Partial fractionsEdexcel International A Level Maths: Subtopic test

1. 1
   The function $f(x)=\frac{10x-1}{(x-1)(2x+1)}$ can be written in the form $\frac{A}{x-1}+\frac{B}{2x+1}$.

   (a)

   Find the value of $A$.

   [1 mark]

   • A$3$
   • B$9$
   • C$4$
   • D$\frac13$

   (b)

   Find the value of $B$.

   [1 mark]

   • A$-6$
   • B$3$
   • C$4$
   • D$-4$

   (c)

   Hence find $\int f(x)\,\mathrm{d}x$.

   [2 marks]

   Total for question 1: 4 marks

2. 2
   The function $g(x)=\frac{3x^2+6}{(x+2)(x-1)^2}$ can be written in the form $\frac{A}{x+2}+\frac{B}{x-1}+\frac{C}{(x-1)^2}$.

   (a)

   Find the value of $A$.

   [1 mark]

   • A$18$
   • B$2$
   • C$-6$
   • D$\frac12$

   (b)

   Find the value of $C$.

   [1 mark]

   • A$9$
   • B$1$
   • C$-3$
   • D$3$

   (c)

   Find the value of $B$.

   [2 marks]

   Total for question 2: 4 marks

3. 3
   The function $h(x)=\frac{3x^2+6x-3}{(x-1)(x+2)}$ is defined for $x>1$.

   (a)

   Express $h(x)$ in the form $P+\frac{Q}{x-1}+\frac{R}{x+2}$, where $P$, $Q$ and $R$ are constants.

   [3 marks]

   (b)

   Hence show that $\int_2^3h(x)\,\mathrm{d}x=3+\ln5$.

   [4 marks]

   Total for question 3: 7 marks

4. 4
   Partial fractions make a rational function easier to expand in a series or to integrate.

   (a)

   Given $f(x)=\frac{5+x}{(1-x)(1+2x)}$:
   (i) express $f(x)$ in partial fractions;
   (ii) hence find the series expansion of $f(x)$ in ascending powers of $x$, up to and including the term in $x^2$.

   [6 marks]

   (b)

   Given $g(x)=\frac{3x^2-6x+9}{(x+1)(x-2)^2}$, find the exact value of $\int_3^5g(x)\,\mathrm{d}x$, giving your answer in the form $p+\ln q$.

   [6 marks]

   Total for question 4: 12 marks

End of questions