Product with quadratic or higher term

Questions where a quadratic term (like 1+x²) or higher polynomial is multiplied by a binomial expansion, requiring more complex term collection.

3 questions · Standard +0.1

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SPS SPS FM 2023 January Q5
7 marks Moderate -0.3
  1. Expand \((2+x)^{-2}\) in ascending powers of \(x\) up to and including the term in \(x^3\), and state the set of values of \(x\) for which the expansion is valid. [5]
  2. Hence find the coefficient of \(x^3\) in the expansion of \(\frac{1+x^2}{(2+x)^2}\). [2]
SPS SPS FM 2023 February Q5
9 marks Standard +0.3
  1. Expand \(\sqrt{1 + 2x}\) in ascending powers of x, up to and including the term in \(x^3\). [4]
  2. Hence expand \(\frac{\sqrt{1 + 2x}}{1 + 9x^2}\) in ascending powers of x, up to and including the term in \(x^3\). [3]
  3. Determine the range of values of x for which the expansion in part (b) is valid. [2]
Pre-U Pre-U 9794/1 2010 June Q7
9 marks Standard +0.3
Let \(f(x) = \frac{1 + x^2}{\sqrt{4 - 3x}}\)
  1. Obtain in ascending powers of \(x\) the first three terms in the expansion of \(\frac{1}{\sqrt{4 - 3x}}\) and state the values of \(x\) for which this expansion is valid. [5]
  2. Hence obtain an approximation to \(f(x)\) in the form \(a + bx + cx^2\) where \(a\), \(b\) and \(c\) are constants. [2]
  3. Use your approximation to estimate \(\int_0^{0.1} f(x) dx\). [2]