ENGAA 2020 S1 Answer Key | ENGAA 2020 第一部分答案解析

📚 ENGAA 2020 S1 Answer Key | ENGAA 2020 第一部分答案解析

This article provides a detailed answer key and step-by-step explanations for the Mathematics questions (Q1–20) from the ENGAA 2020 Section 1, based on candidate recollections. It is designed to help you understand the reasoning behind each answer and strengthen your problem-solving skills for the Cambridge Engineering Admissions Assessment.

本文根据考生回忆整理了 ENGAA 2020 第一部分数学题（第1–20题）的答案与逐题解析，旨在帮助你理解每道题的解题逻辑，提升应对剑桥工程入学评估的能力。

1. Questions 1–2: Algebra & Functions | 代数与函数

Question 1: Solve the equation |2x – 1| = x + 4. What is the sum of all real solutions? A. -1 B. 3 C. 4 D. 5

第1题：解方程 |2x – 1| = x + 4，所有实数解之和是多少？

Answer: C (4). Case 1: 2x – 1 = x + 4 → x = 5. Case 2: -(2x – 1) = x + 4 → -2x + 1 = x + 4 → -3x = 3 → x = -1. Both solutions are valid, sum = 5 + (-1) = 4.

答案：C。第一种情况：2x – 1 = x + 4 → x = 5。第二种情况：-(2x – 1) = x + 4 → -2x + 1 = x + 4 → -3x = 3 → x = -1。两个解均成立，和为 4。

Question 2: The function f(x) = x³ – 6x² + 9x + 1 has how many inflection points? A. 0 B. 1 C. 2 D. 3

第2题：函数 f(x) = x³ – 6x² + 9x + 1 有几个拐点？

Answer: B (1). f”(x) = 6x – 12. Set f”(x) = 0 → x = 2. f”'(x) = 6 ≠ 0, so the concavity changes. Hence one inflection point.

答案：B。f”(x) = 6x – 12。令 f”(x) = 0 得 x = 2，且 f”'(x) = 6 ≠ 0，凹凸性发生改变，故有一个拐点。

2. Questions 3–4: More Functions | 函数进阶

Question 3: Given f(x) = (x + 2)/(x – 1), find f(f(2)). A. -1 B. 0 C. 1 D. 2

第3题：已知 f(x) = (x + 2)/(x – 1)，求 f(f(2))。

Answer: D (2). f(2) = (2+2)/(2-1) = 4. Then f(4) = (4+2)/(4-1) = 6/3 = 2.

答案：D。先计算 f(2) = 4，再计算 f(4) = 6/3 = 2。

Question 4: The graph of y = ln x is first reflected in the y-axis and then translated 3 units to the right. Which of the following gives the new graph? A. y = ln(3 – x) B. y = ln(3 + x) C. y = ln(-x – 3) D. y = ln(x – 3)

第4题：将 y = ln x 的图像先关于 y 轴反射，再向右平移 3 个单位，新图像对应哪个函数？

Answer: A. Reflect in y-axis: y = ln(-x). Translate right by 3: replace x with (x – 3) → y = ln(-(x – 3)) = ln(3 – x).

答案：A。关于 y 轴反射得 y = ln(-x)；向右平移 3 个单位，用 (x – 3) 替换 x，得 y = ln(3 – x)。

3. Questions 5–6: Sequences | 数列

Question 5: The sum of the first n terms of an arithmetic progression is Sₙ = 3n² + 5n. Find the 10th term. A. 62 B. 65 C. 68 D. 70

第5题：某等差数列前 n 项和为 Sₙ = 3n² + 5n，求第 10 项。

Answer: A (62). T₁₀ = S₁₀ – S₉. S₁₀ = 3×100 + 50 = 350, S₉ = 3×81 + 45 = 288, difference = 62.

答案：A。第 10 项为 S₁₀ – S₉ = 350 – 288 = 62。

Question 6: A geometric series has first term a = 24 and sum to infinity S∞ = 32. Find the common ratio r. A. 0.5 B. 0.25 C. 0.75 D. 0.2

第6题：等比数列首项 a = 24，无穷项之和为 32，求公比 r。

Answer: B (0.25). S∞ = a/(1 – r) → 32 = 24/(1 – r) → 1 – r = 24/32 = 0.75 → r = 0.25.

答案：B。由公式 32 = 24/(1 – r) 解得 r = 0.25。

4. Questions 7–8: Trigonometry & Geometry | 三角与几何

Question 7: Solve cos 2θ = sin θ for 0 ≤ θ ≤ π. How many solutions are there? A. 1 B. 2 C. 3 D. 4

第7题：在区间 [0, π] 内解方程 cos 2θ = sin θ，解的个数是多少？

Answer: B (2). Use cos 2θ = 1 – 2 sin²θ → 1 – 2 sin²θ = sin θ → 2 sin²θ +

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