1. 1
   The function $f$ is defined by $f(x) = \frac{2x - 6}{x + 1}$, for $x \ne -1$.

   (a)

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

   [1 mark]

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

   (b)

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

   [1 mark]

   • A$y = 2$
   • B$y = -6$
   • C$x = 2$
   • D$y = 0$

   (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 $g(x) = x^4 - 8x^2 + 3$, for $x \in \mathbb{R}$. The point $(2, -13)$ lies on the graph of $y = g(x)$.

   (a)

   Which other point must also lie on the graph of $y = g(x)$?

   [1 mark]

   • A$(2, 13)$
   • B$(-2, -13)$
   • C$(-2, 13)$
   • D$(-13, 2)$

   (b)

   How many zeros does $g$ have?

   [1 mark]

   • A$0$
   • B$2$
   • C$3$
   • D$4$

   (c)

   Use technology to find the coordinates of the local maximum point of the graph of $y = g(x)$, and write down the range of $g$.

   [2 marks]

   Total for question 2: 4 marks

3. 3
   Two students are learning to touch-type. The typing speed, in words per minute, of the first student after $t$ weeks of practice is modelled by $W(t) = \frac{60t + 20}{t + 2}$, for $t \ge 0$. The typing speed of the second student is modelled by $V(t) = 15 + 4t$, for $0 \le t \le 10$.

   (a)

   Write down the value of $W(0)$ and the equation of the horizontal asymptote of the graph of $y = W(t)$. Interpret the asymptote in context.

   [3 marks]

   (b)

   Use technology to find the values of $t$ at which the two students type at the same speed. Hence state, with a reason, which student is faster after 10 weeks.

   [4 marks]

   Total for question 3: 7 marks

4. 4
   The side profile of a toy car track is modelled by $h(x) = 0.02x^3 - 0.6x^2 + 4.5x$, for $0 \le x \le 20$, where $h$ is the height of the track above the floor, in cm, at a horizontal distance of $x$ cm from the start. A straight support rod runs from the start of the track, $(0, 0)$, to the end of the track, where $x = 20$.

   (a)

   (i) Show that $h(x) = 0.02x(x - 15)^2$.
   (ii) Hence write down the zeros of $h$ and describe what happens to the track at $x = 15$.
   (iii) Use technology to find the coordinates of the local maximum point of the graph of $y = h(x)$, and write down the range of $h$.

   [6 marks]

   (b)

   (i) Show that the equation of the support rod is $y = 0.5x$.
   (ii) Use technology to find the coordinates of the other point, apart from the start, where the rod meets the track, for $0 < x < 20$.
   (iii) Find the greatest vertical distance between the track and the rod for $0 \le x \le 10$.

   [6 marks]

   Total for question 4: 12 marks

End of questions