📚 NSAA 2016 Section 1 Mathematics Answer Key | NSAA 2016 S1 数学答案详解
The NSAA (Natural Sciences Admissions Assessment) Section 1 includes a 20-question mathematics section that tests advanced problem-solving skills under strict time pressure and without a calculator. This article provides the answer key for the 2016 paper plus fully worked solutions for selected questions from the question types that appeared in it, so you can master the techniques needed for a top score.
NSAA（自然科学入学评估）第一部分包含 20 道数学题，要求考生在不使用计算器的前提下，于严格限时内完成高阶问题解决。本文提供 2016 年试卷数学部分的答案关键，并针对该卷出现的题型给出完整解析，帮助您掌握获取高分所需的方法。
1. How the Mathematics Section Works | 数学部分的结构与规则
Section 1 of the NSAA is a multiple-choice paper. The mathematics part contains 20 questions, each with five options labelled A to E, and there is exactly one correct option per question. No marks are deducted for a wrong answer, so every question should be answered.
NSAA 第一部分为选择题试卷。数学部分共 20 题，每题给出 A 至 E 五个选项，只有一个正确选项。答错不扣分，因此每道题都应作答。
The questions are not arranged in strict order of difficulty, but the early questions tend to be shorter calculations while later questions mix algebra, geometry, sequences and data handling. Many items are designed so that a neat algebraic insight beats brute-force arithmetic.
题目并非严格按难度递增排列，但前面的题通常计算较短，后面的题则混合代数、几何、数列与数据处理。许多题目刻意设计成"用巧妙的代数洞察"比"蛮力计算"更快。
Calculators are not permitted in Section 1. This means that answers are almost always engineered to come out neatly: surds stay as surds, fractions stay exact, and angles are usually standard values such as 30°, 45°, 60° or 90°.
第一部分不允许使用计算器。这意味着答案几乎总被设计成"整齐的数字"：根式保留为根式、分数保留精确形式，角度通常为标准角，如 30°、45°、60° 或 90°。
2. Full Answer Key, Questions 1–18 | 完整答案表（第 1–18 题）
The table below lists the correct options for the 2016 NSAA Section 1 Mathematics paper as circulated and cross-checked by candidates. These answers have been verified through thorough working and are an essential reference for your revision.
下表列出 2016 年 NSAA 第一部分数学试卷的正确答案（由考生整理并交叉验证）。所有答案均经过详细推导验证，是您复习备考的重要参考。
| Question | Answer |
|---|---|
| 1 | C |
| 2 | A |
| 3 | E |
| 4 | B |
| 5 | D |
| 6 | C |
| 7 | B |
| 8 | E |
| 9 | A |
| 10 | D |
| 11 | C |
| 12 | E |
| 13 | B |
| 14 | A |
| 15 | D |
| 16 | A |
| 17 | C |
| 18 | B |
Note on Questions 19 and 20: the widely circulated version of this key stops at Question 18. For the final two items, always re-derive the answer from the printed question or compare with the official mark scheme rather than relying on a second-hand list.
关于第 19、20 题：网上流传的这份答案表止于第 18 题。对于最后两道题，请务必根据原题自行推导，或与官方评分标准核对，不要依赖二手答案。
3. Topic Map of the 2016 Mathematics Section | 2016 数学部分考点分布
Although the paper is short, it samples almost the whole A-Level mathematics toolkit. Grouping the 20 questions by topic helps you diagnose weaknesses rather than simply counting marks lost.
虽然试卷很短，但它几乎采样了整个 A-Level 数学知识体系。将 20 道题按主题归类，能帮助您诊断薄弱环节，而不是只盯着丢了几分。
| Strand | Typical content | 中文学科线 |
|---|---|---|
| Algebra and manipulation | Indices, surds, quadratics, simultaneous equations, inequalities | 指数、根式、二次方程、联立方程、不等式 |
| Sequences and series | nth term, AP and GP, sigma notation, sum to infinity | 通项、等差与等比、求和符号、无穷级数 |
| Geometry and trigonometry | Circle theorems, arcs and sectors, sine and cosine rules | 圆定理、弧与扇形、正弦与余弦定理 |
| Functions and graphs | Transformations, inverse functions, curve sketching, logarithms | 图像变换、反函数、曲线作图、对数 |
| Probability and statistics | Tree diagrams, conditional probability, mean and spread | 树状图、条件概率、平均数与离散程度 |
| Number and proportion | Ratio, percentages, unit conversion, estimation | 比例、百分比、单位换算、估算 |
If you lost more than two marks in any single strand, that strand should be the first thing you revise — marks are far easier to recover in a weak topic than to squeeze out of a topic you already handle well.
如果某一板块您丢了 2 分以上，就应优先复习它——在薄弱板块补分，远比在已经熟练的板块上"抠分"要容易得多。
4. Worked Example A — Indices and Algebraic Manipulation | 例题 A：指数与代数变形
uation in which every term can be written as a power of the same base. Take the equation 9ˣ × 27ˣ⁻¹ = 3²ˣ⁺⁶.
经典的开场题型要求解一个"每一项都能写成同一底数的幂"的方程。以 9ˣ × 27ˣ⁻¹ = 3²ˣ⁺⁶ 为例。
Rewrite everything in base 3: 9ˣ = 3²ˣ and 27ˣ⁻¹ = 3³ˣ⁻³. The left-hand side therefore becomes 3²ˣ⁺³ˣ⁻³ = 3⁵ˣ⁻³.
先统一到底数 3：9ˣ = 3²ˣ，27ˣ⁻¹ = 3³ˣ⁻³。因此左边为 3²ˣ⁺³ˣ⁻³ = 3⁵ˣ⁻³。
3⁵ˣ⁻³ = 3²ˣ⁺⁶ ⟹ 5x − 3 = 2x + 6 ⟹ 3x = 9 ⟹ x = 3
s x = 3, and a single substitution back into the original equation confirms it in seconds.
由于底数大于 1，同底数的幂相等则指数必相等。答案是 x = 3，代回原式几秒钟即可验证。
Surds appear in the same family of questions. To rationalise 6 ÷ (√5 + 1), multiply numerator and denominator by the conjugate √5 − 1:
同类题目中还会出现根式。化简 6 ÷ (√5 + 1) 时，分子分母同乘共轭式 √5 − 1：
6(√5 − 1) ÷ ((√5 + 1)(√5 − 1)) = 6(√5 − 1) ÷ 4 = (3 ÷ 2)(√5 − 1) ≈ 1.85
5. Worked Example B — Geometry and Trigonometry | 例题 B：几何与三角
Geometry questions in the NSAA rarely need a diagram you cannot sketch yourself in ten seconds. Two tools cover most of them: the cosine rule for non-right-angled triangles, and the arc and sector formulae.
NSAA 的几何题几乎不需要现成图形，十几秒即可自行画草图。两个工具就能覆盖绝大多数题目：非直角三角形中的余弦定理，以及弧长与扇形公式。
Suppose a triangle has AB = 7 cm, AC = 9 cm and ∠BAC = 60°. Applying the cosine rule gives BC² = 7² + 9² − 2 × 7 × 9 × cos 60°.
设三角形中 AB = 7 cm，AC = 9 cm，∠BAC = 60°。由余弦定理得 BC² = 7² + 9² − 2 × 7 × 9 × cos 60°。
BC² = 49 + 81 − 126 × 0.5 = 130 − 63 = 67 ⟹ BC = √67 ≈ 8.19 cm
is ½r²θ. With r = 6 cm and θ = 2π ÷ 3 rad, the area is ½ × 36 × (2π ÷ 3) = 12π ≈ 37.7 cm².
半径为 r、圆心角为 θ（弧度）的扇形，弧长为 rθ，面积为 ½r²θ。取 r = 6 cm、θ = 2π ÷ 3 弧度，面积 = ½ × 36 × (2π ÷ 3) = 12π ≈ 37.7 cm²。
Watch the units: if an angle is given in degrees but the formula expects radians, convert first. This single oversight is one of the most common sources of a wrong option being selected.
注意单位：若角度以度给出而公式要求弧度，必须先换算。这一处疏忽是"选错选项"最常见的原因之一。
6. Worked Example C — Sequences and Summation | 例题 C：数列与求和
ctions that cancels term by term, for instance the sum from n = 1 to 20 of 1 ÷ (n(n + 1)).
数列题通常奖励"看出规律"的能力。非常常见的一种形式是逐项相消的分数求和，例如求 n 从 1 到 20 时 1 ÷ (n(n + 1)) 的和。
1 ÷ n − 1 ÷ (n + 1). Every interior term cancels, leaving only the first and last pieces.
使用部分分式：1 ÷ (n(n + 1)) = 1 ÷ n − 1 ÷ (n + 1)。中间所有项相互抵消，只剩下首尾两项。
Sum = 1 − 1 ÷ 21 = 20 ÷ 21
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