📚 Advanced Mathematics: ENGAA 2017 S1 Answer Key | 进阶数学：ENGAA 2017 S1 答案解析
This article provides a complete answer key and concise explanations for the Mathematics part (Part A) of the ENGAA 2017 Section 1 paper. The paper tests a wide range of advanced mathematics topics, from algebraic manipulation to calculus and mechanics, all within the format of multiple-choice questions. Use this walkthrough to check your answers, understand the reasoning behind each correct option, and strengthen your problem-solving strategies for the Engineering Admissions Assessment.
本文为ENGAA 2017年Section 1数学部分（Part A）提供完整答案解析和简要思路说明。该部分通过选择题的形式，考查了从代数运算到微积分、力学等广泛的进阶数学知识点。你可以借助这份答案详解核对答案、理解每道题的正确思路，并巩固应对工程入学评估的解题策略。
1. Questions 1–2: Algebraic Manipulation and Quadratic Discriminant | 第1–2题：代数运算与二次方程判别式
The opening questions test essential algebraic skills: expanding products, combining like terms, and applying the discriminant to determine the nature of the roots of a quadratic equation.
前两道题考查核心代数技巧：展开乘积、合并同类项，以及运用判别式判断二次方程根的性质。
Q1 – Simplify the given expression by expanding each product and collecting like terms. The result is x² − x − 3. Always double-check signs when subtracting a bracket. Correct answer: B.
Q1 – 展开每个乘积并合并同类项，化简后得到 x² − x − 3。减去括号时要特别注意变号。正确答案：B。
Q2 – For a quadratic with a parameter to have no real solutions, the discriminant Δ must be strictly negative. Set up the inequality Δ < 0, expand carefully, and solve for the parameter. The solution gives k < 2/3. This tests the condition b² − 4ac < 0. Correct answer: C.
Q2 – 含参数的二次方程无实数解时，判别式 Δ 必须严格小于零。列出 Δ < 0 的不等式，仔细展开并求解参数，得到 k < 2/3。此题考查的是 b² − 4ac < 0 这一条件。正确答案：C。
2. Questions 3–4: Inequalities and Coordinate Geometry | 第3–4题：不等式与坐标几何
These questions move to solving quadratic inequalities and analysing the intersection of a line and a circle, combining algebraic manipulation with geometric interpretation.
接下来是求解二次不等式以及分析直线与圆的交点问题，将代数运算与几何意义相结合。
Q3 – Rearrange the quadratic inequality to bring all terms to one side, factorise the resulting quadratic, and determine the intervals where the product is positive. The solution set is x < −1 or x > 5/3. Always sketch the parabola to confirm sign regions. Correct answer: D.
Q3 – 将二次不等式移项使一边为零，因式分解后判断使乘积为正的区间。解集为 x < −1 或 x > 5/3。最好画出抛物线示意图来验证正负区域。正确答案：D。
Q4 – Substitute the line equation into the circle equation to obtain a quadratic in x (or y). Solve for the intersection points. The coordinates correspond to (2, −1) and (3, 1). This is a typical simultaneous equations problem with one linear and one quadratic equation. Correct answer: A.
Q4 – 将直线方程代入圆的方程，得到一个关于 x（或 y）的二次方程，解出交点坐标 (2, −1) 和 (3, 1)。这是典型的由一个线性方程和一个二次方程组成的联立方程组问题。正确答案：A。
3. Questions 5–6: Trigonometric Equations and Sequences | 第5–6题：三角方程与数列
Question 5 involves solving a simple trigonometric equation, while Question 6 deals with an arithmetic or geometric sequence, demanding clear use of standard formulas.
第5题要求解一个简单的三角方程，第6题则涉及等差或等比数列，需要清晰运用标准公式。
Q5 – Rearrange the equation to a basic form such as sin θ = constant or tan θ = constant. Remember to consider all solutions within the given domain, adding multiples of the period. The correct answer emerges as E.
Q5 – 将方程化为 sin θ = 常数 或 tan θ = 常数 等基本形式。注意在给定范围内找出所有解，补上周期的整数倍。正确答案为 E。
Q6 – Identify the type of sequence (arithmetic or geometric) from the information given. Apply the formula for the nth term or the sum of the first n terms. For an arithmetic series, use Sₙ = n/2 (2a + (n−1)d). The calculation leads to answer A.
Q6 – 根据所给信息判断数列类型（等差或等比），运用通项公式或前 n 项和公式。对于等差数列，使用 Sₙ = n/2 (2a + (n−1)d)。计算结果指向答案 A。
4. Questions 7–8: Exponentials, Logarithms and Differentiation | 第7–8题：指数、对数与微分
These two questions test the laws of logarithms and the application of differentiation to find the equation of a tangent to a curve.
这两道题考查对数运算法则以及通过求导确定曲线切线方程的应用。
Q7 – Use log rules such as logₐ(MN) = logₐM + logₐN and logₐ(M/N) = logₐM − logₐN to simplify the expression. Combine the logarithmic terms into a single log and compare with the given form. The required value leads to answer B.
Q7 – 运用对数法则，如 logₐ(MN) = logₐM + logₐN 和 logₐ(M/N) = logₐM − logₐN 进行化简，将对数项合并为一个对数，再与所给形式比较，求得所需数值。答案为 B。
Q8 – Differentiate the function to find the gradient function dy/dx. Substitute the x-coordinate of the point of tangency to obtain the gradient m. Then use the point-slope form y − y₁ = m(x − x₁) to write the tangent equation. The correct option is C.
Q8 – 对函数求导得到梯度函数 dy/dx，代入切点的 x 坐标求得斜率 m，再利用点斜式 y − y₁ = m(x − x₁) 写出切线方程。正确选项为 C。
5. Questions 9–10: Integration and Vectors | 第9–10题：积分与向量
These questions cover the evaluation of a definite integral to find an area and basic vector operations, both fundamental in engineering mathematics.
这两题涵盖定积分求面积和基础的向量运算，都是工程数学中的基础内容。
Q9 – Set up the definite integral of the given curve between the given bounds. Integrate term by term, evaluate at the upper and lower limits, and subtract. The area is found to be a rational number, matching option E.
Q9 – 对所给曲线在指定区间上设定定积分，逐项积分后代入上下限求差，得出面积为一个有理数，与选项 E 相符。
Q10 – Perform vector addition or subtraction
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