Partial fractions with repeated linear factor

A question is this type if and only if the denominator contains a repeated linear factor (x-a)² requiring partial fractions of the form B/(x-a) + C/(x-a)².

4 questions · Standard +0.7

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CAIE P3 2010 June Q9
10 marks Standard +0.3
  1. Express \(\frac{4 + 5x - x^2}{(1 - 2x)(2 + x)^2}\) in partial fractions. [5]
  2. Hence obtain the expansion of \(\frac{4 + 5x - x^2}{(1 - 2x)(2 + x)^2}\) in ascending powers of \(x\), up to and including the term in \(x^2\). [5]
CAIE P3 2018 November Q8
10 marks Standard +0.3
Let \(f(x) = \frac{7x^2 - 15x + 8}{(1 - 2x)(2 - x)^2}\).
  1. Express \(f(x)\) in partial fractions. [5]
  2. Hence obtain the expansion of \(f(x)\) in ascending powers of \(x\), up to and including the term in \(x^2\). [5]
OCR H240/03 2019 June Q6
10 marks Challenging +1.2
\includegraphics{figure_6} The diagram shows part of the curve \(y = \frac{2x - 1}{(2x + 3)(x + 1)^2}\). Find the exact area of the shaded region, giving your answer in the form \(p + q \ln r\), where \(p\) and \(q\) are positive integers and \(r\) is a positive rational number. [10]
SPS SPS FM Pure 2024 January Q5
13 marks Standard +0.8
Let $$f(x) = \frac{27x^2 + 32x + 16}{(3x + 2)^2(1 - x)}$$
  1. Express \(f(x)\) in terms of partial fractions [5]
  2. Hence, or otherwise, find the series expansion of \(f(x)\), in ascending powers of \(x\), up to and including the term in \(x^2\). Simplify each term. [6]
  3. State, with a reason, whether your series expansion in part (b) is valid for \(x = \frac{1}{2}\). [2]