📚 OCR Year 13 Mathematics: Mock Unit Test Walkthrough | OCR A Level 数学：单元测试模拟卷解析
This walkthrough provides a detailed analysis of a mock unit test for OCR A Level Mathematics, focusing on Core Pure 3 topics typically encountered in Year 13. The paper is designed to mirror the style and difficulty of actual exam questions, covering differentiation, integration, parametric equations, differential equations, and more. Each question is solved step by step with bilingual explanations to help you master both mathematical techniques and exam strategies.
本文对OCR A Level数学核心纯数3单元模拟测试进行详细解析，涵盖Year 13常见的微积分、参数方程、微分方程等考点。模拟卷模仿真题风格与难度，每道题均提供分步解答和中英双语解释，助你掌握数学技巧与应试策略。
1. Q1(a): Product Rule Differentiation | 题目1(a)：乘积法则求导
Question: Differentiate y = x³ e²ˣ with respect to x. | 问题：求 y = x³ e²ˣ 对 x 的导数。
Step 1: Identify the two functions. Let u = x³ and v = e²ˣ. | 步骤1：确定两个函数。设 u = x³, v = e²ˣ。
Step 2: Find their derivatives. u’ = 3x² and v’ = 2e²ˣ (using the chain rule). | 步骤2：求导。u’ = 3x²，v’ = 2e²ˣ（使用链式法则）。
Step 3: Apply the product rule: dy/dx = u’v + uv’. | 步骤3：应用乘积法则：dy/dx = u’v + uv’。
| Product Rule: d/dx (uv) = u’v + uv’ |
Substitute: dy/dx = (3x²)(e²ˣ) + (x³)(2e²ˣ). | 代入：dy/dx = (3x²)(e²ˣ) + (x³)(2e²ˣ)。
Step 4: Simplify to a factorised form. | 步骤4：化简并因式分解。
dy/dx = 3x² e²ˣ + 2x³ e²ˣ = x² e²ˣ (3 + 2x)
This is the final answer. Factoring helps identify stationary points later. | 这是最终答案。因式分解有助于后续寻找驻点。
2. Q1(b): Quotient Rule Differentiation | 题目1(b)：商法则求导
Question: Differentiate y = sin x / (1 + cos x). | 问题：求 y = sin x / (1 + cos x) 的导数。
Step 1: Set u = sin x and v = 1 + cos x. | 步骤1：设 u = sin x, v = 1 + cos x。
Step 2: Differentiate: u’ = cos x, v’ = −sin x. | 步骤2：求导：u’ = cos x, v’ = −sin x。
Step 3: Use the quotient rule: dy/dx = (u’v − uv’) / v². | 步骤3：使用商法则：dy/dx = (u’v − uv’) / v²。
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