Year 10 SQA Advanced Mathematics: Unit Test Mock Paper Walkthrough | SQA 进阶数学：单元测试模拟卷解析

📚 Year 10 SQA Advanced Mathematics: Unit Test Mock Paper Walkthrough | SQA 进阶数学：单元测试模拟卷解析

This article provides a step-by-step walkthrough of a mock unit test for Year 10 SQA Advanced Mathematics. Each question is carefully solved, and the reasoning behind every method is explained. Tips on avoiding common pitfalls are included to help you achieve full marks.

本文详细解析一份为 SQA 进阶数学设计的单元测试模拟卷，逐题展示解题步骤与思路，并穿插易错点提醒，帮助你稳扎稳打拿到高分。

1. Quadratic Equations & Discriminant | 二次方程与判别式

Question: Solve x² – 5x + 6 = 0 and find the value of its discriminant. First, factorise the quadratic: (x – 2)(x – 3) = 0. Setting each bracket to zero gives x = 2 or x = 3. The discriminant Δ = b² – 4ac. Here a = 1, b = –5, c = 6, so Δ = (–5)² – 4×1×6 = 25 – > 0, the equation has two distinct real roots, which matches our factorised solutions.

题目：求解 x² – 5x + 6 = 0 并计算判别式的值。先将二次式因式分解为 (x – 2)(x – 3) = 0，令每个括号为零得到 x = 2 或 x = 3。判别式 Δ = b² – 4ac，其中 a = 1, b = –5, c = 6，代入得 Δ = 25 – 24 = 1。Δ > 0 表明方程有两个不相等的实根，与分解结果一致。

x = 2 or x = 3

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2. Composite & Inverse Functions | 复合函数与逆函数

= 2x² + 3. To find the inverse of g(x), let y = x² and swap x and y: x = y². Solve for y, remembering to restrict the domain if necessary (g is not one-to-one over all reals, but we take the principal branch). Here we write g⁻¹(x) = √x for x ≥ 0. The notation √ denotes the principal square root.

题目：已知 f(x) = 2x + 3，g(x) = x²，求 f(g(x)) 和逆函数 g⁻¹(x)。复合函数将 g(x) 代入 f，得 f(g(x)) = 2(x²) + 3 = 2x² + 3。求反函数时设 y = x²，交换 x 与 y 得 x = y²，解出 y 并限定定义域（g 在整个实数集上不是单射，此处取主值分支），即 g⁻¹(x) = √x，x ≥ 0。

f(g(x)) = 2x² + 3,   g⁻¹(x) = √x (x ≥ 0)

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3. Solving Trigonometric Equations | 解三角方程

0.5. Since sine is positive in the first and second quadrants, the second solution is 180° – ther solutions in the given interval. Always draw the CAST diagram or sine graph to check.

题目：在 0° 至 360° 范围内求解 sin x = 0.5。先找出主值：由 sin 30° = 0.5 得 x = 30°。正弦在第一、第二象限为正，因此第二个解为 180° – 30° = 150°。在给定区间内再无其他解。务必借助 CAST 图或正弦图像避免漏解。

x = 30°, 150°

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4. Exponential & Logarithmic Equations | 指数与对数方程

Question: Solve 3 × 5ˣ = 75. Divide both sides by 3: 5ˣ = 25. Recognise that 25 = 5², so 5ˣ would rewrite it as x = 2³ = 8. The connection between exponentials and logs is essential.

题目：解指数方程 3 × 5ˣ = 75。两边同除以 3 得 5ˣ = 25。注意到 25 = 5²，因此 5ˣ = 5²，故 x = 2。若涉及对数，如 log₂(x) = 3，可改写为 x = 2³ = 8。指数与对数的互化是必考要点。

x = 2

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5. Differentiation & Tangent Lines | 求导与切线方程

Question: Find the derivative of f(x) = x³ – 2, and then determine the equation of the tangent at x = 1. Differentiate term by term: f'(x) = 3x² – 6x. At x = 1, the gradient m = 3(1)² – 6(1) = –3, and the y-coordinate is f(1) = 1 – 3 + 2 = 0, so the point is (1, 0). The tangent line has equation y – 0 = –3(x – 1) → y = –3x + 3.

题目：求 f(x) = x³ – 3x² + 2 的导数，并写出曲线在 x = 1 处的切线方程。逐项求导得 f'(x) = 3x² – 6x。代入 x = 1，斜率 m = 3(1)² – 6(1) = –3；此时 y = f(1) = 1 – 3 + 2 = 0，切点为 (1, 0)。切线方程为 y – 0 = –3(x – 1)，即 y = –3x + 3。

Tangent: y = –3x + 3

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6. Definite Integration & Area | 定积分与面积

Question: Evaluate ∫₀² (3x² – 2x + 1) dx and interpret the result as an area. Find the antiderivative: F(x) = x³ – x² + x. Then use the limits: F(2) – F(0) = (8 – 4 + 2) – [0, 2], the definite integral represents the area under the curve between x = 0 and x = 2.

题目：计算定积分 ∫₀² (3x² – 2x + 1) dx，并说明其面积的几何意义。先求原函数 F(x) = x³ – x² + x。代入上下限：F(2) – F(0) = (8 – 4 + 2) – 0 = 6。因为被积函数在 [0, 2] 上恒正，该定积分即为曲线下方、x 轴上方的面积。

∫₀² (3x² – 2x + 1) dx = 6

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7. Vector Operations & Magnitude | 向量运算与模

Question: Given vectors a = (3, –2) and b = (–1, 4), find 2a + b and the magnitude |a|. Scalar multiplication: 2a = (6, –4). Add b: 2a + b = (6 + (–1), –4 + 4) = (5, 0). The magnitude of a is |a| = √(3² + (–2)²) = √9 + 4 = √13. Vectors are often written in column form, but the arithmetic is the same.

题目：已知向量 a = (3, –2) 和 b = (–1, 4)，求 2a + b 以及 a 的模。数乘得 2a = (6, –4)。向量相加：2a + b = (6 – 1, –4 + 4) = (5, 0)。a 的模 |a| = √(3² + (–2)²) = √9 + 4 = √13。虽然向量常记作列形式，但分量运算规则一致。

2a + b = (5, 0),   |a| = √13

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8. Arithmetic Sequences & Series | 等差数列与求和

Question: An arithmetic sequence begins 5, 9, 13, … the first 20 terms. First term a = 5, common difference d = 4. The nth term is uₙ = a + (n – 1)d = 5 + (n – 1)×4 = 4n + 1. The sum of the first 20 terms is S₂₀ = n/2 [2a + (n – 1)d] = 20/2 × [2×5 + 19×4] = 10 × (10 + 76) = 10 × 86 = 860.

题目：等差数列 5, 9, 13, … ，写出第 n 项公式，并求前 20 项的和。首项 a = 5，公差 d = 4。第 n 项 uₙ = a + (n – 1)d = 5 + (n – 1)×4 = 4n + 1。前 20 项和 S₂₀ = n/2 [2a + (n – 1)d] = 20/2 × (2×5 + 19×4) = 10 × (10 + 76) = 860。

uₙ = 4n + 1,   S₂₀ = 860

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9. Common Mistakes & How to Avoid Them | 常见易错点与规避方法

Several recurring mistakes cost marks in algebra and calculus. When factorising, always expand the factors mentally to check they match the original expression. In trigonometry, forgetting the second angle in the required range is a classic error – always use the CAST diagram. In differentiation, students often mishandle constants: the derivative of a constant is zero, and don't forget to multiply by the power when differentiating xⁿ. Integration requires the constant of integration for indefinite integrals, but not for definite ones. When working with vectors, ensure you add corresponding components correctly. Finally, in sequence problems, double-check whether the question asks for a term or a sum. Reading the question twice saves unnecessary marks lost.

代数与微积分中有一些反复出现的失分点。因式分解后务必展开验证是否与原式一致。解三角方程时最常见的错误是漏掉周期内的其他解，一定要画 CAST 图。求导时常数的导数为 0，对 xⁿ 求导时记得乘上指数再降次。不定积分必须加积分常数 C，而定积分不用。向量运算中要注意对应分量相加。序列题一定要看清题目求的是第几项还是求和。多读一遍题就能避免无谓失分。

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10. Exam Tips & Final Checklist | 考试技巧与最终检查清单

Before the exam, revise essential formulas such as the discriminant, trigonometric exact values, and the sum of arithmetic series. During the test, show all your working clearly – even if the final answer is wrong, method marks can be awarded. Manage your time: don't spend too long on a single question; if stuck, move on and come back later. Always check your answers by substituting them back into the original equation. For graph-related questions, label axes and key points. Finally, ensure your calculator is in the correct mode (degree/radian) and bring a ruler for drawing axes.

考前要熟记判别式、特殊角的三角函数值、等差求和公式等核心知识。考试时务必展示清晰步骤，即使答案出错也能拿到方法分。合理分配时间，遇难题先跳过，回头再补做。答案一定要代回原方程检验。涉及图像时注意标注坐标轴和关键点。最后，确认计算器角度/弧度模式正确，并携带直尺辅助绘图。

Published by TutorHao | Advanced Mathematics Revision Series | aleveler.com

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