Modelling with Trigonometric Functions (A Level only) (OCR A Level Maths: Pure)

Exam Questions

4 hours32 questions
1a
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3 marks

The length of a spring,l cm , at time t seconds, is modelled by the function

l equals 8 plus 2 space sin space t comma t greater or equal than 0.

Write down

(i)
the natural length of the spring,
(ii)
the maximum length of the spring,
(iii)
the minimum length of the spring.
1b
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2 marks
(i)
Find the length of the spring after 5 seconds.
(ii)
Find the time at which the length of the spring first reaches 9.5 cm.
1c
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1 mark

Give one criticism of this model for large values of t.

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2a
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1 mark

A dolphin is swimming such that it is diving in and out of the sea at a constant speed.

The height, h cm, of the dolphin, relative to sea level open parentheses h equals 0 close parentheses, at time t sconds, is to be modelled using the formula  h equals A space sin open parentheses B t close parentheses  where A are B constants.

On each jump and dive the dolphin reaches a height of 70 cm above sea level and a depth of 70 cm below sea level.

Write down the value of a.

2b
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2 marks

Starting at sea level, the dolphin takes straight pi seconds to jump out of the water, dive back under and return to sea level.
Given that 0 less or equal than B less or equal than 2, determine the value of B.

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3
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5 marks

The path of a swing boat fairground ride that swings forwards and backwards is modelled as the arc of a circle, radius 8 space m, as shown in the diagram below.

q3-5-9-modelling-with-trignometric-functions-a-level-only-edexcel-a-level-pure-maths-easy

Ground level is represented by the x-axis.

The value of x represents the horizontal displacement, in metres, of the swing boat relative to the origin.

The value of y represents the height, in metres, of the swing boat above ground level.

The height of the swing boat is modelled using

y equals 12 minus square root of 100 minus x squared end root comma space space space space space space space space space space space minus 8 less or equal than space x less or equal than 8

(i)
Find the height of the boat when it’s horizontal displacement is 6 m.
(ii)
Find the horizontal distance from the origin when the boat is 5 spacem above the ground, giving your answer to three significant figures.
(iii)
Find the maximum height the swing boat reaches.
(iv)
When at its maximum height, find the angle of elevation of the swing boat from the origin.

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4a
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1 mark

The height,h m, of water in a reservoir is modelled by the function

h left parenthesis t right parenthesis equals 6 plus A space sin left parenthesis t right parenthesis comma space t greater or equal than 0 comma

where t is the time in hours after midday.

A is a positive constant.

Write down the height of the water in the reservoir at midday.

4b
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2 marks

The minimum height the water is 3 m.

(i)
Write down the value of A.
(ii)
Find the maximum height of the water.
4c
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3 marks

Find the height of the water at

(i)
2pm,
(ii)
midnight,

giving your answers to two decimal places.

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5a
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3 marks

A Ferris wheel is modelled as a circle with centre open parentheses 0 comma 0 close parentheses and radius 100 m.

There are 32 passenger “pods” which are evenly spaced around the Ferris wheel.

A pod’s position can be determined by the angle,straight theta  radians, which is measured anticlockwise from the positive x-direction, as shown in the diagram below.

q5-5-9-modelling-with-trignometric-functions-a-level-only-edexcel-a-level-pure-maths-easy

The coordinates of a pod,open parentheses x comma y close parentheses , are given by  open parentheses 100 space cos space straight theta comma 100 space sin space straight theta close parentheses.

(i)
Find the angle, in radians, between each pod.

(ii)
Find the coordinates, to one decimal place, of the first pod located anticlockwise above the positive x-axis.
5b
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3 marks
(i)
Write down the angle straight theta for the passenger pod located at the point open parentheses negative 100 comma 0 close parentheses.
(ii)
Determine the angle straight theta for the pod located at the point open parentheses 50 comma 50 square root of 3 close parentheses.

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6a
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3 marks

As part of a quality control test, a lifejacket is thrown into the sea.

The height, h m, of the lifejacket above or below sea level open parentheses h equals 0 close parentheses, at time t seconds after first hitting the water, is modelled by the equation h equals negative e to the power of negative 0.6 t end exponentsin 2 t

(i)
Find the height of the lifejacket after 1.5 seconds.
Is the lifejacket above or below sea level at this point in time?

(ii)
Excluding the case when t equals 0 spacefind the value of t the first time the lifejacket is at sea level.
6b
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2 marks

The lifejacket reaches its furthest point below sea level after 0.64 seconds.
Find the distance below sea level this is, giving your answer to three significant figures.

6c
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1 mark

According to the model, what should happen to the lifejacket as time progresses?

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7a
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3 marks

The number of daylight hours, h, is modelled using the function

h equals 12 plus 5 spacesinopen parentheses d minus 1 close parentheses to the power of degree comma space space space space space space space space space space space space space space space space space space space space space space space space space space space space d greater or equal than 1

where d is the day number on which the model applies.

(i)
Write down the number of daylight hours on day 1.
(ii)
Work out the number of daylight hours on day 136.
7b
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5 marks
(i)
Find the days on which there are 9.5 daylight hours.
(ii)
Hence find the number of days in a year for which there are less than 9.5 daylight hours.
7c
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1 mark

Explain why the model does not quite cover a whole year before repeating itself.

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8a
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3 marks

The alternating voltage,V, in an electrical circuit, t seconds after it is switched on, is modelled by the function

V equals 20 spacecos πt plus 20 square root of 3 space end rootsin πt

where t is measured in seconds.

Use the identity R spacecosopen parentheses πt minus straight alpha close parentheses equals R spacecos alpha spacecos πt plus straight R spacesin alpha spacesin πt to show that

20 spacecos πt plus 20 square root of 3sin πt

can be written as

40 space cos left parenthesis pi t minus straight pi over 3 right parenthesis.

8b
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3 marks
(i)
Write down the maximum voltage in the electrical circuit.
(ii)
Find the voltage at time t equals 0.
(iii)
Find the voltage after two seconds.
8c
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3 marks

After how many seconds does the voltage first equal -20 volts?

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1a
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2 marks

A small spring is extended to its maximum length and released from rest.

The length of the spring, l cm , at time t seconds, where the angle is given in radians, is then modelled by the function

l equals 5 plus 3 spacecos 2 t comma space space space space space space space space t greater or equal than 0

(i)
Write down the natural length of the spring.

(ii)
Write down the maximum extension of the spring.
1b
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3 marks
(i)
Find the length of the spring after 6 seconds.

(ii)
Find the time at which the length of the spring first reaches 4 cm.
1c
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1 mark

State one criticism of this model as time passes.

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2
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3 marks

A dolphin is swimming such that it is diving in and out of the water at a constant speed. On each jump and dive the dolphin reaches a height of 2 m above sea level and a depth of 2 m below sea level.
Starting at sea level, the dolphin takes fraction numerator 2 straight pi over denominator 3 end fraction seconds to jump out of the water, dive back in and return to sea level.  

Write down a model for the height, h m, of the dolphin, relative to sea level, at time t seconds, in the form h equals A spacesinopen parentheses B t close parentheses  where A are B constants to be found.

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3a
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4 marks

The path of a swing boat fairground ride that swings forwards and backwards is modelled as a semi-circle, radius 8 cm, as shown in the diagram below.

q3-5-9-modelling-with-trignometric-functions-a-level-only-edexcel-a-level-pure-maths-medium

Ground level is represented by the x-axis and  represents the height of the boat above ground level.  The path of the boat is given by the formula

y equals 10 minus square root of 64 minus x squared end root space space space space space space space space space space space space minus 8 less or equal than space x less or equal than 8

The boat’s initial position is at the point open parentheses 0 comma 2 close parentheses.

(i)
Find the height of the boat when it is 2 spacem horizontally from its initial position.

(ii)
When the boat is at a height of 6 spacem, find its exact horizontal distance from the origin.
3b
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3 marks

Given that the x-coordinate of the boat is also given by

x equals 8 sin open parentheses straight pi over 6 t close parentheses

where t seconds is the time since the boat was released from its initial position, find the time it takes the boat to swing from one end of the ride to the other.

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4a
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3 marks

The height,h m , of water in a reservoir is modelled by the function

h open parentheses t close parentheses equals A plus B space sin open parentheses straight pi over 6 t close parentheses comma space space t greater or equal than 0

where t is the time in hours after midnight. A and B  are positive constants.

In terms of A and B, write down the natural height of the water in the reservoir, as well as its maximum and minimum heights.

4b
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3 marks

The maximum level of water is 3m higher than its natural level.

The level of water is three times higher at its maximum than at its minimum.

Find the maximum, minimum and natural water levels.

4c
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3 marks
(i)
How many times per day does the water reach its maximum level?

(ii)
Find the times of day when the water level is at its minimum?

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5a
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4 marks

A Ferris wheel with p passenger “pods” is modelled as a circle with centre open parentheses 0 comma 0 close parentheses and radius 50 m.  A pod’s position can be determined by the angle,straight theta radians, which is measured anticlockwise from the positive x-direction, as shown in the diagram below.

q5a-5-9-modelling-with-trignometric-functions-a-level-only-edexcel-a-level-pure-maths-medium

The coordinates of a pod, open parentheses x italic comma y close parentheses, are given by open parentheses A space cos open parentheses straight theta close parentheses comma A space sin open parentheses straight theta close parentheses close parentheses , where A is a positive constant. Ground level is represented by the line with equation y=-60.

(i)
Write down the value of the constant A.
(ii)
The angle between each pod is  begin inline style straight pi over 12 end style radians. Find the value of p.
(iii)
Find the maximum height above the ground of a passenger pod during one complete rotation of the Ferris wheel.
5b
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2 marks

Find, to three significant figures, the angle straight theta for a passenger pod located at the point open parentheses 30 comma 40 close parentheses.

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6a
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2 marks

A lifejacket falls over the side of a boat from a height of 3 m.
The height, h spacem, of the lifejacket above or below sea level open parentheses h equals 0 close parentheses, at time t seconds after falling, is modelled by the equation h equals 3 e to the power of negative 0.7 t end exponentcos 4 t .

The lifejacket reaches its furthest point below sea level after 0.742 seconds.
Find the total distance it has fallen, giving your answer to three significant figures.

6b
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2 marks

Write down the value of t for the first three times the lifejacket is at sea level.

6c
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3 marks
(i)
Find the value of  3 e to the power of negative 0.7 t end exponent  when  t equals 6.2.

(ii)
Hence justify why, from 6.2 seconds on, the lifejacket will always be within 4 centimetres of sea level.

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7a
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5 marks

The number of daylight hours, h, in the UK, during a day d days after the spring equinox (the day in spring when the number of daylight hours is 12), is modelled using the function

h equals 12 plus 9 over 2 sin open parentheses fraction numerator 2 straight pi over denominator 365 end fraction d close parentheses

(i)
Find the number of daylight hours during the day that is 100 days after the spring equinox.

(ii)
Find the number of days after the spring equinox that the two days occur during which the number of daylight hours is closest to 9.
7b
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3 marks

For how many days of the year does the model suggest that the number of daylight hours exceeds 15 hours? Give your answer as a whole number of days.

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8a
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3 marks

The alternating voltage,V , in an electrical circuit, t seconds after it is switched on, is modelled by the function

V equals 55 square root of 3 sin πt over 30 plus 55 space cos space πt over 30

Show that

55 square root of 3 sin πt over 30 plus 55 space cos space πt over 30

can be written as

R spacesinopen parentheses πt over 30 plus alpha close parentheses

where R equals 110 and alpha equals begin inline style straight pi over 6 end style.

8b
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4 marks
(i)
Find the voltage at time t equals 0.

(ii)
Find the voltage after one minute.
8c
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3 marks

After how many seconds does the voltage first equal -55 volts?

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1a
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2 marks

The length of a spring,l cm , at time t seconds, after being released from rest, is modelled by the function

l equals a plus b spacecos 4 t comma space space space space space space space t greater or equal than 0

Describe what the constants a spaceand b spacerepresent in terms of the length of the spring.

1b
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2 marks

Given that the minimum length the spring can attain is 12 spacecm and its maximum length is 30 cm, find the values of a and b.

1c
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3 marks
(i)
A similar spring, with the same values of a and b, has length modelled by

l equals a plus b spacecos 2 t comma space space space space space space space space space t greater or equal than 0
Compare the motion of the two springs.

(ii)
Suggest one way in which the model (for both springs) could be improved.

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2
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3 marks

A hovering helicopter moves up and down at a constant rate between the heights of 200 m and 220 m.  It takes the helicopter begin inline style straight pi over 5 end style seconds to move between these two heights.
Write down a model in the form h equals A plus B spacecosopen parentheses C t close parentheses for the height, h spacem, of the helicopter at time t seconds, where A comma B are C constants to be found.
State the initial height of the helicopter suggested by your model.

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3a
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2 marks

The path of a swing boat fairground ride that swings forwards and backwards is modelled as a semi-circle, radius 10 spacem, as shown in the diagram below.

q3a-5-9-modelling-with-trignometric-functions-a-level-only-edexcel-a-level-pure-maths-hard

At time t seconds, the x-coordinate of the boat is modelled by the function

x open parentheses t close parentheses equals 10 spacesinopen parentheses straight pi over 5 t close parentheses comma space space space space space space space space space space t greater or equal than 0

and the height,h m , of the boat above the ground, at time t seconds, is modelled by

h open parentheses t close parentheses equals 12 minus 10 open vertical bar cos space open parentheses straight pi over 5 t close parentheses close vertical bar comma space space space space space space t greater or equal than 0.

Verify that the initial position of the boat is open parentheses 0 comma 2 close parentheses.

3b
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3 marks
(i)
Write down the coordinates of the boat when it is at its maximum height.

(ii)
Find the time it takes the boat to swing between these two points.
3c
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2 marks

Find the position of the boat when it has swung through an angle of straight pi over 6 anticlockwise from the y-axis, as shown in the diagram above. 
Find the time at which the boat first reaches this position.

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4a
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5 marks

The height, h spacem, of water in a reservoir is modelled by the function

h open parentheses t close parentheses equals A plus Bsinopen parentheses C t close parentheses comma space space space space space space space space space space t greater or equal than 0

where t is the time, in hours, after midnight. A comma B and C  are positive constants.

(i)
Given that the water level rises and falls through one and a half cycles in a 24 hour period, find the value of C.

(ii)
The height of water reaches its minimum of 1 spacem just once per day.
Find the time of day when this occurs.

(iii)
The maximum height of water is 11 spacem. Find the values of A spaceand B.
4b
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3 marks

The reservoir is only capable of holding water to a maximum height of 10 spacem
Should the water level exceed this, an overflow reservoir is available.

During which times of day will the overflow reservoir be in use?
Give your answers to the nearest minute.

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5a
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3 marks

A Ferris wheel with 30 passenger “pods” is modelled as a circle with centre open parentheses 0 comma 0 close parentheses and radius 60 spacem.  A pod’s position can be determined by the angle straight theta radians, which is measured anticlockwise from the positive x-direction, as shown in the diagram below.

q5a-5-9-modelling-with-trignometric-functions-a-level-only-edexcel-a-level-pure-maths-hard

The coordinates of a pod,open parentheses x comma y close parentheses  are given by  open parentheses A space cos open parentheses straight theta close parentheses comma B space sin open parentheses straight theta close parentheses close parentheses  where A spaceand B are positive constants. Ground level is represented by the line with equation y=-62.

(i)
Write down the values of A spaceand B.

(ii)
The pods are evenly distributed around the wheel.
Find the angle between each pod.
5b
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3 marks

Find the height above the ground of a passenger pod when straight theta equals fraction numerator 7 straight pi over denominator 6 end fraction radians.

5c
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2 marks

Find the angle straight theta, to three significant figures, for a passenger pod located at the point open parentheses 48 comma negative 36 close parentheses.

5d
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1 mark

What would you be able to say about the Ferris wheel in the case where A not equal to B?

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6a
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2 marks

A lifejacket falls over the side of a boat from a height of 4 spacem above sea level.
The height,h spacem , of the lifejacket above or below sea level open parentheses h equals 0 close parentheses  at time t seconds after falling, is modelled by the equation h equals A e to the power of negative k t space end exponentcos 2 t, where A are k positive constants.

(i)
Write down the value of A.

(ii)
Briefly explain how the constant k spaceaffects the height of the lifejacket over time.
6b
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4 marks

After 2.054 seconds the lifejacket is 1 m below sea level. Find the value of k and determine whether the lifejacket is rising or sinking.

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7a
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2 marks

The number of daylight hours, h, in the UK, d days after the spring equinox (the day in spring when the number of daylight hours is 12) is modelled using the function

h equals A plus B spacesin open parentheses fraction numerator 2 straight pi over denominator 365 end fraction d close parentheses

where A are B constants.

(i)
Write down the value of A.

(ii)
Given that the maximum number of daylight hours is 16.5, write down the value of B.
7b
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2 marks

For how many days of the year does the number of daylight hours remain below 10? Give your answer as a whole number of days.

7c
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3 marks

If the spring equinox falls on the 21st March, find the dates throughout the year when there are 16 hours of daylight.

7d
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1 mark

The model needs to be adjusted every four years. Suggest a reason why.

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8a
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3 marks

The alternating voltage,V , in an electrical circuit t seconds after it is switched on is modelled by the function

V equals 55 square root of 2 open parentheses sin space πt over 60 plus cos space πt over 60 close parentheses

Express

55 square root of 2 open parentheses sin space πt over 60 plus cos space πt over 60 close parentheses

in the form

R spacesin open parentheses πt over 60 plus alpha close parentheses

where R are alpha constants to be found.R greater than 0  and alpha is  acute.

8b
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2 marks

Find the voltage when the circuit is switched on.

8c
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2 marks
(i)
Write down the maximum voltage and the time at which this first occurs.

(ii)
Find the time it takes the voltage to complete one period (cycle).

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1a
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2 marks

The length of a spring, l spacecm, at time t seconds, after being released from rest, is modelled by the function

l equals a plus b spacecosspace c t comma space space space space space space space space space t greater or equal than 0

where a comma b spaceand c are constants.

(i)
Describe the effect the constant c spacehas on the model.

(ii)
Explain how you know the spring is stretched to its maximum length before  being released.
1b
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3 marks
(i)
It takesspace straight pi over 10 seconds from release until the spring first returns to its starting  length .Find the value of c.

(ii)
Given that the maximum length of the spring is twice its minimum length, find a relationship between a spaceand b.
1c
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1 mark

Explain why the function would not be appropriate for modelling the length of a spring if b greater or equal than a.

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2a
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4 marks

The height above ground,h spacem , of a drone used as part of an air display is modelled by the function h equals A plus B spacesinopen parentheses C t plus D close parentheses , where tis the time in seconds after launch A comma B comma C and D are constants.

The drone is launched upwards from a height of 23 m and straight pi over 6 seconds later it reaches its maximum height of 26 m. The minimum height the drone reaches is 14 m.
Find the value of the constants A,B,C spaceand D given that D is acute.

2b
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3 marks

The drone’s lights switch off when its height drops below 17 m.
Show that the drone’s lights are on for two-thirds of its flight.

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3a
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3 marks

A swing boat fairground ride is modelled as moving forwards and backwards along the path of a semi-circle, radius 18 spacem, as shown in the diagram below.

q3a-5-9-modelling-with-trignometric-functions-a-level-only-edexcel-a-level-pure-maths-veryhard

Show that, for 0 less or equal than theta less or equal than straight pi over 2,

(i)
the x-coordinate of the boat is given by  x equals 18 spacesin theta,

(ii)
the y-coordinate is given by  y equals 20 minus 18 spacecos theta.
3b
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3 marks

The model is refined so that the coordinates of the boat can be calculated from the time, t seconds, after the boat is set in motion.  The x spaceand y coordinates are now given by

x equals 18 spacesin B t space space space space space space space space space space space space space space space space space space space space space space y equals 20 minus 18 open vertical bar cos space B t close vertical bar

where B is a constant.

(i)
Briefly explain why the modulus of cos straight theta is required for the y- coordinate.

(ii)
Given that the time between the boat reaching its maximum height at either end of the ride is 8 seconds, find the value of B.
3c
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3 marks

For 0 less or equal than t less or equal than 4, find the times when the boat is equidistant from the ground and horizontally from the origin.

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4a
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2 marks

The height of water,h m , in a reservoir is modelled by the function

h open parentheses t close parentheses equals A plus B spacesin C t comma space space space space space space space space space space space space t greater or equal than 0

where t is the time in hours after midnight. A comma B spaceand C are positive constants.

Briefly explain how each of the constants A comma B spaceand C spaceaffect the height of the water in the reservoir.

4b
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2 marks

Show that the height of water will first be at its minimum level at time

t equals fraction numerator 3 straight pi over denominator 2 C end fraction

hours after midnight.

4c
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3 marks

Show that the rate of change of the height of water in the reservoir is at its greatest every

fraction numerator k straight pi over denominator C end fraction comma space space space space space space space space space space k element of straight integer numbers subscript 0 superscript plus

hours after midnight.

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5a
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2 marks

Engineers are designing a Ferris wheel with  passenger “pods”.
The wheel is modelled as a circle with centre open parentheses 0 comma a close parentheses and radius r meters.

One of the pods is to be located at the point with coordinates open parentheses 42 comma 136 close parentheses.

q5a-5-9-modelling-with-trignometric-functions-a-level-only-edexcel-a-level-pure-maths-veryhard

The thick lines on the diagram represent two symmetrical ground supports for the Ferris wheel each going from its centre to ground level.

The left-hand support is represented by the equation  4 x minus 3 y plus 240 equals 0.
The x-axis represents ground level.

(i)
Find the equation of the circle.

(ii)
How far from the ground is the lowest point of the Ferris wheel?
5b
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3 marks

The p pods are to be evenly distributed around the wheel.
Ideally the engineers would like no more than three pods to be within the intersection of the supports at any one time. Find the maximum value of p spacethis design approach allows.

5c
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2 marks

For both strength and aesthetic reasons, both the ground supports will be made in two sections. Thinner materials will be used within the wheel so as not to obstruct the view of, and from, the Ferris wheel and thicker material will be used for the lower base supports outside the wheel.

Find the percentage of the thicker material required.

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6a
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2 marks

The height, h spacem, of a helicopter, t seconds after take-off, is modelled by the function

h equals 12 plus 2 tanopen parentheses 1 half t minus straight pi over 2 close parentheses space space space space space space space space space space space 0 less than t less or equal than 6

The time lag between the pilot firing up the helicopter and leaving the ground is accounted for in the model by negative values of h for the period 0 less than t less or equal than 6.

Find the value of α to two significant figures.

6b
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2 marks

Show that the helicopter rises just 4 m between the times of straight pi over 2seconds and  fraction numerator 3 straight pi over denominator 2 end fraction seconds

6c
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2 marks

Find the height of the helicopter at the point at which the model ceases to be valid.

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7a
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2 marks

The number of daylight hours,h , in the UK, d days after the spring equinox (the day in spring when the number of daylight hours is 12) is modelled using the function

h equals 12 plus B space sin open parentheses fraction numerator 2 straight pi over denominator C end fraction d close parentheses

where B spaceand C are constants.

Explain the meaning of the constants B and C in the context of this model.

7b
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3 marks

During a normal year (not a leap year), the maximum number of daylight hours is
16 hours and 38 minutes.

Find the total number of daylight hours in the first half of the year.
(Assume a year in this sense starts on the spring equinox, when d equals 0.
Give your answer to the nearest 10 hours.

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8a
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2 marks

The alternating voltage,V , in a domestic electrical circuit, t seconds after it is switched on is modelled by the function

V equals 115 sin omega t plus 115 square root of 3 space end rootcos omega t

Express

115 spacesin omega t plus 115 square root of 3 spacecos omega t

in the form

R spacesin open parentheses omega t plus alpha close parentheses

where R spaceand alpha are constants to be found.  R greater than 0 and is α acute.

8b
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4 marks

In the UK, domestic electricity runs at a frequency,f , of 50 Hertz (Hz).

The constant omega, is given by omega equals 2 pi f.

(i)
Find the initial voltage when a domestic appliance (such as a kettle or TV) is switched on.

(ii)
Find the time at which the voltage first turns negative.
8c
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2 marks
(i)
Find the period of one cycle of voltage in the UK.

(ii)
In the US, the period of one cycle is  1 over 60seconds
Write down the frequency of US domestic electricity.

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