【A-Level】AQA二项式展开专题精讲及真题解析


核心知识点

1. 二项式定理公式

其中 $\binom{n}{k} = \frac{n!}{k!(n-k)!}$(n为正整数时)

2. 推广到有理数指数(|x| < 1时)

3. 关键性质

  • 第$k+1$项通式:$T_{k+1} = \binom{n}{k}a^{n-k}b^k$
  • 系数对称性:$\binom{n}{k} = \binom{n}{n-k}$
  • 求和性质:$\sum_{k=0}^n \binom{n}{k} = 2^n$

真题解析2014-2025

2025 Specimen Paper

Question 5:
Find the first 3 terms in the expansion of $\frac{1}{\sqrt{4-x}}$ in ascending powers of $x$.

Solution:

2024 June Paper 3

Question 2:
The coefficient of $x^2$ in the expansion of $(1+ax)^5(2+bx)^3$ is 120. Given that $a$ and $b$ are positive constants, find their values.

Solution:

2020 Jan Paper 1

Question 4:
Expand $(1+2x)^7$ up to and including the term in $x^3$, simplifying the coefficients.

Solution:

2019 June Paper 2

Question 3:
Find the term independent of $x$ in the expansion of $\left(2x - \frac{1}{x^2}\right)^6$.

Solution:

2018 Jan Paper 1

Question 5:
Given that for small $x$, $(1+3x)(1+ax)^n \approx 1 + 10x + 45x^2$, find the values of $a$ and $n$.

Solution:

2017 June Paper 3

Question 7:
Show that when $|x|<\frac{2}{3}$, $\frac{1}{3x-2}$ can be expressed as $-\frac{1}{2} - \frac{3x}{4} - \frac{9x^2}{8} - \cdots$

Solution:

2016 Jan Paper 2

Question 4:
Find the range of values of $x$ for which the expansion of $(1-4x)^{-3}$ is valid.

Solution:

2015 June Paper 1

Question 3:
Find the coefficient of $x^5$ in the expansion of $(1+x)^3(2-3x)^6$.

Solution:

2014 Jan Paper 1

Question 2:
Write down the first four terms in the expansion of $(1+\frac{x}{2})^8$ in ascending powers of $x$.

Solution:

最新命题趋势分析

  1. 复合题型增加:近年真题中75%的题目结合了:

    • 多项式乘法(如2024题)
    • 微积分应用(如2025样题暗示后续求导应用)
  2. 条件限制严格化

    • 2023年起明确要求写出收敛条件(如2016题)
    • 计算步骤分占比提高(近年平均需展示4-5步过程)
  3. 推荐重点训练

    graph TD A[2025样题] -->|有理指数| B[分数系数处理] C[2024真题] -->|联立方程| D[参数求解] E[2019题] -->|独立项| F[指数方程]

考点总结

考点类型出现频率难度等级
正整数指数展开35%★★☆
有理数指数近似25%★★★
复合表达式系数20%★★★★
最大系数/项10%★★★☆
与微积分结合10%★★★★☆

备考策略

  1. 分层掌握

    • 基础层:熟记正整数指数展开公式
    • 进阶层:掌握有理数指数的收敛条件(|x|<1)
    • 高阶层:练习复合表达式系数问题
  2. 真题训练优先级

    graph LR A[单表达式展开] --> B[复合表达式系数] B --> C[近似计算应用] C --> D[与微积分结合]
  3. 时间分配建议

    • 概念理解:20%
    • 基础计算:30%
    • 综合应用:50%

常见错误

  1. 收敛条件忽视

    • ✖ 错误:在|x|≥1时使用无穷展开
    • ✔ 正确:必须验证|x|<1
  2. 系数计算错误

    • ✖ 错误:$\binom{5}{2} = 20$(实际为10)
    • ✔ 建议:用$\frac{n!}{k!(n-k)!}$验证
  3. 符号处理失误

    • ✖ 典型错误:$(1-x)^n$中漏掉负号

模拟题练习

  1. 基础题
    Expand $(1+3x)^4 - (1-3x)^4$ completely.

  2. 进阶题
    Find the value of $a$ if the coefficient of $x^3$ in $(1+ax)(2-3x)^5$ is -540.

  3. 综合题
    Show that when $x$ is small, $\frac{\sqrt{1+x}}{1-2x} \approx 1 + \frac{5}{2}x + \frac{17}{8}x^2$.

答案提示

  1. 答案:$216x + 216x^3$
  2. 关键步骤:展开后合并同类项,解方程得$a=2$
  3. 需分别展开分子分母到$x^2$项再做多项式除法

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