Rational expressions and algebraic divisionAQA A-Level Maths: Subtopic test

1. 1
   Let $R(x)=\dfrac{x^2-9}{x^2+x-12}$.

   (a)

   Which expression is equal to $R(x)$, for values of $x$ where both are defined?

   [1 mark]

   • A$\dfrac{-9}{x-12}$
   • B$\dfrac{x-3}{x-4}$
   • C$\dfrac{x+3}{x+4}$
   • D$\dfrac{x+3}{x-4}$

   (b)

   For which value of $x$ is $R(x)$ undefined, although $\dfrac{x+3}{x+4}$ is defined?

   [1 mark]

   • A$x=-4$
   • B$x=-3$
   • C$x=4$
   • D$x=3$

   (c)

   Solve $R(x)=2$.

   [2 marks]

   Total for question 1: 4 marks

2. 2
   Let $S(x)=\dfrac{2}{x+1}+\dfrac{3}{x-2}$.

   (a)

   Which expression is equal to $S(x)$?

   [1 mark]

   • A$\dfrac{5x-1}{(x+1)(x-2)}$
   • B$\dfrac{5x+7}{(x+1)(x-2)}$
   • C$\dfrac{5}{2x-1}$
   • D$\dfrac{6}{(x+1)(x-2)}$

   (b)

   For which value of $x$ is $S(x)=0$?

   [1 mark]

   • A$x=-\dfrac15$
   • B$x=\dfrac15$
   • C$x=5$
   • D$x=-1$ and $x=2$

   (c)

   Find the exact values of $x$ for which $S(x)=1$.

   [2 marks]

   Total for question 2: 4 marks

3. 3
   Let $E(x)=\dfrac{x^2+2x-15}{x^2-4x+3}\div\dfrac{x+5}{x^2-1}$.

   (a)

   Show that $E(x)$ simplifies to $x+1$.

   [3 marks]

   (b)

   The expression $x+1$ is defined for all $x$, but $E(x)$ is not. State the values of $x$ for which $E(x)$ is not defined, giving a reason for each.

   [4 marks]

   Total for question 3: 7 marks

4. 4
   Let $f(x)=\dfrac{2x^3-5x^2+x+7}{x-2}$ and $g(x)=\dfrac{x^2+3x-4}{x^2-1}-\dfrac{2}{x+1}$.

   (a)

   (i) Use algebraic division to express $f(x)$ in the form $Ax^2+Bx+C+\dfrac{D}{x-2}$.
   (ii) Hence solve $f(x)=2x^2-x+1$.

   [6 marks]

   (b)

   Show that $g(x)$ simplifies to $\dfrac{x+2}{x+1}$, and hence solve $g(x)=3$.

   [6 marks]

   Total for question 4: 12 marks

End of questions