1.02y Partial fractions: decompose rational functions

420 questions

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WJEC Unit 3 2018 June Q5
8 marks Moderate -0.3
  1. Show that $$\frac{3x}{(x-1)(x-4)^2} = \frac{A}{(x-1)} + \frac{B}{(x-4)} + \frac{C}{(x-4)^2},$$ where \(A\), \(B\) and \(C\) are constants to be found. [3]
  2. Evaluate \(\int_5^7 \frac{3x}{(x-1)(x-4)^2} \, dx\), giving your answer correct to 3 decimal places. [5]
WJEC Unit 3 2023 June Q3
11 marks Standard +0.3
  1. Express \(\frac{9}{(1-x)(1+2x)^2}\) in terms of partial fractions. [4]
  2. Using your answer from part (a), find the expansion of \(\frac{9}{(1-x)(1+2x)^2}\) in ascending powers of \(x\) as far as the term in \(x^2\). State the values of \(x\) for which the expansion is valid. [7]
WJEC Unit 3 2024 June Q1
11 marks Standard +0.3
The function \(f\) is given by $$f(x) = \frac{25x + 32}{(2x - 5)(x + 1)(x + 2)}.$$
  1. Express \(f(x)\) in terms of partial fractions. [4]
  2. Show that \(\int_1^2 f(x) dx = -\ln P\), where \(P\) is an integer whose value is to be found. [5]
  3. Show that the sign of \(f(x)\) changes in the interval \(x = 2\) to \(x = 3\). Explain why the change of sign method fails to locate a root of the equation \(f(x) = 0\) in this case. [2]
WJEC Further Unit 4 2019 June Q4
16 marks Standard +0.3
  1. Given that \(y = \cot^{-1} x\), show that \(\frac{\mathrm{d}y}{\mathrm{d}x} = \frac{-1}{x^2 + 1}\). [5]
  2. Express \(\frac{6x^2 - 10x - 9}{(2x + 3)(x^2 + 1)}\) in terms of partial fractions. [5]
  3. Hence find \(\int \frac{6x^2 - 8x - 6}{(2x + 3)(x^2 + 1)} \mathrm{d}x\). [5]
  4. Explain why \(\int_{-2}^{5} \frac{6x^2 - 8x - 6}{(2x + 3)(x^2 + 1)} \mathrm{d}x\) cannot be evaluated. [1]
SPS SPS SM Pure 2020 October Q6
5 marks Moderate -0.8
  1. Express \(\frac{x}{(x + 1)(x + 2)}\) in partial fractions. [3]
  2. Hence find \(\int \frac{x}{(x + 1)(x + 2)} dx\). [2]
SPS SPS SM Pure 2022 June Q13
4 marks Moderate -0.3
$$\frac{1 + 11x - 6x^2}{(x - 3)(1 - 2x)} \equiv A + \frac{B}{(x - 3)} + \frac{C}{(1 - 2x)}$$ Find the values of the constants \(A\), \(B\) and \(C\). [4]
SPS SPS FM Pure 2022 February Q1
7 marks Moderate -0.3
  1. Express \(\frac{1}{(2r-1)(2r+1)}\) in partial fractions. [3]
  2. Hence find \(\sum_{r=1}^{n}\frac{1}{(2r-1)(2r+1)}\), expressing the result as a single fraction. [4]
SPS SPS SM 2021 November Q2
6 marks Moderate -0.3
  1. Express \(\frac{5x+7}{(x+3)(x+1)^2}\) in partial fractions. In this question you must show all of your algebraic steps clearly. [3] The function \(f(x) = \frac{2-6x+5x^2}{x^2(1-2x)}\) can be written in the form; $$f(x) = \frac{-2}{x} + \frac{2}{x^2} + \frac{1}{1-2x}$$
  2. Hence find the exact value of \(\int_2^3 \frac{2-6x+5x^2}{x^2(1-2x)} dx\) [3]
SPS SPS FM 2023 January Q3
5 marks Standard +0.8
Express \(\frac{x^2}{(x-1)^2(x-2)}\) in partial fractions. [5]
SPS SPS FM 2023 February Q3
5 marks Standard +0.3
Express \(\frac{(x-7)(x-2)}{(x+2)(x-1)^2}\) in partial fractions. [5]
SPS SPS SM Pure 2023 June Q3
3 marks Standard +0.3
Express in partial fractions, $$\frac{9x^2}{(x-1)^2(2x+1)}$$ [3]
SPS SPS FM Pure 2025 January Q2
8 marks Standard +0.3
  1. Given that $$\frac{3 + 2x^2}{(1 + x)^2(1 - 4x)} = \frac{A}{1 + x} + \frac{B}{(1 + x)^2} + \frac{C}{1 - 4x},$$ where \(A\), \(B\) and \(C\) are constants, find \(B\) and \(C\), and show that \(A = 0\). [4]
  2. Given that \(x\) is sufficiently small, find the first three terms of the binomial expansions of \((1 + x)^{-2}\) and \((1 - 4x)^{-1}\). Hence find the first three terms of the expansion of \(\frac{3 + 2x^2}{(1 + x)^2(1 - 4x)}\). [4]
SPS SPS FM 2025 February Q3
4 marks Moderate -0.3
Express \(\frac{9x^2+43x+8}{(3+x)(1-x)(2x+1)}\) in partial fractions. [4]
OCR H240/03 2018 December Q6
15 marks Challenging +1.2
\includegraphics{figure_6} The diagram shows the curve with parametric equations \(x = \ln(t^2 - 4)\), \(y = \frac{4}{t}\), where \(t > 2\). The shaded region \(R\) is enclosed by the curve, the \(x\)-axis and the lines \(x = \ln 5\) and \(x = \ln 12\).
  1. Show that the area of \(R\) is given by $$\int_a^b \frac{8}{t(t^2 - 4)} dt,$$ where \(a\) and \(b\) are constants to be determined. [4]
  2. In this question you must show detailed reasoning. Hence find the exact area of \(R\), giving your answer in the form \(\ln k\), where \(k\) is a constant to be determined. [8]
  3. Find a cartesian equation of the curve in the form \(y = \text{f}(x)\). [3]
OCR H240/02 2017 Specimen Q4
7 marks Moderate -0.3
  1. Express \(\frac{1}{(x-1)(x+2)}\) in partial fractions [2]
  2. In this question you must show detailed reasoning. Hence find \(\int_2^3 \frac{1}{(x-1)(x+2)} dx\). Give your answer in its simplest form. [5]
Pre-U Pre-U 9794/2 2010 June Q6
10 marks Standard +0.3
  1. Express \(\frac{x-1}{x^2+2x+1}\) in the form \(\frac{A}{x+1} + \frac{B}{(x+1)^2}\), where \(A\) and \(B\) are integers. [2]
  2. Find the quotient and remainder when \(2y^2 + 1\) is divided by \(y + 1\). [2]
  3. A curve in the \(x\)-\(y\) plane passes through the point \((0, 2)\) and satisfies the differential equation $$(2y^2 + 1)(x^2 + 2x + 1)\frac{dy}{dx} = (x - 1)(y + 1).$$ By solving the differential equation find the equation of the curve in implicit form. [6]
Pre-U Pre-U 9795/1 2011 June Q7
11 marks Challenging +1.2
Sketch the curve with equation \(y = \frac{x^2 + 4x}{2x - 1}\), justifying all significant features. [11]
Pre-U Pre-U 9795/1 2015 June Q5
11 marks Standard +0.8
A curve has equation \(y = \frac{2x^2 + 5x - 25}{x - 3}\).
  1. Determine the equations of the asymptotes. [3]
  2. Find the coordinates of the turning points. [5]
  3. Sketch the curve. [3]
Pre-U Pre-U 9794/2 2016 June Q5
7 marks Moderate -0.3
  1. Show that \(\frac{3}{x+2} + \frac{1}{x+1} \equiv \frac{4x+5}{x^2+3x+2}\). [2]
  2. Differentiate \(\frac{4x+5}{x^2+3x+2}\) with respect to \(x\). [3]
  3. Hence show that the function given by $$f(x) = \frac{4x+5}{x^2+3x+2}, \quad x \neq -1, x \neq -2,$$ is a decreasing function. [2]
Pre-U Pre-U 9794/2 Specimen Q6
10 marks Standard +0.8
  1. Express \(y^3 - 3y - 2\) in terms of \(x\), where \(x = y + 1\). [1]
  2. Hence express $$\frac{2y + 5}{y^3 - 3y - 2}$$ in partial fractions. [5]
  3. Find the exact value of $$\int_0^1 \frac{2y + 5}{y^3 - 3y - 2} dy.$$ [4]