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

313 questions

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OCR MEI C4 2011 June Q2
5 marks Moderate -0.8
Find the first three terms in the binomial expansion of \(\sqrt{1 + 3x}\) in ascending powers of \(x\). State the set of values of \(x\) for which the expansion is valid. [5]
OCR MEI C4 2012 June Q2
5 marks Moderate -0.8
Find the first four terms in the binomial expansion of \(\sqrt{1+2x}\). State the set of values of \(x\) for which the expansion is valid. [5]
OCR MEI C4 2013 June Q1
8 marks Moderate -0.3
  1. Express \(\frac{x}{(1 + x)(1 - 2x)}\) in partial fractions. [3]
  2. Hence use binomial expansions to show that \(\frac{x}{(1 + x)(1 - 2x)} = ax + bx^2 + ...\), where \(a\) and \(b\) are constants to be determined. State the set of values of \(x\) for which the expansion is valid. [5]
OCR MEI C4 2014 June Q2
5 marks Moderate -0.3
Find the first three terms in the binomial expansion of \((4+x)^{\frac{1}{2}}\). State the set of values of \(x\) for which the expansion is valid. [5]
Edexcel C4 Q1
6 marks Moderate -0.3
  1. Find the binomial expansion of \((2 - 3x)^{-3}\) in ascending powers of \(x\) up to and including the term in \(x^3\), simplifying each coefficient. [5]
  2. State the set of values of \(x\) for which your expansion is valid. [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]
Edexcel C4 Q8
14 marks Standard +0.3
$$\text{f}(x) = \frac{x(3x-7)}{(1-x)(1-3x)}, \quad |x| < \frac{1}{3}.$$
  1. Find the values of the constants \(A\), \(B\) and \(C\) such that $$\text{f}(x) = A + \frac{B}{1-x} + \frac{C}{1-3x}.$$ [4]
  2. Evaluate $$\int_0^{\frac{1}{4}} \text{f}(x) \, dx,$$ giving your answer in the form \(p + \ln q\), where \(p\) and \(q\) are rational. [5]
  3. Find the series expansion of f(x) in ascending powers of \(x\) up to and including the term in \(x^3\), simplifying each coefficient. [5]
Edexcel C4 Q5
12 marks Standard +0.3
$$f(x) = \frac{5 - 8x}{(1 + 2x)(1 - x)^2}.$$
  1. Express \(f(x)\) in partial fractions. [5]
  2. Find the series expansion of \(f(x)\) in ascending powers of \(x\) up to and including the term in \(x^2\), simplifying each coefficient. [6]
  3. State the set of values of \(x\) for which your expansion is valid. [1]
OCR C4 Q4
7 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 coefficient. [4]
  2. Hence, or otherwise, show that for small \(x\), $$\left(\frac{2-x}{1-3x}\right)^2 \approx 4 + 20x + 85x^2 + 330x^3.$$ [3]
OCR C4 Q3
9 marks Standard +0.3
$$f(x) = 3 - \frac{x-1}{x-3} + \frac{x+11}{2x^2-5x-3}, \quad |x| < \frac{1}{2}.$$
  1. Show that $$f(x) = \frac{4x-1}{2x+1}.$$ [4]
  2. Find the series expansion of \(f(x)\) in ascending powers of \(x\) up to and including the term in \(x^3\), simplifying each coefficient. [5]
OCR C4 Q5
8 marks Standard +0.8
  1. Express \(\frac{2 + 20x}{1 + 2x - 8x^2}\) as a sum of partial fractions. [3]
  2. Hence find the series expansion of \(\frac{2 + 20x}{1 + 2x - 8x^2}\), \(|x| < \frac{1}{4}\), in ascending powers of \(x\) up to and including the term in \(x^3\), simplifying each coefficient. [5]
OCR MEI C4 Q5
7 marks Standard +0.3
  1. Find the first three non-zero terms of the binomial expansion of \(\frac{1}{\sqrt{4-x^2}}\) for \(|x| < 2\). [4]
  2. Use this result to find an approximation for \(\int_0^1 \frac{1}{\sqrt{4-x^2}} dx\), rounding your answer to 4 significant figures. [2]
  3. Given that \(\int \frac{1}{\sqrt{4-x^2}} dx = \arcsin\left(\frac{1}{2}x\right) + c\), evaluate \(\int_0^1 \frac{1}{\sqrt{4-x^2}} dx\), rounding your answer to 4 significant figures. [1]
Edexcel AEA 2002 June Q2
9 marks Challenging +1.8
In the binomial expansion of $$(1 - 4x)^p, \quad |x| < \frac{1}{4},$$ the coefficient of \(x^2\) is equal to the coefficient of \(x^4\) and the coefficient of \(x^3\) is positive. Find the value of \(p\). [9]
Edexcel AEA 2004 June Q2
10 marks Challenging +1.3
  1. For the binomial expansion of \(\frac{1}{(1-x)^2}\), \(|x| < 1\), in ascending powers of \(x\),
    1. find the first four terms,
    2. write down the coefficient of \(x^n\). [2]
  2. Hence, show that, for \(|x| < 1\), \(\sum_{n=1}^{\infty} nx^n = \frac{x}{(1-x)^2}\). [2]
  3. Prove that, for \(|x| < 1\), \(\sum_{n=1}^{\infty} (an+1)x^n = \frac{(a+1)x-x^2}{(1-x)^2}\), where \(a\) is a constant. [4]
  4. Hence evaluate \(\sum_{n=1}^{\infty} \frac{5n+1}{2^{3n}}\). [2]
OCR H240/02 2020 November Q3
7 marks Moderate -0.3
In this question you should assume that \(-1 < x < 1\).
  1. For the binomial expansion of \((1 - x)^{-2}\)
    1. find and simplify the first four terms, [2]
    2. write down the term in \(x^n\). [1]
  2. Write down the sum to infinity of the series \(1 + x + x^2 + x^3 + \ldots\). [1]
  3. Hence or otherwise find and simplify an expression for \(2 + 3x + 4x^2 + 5x^3 + \ldots\) in the form \(\frac{a - x}{(b - x)^2}\) where \(a\) and \(b\) are constants to be determined. [3]
AQA Paper 2 2019 June Q9
9 marks Standard +0.3
  1. Show that the first two terms of the binomial expansion of \(\sqrt{4 - 2x^2}\) are $$2 - \frac{x^2}{2}$$ [2 marks]
  2. State the range of values of \(x\) for which the expansion found in part (a) is valid. [2 marks]
  3. Hence, find an approximation for $$\int_0^{0.4} \sqrt{\cos x} \, dx$$ giving your answer to five decimal places. Fully justify your answer. [4 marks]
  4. A student decides to use this method to find an approximation for $$\int_0^{1.4} \sqrt{\cos x} \, dx$$ Explain why this may not be a suitable method. [1 mark]
AQA Paper 2 2024 June Q9
13 marks Standard +0.3
    1. Find the binomial expansion of \((1 + 3x)^{-1}\) up to and including the term in \(x^2\) [2 marks]
    2. Show that the first three terms in the binomial expansion of $$\frac{1}{2 - 3x}$$ form a geometric sequence and state the common ratio. [5 marks]
  2. It is given that $$\frac{36x}{(1 + 3x)(2 - 3x)} = \frac{P}{(2 - 3x)} + \frac{Q}{(1 + 3x)}$$ where \(P\) and \(Q\) are integers. Find the value of \(P\) and the value of \(Q\) [3 marks]
    1. Using your answers to parts (a) and (b), find the binomial expansion of $$\frac{12x}{(1 + 3x)(2 - 3x)}$$ up to and including the term in \(x^2\) [2 marks]
    2. Find the range of values of \(x\) for which the binomial expansion of $$\frac{12x}{(1 + 3x)(2 - 3x)}$$ is valid. [1 mark]
AQA Paper 2 Specimen Q1
1 marks Easy -1.8
State the values of \(|x|\) for which the binomial expansion of \((3 + 2x)^{-4}\) is valid. Circle your answer. [1 mark] \(|x| < \frac{2}{3}\) \(\quad\) \(|x| < 1\) \(\quad\) \(|x| < \frac{3}{2}\) \(\quad\) \(|x| < 3\)
AQA Paper 3 2021 June Q4
5 marks Moderate -0.8
  1. Show that the first three terms, in descending powers of \(x\), of the expansion of $$(2x - 3)^{10}$$ are given by $$1024x^{10} + px^9 + qx^8$$ where \(p\) and \(q\) are integers to be found. [3 marks]
  2. Find the constant term in the expansion of $$\left(2x - \frac{3}{x}\right)^{10}$$ [2 marks]
AQA Paper 3 2022 June Q1
1 marks Easy -1.8
State the range of values of \(x\) for which the binomial expansion of $$\sqrt{1 - \frac{x}{4}}$$ is valid. Circle your answer. [1 mark] \(|x| < \frac{1}{4}\) \quad\quad \(|x| < 1\) \quad\quad \(|x| < 2\) \quad\quad \(|x| < 4\)
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]
SPS SPS FM Pure 2021 June Q6
5 marks Moderate -0.3
  1. Use the binomial expansion, in ascending powers of \(x\), to show that $$\sqrt{4-x} = 2 - \frac{1}{4}x + kx^2 + ...$$ where \(k\) is a rational constant to be found. [4] A student attempts to substitute \(x = 1\) into both sides of this equation to find an approximate value for \(\sqrt{3}\).
  2. State, giving a reason, if the expansion is valid for this value of \(x\). [1]
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]