Answers to Exercises – Further Pure 3 | 练习题答案 – 进阶纯数3

📚 Answers to Exercises – Further Pure 3 | 练习题答案 – 进阶纯数3

Welcome to this AQA Further Pure 3 answer guide. The solutions below cover typical FP3 topics: polar coordinates, hyperbolic functions, complex numbers, loci, differential equations, vectors and matrices. Each exercise is paired with a concise explanation so you can check both your method and your final answer.

欢迎使用本 AQA 进阶纯数3 答案指南。以下解答涵盖 FP3 常见主题：极坐标、双曲函数、复数、轨迹、微分方程、向量和矩阵。每道练习都配有简明解析，既核对最终答案，也检查解题方法。

1. Polar Coordinates – Area and Tangents | 极坐标 – 面积与切线

This section checks polar area and tangent slope. Remember that x = r cos θ and y = r sin θ, and use symmetry where possible.

本节检查极坐标面积与切线斜率。记住 x = r cos θ 且 y = r sin θ，并尽量利用对称性。

Exercise 1. The curve C has polar equation r = 1 + cos θ, 0 ≤ θ < 2π. Find the area enclosed by C.

练习 1. 曲线 C 的极坐标方程为 r = 1 + cos θ，0 ≤ θ < 2π。求 C 围成的面积。

Solution. Expand the integrand using cos² θ = (1 + cos 2θ)/2. The area integral is:

解答. 利用 cos² θ = (1 + cos 2θ)/2 展开被积函数。面积积分为：

A = ½ ∫₀²π (1 + cos θ)² dθ = ½ ∫₀²π (1 + 2 cos θ + cos² θ) dθ

A = ½ [θ + 2 sin θ + θ/2 + sin 2θ/4]₀²π = ½ × 3π = 3π/2

Therefore the enclosed area is 3π/2 square units.

因此围成的面积为 3π/2 平方单位。

Exercise 2. For the same curve, find dy/dx at θ = π/3.

练习 2. 对同一曲线，求 θ = π/3 处的 dy/dx。

Solution. First write x and y in terms of θ:

解答. 先将 x 和 y 写成关于 θ 的表达式：

x = (1 + cos θ) cos θ = cos θ + cos² θ, y = (1 + cos θ) sin θ = sin θ + sin θ cos θ

Differentiating gives dx/dθ = -sin θ – sin 2θ and dy/dθ = cos θ + cos 2θ. At θ = π/3, dx/dθ = -√3/2 – √3/2 = -√3 and dy/dθ = 1/2 – 1/2 = 0.

求导得 dx/dθ = -sin θ – sin 2θ 且 dy/dθ = cos θ + cos 2θ。在 θ = π/3 处，dx/dθ = -√3/2 – √3/2 = -√3，dy/dθ = 1/2 – 1/2 = 0。

dy/dx = (dy/dθ)/(dx