Level 1

[Maximum mark: 6]

The number of daily visitors to the famous 'Bondi Beach' in Sydney, Australia, and the daily temperature on those days, were recorded for eight days in February. The table below shows this data and has been ordered in ascending order by temperature.

The number of visitors to Bondi Beach varies linearly with the temperature.

1. Find

   1. Pearson's product-moment correlation coefficient, $r$ ;

   2. the equation of the regression line $y$ on $x$. [4]
      

2. Use the equation of the regression line $y$ on $x$ to estimate the number of visitors to Bondi Beach during a day the temperature is $26^\circ$C. [2]
   

[Maximum mark: 6]

A lego storage box is left with $4$ blue and $7$ red bricks. Max takes a brick from the box at random and attaches it to his new castle. He then chooses another brick from the same box at random.

1. Complete the following tree diagram. [3]
   

   

2. Calculate the probability that both bricks chosen by Max are of the same color. [3]
   

[Maximum mark: 6]

A population of $50$ hamsters was introduced to a new town. One month later, the number of hamsters was $62$. The number of hamsters, $P$, can be modelled by the function

where $t$ is the time, in months, since the hamsters were introduced to the town.

1. Find the value of $b$. [2]
   

2. Calculate the number of hamsters in the town after $6$ months. [2]
   

A wildlife specialist estimates that the town has enough drink and food to support a maximum population of $2000$ hamsters.

3. Calculate the number of months it takes for the hamster population to reach this maximum. [2]
   

[Maximum mark: 6]

On the first day of September, $2019$, Gloria planted $5$ flowers in her garden. The number of flowers, which she plants at every day of the month, forms an arithmetic sequence. The number of flowers she is going to plant in the last day of September is $63$.

1. Find the common difference of the sequence. [2]
   

2. Find the total number of flowers Gloria is going to plant during September.[2]
   

3. Gloria estimated she would plant $1000$ flowers in the month of September. Calculate the percentage error in Gloria's estimate. [2]
   

[Maximum mark: 6]

The Voronoi diagram below shows the location of fire stations in outback Queensland, a North-Eastern state of Australia.

Horizontal scale: $1$ unit represents $100$ km.

Vertical scale: $1$ unit represents $100$ km.

1. Identify the nearest fire-station to houses located at

   1. $(300,300)$

   2. $(-200,100)$ [2]
      

A house located at $(200,200)$ has an internal fire and requires assistance.

2. Determine which fire-stations the house is closest to. [2]
   

3. Determine how far the house is from these two fire-stations. Give your answer to the nearest kilometre. [2]
   

[Maximum mark: 6]

The following diagram shows an archery target which is divided into three regions A, B and C.

A contest consists of an archer shooting one arrow at the target. The probability of hitting each region is given in the following table.

1. Find the probability that the arrow does not hit the target. [2]
   

An archer scores points on the contest as shown in the following table.

2. Given that the contest is fair, find the value of $k$. [4]
   

[Maximum mark: 6]

The water depth, $D$, in metres, in a harbour on a particular day can be modelled by the equation

where $t$ is the elapsed time, in hours, since midnight.

1. Draw the graph of $D$ versus $t$ on the grid below. [2]
   

   

2. Find the lowest and highest depths of water in the harbour, and the times when they occur. [2]
   

3. A large yacht has a draught of $3$ metres. Decide whether the yacht will be able to enter the harbour at $5$ pm. [2]
   

[Maximum mark: 5]

The daily cost of production of a company producing lithium batteries is modelled by the cubic function

where $x$ is the number of lithium batteries produced and $C$ the cost in USD.

1. Write down the daily cost to the company if no lithium batteries are produced.[1]
   

The marginal cost of production is the cost of producing one additional unit. This can be approximated by the gradient of the cost function.

2. Find an expression for the marginal cost, $C'(x)$, of producing $x$ lithium batteries. [2]
   

3. Find the marginal cost of producing

   1. $50$ lithium batteries;

   2. $100$ lithium batteries. [2]
      

[Maximum mark: 5]

A military helicopter flies from an airbase at A to a battlefield at B and then to a hospital at H. The routes taken by the helicopter are given by the vectors

Distances are measured in kilometres.

1. Determine the vector $\vv{\text{AH}}$. [2]
   

After evacuating wounded soldiers from the battlefield to the front-line hospital, the helicopter returns back to the airbase.

2. Write down the vector that describes this return flight. [1]
   

3. Find the direct distance from the hospital to the airbase. [2]
   

[Maximum mark: 6]

The following diagram shows the parabolic shape of a gateway arch that has a span of $12$ metres and a maximum height of $8$ metres.

The curve has an equation in the form $y = k(x - 6)^2 + 8$.

1. Determine the value of $k$. [2]
   

2. Write down an integral that represents the cross sectional area under the arch shown as OMN. [2]
   

3. Find the cross sectional area under the arch. [2]