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

313 questions

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Edexcel C4 2014 June Q2
5 marks Moderate -0.3
Given that the binomial expansion of \((1 + kx)^{-4}\), \(|kx| < 1\), is $$1 - 6x + Ax^2 + \ldots$$
  1. [(a)] find the value of the constant \(k\), \hfill [2]
  2. [(b)] find the value of the constant \(A\), giving your answer in its simplest form. \hfill [3]
CAIE P3 2024 November Q1
4 marks Moderate -0.8
Expand \((9 - 3x)^{\frac{1}{2}}\) in ascending powers of \(x\), up to and including the term in \(x^2\), simplifying the coefficients. [4]
CAIE P3 2006 June Q9
10 marks Standard +0.3
  1. Express \(\frac{10}{(2-x)(1+x^2)}\) in partial fractions. [5]
  2. Hence, given that \(|x| < 1\), obtain the expansion of \(\frac{10}{(2-x)(1+x^2)}\) in ascending powers of \(x\), up to and including the term in \(x^3\), simplifying the coefficients. [5]
CAIE P3 2010 June Q9
10 marks Standard +0.3
  1. Express \(\frac{4 + 5x - x^2}{(1 - 2x)(2 + x)^2}\) in partial fractions. [5]
  2. Hence obtain the expansion of \(\frac{4 + 5x - x^2}{(1 - 2x)(2 + x)^2}\) in ascending powers of \(x\), up to and including the term in \(x^2\). [5]
CAIE P3 2017 June Q8
10 marks Standard +0.3
Let \(\mathrm{f}(x) = \frac{5x^2 - 7x + 4}{(3x + 2)(x^2 + 5)}\).
  1. Express \(\mathrm{f}(x)\) in partial fractions. [5]
  2. Hence obtain the expansion of \(\mathrm{f}(x)\) in ascending powers of \(x\), up to and including the term in \(x^2\). [5]
CAIE P3 2013 November Q7
10 marks Standard +0.3
Let \(f(x) = \frac{2x^2 - 7x - 1}{(x-2)(x^2+3)}\).
  1. Express \(f(x)\) in partial fractions. [5]
  2. Hence obtain the expansion of \(f(x)\) in ascending powers of \(x\), up to and including the term in \(x^2\). [5]
Edexcel P4 2024 June Q8
8 marks Standard +0.8
$$f(x) = (8 - 3x)^{\frac{4}{3}} \quad 0 < x < \frac{8}{3}$$
  1. Show that the binomial expansion of \(f(x)\) in ascending powers of \(x\) up to and including the term in \(x^3\) is $$A - 8x + \frac{x^2}{2} + Bx^3 + ...$$ where \(A\) and \(B\) are constants to be found. [4]
  2. Use proof by contradiction to prove that the curve with equation $$y = 8 + 8x - \frac{15}{2}x^2$$ does not intersect the curve with equation $$y = A - 8x + \frac{x^2}{2} + Bx^3 \quad 0 < x < \frac{8}{3}$$ where \(A\) and \(B\) are the constants found in part (a). (Solutions relying on calculator technology are not acceptable.) [4]
Edexcel P4 2022 October Q4
8 marks Standard +0.3
$$g(x) = \frac{1}{\sqrt{4-x^2}}$$
  1. Find, in ascending powers of \(x\), the first four non-zero terms of the binomial expansion of \(g(x)\). Give each coefficient in simplest form. [5]
  2. State the range of values of \(x\) for which this expansion is valid. [1]
  3. Use the expansion from part (a) to find a fully simplified rational approximation for \(\sqrt{3}\) Show your working and make your method clear. [2]
Edexcel C4 Q1
5 marks Moderate -0.3
Use the binomial theorem to expand $$\sqrt{(4-9x)}, \quad |x| < \frac{4}{9},$$ in ascending powers of \(x\), up to and including the term in \(x^3\), simplifying each term. [5]
Edexcel C4 2013 June Q1
8 marks Moderate -0.3
  1. Find the binomial expansion of $$\sqrt{(9 + 8x)}, \quad |x| < \frac{9}{8}$$ in ascending powers of \(x\), up to and including the term in \(x^2\). Give each coefficient as a simplified fraction. [5]
  2. Use your expansion to estimate the value of \(\sqrt{11}\), giving your answer as a single fraction. [3]
Edexcel C4 2015 June Q1
8 marks Moderate -0.3
  1. Find the binomial expansion of $$(4 + 5x)^{\frac{1}{2}}, \quad |x| < \frac{4}{5}$$ in ascending powers of \(x\), up to and including the term in \(x^2\). Give each coefficient in its simplest form. [5]
  2. Find the exact value of \((4 + 5x)^{\frac{1}{2}}\) when \(x = \frac{1}{10}\) Give your answer in the form \(k\sqrt{2}\), where \(k\) is a constant to be determined. [1]
  3. Substitute \(x = \frac{1}{10}\) into your binomial expansion from part (a) and hence find an approximate value for \(\sqrt{2}\) Give your answer in the form \(\frac{p}{q}\) where \(p\) and \(q\) are integers. [2]
Edexcel C4 Q3
13 marks Standard +0.3
$$f(x) = \frac{1 + 14x}{(1 - x)(1 + 2x)}, \quad |x| < \frac{1}{2}.$$
  1. Express \(f(x)\) in partial fractions. [3]
  2. Hence find the exact value of \(\int_{-\frac{1}{6}}^{\frac{1}{4}} f(x) \, dx\), giving your answer in the form \(\ln p\), where \(p\) is rational. [5]
  3. Use the binomial theorem to expand \(f(x)\) in ascending powers of \(x\), up to and including the term in \(x^5\), simplifying each term. [5]
Edexcel C4 Q4
8 marks Moderate -0.3
$$f(x) = (1 + 3x)^{-1}, \quad |x| < \frac{1}{3}.$$
  1. Expand \(f(x)\) in ascending powers of \(x\) up to and including the term in \(x^3\). [3]
  2. Hence show that, for small \(x\), $$\frac{1 + x}{1 + 3x} \approx 1 - 2x + 6x^2 - 18x^3.$$ [2]
  3. Taking a suitable value for \(x\), which should be stated, use the series expansion in part \((b)\) to find an approximate value for \(\frac{101}{103}\), giving your answer to 5 decimal places. [3]
Edexcel C4 Q11
8 marks Moderate -0.3
Given that $$\frac{10(2 - 3x)}{(1 - 2x)(2 + x)} \equiv \frac{A}{1 - 2x} + \frac{B}{2 + x},$$
  1. find the values of the constants \(A\) and \(B\). [3]
  2. Hence, or otherwise, find the series expansion in ascending powers of \(x\), up to and including the term in \(x^3\), of \(\frac{10(2 - 3x)}{(1 - 2x)(2 + x)}\), for \(|x| < \frac{1}{2}\). [5]
Edexcel C4 Q21
10 marks Standard +0.3
  1. Prove that, when \(x = \frac{1}{12}\), the value of \((1 + 5x)^{-\frac{1}{2}}\) is exactly equal to \(\sin 60°\). [3]
  2. Expand \((1 + 5x)^{-\frac{1}{2}}\), \(|x| < 0.2\), in ascending powers of \(x\) up to and including the term in \(x^3\), simplifying each term. [4]
  3. Use your answer to part \((b)\) to find an approximation for \(\sin 60°\). [2]
  4. Find the difference between the exact value of \(\sin 60°\) and the approximation in part \((c)\). [1]
Edexcel C4 Q29
8 marks Moderate -0.3
  1. Expand \((1 + 3x)^{-2}\), \(|x| < \frac{1}{3}\), in ascending powers of \(x\) up to and including the term in \(x^3\), simplifying each term. [4]
  2. Hence, or otherwise, find the first three terms in the expansion of \(\frac{x + 4}{(1 + 3x)^2}\) as a series in ascending powers of \(x\). [4]
AQA C4 2010 June Q4
7 marks Moderate -0.3
    1. Find the binomial expansion of \((1 + x)^{\frac{3}{2}}\) up to and including the term in \(x^2\). [2 marks]
    2. Find the binomial expansion of \((16 + 9x)^{\frac{3}{2}}\) up to and including the term in \(x^2\). [3 marks]
  2. Use your answer to part (a)(ii) to show that \(13^{\frac{3}{2}} \approx 46 + \frac{a}{b}\), stating the values of the integers \(a\) and \(b\). [2 marks]
AQA C4 2016 June Q1
11 marks Moderate -0.3
  1. Express \(\frac{19x - 3}{(1 + 2x)(3 - 4x)}\) in the form \(\frac{A}{1 + 2x} + \frac{B}{3 - 4x}\). [3 marks]
    1. Find the binomial expansion of \(\frac{19x - 3}{(1 + 2x)(3 - 4x)}\) up to and including the term in \(x^2\). [7 marks]
    2. State the range of values of \(x\) for which this expansion is valid. [1 mark]
Edexcel C4 Q7
16 marks Standard +0.8
$$\text{f}(x) = \frac{25}{(3 + 2x)^2(1 - x)}, \quad |x| < 1.$$
  1. Express f(x) as a sum of partial fractions. [4]
  2. Hence find \(\int \text{f}(x) \, dx\). [5]
  3. Find the series expansion of f(x) in ascending powers of \(x\) up to and including the term in \(x^2\). Give each coefficient as a simplified fraction. [7]
Edexcel C4 Q6
8 marks Standard +0.3
When \((1 + ax)^n\) is expanded as a series in ascending powers of \(x\), the coefficients of \(x\) and \(x^2\) are \(-6\) and \(27\) respectively.
  1. Find the value of \(a\) and the value of \(n\). [5]
  2. Find the coefficient of \(x^3\). [2]
  3. State the set of values of \(x\) for which the expansion is valid. [1]
Edexcel C4 Q5
10 marks Moderate -0.3
  1. Prove that, when \(x = \frac{1}{15}\), the value of \((1 + 5x)^{-\frac{1}{3}}\) is exactly equal to \(\sin 60°\). [3]
  2. Expand \((1 + 5x)^{-\frac{1}{3}}\), \(|x| < 0.2\), in ascending powers of \(x\) up to and including the term in \(x^3\), simplifying each term. [4]
  3. Use your answer to part (b) to find an approximation for \(\sin 60°\). [2]
  4. Find the difference between the exact value of \(\sin 60°\) and the approximation in part (c). [1]
OCR C4 2007 January Q5
7 marks Moderate -0.3
  1. Expand \((1 - 3x)^{-\frac{1}{2}}\) in ascending powers of \(x\), up to and including the term in \(x^3\). [4]
  2. Hence find the coefficient of \(x^3\) in the expansion of \(\left(1 - 3(x + x^3)\right)^{-\frac{1}{2}}\). [3]
OCR C4 2005 June Q8
11 marks Standard +0.3
  1. Given that \(\frac{3x + 4}{(1 + x)(2 + x)^2} \equiv \frac{A}{1 + x} + \frac{B}{2 + x} + \frac{C}{(2 + x)^2}\), find \(A\), \(B\) and \(C\). [5]
  2. Hence or otherwise expand \(\frac{3x + 4}{(1 + x)(2 + x)^2}\) in ascending powers of \(x\), up to and including the term in \(x^2\). [5]
  3. State the set of values of \(x\) for which the expansion in part (ii) is valid. [1]
OCR C4 2006 June Q2
7 marks Moderate -0.8
  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]
OCR MEI C4 2012 January Q6
6 marks Standard +0.3
Given the binomial expansion \((1 + qx)^p = 1 - x + 2x^2 + \ldots\), find the values of \(p\) and \(q\). Hence state the set of values of \(x\) for which the expansion is valid. [6]