1. 1
   The height, in metres, of a ball thrown into the air is modelled by $h = -5t^2 + 20t$, where $t$ is the time in seconds after it is thrown.

   (a)

   What is the height of the ball when $t = 1$?

   [1 mark]

   • A5 m
   • B15 m
   • C20 m
   • D25 m

   (b)

   At what value of $t$ (other than $t = 0$) does the ball land ($h = 0$)?

   [1 mark]

   • A$t = 2$
   • B$t = 5$
   • C$t = 4$
   • D$t = 20$

   (c)

   At what value of $t$ does the ball reach its maximum height?

   [1 mark]

   • A$t = 1$
   • B$t = 2$
   • C$t = 4$
   • D$t = 10$

   Total for question 1: 3 marks

2. 2
   The graphs of $y = \frac{12}{x}$ and $y = x - 4$ intersect at two points.

   (a)

   Show that the $x$-coordinates of the points of intersection satisfy the equation $x^2 - 4x - 12 = 0$.

   [2 marks]

   (b)

   Hence find the coordinates of the two points of intersection.

   [2 marks]

   Total for question 2: 4 marks

3. 3
   Curve C has equation $y = x^3 - 4x$ and line L has equation $y = 3x$.

   (a)

   Find all the $x$-coordinates of the points of intersection of C and L by solving $x^3 - 4x = 3x$.

   [3 marks]

   (b)

   Find the $y$-coordinate of the intersection point where $x = \sqrt{7}$, giving your answer as a surd in its simplest form.

   [3 marks]

   Total for question 3: 6 marks

4. 4
   The number of bacteria, $N$ (in thousands), in a sample after $t$ hours is modelled by $N = t^3 - 6t^2 + 9t + 40$ for $0 \leq t \leq 6$.

   (a)

   Calculate $N$ when $t = 4$, and state what this value represents in the context of the model.

   [4 marks]

   (b)

   Determine whether the bacteria population is increasing or decreasing between $t = 4$ and $t = 5$, by calculating $N$ at $t = 5$ and comparing it with your answer to part (a).

   [4 marks]

   (c)

   Find the value(s) of $t$ in the given domain, other than $t = 0$, for which $N = 40$, by solving the equation $t^3 - 6t^2 + 9t + 40 = 40$.

   [5 marks]

   Total for question 4: 13 marks

End of questions