[Maximum number: 4]

A suitable site for the landing of a spacecraft on the planet Mars is identified at a point, A. The shortest time from sunrise to sunset at point A must be found.
Radians should be used throughout this question. All values given in the question should be treated as exact.
Mars completes a full orbit of the Sun in 669 Martian days, which is one Martian year.

On day t, where $t \in \mathbb{Z}$ , the length of time, in hours, from the start of the Martian day until sunrise at point A can be modelled by a function, R(t), where

R(t)=a \sin (b t)+c, t \in \mathbb{R} .

The graph of R is shown for one Martian year.

(a)

(i)

Hence or otherwise find an equation for L in the form $L(t)=p \sin (q t+r)+d$ , where $p, q, r, d \in \mathbb{R}$ .

[ 4 ]

[Maximum number: 13]

George is researching the growth in the number of electric vehicles (EVs) in the European Union in order to investigate some of the difficulties that might arise if the target of banning sales of all petrol and diesel cars in 2035 is to be met.
George begins his research by predicting how many electric vehicles (EVs) will be in the European Union in 2035.
The number of EVs in the European Union, N, measured in millions, is shown in the following table. The time t is measured in years from the beginning of 2016 where $t \in \mathbb{R}$ .

George models this data set using the logistic function $N=\frac{310}{1+C \mathrm{e}^{-k t}}$ , where $C, k \in \mathbb{R}^{+}$ .

(a)

In context, explain the significance of 310 in George's logistic function.

[ 1 ]

(b)

(i)

Use the value of N at t=1 to show that $C=1549 \mathrm{e}^{k}$ .

[ 2 ]

(ii)

Use the value of N at t=7 to find a second expression for C.

[ 3 ]

(c)

Use your answers to part (b) to find a value for

[ 3 ]

(i)

$\quad k$ .

[ 2 ]

(ii)

C.

George uses his model to predict values of N at the end of the years 2017 to 2021. These values are shown in the following table, correct to one decimal place.

[ 1 ]

(d)

Use the model to predict the number of EVs in the European Union at the end of 2035.

One of the main difficulties in meeting the European Union's target is the provision of sufficient public charging points.

The European Union estimates that 80 % of car owners will be able to charge their cars at home and the remaining 20 % will require public charging points. It is planned to have one public charging point for every 10 cars that require them.

[ 2 ]

(e)

Use George's model to find an expression for the total number of public charging points required at time t.

At the end of 2020 there were 0.22 million public charging points in the European Union and at the end of 2022 there were 0.54 million.

[ 2 ]

[Maximum number: 1]

A cat runs inside a circular exercise wheel, making the wheel spin at a constant rate in an anticlockwise direction. The height, $h \mathrm{~cm}$ , of a fixed point, P , on the wheel can be modelled by $h(t)=a \sin (b t)+c$ where t is the time in seconds and $a, b, c \in \mathbb{R}^{+}$ .

When t=0, point P is at a height of 78 cm .

(a)

Write down the new value of b.

[ 1 ]

[Maximum number: 11]

In this question researchers are trying to find the most accurate model to use when modelling a population of wolves.
Historically, a population of wolves in an area had a stable size of 200 . After some years of disruption, the population was reduced to 40 wolves. At this point, the area became a protected space and the population began to grow again.
Researchers in the area wish to model the size of the wolf population, x, as a function of t, where t is the time, in years, since the area became protected.

(a)

Initially, the researchers consider using the logistic model

x=\frac{L}{1+C \mathrm{e}^{-k t}}, \text { where } L, C, k \in \mathbb{R}^{+}

The researchers decide to let L=200.

[ 7 ]

(i)

State the assumption being made in assuming L=200.

At t=0, the population of wolves is 40 .

[ 1 ]

(ii)

Find the value of C.

At t=5, the population of wolves is found to have increased to 70 .

[ 2 ]

(iii)

Find the value of k.

[ 2 ]

(iv)

Use your model to predict the size of the wolf population in the area 10 years after it became protected. Give your answer correct to the nearest whole number.

[ 2 ]

(b)

An alternative model for population growth is called the Gompertz model. When applied by the researchers to the wolf population, this model satisfies the differential equation

\frac{\mathrm{d} x}{\mathrm{~d} t}=a x \ln \left(\frac{200}{x}\right), a \in \mathbb{R}^{+} .

[ 3 ]

(i)

Use the Gompertz model to predict the size of the wolf population at t=10. Give your answer correct to the nearest whole number.

After 10 years, the wolf population is measured and is found to be 85 .

[ 3 ]

(c)

Comment on the predictions made by the two models.

By tracking individual wolves, the researchers find that about 3 % of the wolf population emigrate from the protected area each year.

They decide to adapt the Gompertz model to allow for this. The new model will satisfy the differential equation

\frac{\mathrm{d} x}{\mathrm{~d} t}=0.0855 x \ln \left(\frac{200}{x}\right)-0.03 x

[ 1 ]

[Maximum number: 1]

Ethan and Avery are researching population data for the city of Los Angeles to create a model predicting population values. They collect the following data.

Population data for the city of Los Angeles

Ethan proposes the population can be modelled using quadratic regression to find a function of the form $f(x)=a x^{2}+b x+c$ , where x is the number of years after 1900 .

(a)

State a reason why it may be valid to use Avery's proposal to predict the future population.

[ 1 ]

[Maximum number: 7]

A student investigating the relationship between chemical reactions and temperature finds the Arrhenius equation on the internet.

k=A \mathrm{e}^{-\frac{c}{T}}

This equation links a variable k with the temperature T, where A and c are positive constants and T>0.

(a)

Write down

[ 4 ]

(i)

the gradient of this line in terms of c;

(ii)

the y-intercept of this line in terms of A.

The following data are found for a particular reaction, where T is measured in Kelvin and k is measured in $\mathrm{cm}^{3} \mathrm{~mol}^{-1} \mathrm{~s}^{-1}$ :

[ 4 ]

(b)

Find an estimate of

[ 3 ]

(i)

c;

(ii)

A.

It is not required to state units for these values.

[ 3 ]

[Maximum number: 4]

Walkea Tree is a furniture store. They offer a 30 % discount on all chairs priced over 250 Canadian dollars (CAD).
A different furniture store, LuxeCraft, offer customers a reduction of 150 CAD on all chairs priced over 250 CAD .

(a)

Describe the meaning of $(f \circ g)(x)$ , in context.

[ 2 ]

(b)

State, with justification, whether $(f \circ g)(x)$ or $(g \circ f)(x)$ is better for the customer when they buy a chair with an original price of 450 CAD .

[ 2 ]

[Maximum number: 4]

The function f has domain $-4 \leq x \leq 4$ .
The graph of y=f(x) has endpoints (-4,1) and (4,0), and a local maximum at (-1,3). This is shown in each of the following diagrams.

(a)

On the axes provided, sketch the graph of y=2 f(x)-1.

[ 2 ]

(b)

On the axes provided, sketch the graph of y=f(2 x-2).

[ 2 ]

[Maximum number: 3]

Gloria wants to model the curved edge of a butterfly wing. She inserts a photo of the wing into her graphing software and finds the coordinates of four points on the edge of the wing.

Gloria thinks a cubic curve will be a good model for the butterfly wing.

(a)

Find the equation of the cubic curve that models the life-size wing.

[ 3 ]

[Maximum number: 8]

A satellite orbiting the Earth experiences a gravitational force, $F(\mathrm{~N})$ , which varies inversely as the square of its distance, d (kilometres), from the centre of the Earth.
When d=7000, F=40000.

(a)

(i)

Identify which two of the following graphs would form a straight line.

A

F against d

B

F against $d^{2}$

C

F against $d^{-2}$

D

$\log _{10} F$ against $\log _{10} d$ E. $\quad \log _{10} F$ against d F. $\quad F$ against $\log _{10} d$

[ 4 ]

(ii)

For each of these two graphs, find the value of the gradient of the line formed.

[ 4 ]