1. 1
   A company's monthly profit, $P$ (in thousands of dollars), from selling $x$ hundred units is modelled by $P = -x^2 + 8x - 12$, for $0 \leq x \leq 8$.

   (a)

   Find the value of $P$ when $x = 3$.

   [1 mark]

   • A$3$
   • B$5$
   • C$-3$
   • D$9$

   (b)

   The company breaks even (that is, $P = 0$) at two values of $x$. What is the sum of these two values?

   [1 mark]

   • A$2$
   • B$6$
   • C$8$
   • D$12$

   (c)

   What is the equation of the vertical line of symmetry of the graph of $P = -x^2 + 8x - 12$?

   [1 mark]

   • A$x = -4$
   • B$x = 3$
   • C$x = 8$
   • D$x = 4$

   Total for question 1: 3 marks

2. 2
   A stone is thrown from a cliff above the sea. Its height above sea level, $h$ metres, after $t$ seconds is given by $h = -5t^2 + 20t + 25$.

   (a)

   Find the height of the stone, $h$, when $t = 2$.

   [2 marks]

   (b)

   Find the time, $t$, at which the stone hits the sea (that is, when $h = 0$).

   [2 marks]

   Total for question 2: 4 marks

3. 3
   A farmer has 40 m of fencing to enclose a rectangular plot against an existing straight wall, so only three sides need fencing. If the width perpendicular to the wall is $x$ metres, the enclosed area $A$ m$^2$ is given by $A = x(40 - 2x)$.

   (a)

   Show that $A = -2x^2 + 40x$, and find the value of $A$ when $x = 12$.

   [3 marks]

   (b)

   By completing the square, find the value of $x$ that gives the maximum area, and state the maximum area.

   [3 marks]

   Total for question 3: 6 marks

4. 4
   A bridge cable forms a parabolic arch. Measuring horizontal distance $x$ metres from the left support, the height of the cable above the road, $y$ metres, is modelled by $y = 0.05x^2 - 2x + 25$, for $0 \leq x \leq 40$.

   (a)

   Calculate the height of the cable above the road at the left support ($x = 0$) and at the right support ($x = 40$).

   [4 marks]

   (b)

   By completing the square, find the minimum height of the cable and the horizontal distance at which it occurs.

   [4 marks]

   (c)

   Find the two values of $x$ at which the cable is exactly $12.5$ m above the road, giving your answers correct to 2 decimal places.

   [5 marks]

   Total for question 4: 13 marks

End of questions