[Maximum number: 9]

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)

Use your answers to parts (b) and (c) to find

[ 3 ]

(i)

the maximum value of R(t);

[ 2 ]

(ii)

the minimum value of R(t).

[ 1 ]

(b)

Find the value of c.

Let S(t) be the length of time, in hours, from the start of the Martian day until sunset at point A on day t . S(t) can be modelled by the function

S(t)=1.5 \sin (0.00939 t+2.83)+18.65

The length of time between sunrise and sunset at point A, L(t), can be modelled by the function

L(t)=1.5 \sin (0.00939 t+2.83)-1.6 \sin (0.00939 t)+d

[ 2 ]

(c)

Find the value of d.

Let $f(t)=1.5 \sin (0.00939 t+2.83)-1.6 \sin (0.00939 t)$ and hence L(t)=f(t)+d.
f(t) can be written in the form $\operatorname{Im}\left(z_{1}-z_{2}\right)$ , where $z_{1}$ and $z_{2}$ are complex functions of t.

[ 2 ]

(d)

(i)

Find, in hours, the shortest time from sunrise to sunset at point A that is predicted by this model.

[ 2 ]

[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: 10]

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 ]

[Maximum number: 11]

In this question you will use a historic method of calculating the cost of a barrel of wine to determine which shape of barrel gives the best value for money.
In Austria in the 17th century, one method for measuring the volume of a barrel of wine, and hence determining its cost, was by inserting a straight stick into a hole in the side, as shown in the following diagram, and measuring the length SD . The longer the length, the greater the cost to the customer.

Let SD be d metres and the cost be C gulden (the local currency at the time). When the length of SD was 0.5 metres, the cost was 0.80 gulden.

(a)

Given that C was directly proportional to d, find an equation for C in terms of d.

A particular barrel of wine cost 0.96 gulden.

[ 3 ]

(b)

Show that d=0.6.

[ 1 ]

(c)

Find the equation of the quadratic curve, ASB .

[ 6 ]

(d)

State one assumption, not already given, that has been made in using these models to find the shape of the barrel that gives the best value.

[ 1 ]

[Maximum number: 7]

A player throws a basketball. The height of the basketball is modelled by

h(t)=-4.75 t^{2}+8.75 t+1.5, t \geq 0

where h is the height of the basketball above the ground, in metres, and t is the time, in seconds, after it was thrown.

(a)

Find how long it takes for the basketball to reach its maximum height.

[ 2 ]

(b)

Assuming that no player catches the basketball, find how long it would take for the basketball to hit the ground.

Another player catches the basketball when it is at a height of 1.2 metres.

[ 2 ]

(c)

Find the value of t when this player catches the basketball.

[ 2 ]

(d)

Write down one limitation of using h(t) to model the height of the basketball.

[ 1 ]

[Maximum number: 5]

This question considers the optimal route between two points, separated by several regions where different speeds are possible.
Huw lives in a house, H , and he attends a school, S , where H and S are marked on the following diagram. The school is situated 1.2 km south and 4 km east of Huw's house. There is a boundary [MN], going from west to east, 0.4 km south of his house. The land north of [MN] is a field over which Huw runs at 15 kilometres per hour $\left(\mathrm{km} \mathrm{h}^{-1}\right)$ . The land south of [MN] is rough ground over which Huw walks at $5 \mathrm{~km} \mathrm{~h}^{-1}$ . The two regions are shown in the following diagram.

diagram not to scale

(a)

Huw realizes that his journey time could be reduced by taking a less direct route. He therefore defines a point P on [MN] that is $x \mathrm{~km}$ east of M. Huw decides to run from H to P and then walk from P to S . Let T(x) represent the time, in hours, taken by Huw to complete the journey along this route.

[ 5 ]

(i)

Show that $T(x)=\frac{\sqrt{0.4^{2}+x^{2}}+3 \sqrt{0.8^{2}+(4-x)^{2}}}{15}$ .

[ 3 ]

(ii)

Sketch the graph of y=T(x).

[ 2 ]

[Maximum number: 6]

Dilara is designing a kite ABCD on a set of coordinate axes in which one unit represents 10 cm .
The coordinates of A, B and C are ( 2,0 ), ( 0,4 ) and ( 4,6 ) respectively. Point D lies on the x-axis. $[\mathrm{AC}]$ is perpendicular to $[\mathrm{BD}]$ . This information is shown in the following diagram.

(a)

Find the gradient of the line through A and C .

[ 2 ]

(b)

Write down the gradient of the line through B and D .

[ 1 ]

(c)

Find the equation of the line through B and D . Give your answer in the form a x+b y+d=0, where a, b and d are integers.

[ 2 ]

(d)

Write down the x-coordinate of point D .

[ 1 ]

[Maximum number: 14]

Mai is at an amusement park. A map of part of the amusement park is represented on the following coordinate axes.
Mai's favourite three attractions are positioned at A(0,16), B(12,20) and C(12,0). All measurements are in metres.

(a)

(i)

Write down the gradient of [AC].

[Maximum number: 5]

The company Fred Express delivers packages. From past experience, the time taken, T, to deliver a package follows a normal distribution with mean 64 hours and standard deviation 12 hours.

(a)

Write down an expression for the amount charged to deliver a package of weight $x \mathrm{~kg}$ , where x>1.

[ 2 ]

(b)

Find the amount Fred Express charges for a 5.3 kg package.

Meiling is charged $7.20 for the delivery of a package.

[ 1 ]

(c)

Find the weight of Meiling's package.

[ 2 ]

[Maximum number: 3]

This question is about comparing the academic performance of two schools.
At age 18, all students in school A and school B take the same final exam. Augustin is studying the results in these schools.
Augustin chooses to take a representative sample of size six from each school. For each student in the sample, he will conduct an interview.

(a)

Give, in context, an interpretation of the gradient 0.37 in the model.

[ 1 ]

(b)

For each student, Augustin uses the value $y-\hat{y}$ rounded to one decimal place to measure the extent to which a school helped to improve a student's results. He calls this "school value added". This is shown in Table 2.

Table 2

[ 2 ]

(i)

Show that the value of q in Table 2 is 0.6.

[ 2 ]

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