1.02y Partial fractions: decompose rational functions

420 questions

Sort by: Default | Easiest first | Hardest first
Edexcel P4 2024 June Q5
13 marks Standard +0.3
\includegraphics{figure_2} Figure 2 shows a sketch of the curve defined by the parametric equations $$x = t^2 + 2t \quad y = \frac{2}{t(3-t)} \quad a \leq t \leq b$$ where \(a\) and \(b\) are constants. The ends of the curve lie on the line with equation \(y = 1\)
  1. Find the value of \(a\) and the value of \(b\) [2]
The region \(R\), shown shaded in Figure 2, is bounded by the curve and the line with equation \(y = 1\)
  1. Show that the area of region \(R\) is given by $$M - k \int_a^b \frac{t+1}{t(3-t)} dt$$ where \(M\) and \(k\) are constants to be found. [5]
    1. Write \(\frac{t+1}{t(3-t)}\) in partial fractions.
    2. Use algebraic integration to find the exact area of \(R\), giving your answer in simplest form. [6]
Edexcel P4 2022 October Q2
7 marks Moderate -0.3
  1. Express \(\frac{3x}{(2x-1)(x-2)}\) in partial fraction form. [3]
  2. Hence show that $$\int_5^{25} \frac{3x}{(2x-1)(x-2)} \, dx = \ln k$$ where \(k\) is a fully simplified fraction to be found. (Solutions relying entirely on calculator technology are not acceptable.) [4]
Edexcel C4 Q3
8 marks Moderate -0.3
  1. Express \(\frac{5x + 3}{(2x - 3)(x + 2)}\) in partial fractions. [3]
  2. Hence find the exact value of \(\int_0^1 \frac{5x + 3}{(2x - 3)(x + 2)} dx\), giving your answer as a single logarithm. [5]
Edexcel C4 2013 June Q6
11 marks Moderate -0.3
    1. Express \(\frac{7x}{(x + 3)(2x - 1)}\) in partial fractions. [3]
    2. Given that \(x > \frac{1}{2}\), find $$\int \frac{7x}{(x + 3)(2x - 1)} \, dx$$ [3]
  2. Using the substitution \(u^3 = x\), or otherwise, find $$\int \frac{1}{x + x^3} \, dx, \quad x > 0$$ [5]
Edexcel C4 2015 June Q7
13 marks Standard +0.8
  1. Express \(\frac{2}{P(P-2)}\) in partial fractions. [3]
A team of biologists is studying a population of a particular species of animal. The population is modelled by the differential equation $$\frac{dP}{dt} = \frac{1}{2}P(P-2)\cos 2t, \quad t \geqslant 0$$ where \(P\) is the population in thousands, and \(t\) is the time measured in years since the start of the study. Given that \(P = 3\) when \(t = 0\),
  1. solve this differential equation to show that $$P = \frac{6}{3 - e^{\frac{1}{2}\sin 2t}}$$ [7]
  2. find the time taken for the population to reach 4000 for the first time. Give your answer in years to 3 significant figures. [3]
Edexcel C4 Q3
13 marks Standard +0.3
$$f(x) = \frac{1 + 14x}{(1 - x)(1 + 2x)}, \quad |x| < \frac{1}{2}.$$
  1. Express \(f(x)\) in partial fractions. [3]
  2. Hence find the exact value of \(\int_{-\frac{1}{6}}^{\frac{1}{4}} f(x) \, dx\), giving your answer in the form \(\ln p\), where \(p\) is rational. [5]
  3. Use the binomial theorem to expand \(f(x)\) in ascending powers of \(x\), up to and including the term in \(x^5\), simplifying each term. [5]
Edexcel C4 Q11
8 marks Moderate -0.3
Given that $$\frac{10(2 - 3x)}{(1 - 2x)(2 + x)} \equiv \frac{A}{1 - 2x} + \frac{B}{2 + x},$$
  1. find the values of the constants \(A\) and \(B\). [3]
  2. Hence, or otherwise, find the series expansion in ascending powers of \(x\), up to and including the term in \(x^3\), of \(\frac{10(2 - 3x)}{(1 - 2x)(2 + x)}\), for \(|x| < \frac{1}{2}\). [5]
Edexcel C4 Q20
9 marks Standard +0.3
\includegraphics{figure_1} Figure 1 shows part of the curve with equation \(y = f(x)\), where $$f(x) = \frac{x^2 + 1}{(1 + x)(3 - x)}, \quad 0 \leq x < 3.$$
  1. Given that \(f(x) = A + \frac{B}{1 + x} + \frac{C}{3 - x}\), find the values of the constants \(A\), \(B\) and \(C\). [4]
The finite region \(R\), shown in Fig. 1, is bounded by the curve with equation \(y = f(x)\), the \(x\)-axis, the \(y\)-axis and the line \(x = 2\).
  1. Find the area of \(R\), giving your answer in the form \(p + q \ln r\), where \(p\), \(q\) and \(r\) are rational constants to be found. [5]
Edexcel C4 Q28
6 marks Standard +0.3
The function f is given by $$f(x) = \frac{3(x + 1)}{(x + 2)(x - 1)}, \quad x \in \mathbb{R}, x \neq -2, x \neq 1.$$
  1. Express \(f(x)\) in partial fractions. [3]
  2. Hence, or otherwise, prove that \(f'(x) < 0\) for all values of \(x\) in the domain. [3]
Edexcel FP2 Q38
10 marks Standard +0.3
  1. Express \(\frac{1}{r(r + 2)}\) in partial fractions. [2]
  2. Hence prove, by the method of differences, that $$\sum_{r=1}^{n} \frac{4}{r(r + 2)} = \frac{n(3n + 5)}{(n + 1)(n + 2)}.$$ [5]
  3. Find the value of \(\sum_{r=50}^{100} \frac{4}{r(r + 2)}\), to 4 decimal places. [3]
Edexcel FP2 Q41
5 marks Standard +0.3
  1. By expressing \(\frac{2}{4r^2 - 1}\) in partial fractions, or otherwise, prove that $$\sum_{r=1}^{n} \frac{2}{4r^2 - 1} = 1 - \frac{1}{2n + 1}.$$ [3]
  2. Hence find the exact value of \(\sum_{r=11}^{20} \frac{2}{4r^2 - 1}\). [2]
Edexcel C3 Q3
6 marks Moderate -0.3
  1. Express as a fraction in its simplest form $$\frac{2}{x - 3} + \frac{13}{x^2 + 4x - 21}.$$ [3]
  2. Hence solve $$\frac{2}{x - 3} + \frac{13}{x^2 + 4x - 21} = 1.$$ [3]
Edexcel C3 Q4
9 marks Standard +0.3
  1. Express $$\frac{x-10}{(x-3)(x+4)} - \frac{x-8}{(x-3)(2x-1)}$$ as a single fraction in its simplest form. [5]
  2. Hence, show that the equation $$\frac{x-10}{(x-3)(x+4)} - \frac{x-8}{(x-3)(2x-1)} = 1$$ has no real roots. [4]
AQA C4 2010 June Q3
8 marks Moderate -0.3
    1. Express \(\frac{7x - 3}{(x + 1)(3x - 2)}\) in the form \(\frac{A}{x + 1} + \frac{B}{3x - 2}\). [3 marks]
    2. Hence find \(\int \frac{7x - 3}{(x + 1)(3x - 2)} dx\). [2 marks]
  2. Express \(\frac{6x^2 + x + 2}{2x^2 - x + 1}\) in the form \(P + \frac{Qx + R}{2x^2 - x + 1}\). [3 marks]
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]
AQA C4 2016 June Q3
8 marks Standard +0.3
  1. Express \(\frac{3 + 13x - 6x^2}{2x - 3}\) in the form \(Ax + B + \frac{C}{2x - 3}\). [4 marks]
  2. Show that \(\int_3^6 \frac{3 + 13x - 6x^2}{2x - 3} \, dx = p + q \ln 3\), where \(p\) and \(q\) are rational numbers. [4 marks]
Edexcel C4 Q7
16 marks Standard +0.8
$$\text{f}(x) = \frac{25}{(3 + 2x)^2(1 - x)}, \quad |x| < 1.$$
  1. Express f(x) as a sum of partial fractions. [4]
  2. Hence find \(\int \text{f}(x) \, dx\). [5]
  3. Find the series expansion of f(x) in ascending powers of \(x\) up to and including the term in \(x^2\). Give each coefficient as a simplified fraction. [7]
Edexcel C4 Q2
9 marks Standard +0.3
\includegraphics{figure_1} Figure 1 shows part of the curve with equation \(y = f(x)\), where $$f(x) = \frac{x^2 + 1}{(1 + x)(3 - x)}, \quad 0 \leq x < 3.$$
  1. Given that \(f(x) = A + \frac{B}{1 + x} + \frac{C}{3 - x}\), find the values of the constants \(A\), \(B\) and \(C\). [4]
The finite region \(R\), shown in Fig. 1, is bounded by the curve with equation \(y = f(x)\), the \(x\)-axis, the \(y\)-axis and the line \(x = 2\).
  1. Find the area of \(R\), giving your answer in the form \(p + q \ln r\), where \(p\), \(q\) and \(r\) are rational constants to be found. [5]
OCR C4 2007 January Q6
7 marks Moderate -0.3
  1. Express \(\frac{2x + 1}{(x - 3)^2}\) in the form \(\frac{A}{x - 3} + \frac{B}{(x - 3)^2}\), where \(A\) and \(B\) are constants. [3]
  2. Hence find the exact value of \(\int_4^{10} \frac{2x + 1}{(x - 3)^2} \, dx\), giving your answer in the form \(a + b \ln c\), where \(a\), \(b\) and \(c\) are integers. [4]
OCR C4 2005 June Q8
11 marks Standard +0.3
  1. Given that \(\frac{3x + 4}{(1 + x)(2 + x)^2} \equiv \frac{A}{1 + x} + \frac{B}{2 + x} + \frac{C}{(2 + x)^2}\), find \(A\), \(B\) and \(C\). [5]
  2. Hence or otherwise expand \(\frac{3x + 4}{(1 + x)(2 + x)^2}\) in ascending powers of \(x\), up to and including the term in \(x^2\). [5]
  3. State the set of values of \(x\) for which the expansion in part (ii) is valid. [1]
OCR C4 2006 June Q3
8 marks Moderate -0.3
  1. Express \(\frac{3 - 2x}{x(3 - x)}\) in partial fractions. [3]
  2. Show that \(\int_1^2 \frac{3 - 2x}{x(3 - x)} dx = 0\). [4]
  3. What does the result of part (ii) indicate about the graph of \(y = \frac{3 - 2x}{x(3 - x)}\) between \(x = 1\) and \(x = 2\)? [1]
OCR MEI C4 2012 January Q1
5 marks Moderate -0.5
Express \(\frac{x+1}{x^2(2x-1)}\) in partial fractions. [5]
OCR MEI C4 2009 June Q2
7 marks Moderate -0.3
Using partial fractions, find \(\int \frac{x}{(x+1)(2x+1)} \, dx\). [7]
OCR MEI C4 2011 June Q1
5 marks Moderate -0.5
Express \(\frac{1}{(2x + 1)(x^2 + 1)}\) in partial fractions. [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]