Direct single expansion substitution

Questions that expand a single binomial expression (e.g., (1+x)^n or (a+bx)^n) and substitute a specific value to approximate a surd or root directly.

11 questions · Moderate -0.2

1.04c Extend binomial expansion: rational n, |x|<1
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Edexcel C4 2013 June Q4
9 marks Standard +0.3
4. (a) Find the binomial expansion of $$\sqrt [ 3 ] { ( 8 - 9 x ) , \quad } \quad | x | < \frac { 8 } { 9 }$$ in ascending powers of \(x\), up to and including the term in \(x ^ { 3 }\). Give each coefficient as a simplified fraction.
(b) Use your expansion to estimate an approximate value for \(\sqrt [ 3 ] { 7100 }\), giving your answer to 4 decimal places. State the value of \(x\), which you use in your expansion, and show all your working.
OCR C4 Q6
9 marks Standard +0.3
6. $$f ( x ) = \frac { 3 } { \sqrt { 1 - x } } , | x | < 1$$
  1. Show that \(\mathrm { f } \left( \frac { 1 } { 10 } \right) = \sqrt { 10 }\).
  2. Expand \(\mathrm { f } ( x )\) in ascending powers of \(x\) up to and including the term in \(x ^ { 3 }\), simplifying each coefficient.
  3. Use your expansion to find an approximate value for \(\sqrt { 10 }\), giving your answer to 8 significant figures.
  4. Find, to 1 significant figure, the percentage error in your answer to part (c).
Edexcel Paper 1 2020 October Q1
5 marks Moderate -0.8
  1. (a) Find the first four terms, in ascending powers of \(x\), of the binomial expansion of
$$( 1 + 8 x ) ^ { \frac { 1 } { 2 } }$$ giving each term in simplest form.
(b) Explain how you could use \(x = \frac { 1 } { 32 }\) in the expansion to find an approximation for \(\sqrt { 5 }\) There is no need to carry out the calculation.
Edexcel Paper 1 Specimen Q2
8 marks Moderate -0.8
  1. (a) Show that the binomial expansion of
$$( 4 + 5 x ) ^ { \frac { 1 } { 2 } }$$ in ascending powers of \(x\), up to and including the term in \(x ^ { 2 }\) is $$2 + \frac { 5 } { 4 } x + k x ^ { 2 }$$ giving the value of the constant \(k\) as a simplified fraction.
(b) (i) Use the expansion from part (a), with \(x = \frac { 1 } { 10 }\), to find an approximate value for \(\sqrt { 2 }\) Give your answer in the form \(\frac { p } { q }\) where \(p\) and \(q\) are integers.
(ii) Explain why substituting \(x = \frac { 1 } { 10 }\) into this binomial expansion leads to a valid approximation.
Edexcel C4 Q2
8 marks Moderate -0.3
2. $$f ( x ) = \frac { 3 } { \sqrt { 1 - x } } , | x | < 1$$
  1. Show that \(\mathrm { f } \left( \frac { 1 } { 10 } \right) = \sqrt { 10 }\).
  2. Expand \(\mathrm { f } ( x )\) in ascending powers of \(x\) up to and including the term in \(x ^ { 3 }\), simplifying each coefficient.
  3. Use your expansion to find an approximate value for \(\sqrt { 10 }\), giving your answer to 8 significant figures.
  4. Find, to 1 significant figure, the percentage error in your answer to part (c).
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 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]
Edexcel C4 Q3
9 marks Standard +0.3
  1. Show that \((1 + \frac{1}{24})^{-\frac{1}{2}} = k\sqrt{6}\), where \(k\) is rational. [2]
  2. Expand \((1 + \frac{1}{4}x)^{-\frac{1}{2}}\), \(|x| < 2\), in ascending powers of \(x\) up to and including the term in \(x^3\), simplifying each coefficient. [4]
  3. Use your answer to part \((b)\) with \(x = \frac{1}{6}\) to find an approximate value for \(\sqrt{6}\), giving your answer to 5 decimal places. [3]
AQA Paper 3 Specimen Q5
11 marks Moderate -0.3
  1. Find the first three terms, in ascending powers of \(x\), in the binomial expansion of \((1 + 6x)^{\frac{1}{3}}\) [2 marks]
  2. Use the result from part (a) to obtain an approximation to \(\sqrt[3]{1.18}\) giving your answer to 4 decimal places. [2 marks]
  3. Explain why substituting \(x = \frac{1}{2}\) into your answer to part (a) does not lead to a valid approximation for \(\sqrt[3]{4}\). [1 mark]
WJEC Unit 3 2018 June Q6
5 marks Moderate -0.3
Write down the first three terms in the binomial expansion of \((1-4x)^{-\frac{1}{2}}\) in ascending powers of \(x\). State the range of values of \(x\) for which the expansion is valid. By writing \(x = \frac{1}{13}\) in your expansion, find an approximate value for \(\sqrt{13}\) in the form \(\frac{a}{b}\), where \(a\), \(b\) are integers. [5]
WJEC Unit 3 Specimen Q4
4 marks Moderate -0.8
  1. Expand \((1-x)^{-\frac{1}{2}}\) in ascending power of \(x\) as far as the term in \(x^2\). State the range of \(x\) for which the expansion is valid. [2]
  2. By taking \(x = \frac{1}{10}\), find an approximation for \(\sqrt{10}\) in the form \(\frac{a}{b}\), where \(a\) and \(b\) are to be determined. [2]