Rational function graphsAQA A-Level Further Maths: Subtopic test

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

   (a)

   Which line is the vertical asymptote of $C$?

   [1 mark]

   • A$x=-3$
   • B$x=3$
   • C$x=-\frac12$
   • D$x=2$

   (b)

   Which line is the horizontal asymptote of $C$?

   [1 mark]

   • A$y=3$
   • B$y=\frac12$
   • C$y=0$
   • D$y=2$

   (c)

   Find the coordinates of the points where $C$ meets the coordinate axes.

   [2 marks]

   Total for question 1: 4 marks

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

   (a)

   Which line is the horizontal asymptote of $C$?

   [1 mark]

   • A$y=3$
   • B$y=2$
   • C$y=\frac13$
   • D$y=0$

   (b)

   Find the $x$-coordinates of the points where $C$ meets the line $y=x$.

   [1 mark]

   • A$1$ only
   • B$0$ and $-1$
   • C$0$ and $1$
   • D$0$ and $-5$

   (c)

   Determine whether $C$ lies above or below its horizontal asymptote when $x$ is large and positive.

   [2 marks]

   Total for question 2: 4 marks

3. 3
   The curve $C$ has equation $y=\dfrac{x+3}{x-2}$ and the line $L$ has equation $y=5x-9$.

   (a)

   Show that $C$ and $L$ meet where $x=1$ and where $x=3$.

   [3 marks]

   (b)

   Hence solve the inequality $\dfrac{x+3}{x-2}>5x-9$.

   [4 marks]

   Total for question 3: 7 marks

4. 4
   The curve $C$ has equation $y=\dfrac{x^2}{x^2+2x-3}$.

   (a)

   (i) Write down the equations of the three asymptotes of $C$.
   (ii) Find the coordinates of the point where $C$ meets the coordinate axes.
   (iii) State whether $C$ approaches its horizontal asymptote from above or below as $x\to+\infty$.

   [6 marks]

   (b)

   By forming a quadratic equation in $x$ and using its discriminant, show that $y\le0$ or $y\ge\frac34$ for all points on $C$, and hence find the coordinates of the stationary points of $C$.

   [6 marks]

   Total for question 4: 12 marks

End of questions