Oblique asymptotes of rational functionsAQA A-Level Further Maths: Subtopic test

1. 1
   The curve $C$ has equation $y=\dfrac{x^2+x-6}{x-1}$.

   (a)

   Which line is the vertical asymptote of $C$?

   [1 mark]

   • A$x=1$
   • B$x=-1$
   • C$x=2$
   • D$x=-3$

   (b)

   Which line is the oblique asymptote of $C$?

   [1 mark]

   • A$y=x-2$
   • B$y=x+2$
   • C$y=x$
   • D$y=x+6$

   (c)

   Show that $C$ lies below its oblique asymptote when $x>1$.

   [2 marks]

   Total for question 1: 4 marks

2. 2
   The curve $C$ has equation $y=\dfrac{2x^2-3x+1}{x+2}$.

   (a)

   Which line is the oblique asymptote of $C$?

   [1 mark]

   • A$y=2x+1$
   • B$y=2x-3$
   • C$y=2x-7$
   • D$y=2x+7$

   (b)

   Which statement about $C$ and its oblique asymptote is correct?

   [1 mark]

   • A$C$ crosses the asymptote at $x=-2$
   • B$C$ is above the asymptote for all $x$
   • C$C$ is below the asymptote for $x>-2$
   • D$C$ is above the asymptote for $x>-2$ and below it for $x<-2$

   (c)

   Show that $C$ does not meet its oblique asymptote.

   [2 marks]

   Total for question 2: 4 marks

3. 3
   The curve $C$ has equation $y=\dfrac{2x^2+3}{x-1}$.

   (a)

   Find the equations of all the asymptotes of $C$.

   [3 marks]

   (b)

   The line $y=2x$ meets $C$. Find the coordinates of the point of intersection, and explain why there is only one.

   [4 marks]

   Total for question 3: 7 marks

4. 4
   The curve $C$ has equation $y=\dfrac{x^2-x-6}{x+1}$. The curve $D$ has equation $y=\dfrac{x^2+ax+b}{x-1}$, where $a$ and $b$ are constants.

   (a)

   (i) Find the equations of the vertical and oblique asymptotes of $C$.
   (ii) Find the coordinates of the points where $C$ meets the coordinate axes.

   [6 marks]

   (b)

   $D$ has oblique asymptote $y=x+3$ and passes through the point $(0,-2)$.
   (i) Find the values of $a$ and $b$.
   (ii) Determine whether $D$ lies above or below its oblique asymptote when $x>1$.

   [6 marks]

   Total for question 4: 12 marks

End of questions