Year 11 Edexcel Further Maths Unit Test Mock Paper Breakdown | Year 11 Edexcel 进阶数学：单元测试模拟卷解析

📚 Year 11 Edexcel Further Maths Unit Test Mock Paper Breakdown | Year 11 Edexcel 进阶数学：单元测试模拟卷解析

This mock paper breakdown covers typical questions from a Year 11 Edexcel Further Maths unit test, including logarithms, polynomials, matrices, trigonometry, series, differentiation, integration, and vectors. Each question is solved step by step with key commentary in both English and Chinese, highlighting exam techniques and common mistakes. Use this guide to refine your understanding and boost your confidence.

本模拟卷解析涵盖 Year 11 Edexcel 进阶数学单元测试中的典型题目，包括对数、多项式、矩阵、三角学、级数、微分、积分和向量。每道题均分步解答，并附有中英双语重点评注，突出考试技巧与常见错误。请利用本指南完善你的理解并增强信心。

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1. Logarithmic Equations | 对数方程

Question: Solve log₂(x+1) – log₂(x-2) = 3.

题目：求解 log₂(x+1) – log₂(x-2) = 3。

Combine the two logarithms using the quotient rule: logₐ M – logₐ N = logₐ(M/N).

使用商规则合并两个对数：logₐ M – logₐ N = logₐ(M/N)。

log₂((x+1)/(x-2)) = 3

Rewrite the logarithmic equation in exponential form: if log₂ A = 3, then A = 2³ = 8.

将对数方程改写为指数形式：若 log₂ A = 3，则 A = 2³ = 8。

(x+1)/(x-2) = 8

Solve the resulting rational equation: x+1 = 8(x-2) → x+1 = 8x – 16 → 7x = 17 → x = 17/7.

解所得有理方程：x+1 = 8(x-2) → x+1 = 8x – 16 → 7x = 17 → x = 17/7。

Always check the domain restrictions for logarithms: arguments must be positive. Here x+1 > 0 and x-2 > 0 ⇒ x > 2. Since 17/7 ≈ 2.43 > 2, the solution is valid.

务必检查对数的定义域限制：真数必须为正。此处 x+1 > 0 且 x-2 > 0 ⇒ x > 2。因为 17/7 ≈ 2.43 > 2，该解有效。

Final answer: x = 17/7. A common error is to skip the domain check and accept an extraneous root.

最终答案：x = 17/7。常见错误是跳过定义域检查而接受增根。

Key takeaway: Always isolate the logarithm before converting to exponential form, and write the domain check explicitly to secure full marks.

关键要点：在转换为指数形式前，始终先合并或隔离对数，并明确写出定义域检查以获得满分。

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2. Remainder Theorem | 余数定理

Question: Given f(x) = 2x³ – 5x² + 3x + 2, find the remainder when f(x) is divided by (x-2).

题目：已知 f(x) = 2x³ – 5x² + 3x + 2，求 f(x) 除以 (x-2) 的余数。

The Remainder Theorem states that when a polynomial f(x) is divided by (x – a), the remainder is f(a). Here a = 2.

余数定理指出，多项式 f(x) 除以 (x – a) 时，余数等于 f(a)。此处 a = 2。

Substitute x = 2 into the polynomial: f(2) = 2(2)³ – 5(2)² + 3(2) + 2.

将 x = 2 代入多项式：f(2) = 2(2)³ – 5(2)² + 3(2) + 2。

f(2) = 2×8 – 5×4 + 6 + 2 = 16 – 20 + 6 + 2 = 4

The remainder is therefore 4. Polynomial long division would yield the same result but is far more time-consuming.

因此余数为 4。多项式长除法会得到相同结果，但耗时得多。

Common pitfall: Misapplying signs when substituting or forgetting to include constant term. Write f(2) clearly and check arithmetic.

常见陷阱：代入时符号错误，或忘记常数项。清晰写出 f(2) 并检查运算。

Key takeaway: Remember that (x – a) gives a remainder f(a), while (ax – b) requires the remainder to be f(b/a).

关键要点：记住 (x – a) 的余数是 f(a)，而 (ax – b) 的余数是 f(b/a)。

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3. Inverse of a 2×2 Matrix | 二阶矩阵的逆

Question: Find the inverse of the matrix A = [2 1; 5 3].

题目：求矩阵 A = [2 1; 5 3] 的逆矩阵。

First compute the determinant: det A = (2)(3) – (1)(5) = 6 – 5 = 1. Since the determinant is non-zero, the inverse exists.

先计算行列式：det A = (2)(3) – (1)(5) = 6 – 5 = 1。因为行列式不为零，逆矩阵存在。

The standard inverse formula for a 2×2 matrix [a b; c d] is (1/det) [d -b; -c a].

二阶矩阵 [a b; c d]Published by TutorHao | Year 11 进阶数学 Revision Series | aleveler.com

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