1. 1
   A shop's stock of a viral toy decays according to $N = 800(0.5)^t$, where $N$ is the number remaining and $t$ is the number of weeks since launch.

   (a)

   Find $N$ when $t = 2$.

   [1 mark]

   • A$200$
   • B$400$
   • C$100$
   • D$600$

   (b)

   What is the value of $N$ at $t = 0$, and what does this represent?

   [1 mark]

   • A$0$, the final stock
   • B$800$, the initial stock
   • C$400$, the average stock
   • D$1$, a scaling constant

   (c)

   As $t$ increases without bound, what is the horizontal asymptote of the graph of $N = 800(0.5)^t$?

   [1 mark]

   • AThere is no asymptote
   • B$N = 800$
   • C$N = 0.5$
   • D$N = 0$

   Total for question 1: 3 marks

2. 2
   An electrical circuit has current $y$ amps and resistance $x$ ohms related by $y = \frac{12}{x}$, for $x > 0$.

   (a)

   Find the value of $y$ when $x = 2$, and describe what happens to $y$ as $x$ approaches $0$ from above.

   [2 marks]

   (b)

   Explain why the graph of $y = \frac{12}{x}$ has no value of $y$ when $x = 0$, and state the vertical asymptote.

   [2 marks]

   Total for question 2: 4 marks

3. 3
   A cyclist's distance travelled, $y$ metres, on a hilly track after $x$ seconds is modelled by $y = 2x^3 - 9x^2 + 12x$, for $0 \leq x \leq 4$.

   (a)

   Find the value of $y$ when $x = 1$ and when $x = 3$, and use these to estimate the gradient of the curve between these two points.

   [3 marks]

   (b)

   By drawing a tangent at $x = 2$ (using values of $y$ close to $x=2$), estimate the gradient of the curve at $x = 2$ and interpret this value in context.

   [3 marks]

   Total for question 3: 6 marks

4. 4
   A factory machine depreciates in value according to $V = 20000(0.8)^t$, where $V$ is its value in dollars and $t$ is the number of years since purchase.

   (a)

   Find the value of $V$ when $t = 0$ and when $t = 4$, and state what $t=0$ represents in context.

   [4 marks]

   (b)

   Find the number of complete years after which the machine's value first falls below $\$6000$.

   [4 marks]

   (c)

   The factory also owns a second, older machine whose value is modelled by $W = 9000(0.9)^t$. Find the value of $t$ (to the nearest whole year) at which the two machines have approximately equal value, by evaluating both models at successive integer values of $t$ and comparing.

   [5 marks]

   Total for question 4: 13 marks

End of questions