Partial fractionsEdexcel A-Level Maths: Subtopic test

1. 1
   $\frac{5x+1}{(x-1)(x+2)}\equiv\frac{A}{x-1}+\frac{B}{x+2}$, where $A$ and $B$ are constants.

   (a)

   Find the value of $A$.

   [1 mark]

   • A$2$
   • B$3$
   • C$6$
   • D$-2$

   (b)

   Find the value of $B$.

   [1 mark]

   • A$2$
   • B$3$
   • C$-3$
   • D$9$

   (c)

   Hence find $\int\frac{5x+1}{(x-1)(x+2)}\,dx$.

   [2 marks]

   Total for question 1: 4 marks

2. 2
   $f(x)=\frac{7x+4}{(x+1)(2x-1)}$, $x\neq-1$, $x\neq\frac12$. It is given that $f(x)\equiv\frac{A}{x+1}+\frac{B}{2x-1}$.

   (a)

   Find the value of $B$.

   [1 mark]

   • A$\frac{15}{2}$
   • B$\frac52$
   • C$1$
   • D$5$

   (b)

   Find $f'(0)$.

   [1 mark]

   • A$-6$
   • B$-11$
   • C$9$
   • D$11$

   (c)

   Find the coefficient of $x^2$ in the series expansion of $f(x)$ in ascending powers of $x$, valid for $|x|<\frac12$.

   [2 marks]

   Total for question 2: 4 marks

3. 3
   $g(x)=\frac{3x^2+7x+5}{(x+1)(x+2)^2}$, $x>-1$.

   (a)

   Find the values of the constants $A$, $B$ and $C$ such that $g(x)\equiv\frac{A}{x+1}+\frac{B}{x+2}+\frac{C}{(x+2)^2}$.

   [3 marks]

   (b)

   Hence find the exact value of $\int_0^1g(x)\,dx$.

   [4 marks]

   Total for question 3: 7 marks

4. 4
   $f(x)=\frac{5-x}{(1+x)(2-x)}$, $|x|<1$.

   (a)

   (i) Find the constants $A$ and $B$ such that $f(x)\equiv\frac{A}{1+x}+\frac{B}{2-x}$.
   (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)

   (i) Find the gradient of the curve $y=f(x)$ at the point where $x=0$.
   (ii) Find the exact area of the region bounded by the curve, the coordinate axes and the line $x=1$.

   [6 marks]

   Total for question 4: 12 marks

End of questions