Practice 2

Question Paper

87 marks

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Question 1
The triangle above has vertices A(2, 5), B (6, 2), and C (8, 7). Use the diagram to
answer the following questions.

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1. Calculate the midpoint of the side (AB) [2]

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2. Find the slope of side (AB). [2]

3. Determine the equation of the perpendicular bisector of side (AB) in slope- [4]
intercept form (y = mx + c).

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4. If the perpendicular bisector intersects the line (y = 2x - 1), find the point of [4]
intersection.

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5. Explain what the intersection of the perpendicular bisector with the line y = [3]
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2x − 1 represents geometrically.
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Question 2
The diagram below shows a helicopter hovering at point H, 380 m vertically above a
lake. Point A is the point on the surface of the lake, directly below the helicopter.

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Minta is swimming at a constant speed in the direction of point Minta observes the
helicopter from point C as she looks upward at an angle of 25°. After 15 minutes,
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Minta is at point B and she observes the same helicopter at an angle of 40°.
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1. Write down the size of the angle of depression from H to C. [1]

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2. Find the distance from A to C. [2]

3. Find the distance from B to C. [3]

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4. Find Minta's speed, in metres per hour. [1]

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Question 3
A solar energy company is planning to install solar panels on the roof of a community
center. The roofline can be modeled by the equation y =
2

3
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x − 5 . To maximize
sunlight exposure, the solar panels must be installed along a line that is
perpendicular to the roofline.

1. Find the equation of the line along which the solar panels should be [2]
installed, passing through the point (3, 1) and perpendicular to the roofline.

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2. Write the equation of the solar panel line in normal form. [2]
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Question 4

The point A has coordinates (4 , −8) and the point B has coordinates (−2 , 4).

The point D has coordinates (−3 , 1).

1. [2]
Write down the coordinates of C, the midpoint of line segment AB.

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2. [2]
Find the gradient of the line DC.
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3. [2]
Find the equation of the line DC. Write your answer in the form ax + by
+ d = 0 where a , b and d are integers.

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Question 5
You are standing on a bridge 30 feet above a river, watching a log floating towards
you. The angle with the horizontal to the front of the log is 16.7°, and the angle with
the horizontal to the back of the log is 14°.

1. Calculate the distance from the bridge to the front of the log based on the [2]
given angle.

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2. Determine the distance from the bridge to the back of the log using the angle [2]
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provided.
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3. Calculate the length of the log by finding the difference between the two [2]
distances.
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Question 6
A rescue helicopter is hovering at an altitude of 1500 meters and needs to lower a
rescue line to a person on a mountainside. The angle of depression from the
helicopter to the person is 42∘.

1. Calculate the length of the rescue line needed to reach the person on the [4]
mountainside.

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2. If the helicopter moves horizontally by 500 meters to improve its position, [4]
calculate the new angle of depression to the person.

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Question 7
A lifeguard tower is built on a right-angled triangular base for stability. One of the
angles of the base measures 30°, and the hypotenuse of the triangular base is 12
cm.

1. Calculate the length of the side opposite the 30° angle to understand the [2]
width of the base.

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2. Calculate the length of the remaining side using the Pythagorean theorem. [3]
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This will provide the full dimensions of the tower’s base.

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Question 8
A newly developed residential area includes four gyms, positioned at specific
locations on a map with a Voronoi diagram overlay. The coordinates of the gyms are
given as:

Gym A(−2, −7), Gym B(1, 4), Gym C(−4, −1), Gym D(5, 0)

Emma's house is located at (−1, 1), and each unit on the map represents 1
kilometer. Answer the following questions based on this information.

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1. Identify the closest gyms to Emma’s house. Calculate the distances from [3]
Emma’s house at (−1,1) to each gym location and determine the closest
gyms.

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2. Calculate the exact distance from Emma’s house to the two closest gyms.
Use your results from part (a) to find the distances to the two nearest gyms,
[4]
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expressing the answers in exact form.
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3. Determine the minimum distance from Emma’s house to the line equidistant [5]
between Gyms A and C. Find the equation of the line equidistant from A and
C (the perpendicular bisector), then calculate the minimum distance from
Emma’s house to this line.

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Question 9
In Lucy's music academy, eight students took their piano diploma examination and
achieved scores out of 150. For her records, Lucy decided to record the average
number of hours per week each student reported practising in the weeks prior to
their examination. These results are summarized in the table below.

Average weekly practice time (h) 28 13 45 33 17 29 39 36

Diploma score (D) 115 82 120 116 79 101 110 121

1. Find Pearson's product-moment correlation coefficient, r, for these data. [2]

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2. The relationship between the variables can be modelled by the regression [1]
equation D = ah + b. Write down the value of a and the value of b.
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3. One of these eight students was disappointed with her result and wished she [2]
had practised more. Based on the given data, determine how her score
could have been expected to alter had she practised an extra five hours per
week.

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Question 10
A rectangular plot of land has a sloping boundary line modeled by the equation y =

2
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x + 3 .

1. Calculate the slope of the boundary line and interpret its meaning. [2]

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2. If the length of the land along the x-axis is 100 meters, calculate the height [2]
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difference between the start and end of the boundary line.
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3. Determine the equation of a line perpendicular to the boundary that [3]
intersects it at the point (50, 28).

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4. If a second boundary needs to run parallel to the first and 20 meters higher, [3]
find its equation in intercept form. .c
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Question 11
A landscaper is designing a triangular garden △XYZ with side lengths x=9cm and
y=13cm, and an angle Z=120° between them.

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1. Use the cosine rule to calculate the length of side z, giving the landscaper a [4]
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complete set of measurements for the garden.

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2. Calculate the area of the triangular garden, which will help in estimating the [3]
amount of soil needed.

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3. Discuss how the obtuse angle (120°) affects the shape of the garden and [2]
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why this might influence the types of plants that could be grown in the
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garden.
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