Expand and state validity

Questions that ask to expand a binomial expression (with fractional or negative exponent) and then state the range/set of values for which the expansion is valid.

33 questions · Moderate -0.5

1.04c Extend binomial expansion: rational n, |x|<11.04d Binomial expansion validity: convergence conditions
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OCR MEI Paper 2 2020 November Q6
4 marks Moderate -0.8
6
  1. Find the first three terms in ascending powers of \(x\) of the binomial expansion of \(( 1 + 4 x ) ^ { \frac { 1 } { 2 } }\).
  2. State the range of values of \(x\) for which this expansion is valid.
OCR MEI Paper 3 Specimen Q2
4 marks Moderate -0.5
2 Find the first four terms of the binomial expansion of \(( 1 - 2 x ) ^ { \frac { 1 } { 2 } }\). State the set of values of \(x\) for which the expansion is valid.
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]
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]
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]
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]
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]