NSAA 2019 S1 Answer Key: Advanced Mathematics | NSAA 2019 S1 进阶数学答案与解析

📚 NSAA 2019 S1 Answer Key: Advanced Mathematics | NSAA 2019 S1 进阶数学答案与解析

Welcome to our detailed answer key and analysis for the NSAA 2019 Section 1 Mathematics paper, focusing on advanced mathematics topics. This guide provides the correct answers accompanied by step-by-step explanations to help you master the key concepts tested. All questions are reconstructed in the style of the original assessment to illustrate the core techniques required.

欢迎阅读 NSAA 2019 Section 1 数学部分的详尽答案解析，聚焦进阶数学考点。本文提供正确答案并配有逐步解析，助你掌握所测核心概念。所有题目均依据原卷风格重构，旨在展示所需的解题技巧。

1. Algebra and Polynomials | 代数与多项式

This section covers two NSAA 2019 S1 questions on polynomial remainder and quadratic roots.

本节涵盖 NSAA 2019 S1 中两道关于多项式余数和二次方程根的题目。

Question 1 (Remainder Theorem): Given f(x) = 2x³ − x² + 3x − 4, find the remainder when f(x) is divided by (x − 3). Options: A) 44, B) 46, C) 48, D) 50, E) 52. Answer: D) 50. By the Remainder Theorem, f(3) = 2(27) − 9 + 9 − 4 = 54 − 4 = 50.

问题1（余数定理）：给定 f(x)=2x³−x²+3x−4，求除以 (x−3) 的余数。选项：A)44 B)46 C)48 D)50 E)52。答案D)50。由余数定理，f(3)=2×27−9+9−4=54−4=50。

Question 2 (Roots of quadratic): If α and β are roots of x² + p x + q = 0, and α = 3, β = -5, find the value of p + q. Options: A) -17, B) -15, C) -13, D) -11, E) -9. Answer: C) -13. Sum α+β = -p = -2 → p=2; product αβ = q = -15; p+q = -13.

问题2（二次方程根）：若 α 和 β 为 x²+px+q=0 的根，且 α=3, β=-5，求 p+q 的值。选项：A)-17 B)-15 C)-13 D)-11 E)-9。答案C)-13。根之和 α+β=−p=−2 ⇒ p=2；积 αβ=q=−15；p+q=−13。

2. Functions and Graphs | 函数与图像

Here we examine composite functions and graphical transformations that appeared in the NSAA 2019 S1.

此处我们分析 NSAA 2019 S1 中出现的复合函数与图像变换问题。

Question 3 (Composite function): Let f(x) = 2x + 3 and g(x) = x² − 1. Find fg(2). Options: A) 5, B) 9, C) 11, D) 13, E) 15. Answer: B) 9. First g(2) = 4 − 1 = 3, then f(3) = 2×3 + 3 = 9.

问题3（复合函数）：设 f(x)=2x+3，g(x)=x²−1，求 fg(2)。选项：A)5 B)9 C)11 D)13 E)15。答案B)9。先求 g(2)=4−1=3，再求 f(3)=2×3+3=9。

Question 4 (Graph transformation): The point (2,3) lies on y = f(x). After transforming to y = 2f(x−1), what are the new coordinates? Options: A) (1,6), B) (3,6), C) (1,5), D) (3,5), E) (2,6). Answer: B) (3,6). Horizontal shift right by 1 gives (3,3); vertical stretch by factor 2 gives y = 2×3 = 6.

问题4（图像变换）：点 (2,3) 在 y=f(x) 上。变换为 y=2f(x−1) 后，新坐标是多少？选项：A)(1,6) B)(3,6) C)(1,5) D)(3,5) E)(2,6)。答案B)(3,6)。水平右移1得 (3,3)；纵向拉伸2倍得 y=6。

3. Trigonometry | 三角学

Trigonometric equations and the cosine rule feature in these NSAA 2019 S1 advanced mathematics items.

NSAA 2019 S1 进阶数学中考查了三角方程与余弦定理。

Question 5 (Trigonometric equation): Solve 2sin²θ − cosθ − 1 = 0 for 0 ≤ θ ≤ 2π. Options: A) π/3, π only; B) π/3, 5π/3 only; C) π/3, π, 5π/3; D) π/3, 2π/3, 5π/3; E) π, 5π/3 only. Answer: C) π/3, π, 5π/3. Substitute sin²θ = 1−cos²θ → 2(1−cos²θ)−cosθ−1=0 → 2cos²θ+cosθ−1=0 → (2cosθ−1)(cosθ+1)=0 → cosθ=½ → θ=π/3, 5π/3; cosθ=−1 → θ=π.

问题5（三角方程）：求解 2sin²θ − cosθ − 1 = 0，0 ≤ θ ≤ 2π。选项：A)仅 π/3, π；B)仅 π/3, 5π/3；C) π/3, π, 5π/3；D) π/3, 2π/3, 5π/3；E)仅 π, 5π/3。答案C) π/3, π, 5π/3。代入 sin²θ

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