Direct quotient expansion

Expand a quotient √((1±ax)/(1±bx)) directly by expanding numerator and denominator separately, then multiplying the series.

8 questions · Standard +0.1

1.04c Extend binomial expansion: rational n, |x|<11.04d Binomial expansion validity: convergence conditions
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CAIE P3 2012 June Q3
5 marks Moderate -0.3
3 Expand \(\sqrt { } \left( \frac { 1 - x } { 1 + x } \right)\) in ascending powers of \(x\), up to and including the term in \(x ^ { 2 }\), simplifying the coefficients.
CAIE P3 2022 November Q2
5 marks Standard +0.3
2 Expand \(\sqrt { \frac { 1 + 2 x } { 1 - 2 x } }\) in ascending powers of \(x\), up to and including the term in \(x ^ { 2 }\), simplifying the coefficients.
Edexcel C34 2018 October Q6
9 marks Standard +0.3
6. (a) Use binomial expansions to show that, for \(| x | < \frac { 1 } { 2 }\) (b) Find the exact value of \(\sqrt { \frac { 1 + 2 x } { 1 - x } }\) when \(x = \frac { 1 } { 10 }\) Give your answer in the form \(k \sqrt { 3 }\), where \(k\) is a constant to be determined.
(c) Substitute \(x = \frac { 1 } { 10 }\) into the expansion given in part (a) and hence find an approximate value for \(\sqrt { 3 }\) Give your answer in the form \(\frac { a } { b }\) where \(a\) and \(b\) are integers. $$\sqrt { \frac { 1 + 2 x } { 1 - x } } \approx 1 + \frac { 3 } { 2 } x + \frac { 3 } { 8 } x ^ { 2 }$$
Edexcel C4 2013 June Q2
9 marks Moderate -0.3
  1. Use the binomial expansion to show that $$\left. \sqrt { ( } \frac { 1 + x } { 1 - x } \right) \approx 1 + x + \frac { 1 } { 2 } x ^ { 2 } , \quad | x | < 1$$
  2. Substitute \(x = \frac { 1 } { 26 }\) into $$\sqrt { \left( \frac { 1 + x } { 1 - x } \right) = 1 + x + \frac { 1 } { 2 } x ^ { 2 } }$$ to obtain an approximation to \(\sqrt { } 3\) Give your answer in the form \(\frac { a } { b }\) where \(a\) and \(b\) are integers.
OCR C4 2008 June Q5
8 marks Standard +0.3
5
  1. Show that \(\sqrt { \frac { 1 - x } { 1 + x } } \approx 1 - x + \frac { 1 } { 2 } x ^ { 2 }\), for \(| x | < 1\).
  2. By taking \(x = \frac { 2 } { 7 }\), show that \(\sqrt { 5 } \approx \frac { 111 } { 49 }\).
SPS SPS SM Pure 2021 May Q5
8 marks Standard +0.3
  1. Show that \(\sqrt{\frac{1-x}{1+x}} \approx 1 - x + \frac{1}{2}x^2\), for \(|x| < 1\). [5]
  2. By taking \(x = \frac{2}{7}\), show that \(\sqrt{5} \approx \frac{111}{49}\). [3]
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 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]