[Maximum number: 3]

Gabriel is investigating the shape of model airplane wings. A cross-section of one of the wings is shown, graphed on the coordinate axes.

The shaded part of the cross-section is the area between the x-axis and the curve with equation

y=2 \sqrt{x}-\frac{x}{5}+1, \text { for } 0 \leq x \leq 100

where x is the distance, in cm , from the front of the wing and y is the height, in cm , above the horizontal axis through the wing, as shown in the diagram.

(a)

Find the values of a, b and c, shown in the table.

Gabriel uses the trapezoidal rule with four intervals to estimate the shaded area of the cross-section of the wing.

[ 3 ]

[Maximum number: 12]

The mean annual temperatures for Earth, recorded at fifty-year intervals, are shown in the table.

Tami creates a linear model for this data by finding the equation of the straight line passing through the points with coordinates (1708,8.73) and (1958,9.45).

(a)

Calculate the gradient of the straight line that passes through these two points.

[ 2 ]

(b)

(i)

Interpret the meaning of the gradient in the context of the question.

[ 2 ]

(ii)

State appropriate units for the gradient.

[ 2 ]

(c)

Find the equation of this line giving your answer in the form y=m x+c.

[ 2 ]

(d)

Use Tami's model to estimate the mean annual temperature in the year 2000.

Thandizo uses linear regression to obtain a model for the data.

[ 2 ]

(e)

State two reasons why Thandizo's prediction may not be valid.

[ 2 ]

[Maximum number: 7]

In a given week, the number of students in a particular primary school that were absent due to headlice (H), influenza (I) and/or chickenpox (C) were recorded as follows.
The primary school has 500 students.
35 students had headlice only
20 students had influenza only
5 students had chickenpox only
4 students had headlice and influenza but not chickenpox
2 students had headlice and chickenpox but not influenza
3 students had influenza and chickenpox but not headlice
1 student had headlice, influenza and chickenpox

(a)

Use Diego's model to calculate the number of students who started the school year with influenza.

It is known that 130 students have had influenza during the first 10 days of the school year.

[ 2 ]

(b)

Find the value of k.

[ 2 ]

(c)

Using this model, calculate how many days it will take for 200 students to have had influenza since the start of the school year.

By the last day of the school year, it is known that 300 students have had influenza.

[ 2 ]

(d)

Comment on the appropriateness of Diego's model.

[ 1 ]

[Maximum number: 12]

Boris recorded the number of daylight hours on the first day of each month in a northern hemisphere town.
This data was plotted onto a scatter diagram. The points were then joined by a smooth curve, with minimum point (0,8) and maximum point (6,16) as shown in the following diagram.

Let the curve in the diagram be y=f(t), where t is the time, measured in months, since Boris first recorded these values.

Boris thinks that f(t) might be modelled by a quadratic function.

(a)

Write down one reason why a quadratic function would not be a good model for the number of hours of daylight per day, across a number of years.

Paula thinks that a better model is $f(t)=a \cos (b t)+d, t \geq 0$ , for specific values of a, b and d.

[ 1 ]

(b)

For Paula's model, use the diagram to write down

[ 4 ]

(i)

the amplitude.

(ii)

the period.

(iii)

the equation of the principal axis.

[ 4 ]

(c)

Hence or otherwise find the equation of this model in the form:

f(t)=a \cos (b t)+d

[ 3 ]

(d)

For the first year of the model, find the length of time when there are more than 10 hours and 30 minutes of daylight per day.

The true maximum number of daylight hours was 16 hours and 14 minutes.

[ 4 ]

[Maximum number: 5]

The height of a baseball after it is hit by a bat is modelled by the function

h(t)=-4.8 t^{2}+21 t+1.2

where h(t) is the height in metres above the ground and t is the time in seconds after the ball was hit.

(a)

Write down the height of the ball above the ground at the instant it is hit by the bat.

[ 1 ]

(b)

Find the value of t when the ball hits the ground.

[ 2 ]

(c)

State an appropriate domain for t in this model.

[ 2 ]

[Maximum number: 6]

The number of words that a child understands is modelled by the equation

y=-167+6264 \log _{10} x, \text { for } x \in \mathbb{R}^{+}

where x is the number of years since the child was born, and y is the number of words the child understands.

(a)

Use the model to predict the number of words that a child understands when x=10.

To be fluent in a language, a child must understand at least 5000 words.

[ 2 ]

(b)

Use the model to predict how long it takes for a child to become fluent. Give your answer correct to four significant figures.

[ 2 ]

(c)

Determine if the model is realistic for a child on their first birthday. Give a reason to support your decision.

[ 2 ]

[Maximum number: 5]

DeVaughn throws a javelin in a school track and field competition.

The height, h, of the front tip of the javelin above the ground, in metres, is modelled by the following quadratic function,

h(t)=-3.6 t^{2}+10.8 t+1.8, t \geq 0

where t is the time in seconds after the javelin is thrown.

(a)

Write down the height of the front tip of the javelin at the time it is thrown.

[ 1 ]

(b)

Find the value of t when the front tip of the javelin reaches its maximum height.

[ 2 ]

(c)

Find the value of t when the front tip of the javelin strikes the ground.

[ 2 ]

[Maximum number: 9]

The water level, h, in metres, in a water tank after t hours of irrigation is modelled by the following function.

h(t)=\frac{20}{2 t+5}, t \geq 0

(a)

Find the value of h(0.5).

[ 2 ]

(b)

(i)

Find the value of $h^{-1}(2.5)$ .

[ 3 ]

(ii)

Interpret this value in context.

[ 3 ]

(c)

Write down the range of $h^{-1}$ .

[ 1 ]

[Maximum number: 8]

TurboEats is a food delivery company that charges customers $ 0.80 per kilometre plus a fixed amount of $ 1.50. Their delivery charge, T in $, is represented by the function

T(x)=0.80 x+1.50, \text { for } x \geq 0,

where x is the distance in kilometres from where the food is picked up to where it is delivered.
Lorena's home is 5.1 km from a pizza café.

(a)

Find the charge for TurboEats to deliver a pizza from the café to her home.

[ 2 ]

(b)

A second food delivery company, GogoDart, charges customers according to the function G(x)=a x+b, for $x \geq 0$ . Their charges for the first 6 km are shown in the following graph of y=G(x). The line passes through integer grid points.

[ 4 ]

(i)

Write down the value of a.

[ 2 ]

(ii)

Write down the value of b.

[ 2 ]

(c)

Explain why TurboEats is always cheaper than GogoDart.

[ 2 ]

[Maximum number: 1]

Consider the graph of the following function:

f(x)=\frac{16}{x}+\frac{x^{2}}{8}, \text { for } x \neq 0

(a)

Write down the equation of the vertical asymptote of f(x).

[ 1 ]