A-level · Algebra & functions

Partial fractionsExam-style questions

Splitting one fraction into a sum of simpler ones. Partial fractions as adding fractions run backwards, and why the pieces are worth having; distinct linear factors, found by substituting the values that make brackets vanish; factorising the denominator first, including factors like (2x + 1); the extra piece a repeated factor needs, and equating coefficients for it; and dividing out a top-heavy fraction first.

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26 questions · A-level level · mark scheme and worked solution on each · no sign-up to try

DifficultyAll Level 2 Level 3 Level 4 Level 5

1
Express $\dfrac{2x + 5}{(x + 2)^{2}}$ in the form $\dfrac{A}{x + 2} + \dfrac{B}{(x + 2)^{2}}$ . (4 marks)
Level 4 · 4 marksTry it
2
Express $\dfrac{4x - 5}{(x - 1)(x - 2)}$ in partial fractions.
Level 4 · 3 marksTry it
3
In this question you must show detailed reasoning.

Show that

$\int_0^1 \dfrac{3x + 5}{(x + 1)(x + 3)}\;\mathrm{d}x = \ln\dfrac{32}{9}$

(7 marks)
Level 4 · 7 marksTry it
4
Express $\dfrac{2x + 1}{x^2 + x - 6}$ in partial fractions.
Level 4 · 4 marksTry it
5
In this question you must show all stages of your working.

Solutions relying entirely on calculator technology are not acceptable.

$\text{f}(x) = \frac{kx + 3}{(x + 1)(x + 2)} \qquad x > -1$

where $k$ is a positive constant.
(a) Express $\text{f}(x)$ in partial fractions, giving your answer in terms of $k$ .

(3 marks)
(b) Hence find the exact value of $k$ for which

$\int_0^1 \text{f}(x)\,dx = 2\ln 2$

(4 marks)
Level 4 · 7 marksTry it

4 more questions on partial fractions

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