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Date November 2021 Marks available 2 Reference code 21N.2.AHL.TZ0.2
Level Additional Higher Level Paper Paper 2 Time zone Time zone 0
Command term Calculate Question number 2 Adapted from N/A

Question

A wind turbine is designed so that the rotation of the blades generates electricity. The turbine is built on horizontal ground and is made up of a vertical tower and three blades.

The point A is on the base of the tower directly below point B at the top of the tower. The height of the tower, AB, is 90m. The blades of the turbine are centred at B and are each of length 40m. This is shown in the following diagram.

The end of one of the blades of the turbine is represented by point C on the diagram. Let h be the height of C above the ground, measured in metres, where h varies as the blade rotates.

Find the

The blades of the turbine complete 12 rotations per minute under normal conditions, moving at a constant rate.

The height, h, of point C can be modelled by the following function. Time, t, is measured from the instant when the blade [BC] first passes [AB] and is measured in seconds.

ht=90-40cos72t°, t0

maximum value of h.

[1]
a.i.

minimum value of h.

[1]
a.ii.

Find the time, in seconds, it takes for the blade [BC] to make one complete rotation under these conditions.

[1]
b.i.

Calculate the angle, in degrees, that the blade [BC] turns through in one second.

[2]
b.ii.

Write down the amplitude of the function.

[1]
c.i.

Find the period of the function.

[1]
c.ii.

Sketch the function h(t) for 0t5, clearly labelling the coordinates of the maximum and minimum points.

[3]
d.

Find the height of C above the ground when t=2.

[2]
e.i.

Find the time, in seconds, that point C is above a height of 100 m, during each complete rotation.

[3]
e.ii.

The wind speed increases and the blades rotate faster, but still at a constant rate.

Given that point C is now higher than 110 m for 1 second during each complete rotation, find the time for one complete rotation.

[5]
f.

Markscheme

maximum h=130 metres             A1

 

[1 mark]

a.i.

minimum h=50 metres             A1

 

[1 mark]

a.ii.

60÷12=  5 seconds             A1

 

[1 mark]

b.i.

360÷5            (M1)


Note: Award (M1) for 360 divided by their time for one revolution.

=72°             A1

 

[2 marks]

b.ii.

(amplitude =)  40         A1

 

[1 mark]

c.i.

(period =36072=5         A1

 

[1 mark]

c.ii.

Maximum point labelled with correct coordinates.         A1

At least one minimum point labelled. Coordinates seen for any minimum points must be correct.         A1

Correct shape with an attempt at symmetry and “concave up" evident as it approaches the minimum points. Graph must be drawn in the given domain.         A1

 

[3 marks]

d.

h=90-40cos144°           (M1)

h= 122m  122.3606           A1

 

[2 marks]

e.i.

evidence of h=100 on graph  OR  100=90-40cos72t           (M1)

t coordinates 3.55 (3.54892...)  OR  1.45 (1.45107...) or equivalent           (A1)


Note: Award A1 for either t-coordinate seen.


=2.10 seconds  2.09784           A1

 

[3 marks]

e.ii.

METHOD 1

90-40cosat°=110           (M1)

cosat°=-0.5

at°=120, 240           (A1)

1=240a-120a           (M1)

a=120           (A1)

period =360120=3 seconds           A1

 

METHOD 2

attempt at diagram           (M1)

cosα=2040 (or recognizing special triangle)           (M1)

angle made by C2α=120°           (A1)

one third of a revolution in 1 second           (M1)

hence one revolution =3 seconds           A1

 

METHOD 3

considering ht=110 on original function           (M1)

t=53  or  103           (A1)

103-53=53           (A1)


Note: Accept t=1.67 or equivalent.


so period is 35 of original period           (R1)

so new period is 3 seconds           A1

 

[5 marks]

f.

Examiners report

This was perhaps the question with the best responses on the paper. Many candidates got close to full marks on this problem. The issues associated with the question were sometimes due to a lack of understanding of the definitions of amplitude and period. A good number of candidates solved both parts of part (e) suggesting that they had a good understanding of the concept of a function and how it can be applied to mathematical models. Part (f) was also well done by a surprisingly large number of candidates using a variety of approaches. This is evidence that candidates had good problem-solving skills.

a.i.

This was perhaps the question with the best responses on the paper. Many candidates got close to full marks on this problem. The issues associated with the question were sometimes due to a lack of understanding of the definitions of amplitude and period. A good number of candidates solved both parts of part (e) suggesting that they had a good understanding of the concept of a function and how it can be applied to mathematical models. Part (f) was also well done by a surprisingly large number of candidates using a variety of approaches. This is evidence that candidates had good problem-solving skills.

a.ii.

This was perhaps the question with the best responses on the paper. Many candidates got close to full marks on this problem. The issues associated with the question were sometimes due to a lack of understanding of the definitions of amplitude and period. A good number of candidates solved both parts of part (e) suggesting that they had a good understanding of the concept of a function and how it can be applied to mathematical models. Part (f) was also well done by a surprisingly large number of candidates using a variety of approaches. This is evidence that candidates had good problem-solving skills.

b.i.

This was perhaps the question with the best responses on the paper. Many candidates got close to full marks on this problem. The issues associated with the question were sometimes due to a lack of understanding of the definitions of amplitude and period. A good number of candidates solved both parts of part (e) suggesting that they had a good understanding of the concept of a function and how it can be applied to mathematical models. Part (f) was also well done by a surprisingly large number of candidates using a variety of approaches. This is evidence that candidates had good problem-solving skills.

b.ii.

This was perhaps the question with the best responses on the paper. Many candidates got close to full marks on this problem. The issues associated with the question were sometimes due to a lack of understanding of the definitions of amplitude and period. A good number of candidates solved both parts of part (e) suggesting that they had a good understanding of the concept of a function and how it can be applied to mathematical models. Part (f) was also well done by a surprisingly large number of candidates using a variety of approaches. This is evidence that candidates had good problem-solving skills.

c.i.

This was perhaps the question with the best responses on the paper. Many candidates got close to full marks on this problem. The issues associated with the question were sometimes due to a lack of understanding of the definitions of amplitude and period. A good number of candidates solved both parts of part (e) suggesting that they had a good understanding of the concept of a function and how it can be applied to mathematical models. Part (f) was also well done by a surprisingly large number of candidates using a variety of approaches. This is evidence that candidates had good problem-solving skills.

c.ii.

This was perhaps the question with the best responses on the paper. Many candidates got close to full marks on this problem. The issues associated with the question were sometimes due to a lack of understanding of the definitions of amplitude and period. A good number of candidates solved both parts of part (e) suggesting that they had a good understanding of the concept of a function and how it can be applied to mathematical models. Part (f) was also well done by a surprisingly large number of candidates using a variety of approaches. This is evidence that candidates had good problem-solving skills.

d.

This was perhaps the question with the best responses on the paper. Many candidates got close to full marks on this problem. The issues associated with the question were sometimes due to a lack of understanding of the definitions of amplitude and period. A good number of candidates solved both parts of part (e) suggesting that they had a good understanding of the concept of a function and how it can be applied to mathematical models. Part (f) was also well done by a surprisingly large number of candidates using a variety of approaches. This is evidence that candidates had good problem-solving skills.

e.i.

This was perhaps the question with the best responses on the paper. Many candidates got close to full marks on this problem. The issues associated with the question were sometimes due to a lack of understanding of the definitions of amplitude and period. A good number of candidates solved both parts of part (e) suggesting that they had a good understanding of the concept of a function and how it can be applied to mathematical models. Part (f) was also well done by a surprisingly large number of candidates using a variety of approaches. This is evidence that candidates had good problem-solving skills.

e.ii.

This was perhaps the question with the best responses on the paper. Many candidates got close to full marks on this problem. The issues associated with the question were sometimes due to a lack of understanding of the definitions of amplitude and period. A good number of candidates solved both parts of part (e) suggesting that they had a good understanding of the concept of a function and how it can be applied to mathematical models. Part (f) was also well done by a surprisingly large number of candidates using a variety of approaches. This is evidence that candidates had good problem-solving skills.

f.

Syllabus sections

Topic 2—Functions » SL 2.3—Graphing
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