Rational expressionsEdexcel A-Level Maths: Subtopic test

1. 1
   The expression $\frac{x^2-9}{x^2+x-6}$ is defined for all real $x$ except where the denominator is zero.

   (a)

   Which expression is equal to $\frac{x^2-9}{x^2+x-6}$ for values of $x$ where it is defined?

   [1 mark]

   • A$\frac{x-3}{x+2}$
   • B$\frac{x+3}{x-2}$
   • C$\frac{-9}{x-6}$
   • D$\frac{x-3}{x-2}$

   (b)

   For which values of $x$ is the original expression $\frac{x^2-9}{x^2+x-6}$ undefined?

   [1 mark]

   • A$x=-3$ and $x=2$
   • B$x=2$ only
   • C$x=3$ and $x=-3$
   • D$x=-3$ only

   (c)

   Hence write $\frac{x^2-9}{x^2+x-6}$ in the form $A+\frac{B}{x-2}$, where $A$ and $B$ are constants.

   [2 marks]

   Total for question 1: 4 marks

2. 2
   The expression $\frac{x^3+8}{x^2-4}$ is considered for values of $x$ where it is defined.

   (a)

   Which expression is equal to $\frac{x^3+8}{x^2-4}$?

   [1 mark]

   • A$\frac{x+2}{x-2}$
   • B$\frac{x^2+4}{x-2}$
   • C$\frac{x^2-2x+4}{x-2}$
   • D$\frac{x^2+2x+4}{x-2}$

   (b)

   What is the value of $\frac{x^3+8}{x^2-4}$ when $x=3$?

   [1 mark]

   • A$5$
   • B$7$
   • C$35$
   • D$\frac{35}{13}$

   (c)

   Hence write $\frac{x^3+8}{x^2-4}$ in the form $x+\frac{k}{x-2}$, where $k$ is a constant.

   [2 marks]

   Total for question 2: 4 marks

3. 3
   The function $f(x)=\frac{3x^2+x-2}{2x^2-x-3}$ is defined for all real $x$ except where its denominator is zero.

   (a)

   Simplify $f(x)$.

   [3 marks]

   (b)

   Hence solve $f(x)=2$, and write $f(x)$ in the form $p+\frac{q}{2x-3}$, where $p$ and $q$ are constants.

   [4 marks]

   Total for question 3: 7 marks

4. 4
   Let $A=\frac{x^2-5x+6}{x^2-4}$ and $B=\frac{x^2+5x+6}{x^2+4x+3}$.

   (a)

   (i) Simplify $A$.
   (ii) Simplify $B$.
   (iii) Hence show that $A\times B=\frac{x-3}{x+1}$.

   [6 marks]

   (b)

   Let $E=A\times B=\frac{x-3}{x+1}$.
   (i) Write $E$ in the form $1+\frac{k}{x+1}$.
   (ii) Solve $E=\frac12$.
   (iii) Explain why $E$ can never equal $1$.

   [6 marks]

   Total for question 4: 12 marks

End of questions