1. 1
   A quadratic function is defined by $f(x) = x^2 - 4x - 5$ for all real values of $x$.

   (a)

   What are the roots of $f(x) = 0$?

   [1 mark]

   • A$x = -1$ and $x = 5$
   • B$x = 1$ and $x = -5$
   • C$x = -1$ and $x = -5$
   • D$x = 1$ and $x = 5$

   (b)

   What is the $x$-coordinate of the turning point of the graph of $f(x)$?

   [1 mark]

   • A$x = 1$
   • B$x = 4$
   • C$x = -2$
   • D$x = 2$

   (c)

   What is the $y$-intercept of the graph of $f(x)$?

   [1 mark]

   • A$-4$
   • B$5$
   • C$-5$
   • D$4$

   Total for question 1: 3 marks

2. 2
   A cubic function is given by $g(x) = x^3 - 3x^2 - 4x$ for all real values of $x$.

   (a)

   Show that $g(x)$ can be written in the fully factorised form $x(x-4)(x+1)$.

   [2 marks]

   (b)

   Find the three roots of $g(x) = 0$.

   [2 marks]

   Total for question 2: 4 marks

3. 3
   The value of a rare coin, in pounds, $t$ years after purchase is modelled by the function $V(t) = 200(1.15)^t$ for $t \geq 0$.

   (a)

   Calculate the value of the coin 5 years after purchase, giving your answer to the nearest pound.

   [3 marks]

   (b)

   Explain why $V(t)$ can never be zero or negative for any value of $t$, and describe what happens to $V(t)$ as $t$ increases without bound.

   [3 marks]

   Total for question 3: 6 marks

4. 4
   The height, in metres, of a passenger on a Ferris wheel above the ground $t$ minutes after the ride starts is modelled by $h(t) = 10 + 8\sin(30t)^\circ$, for $0 \leq t \leq 12$.

   (a)

   Calculate the height of the passenger after 3 minutes, showing your method.

   [4 marks]

   (b)

   Find the maximum and minimum heights reached by the passenger during the ride, explaining your reasoning.

   [4 marks]

   (c)

   Show that the passenger reaches a height of 14 metres at some time between $t = 0$ and $t = 3$ minutes, and find this value of $t$.

   [5 marks]

   Total for question 4: 13 marks

End of questions