1.04c Extend binomial expansion: rational n, |x|<1

313 questions

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SPS SPS FM 2022 February Q8
7 marks Moderate -0.3
  1. Expand \((1 - 3x)^{-2}\) in ascending powers of \(x\), up to and including the term in \(x^2\). [3]
  2. Find the coefficient of \(x^2\) in the expansion of \(\frac{(1 + 2x)^2}{(1 - 3x)^2}\) in ascending powers of \(x\). [4]
SPS SPS FM 2023 January Q5
7 marks Moderate -0.3
  1. Expand \((2+x)^{-2}\) in ascending powers of \(x\) up to and including the term in \(x^3\), and state the set of values of \(x\) for which the expansion is valid. [5]
  2. Hence find the coefficient of \(x^3\) in the expansion of \(\frac{1+x^2}{(2+x)^2}\). [2]
SPS SPS FM 2023 February Q5
9 marks Standard +0.3
  1. Expand \(\sqrt{1 + 2x}\) in ascending powers of x, up to and including the term in \(x^3\). [4]
  2. Hence expand \(\frac{\sqrt{1 + 2x}}{1 + 9x^2}\) in ascending powers of x, up to and including the term in \(x^3\). [3]
  3. Determine the range of values of x for which the expansion in part (b) is valid. [2]
SPS SPS FM Pure 2023 June Q9
8 marks Standard +0.3
  1. Use the binomial expansion to show that \((1 - 2x)^{-\frac{1}{4}} \approx 1 + x + \frac{5}{8}x^2\) for sufficiently small values of \(x\). [2]
  2. For what values of \(x\) is the expansion valid? [1]
  3. Find the expansion of \(\sqrt{\frac{1+2x}{1-2x}}\) in ascending powers of \(x\) as far as the term in \(x^2\). [3]
  4. Use \(x = \frac{1}{20}\) in your answer to part (iii) to find an approximate value for \(\sqrt{11}\). [2]
SPS SPS SM Pure 2023 September Q1
6 marks Moderate -0.8
In all questions you must show all stages of your working, justifying solutions and not relying solely on calculator technology.
  1. Find the first four terms, in ascending powers of \(x\), of the binomial expansion of \(\left(1+\frac{x}{2}\right)^7\), giving each coefficient in exact simplified form. [3]
  2. Hence determine the coefficient of \(x\) in the expansion of $$\left(1+\frac{2}{x}\right)^2\left(1+\frac{x}{2}\right)^7.$$ [3]
SPS SPS FM Pure 2025 January Q2
8 marks Standard +0.3
  1. Given that $$\frac{3 + 2x^2}{(1 + x)^2(1 - 4x)} = \frac{A}{1 + x} + \frac{B}{(1 + x)^2} + \frac{C}{1 - 4x},$$ where \(A\), \(B\) and \(C\) are constants, find \(B\) and \(C\), and show that \(A = 0\). [4]
  2. Given that \(x\) is sufficiently small, find the first three terms of the binomial expansions of \((1 + x)^{-2}\) and \((1 - 4x)^{-1}\). Hence find the first three terms of the expansion of \(\frac{3 + 2x^2}{(1 + x)^2(1 - 4x)}\). [4]
SPS SPS FM 2025 February Q2
5 marks Moderate -0.3
  1. Find the first three terms in the expansion of \((1-2x)^{-1}\) in ascending powers of \(x\), where \(|x| < \frac{1}{2}\). [3]
  2. Hence find the coefficient of \(x^2\) in the expansion of \(\frac{x+3}{\sqrt{1-2x}}\). [2]
SPS SPS FM Pure 2025 June Q2
10 marks Standard +0.3
  1. Use binomial expansions to show that \(\sqrt{\frac{1 + 4x}{1 - x}} \approx 1 + \frac{5}{2}x - \frac{5}{8}x^2\) [6]
A student substitutes \(x = \frac{1}{2}\) into both sides of the approximation shown in part (a) in an attempt to find an approximation to \(\sqrt{6}\)
  1. Give a reason why the student should not use \(x = \frac{1}{2}\) [1]
  2. Substitute \(x = \frac{1}{11}\) into $$\sqrt{\frac{1 + 4x}{1 - x}} = 1 + \frac{5}{2}x - \frac{5}{8}x^2$$ to obtain an approximation to \(\sqrt{6}\). Give your answer as a fraction in its simplest form. [3]
OCR H240/03 2017 Specimen Q5
8 marks Standard +0.3
  1. Find the first three terms in the expansion of \((1 + px)^{\frac{1}{3}}\) in ascending powers of \(x\). [3]
  2. The expansion of \((1 + qx)(1 + px)^{\frac{1}{3}}\) is \(1 + x - \frac{2}{9}x^2 + ...\) Find the possible values of the constants \(p\) and \(q\). [5]
Pre-U Pre-U 9794/1 2010 June Q7
9 marks Standard +0.3
Let \(f(x) = \frac{1 + x^2}{\sqrt{4 - 3x}}\)
  1. Obtain in ascending powers of \(x\) the first three terms in the expansion of \(\frac{1}{\sqrt{4 - 3x}}\) and state the values of \(x\) for which this expansion is valid. [5]
  2. Hence obtain an approximation to \(f(x)\) in the form \(a + bx + cx^2\) where \(a\), \(b\) and \(c\) are constants. [2]
  3. Use your approximation to estimate \(\int_0^{0.1} f(x) dx\). [2]
Pre-U Pre-U 9794/1 2011 June Q8
8 marks Standard +0.3
  1. Find and simplify the first three terms in the expansion of \((1 - 4a)^{\frac{1}{2}}\) in ascending powers of \(a\), where \(|a| < \frac{1}{4}\). [4]
  2. Hence show that the roots of the quadratic equation \(x^2 - x + a = 0\) are approximately \(1 - a - a^2\) and \(a + a^2\), where \(a\) is small. [4]
Edexcel AEA 2014 June Q4
13 marks Hard +2.3
Given that $$(1 + x)^n = 1 + \sum_{r=1}^{\infty} \frac{n(n-1)...(n-r+1)}{1 \times 2 \times ... \times r} x^r \quad (|x| < 1, x \in \mathbb{R}, n \in \mathbb{R})$$
  1. show that $$(1 - x)^{-\frac{1}{2}} = \sum_{r=0}^{\infty} \binom{2r}{r}\left(\frac{x}{4}\right)^r$$ [5]
  2. show that \((9 - 4x^2)^{-\frac{1}{2}}\) can be written in the form \(\sum_{r=0}^{\infty} \binom{2r}{r} \frac{x^{2r}}{3^{r}}\) and give \(q\) in terms of \(r\). [3]
  3. Find \(\sum_{r=1}^{\infty} \binom{2r}{r} \times \frac{2r}{9} \times \left(\frac{x}{3}\right)^{2r-1}\) [3]
  4. Hence find the exact value of $$\sum_{r=1}^{\infty} \binom{2r}{r} \times \frac{2r\sqrt{5}}{9} \times \frac{1}{5^r}$$ giving your answer as a rational number. [2]
Edexcel AEA 2015 June Q4
15 marks Challenging +1.8
  1. Find the binomial series expansion for \((4 + y)^{\frac{1}{2}}\) in ascending powers of \(y\) up to and including the term in \(y^3\). Simplify the coefficient of each term. [3]
  2. Hence show that the binomial series expansion for \((4 + 5x + x^2)^{\frac{1}{2}}\) in ascending powers of \(x\) up to and including the term in \(x^3\) is $$2 + \frac{5x}{4} - \frac{9x^2}{64} + \frac{45x^3}{512}$$ [3]
  3. Show that the binomial series expansion of \((4 + 5x + x^2)^{\frac{1}{2}}\) will converge for \(-\frac{1}{2} < x \leq \frac{1}{2}\) [6]
  4. Use the result in part (b) to estimate $$\int_{-\frac{1}{2}}^{\frac{1}{2}} \sqrt{4 + 5x + x^2} \, dx$$ Give your answer as a single fraction. [3]