adient between (1, 2) and (4, 8), use m = (y₂ - y₁)/(x₂ - x₁) = (8 - 2)/(4 - 1) = 6/3 = 2.
求 (1, 2) 和 (4, 8) 之间的斜率，使用 m = (y₂ - y₁)/(x₂ - x₁) = (8 - 2)/(4 - 1) = 6/3 = 2。
= 2(x - 1), giving y = 2x. Always simplify to the required form, such as ax + by + c = 0.
过 (1, 2) 且斜率为 2 的直线方程：y - 2 = 2(x - 1)，得到 y = 2x。始终化简为题目要求的形式，如 ax + by + c = 0。
product of gradients is -1. Example: a line perpendicular to y = 2x + 3 has gradient -1/2.
对于平行线，斜率相等；对于垂直线，斜率之积为 -1。示例：与 y = 2x + 3 垂直的直线斜率为 -1/2。
To find the midpoint of two points, average the x-coordinates and y-coordinates: midpoint of (1, 4) and (7, 10) is (4, 7).
求两点中点时，对 x 坐标和 y 坐标分别取平均：(1, 4) 和 (7, 10) 的中点为 (4, 7)。
uses Pythagoras: distance = √[(x₂ - x₁)² + (y₂ - y₁)²]. For (1, 2) and (4, 6), distance = √9 + 16 = 5.
两点之间的距离使用勾股定理：距离 = √[(x₂ - x₁)² + (y₂ - y₁)²]。对于 (1, 2) 和 (4, 6)，距离 = √9 + 16 = 5。
5. Trigonometry in 9H Prelims | 三角学
/hypotenuse, tan θ = opposite/adjacent. Label the sides before choosing the correct ratio.
在直角三角形中，sin θ = 对边/斜边，cos θ = 邻边/斜边，tan θ = 对边/邻边。选择正确比例前先标出各边。
Example: If sin θ = 3/5, the missing side is √5² - 3² = 4, so cos θ = 4/5 and tan θ = 3/4.
示例：若 sin θ = 3/5，则缺失边为 √5² - 3² = 4，因此 cos θ = 4/5，tan θ = 3/4。
Use exact values for 30°, 45° and 60°: sin 45° = √2/2, cos 60° = 1/2, tan 30° = 1/√3 or √3/3.
使用 30°、45
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