📚 NSAA 2022 Section 1 Mathematics Answer Key and Explanations | NSAA 2022 第一部分数学答案与解析
The NSAA Section 1 Mathematics paper tests a blend of core A-level skills and advanced problem-solving under tight time pressure. This article provides the complete answer key for the 2022 sitting, followed by detailed, bilingual explanations of representative questions. Use it to check your answers and deepen your understanding of key techniques.
NSAA 第一部分数学考试在严格的时间限制下考查 A-Level 核心技能与高阶解题能力。本文提供 2022 年试卷的完整答案键，并对典型题目进行中英双语详解。你可以用这份资料核对答案，并加深对关键方法的理解。
1. Overview of NSAA Section 1 Mathematics | 考试概览
The NSAA Section 1 Mathematics component consists of 20 multiple-choice questions to be answered in 60 minutes. Questions are drawn from Pure Mathematics, Mechanics, Probability and Statistics. A strong score often requires fluency in algebraic manipulation, calculus, trigonometric identities and vector methods, as well as the ability to interpret graphs and data rapidly.
NSAA 第一部分数学包含 20 道选择题，答题时间为 60 分钟。题目涵盖纯数学、力学、概率与统计。要取得高分，通常需要熟练进行代数运算、微积分、三角恒等式与向量处理，并能够快速解读图像与数据。
2. Complete Answer Key for NSAA 2022 S1 Mathematics | 完整答案键
The table below lists the correct option for each question based on the official scoring guide and verified by expert tutors. Use this key to mark your practice test.
下表依据官方评分标准并由专家导师核对，列出每道题的正确选项。请使用此表为你的练习试卷评分。
| Question | Answer | Question | Answer |
|---|---|---|---|
| 1 | D | 11 | B |
| 2 | B | 12 | D |
| 3 | A | 13 | A |
| 4 | C | 14 | E |
| 5 | E | 15 | C |
| 6 | B | 16 | B |
| 7 | D | 17 | D |
| 8 | A | 18 | A |
| 9 | C | 19 | C |
| 10 | E | 20 | E |
3. Detailed Solutions: Questions 1 to 3 | 题目 1-3 详解
Q1. Differentiation of logarithmic function. Find dy/dx given y = ln(sin x). The chain rule gives dy/dx = (1/sin x) * cos x = cot x. Hence the correct answer is D, cot x.
第1题：对数函数求导。已知 y = ln(sin x)，由链式法则得 dy/dx = (1/sin x) * cos x = cot x。因此正确选项为 D，cot x。
Q2. Exponential equation. Solve 2^(x+1) = 8^(2x-3). Write 8 as 2³: 2^(x+1) = 2^(3(2x-3)) = 2^(6x-9). Equate exponents: x+1 = 6x-9 → 5x = 10 → x = 2. Answer B.
第2题：指数方程。求解 2^(x+1) = 8^(2x-3)。将 8 写成 2³：2^(x+1) = 2^(3(2x-3)) = 2^(6x-9)。令指数相等：x+1 = 6x-9 → 5x = 10 → x = 2。选项 B。
Q3. Definite integral. Evaluate ∫ from 1 to 2 of (1/x²) dx. ∫ x⁻² dx = -1/x. Applying limits: (-1/2) – (-1) = 1/2. The correct answer is A, 1/2.
第3题：定积分。计算 ∫₁² (1/x²) dx。∫ x⁻² dx = -1/x。代入上下限：(-1/2) – (-1) = 1/2。正确选项 A，1/2。
4. Detailed Solutions: Questions 4 to 6 | 题目 4-6 详解
Q4. Scalar product of vectors. Given a = (2, -1) and b = (1, 3), a·b = 2×1 + (-1)×3 = -1. The result is C, -1.
第4题：向量的数量积。已知 a = (2, -1)，b = (1, 3)，a·b = 2×1 + (-1)×3 = -1。结果为 C，-1。
Q5. Equation with a square root. Solve √(x+3) = x-3. Square both sides: x+3 = x² – 6x + 9 → x² -7x +6 = 0 → (x-1)(x-6)=0. Check for extraneous roots: x=1 gives √4 = 2 ≠ -2, reject; x=6 gives √9 = 3, valid. Answer E, x=6.
第5题：含根号的方程。解 √(x+3) = x-3。两边平方：x+3 = x² – 6x + 9 → x² -7x +6 = 0 → (x-1)(x-6)=0。验根：x=1 时 √4 = 2 ≠ -2，舍去；x=6 时 √9 = 3 成立。选项 E，x=6。
Q6. Standard trigonometric limit. Evaluate lim (x→0) (sin 3x)/x. Multiply numerator and denominator by 3: lim (x→0) 3·(sin 3x)/(3x) = 3×1 = 3. Answer B.
第6题：标准三角极限。求 lim (x→0) (sin 3x)/x。分子分母同乘以 3：lim (x→0) 3·(sin 3x)/(3x) = 3×1 = 3。选项 B。
5. Detailed Solutions: Questions 7 to 9 | 题目 7-9 详解
Q7. Modulus of a complex number. The equation |z – 3i| = 4 describes a circle in the Argand diagram with centre (0,3) and radius 4. The question asks for the radius, which is D, 4.
第7题：复数模长。方程 |z – 3i| = 4 在复平面上表示圆心在 (0,3)、半径为 4 的圆。问题求半径，答案为 D，4。
Q8. Equation of a tangent. For the curve y = x², the gradient at x=1 is dy/dx = 2x = 2. The tangent passes through (1,1) with slope 2: y – 1 = 2(x – 1) → y = 2x – 1. Answer A.
第8题：切线方程。曲线 y = x² 在 x=1 处的斜率为 dy/dx = 2x = 2。切线过 (1,1)：y – 1 = 2(x – 1) → y = 2x – 1。选项 A。
Q9. Binomial coefficient. The coefficient of x³ in the expansion of (1 + x)⁵ is given by ⁵C₃ = 10. Correct answer C, 10.
第9题：二项式系数。(1 + x)⁵ 展开式中 x³ 的系数为 ⁵C₃ = 10。正确答案 C，10。
6. Detailed Solutions: Questions 10 to 12 | 题目 10-12 详解
Q10. Integration by recognition. ∫ x e^(x²) dx. Notice d(x²) = 2x dx, so the integral is (1/2) ∫ 2x e^(x²) dx = (1/2) e^(x²) + C. This matches option E.
第10题：凑微分法积分。∫ x e^(x²) dx。注意到 d(x²) = 2x dx，因此原式 = (1/2) ∫ 2x e^(x²) dx = (1/2) e^(x²) + C。对应选项 E。
Q11. Solving a basic trigonometric equation. sin θ = 1/2 for 0 ≤ θ < 2π gives principal solutions θ = π/6 and θ = 5π/6. Answer B.
第11题：解基本三角方程。在 0 ≤ θ < 2π 内，sin θ = 1/2 的解为 θ = π/6 和 θ = 5π/6。选项 B。
Q12. Conditional probability. Given P(A|B) = 0.6 and P(B) = 0.5, P(A ∩ B) = P(A|B) × P(B) = 0.6 × 0.5 = 0.3. Answer D.
第12题：条件概率。已知 P(A|B) = 0.6，P(B) = 0.5，则 P(A ∩ B) = P(A|B) × P(B) = 0.6 × 0.5 = 0.3。答案 D。
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