Partial fractionsAQA A-Level Maths: Subtopic test

1. 1
   It is given that $\dfrac{7x+1}{(x-1)(x+2)}\equiv\dfrac{A}{x-1}+\dfrac{B}{x+2}$ for all $x\neq1,-2$, where $A$ and $B$ are constants.

   (a)

   Find the value of $A$.

   [1 mark]

   • A$\dfrac83$
   • B$\dfrac{13}{3}$
   • C$8$
   • D$-\dfrac83$

   (b)

   Find the value of $B$.

   [1 mark]

   • A$\dfrac83$
   • B$-13$
   • C$\dfrac{13}{3}$
   • D$-\dfrac{13}{3}$

   (c)

   Verify that the partial fractions you found for $A$ and $B$ are correct by substituting $x=0$ into both sides of the identity.

   [2 marks]

   Total for question 1: 4 marks

2. 2
   It is given that $\dfrac{5x+3}{(x+1)^2}\equiv\dfrac{P}{x+1}+\dfrac{Q}{(x+1)^2}$ for all $x\neq-1$, where $P$ and $Q$ are constants.

   (a)

   Find the value of $P$.

   [1 mark]

   • A$-2$
   • B$3$
   • C$8$
   • D$5$

   (b)

   Find the value of $Q$.

   [1 mark]

   • A$5$
   • B$-2$
   • C$2$
   • D$3$

   (c)

   Use the partial fractions to evaluate $\dfrac{5x+3}{(x+1)^2}$ when $x=1$.

   [2 marks]

   Total for question 2: 4 marks

3. 3
   Let $\mathrm{f}(x)=\dfrac{4x+9}{(x+2)(x+3)}$ and $\mathrm{g}(x)=\dfrac{5x+4}{(x-1)(x+2)^2}$, defined for values of $x$ where the denominators are non-zero.

   (a)

   Express $\mathrm{f}(x)$ in partial fractions.

   [3 marks]

   (b)

   Express $\mathrm{g}(x)$ in partial fractions.

   [4 marks]

   Total for question 3: 7 marks

4. 4
   Let $\mathrm{h}(x)=\dfrac{30}{(x+1)(x-2)(x+3)}$ and $\mathrm{k}(x)=\dfrac{3x-6}{(2x-1)(x+1)^2}$, defined for values of $x$ where the denominators are non-zero.

   (a)

   (i) Express $\mathrm{h}(x)$ in partial fractions.
   (ii) Hence evaluate $\mathrm{h}(1)$ using your partial fractions, and check your answer against the original expression.

   [6 marks]

   (b)

   Express $\mathrm{k}(x)$ in partial fractions.

   [6 marks]

   Total for question 4: 12 marks

End of questions