1. 1
   The function $f$ is defined by $f(x) = \dfrac{2x + 3}{x^{2} - x - 6}$, for $x \in \mathbb{R}$, $x \ne -2$, $x \ne 3$.

   (a)

   Write down the equations of the vertical asymptotes of the graph of $y = f(x)$.

   [1 mark]

   • A$x = -3$ and $x = 2$
   • B$x = 3$ and $x = -2$
   • C$x = -\frac{3}{2}$
   • D$y = 3$ and $y = -2$

   (b)

   Write down the equation of the horizontal asymptote of the graph of $y = f(x)$.

   [1 mark]

   • A$y = 0$
   • B$y = 2$
   • C$y = -\frac{1}{2}$
   • D$y = 2x$

   (c)

   Find the coordinates of the points where the graph of $y = f(x)$ meets the axes.

   [2 marks]

   Total for question 1: 4 marks

2. 2
   The function $g$ is defined by $g(x) = \dfrac{x^{2} + 3x - 1}{x - 2}$, for $x \in \mathbb{R}$, $x \ne 2$.

   (a)

   Find the equation of the oblique asymptote of the graph of $y = g(x)$.

   [1 mark]

   • A$y = x + 3$
   • B$y = x + 1$
   • C$y = x + 5$
   • D$y = x - 5$

   (b)

   Find the coordinates of the $y$-intercept of the graph of $y = g(x)$.

   [1 mark]

   • A$\left(0, -\frac{1}{2}\right)$
   • B$(0, -1)$
   • C$\left(0, -\frac{9}{2}\right)$
   • D$\left(0, \frac{1}{2}\right)$

   (c)

   Find the exact $x$-coordinates of the points where the graph of $y = g(x)$ meets the $x$-axis.

   [2 marks]

   Total for question 2: 4 marks

3. 3
   The function $h$ is defined by $h(x) = \dfrac{ax + b}{x^{2} + cx - 8}$, where $a, b, c \in \mathbb{R}$. The graph of $y = h(x)$ has vertical asymptotes $x = 2$ and $x = -4$, an $x$-intercept at $(-2, 0)$ and a $y$-intercept at $\left(0, -\frac{1}{2}\right)$.

   (a)

   Find the values of $a$, $b$ and $c$.

   [3 marks]

   (b)

   Show that the equation $h(x) = k$ has at least one real solution for every $k \in \mathbb{R}$. State what this tells you about the range of $h$.

   [4 marks]

   Total for question 3: 7 marks

4. 4
   A courier company models the mean cost per parcel, $C$ dollars, when a van delivers $x$ hundred parcels in a day by $C(x) = \dfrac{x^{2} + 4x + 68}{x + 2}$, for $x > 0$.

   (a)

   (i) Show that $C(x) = x + 2 + \dfrac{64}{x + 2}$.
   (ii) Write down the equations of the asymptotes of the graph of $y = \dfrac{x^{2} + 4x + 68}{x + 2}$, $x \in \mathbb{R}$, $x \ne -2$.
   (iii) Explain why this graph does not meet the $x$-axis.
   (iv) Interpret the oblique asymptote in the context of the model.

   [6 marks]

   (b)

   Find the number of parcels that minimises the mean cost per parcel, and the minimum mean cost. The company regards a day as efficient if the mean cost per parcel is at most 20 dollars. Find the range of the number of parcels delivered on an efficient day.

   [6 marks]

   Total for question 4: 12 marks

End of questions