4.05c Partial fractions: extended to quadratic denominators

48 questions

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Pre-U Pre-U 9795 Specimen Q1
Standard +0.3
1 The \(n\)th triangular number, \(T _ { n }\), is given by the formula \(T _ { n } = \frac { 1 } { 2 } n ( n + 1 )\).
  1. Express \(\frac { 1 } { T _ { n } }\) in terms of partial fractions.
  2. Hence, using the method of differences, show that \(\sum _ { n = 1 } ^ { N } \left( \frac { 1 } { T _ { n } } \right) = \frac { 2 N } { N + 1 }\).
CAIE P3 2017 June Q8
10 marks Standard +0.3
Let \(\mathrm{f}(x) = \frac{5x^2 - 7x + 4}{(3x + 2)(x^2 + 5)}\).
  1. Express \(\mathrm{f}(x)\) in partial fractions. [5]
  2. Hence obtain the expansion of \(\mathrm{f}(x)\) in ascending powers of \(x\), up to and including the term in \(x^2\). [5]
CAIE FP1 2003 November Q11
28 marks Challenging +1.2
Answer only one of the following two alternatives. EITHER The curve \(C\) has equation \(y = \frac{5(x-1)(x+2)}{(x-2)(x+3)}\).
  1. Express \(y\) in the form \(P + \frac{Q}{x-2} + \frac{R}{x+3}\). [3]
  2. Show that \(\frac{dy}{dx} = 0\) for exactly one value of \(x\) and find the corresponding value of \(y\). [4]
  3. Write down the equations of all the asymptotes of \(C\). [3]
  4. Find the set of values of \(k\) for which the line \(y = k\) does not intersect \(C\). [4]
OR A curve has equation \(y = \frac{5}{3}x^{\frac{3}{2}}\), for \(x \geq 0\). The arc of the curve joining the origin to the point where \(x = 3\) is denoted by \(R\).
  1. Find the length of \(R\). [4]
  2. Find the \(y\)-coordinate of the centroid of the region bounded by the \(x\)-axis, the line \(x = 3\) and \(R\). [5]
  3. Show that the area of the surface generated when \(R\) is rotated through one revolution about the \(y\)-axis is \(\frac{232\pi}{15}\). [5]
Edexcel FP2 Q1
6 marks Standard +0.8
  1. Express \(\frac{1}{t(t+2)}\) in partial fractions. [1]
  2. Hence show that \(\sum_{n=1}^{\infty} \frac{4}{n(n+2)} = \frac{n(3n+5)}{(n+1)(n+2)}\) [5]
Edexcel FP2 Q1
7 marks Standard +0.3
  1. Express \(\frac{3}{(3r-1)(3r+2)}\) in partial fractions. [2]
  2. Using your answer to part (a) and the method of differences, show that $$\sum_{r=1}^n \frac{3}{(3r-1)(3r+2)} = \frac{3n}{2(3n+2)}$$ [3]
  3. Evaluate \(\sum_{r=1}^{30} \frac{3}{(3r-1)(3r+2)}\), giving your answer to 3 significant figures. [2]
Edexcel FP2 Q6
21 marks Standard +0.3
  1. Express \(\frac{1}{r(r + 2)}\) in partial fractions. [2]
  2. Hence prove, by the method of differences, that $$\sum_{r=1}^{2n} \frac{1}{r(r + 2)} = \frac{n(4n + 5)}{4(n + 1)(n + 2)},$$ [6]
The point \(Q\) represents a complex number \(z\) on an Argand diagram such that $$|z - 6| = 2|z - 3|.$$
  1. Show that, as \(z\) varies, the locus of \(P\) is a circle, stating the radius and the coordinates of the centre of this circle. [6]
where \(a\) and \(b\) are constants to be found.
  1. Hence show that $$\sum_{r=1}^{2n} \frac{1}{r(r + 2)} = \frac{n(4n + 5)}{4(n + 1)(n + 2)},$$ [3]
  2. Find the complex number for which both \(|z - 6| = 2|z - 3|\) and \(\arg(z - 6) = -\frac{3\pi}{4}\). [4]
Edexcel FP2 Q3
8 marks Standard +0.3
  1. Express \(\frac{2}{(r + 1)(r + 3)}\) in partial fractions. [2]
  2. Hence show that $$\sum_{r=1}^{\infty} \frac{2}{(r + 1)(r + 3)} = \frac{n(5n + 13)}{6(n + 2)(n + 3)}$$ [4]
  3. Evaluate \(\sum_{r=1}^{30} \frac{2}{(r + 1)(r + 3)}\), giving your answer to 3 significant figures. [2]
Edexcel FP2 Q1
5 marks Moderate -0.3
  1. Express \(\frac{2}{(2r + 1)(2r + 3)}\) in partial fractions. [2]
  2. Using your answer to (a), find, in terms of \(n\), $$\sum_{r=1}^n \frac{2}{(2r + 1)(2r + 3)}$$ [3]
Give your answer as a single fraction in its simplest form.
Edexcel FP2 Q12
7 marks Standard +0.8
  1. Express \(\frac{2}{(r + 1)(r + 3)}\) in partial fractions. [2]
  2. Hence prove that \(\sum_{r=1}^{n} \frac{2}{(r + 1)(r + 3)} = \frac{n(5n + 13)}{6(n + 2)(n + 3)}\). [5]
AQA Further AS Paper 1 2019 June Q11
8 marks Challenging +1.2
  1. Curve \(C\) has equation $$y = \frac{x^2 + px - q}{x^2 - r}$$ where \(p\), \(q\) and \(r\) are positive constants. Write down the equations of its asymptotes. [2 marks]
  2. Find the set of possible \(y\)-coordinates for the graph of $$y = \frac{x^2 + x - 6}{x^2 - 1}, \quad x \neq \pm 1$$ giving your answer in exact form. No credit will be given for solutions based on differentiation. [6 marks]
AQA Further Paper 1 Specimen Q3
6 marks Standard +0.8
\begin{enumerate}[label=(\alph*)] \item Given that $$\frac{2}{(r + 1)(r + 2)(r + 3)} \equiv \frac{A}{(r + 1)(r + 2)} + \frac{B}{(r + 2)(r + 3)}$$ find the values of the integers \(A\) and \(B\) [2 marks] \item Use the method of differences to show clearly that $$\sum_{r=4}^{97} \frac{1}{(r + 1)(r + 2)(r + 3)} = \frac{89}{19800}$$ [4 marks]
AQA Further Paper 2 Specimen Q10
8 marks Challenging +1.8
Evaluate the improper integral \(\int_0^{\infty} \frac{4x - 30}{(x^2 + 5)(3x + 2)} \, dx\), showing the limiting process used. Give your answer as a single term. [8 marks]
OCR Further Pure Core 2 Specimen Q4
5 marks Challenging +1.2
It is given that \(\frac{5x^2+x+12}{x^2+kx} = \frac{A}{x} + \frac{Bx+C}{x^2+k}\) where \(k\), \(A\), \(B\) and \(C\) are positive integers. Determine the set of possible values of \(k\). [5]
OCR MEI Further Pure Core Specimen Q5
7 marks Standard +0.8
  1. Express \(\frac{2}{(r+1)(r+3)}\) in partial fractions. [2]
  2. Hence find \(\sum_{r=1}^{n} \frac{1}{(r+1)(r+3)}\), expressing your answer as a single fraction. [5]
WJEC Further Unit 4 2022 June Q14
10 marks Challenging +1.8
Evaluate the integral $$\int_2^4 \frac{6x^2 + 2x + 16}{x^3 - x^2 + 3x - 3} dx,$$ giving your answer correct to three decimal places. [10]
WJEC Further Unit 4 2023 June Q11
14 marks Challenging +1.2
Evaluate the integrals
  1. \(\int_{-2}^{0} e^{2x} \sinh x \, \mathrm{d}x\), [5]
  2. \(\int_{\frac{1}{2}}^{3} \frac{5}{(x-1)(x^2+9)} \, \mathrm{d}x\). [9]
WJEC Further Unit 4 2024 June Q5
14 marks Challenging +1.8
Find each of the following integrals.
  1. \(\int \frac{3-x}{x(x^2+1)} \mathrm{d}x\) [8]
  2. \(\int \frac{\sinh 2x}{\sqrt{\cosh^4 x - 9\cosh^2 x}} \mathrm{d}x\) [6]
WJEC Further Unit 4 Specimen Q7
10 marks Standard +0.8
The function \(f\) is defined by $$f(x) = \frac{8x^2 + x + 5}{(2x + 1)(x^2 + 3)}.$$
  1. Express \(f(x)\) in partial fractions. [4]
  2. Hence evaluate $$\int_2^5 f(x)dx,$$ giving your answer correct to three decimal places. [6]
SPS SPS FM Pure 2021 May Q1
7 marks Standard +0.3
In this question you must show detailed reasoning.
  1. By using partial fractions show that \(\sum_{r=1}^{\infty} \frac{1}{r^2 + 3r + 2} = \frac{1}{2} - \frac{1}{n+2}\). [5]
  2. Hence determine the value of \(\sum_{r=1}^{\infty} \frac{1}{r^2 + 3r + 2}\). [2]
SPS SPS FM Pure 2021 May Q5
5 marks Moderate -0.3
Express \(\frac{5x^2+x+12}{x^3+4x}\) in partial fractions. [5]
SPS SPS FM Pure 2023 February Q10
10 marks Challenging +1.3
  1. Find the general solution of the differential equation $$\frac{dy}{dx} + \frac{2y}{x} = \frac{x+3}{x(x-1)(x^2+3)} \quad (x > 1)$$ [8]
  2. Find the particular solution for which \(y = 0\) when \(x = 3\). Give your answer in the form \(y = f(x)\). [2]
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]
OCR Further Pure Core 2 2018 March Q8
12 marks Challenging +1.8
In this question you must show detailed reasoning. Show that \(\int_0^2 \frac{2x^2 + 3x - 1}{x^3 - 3x^2 + 4x - 12} dx = \frac{3}{8}\pi - \ln 9\). [12]