Modelling involving Numerical Methods (Edexcel A Level Maths: Pure): Exam Questions

4 hours36 questions
1a
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3 marks

A bypass is to be built around a village. On the graph below the road through the village is modelled by the line  y equals x. The bypass is modelled by the equation y equals square root of fraction numerator 40 x over denominator x plus 1 end fraction end root

q1a-10-2-modelling-involving-numerical-methods-easy-a-level-maths-pure-screenshots

The bypass runs from the origin to the point P open parentheses p comma space p close parentheses.

Use the iteration formula 

x subscript n plus 1 end subscript equals square root of fraction numerator space 40 x subscript n space over denominator x subscript n plus 1 end fraction end root

withspace x subscript 0 equals 5 spaceto find the value of p, correct to 3 significant figures.

1b
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2 marks

Use the interval open square brackets 5.835 comma space 5.845 close square brackets and the function

space straight f open parentheses x close parentheses equals x minus space square root of fraction numerator 40 x over denominator x plus 1 end fraction end root

to show that your answer to part (a) is correct to 3 significant figures.

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2a
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2 marks

Scientists are analysing the infection rate, R, at which a virus spreads amongst a population.  They model R using the function

space space R open parentheses t close parentheses equals 1.1 plus cos space t space space space space space space space space space space space space space space space space space space space left parenthesis t greater or equal than 0 right parenthesis

where t is the number of months since the first case of the virus was discovered.

The diagram below shows the graphs ofy equals R left parenthesis t right parenthesis and y equals 1.

q2a-10-2-modelling-involving-numerical-methods-easy-a-level-maths-pure-screenshots

(i) Show that the equation R open parentheses t close parentheses equals 1 can be written in the form  cos space t space plus 0.1 equals 0.

(ii) Find the derivative of  cos space t space plus 0.1.

2b
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3 marks

Apply the Newton-Raphson method with t subscript 0 equals 1.5, along with your answers to part (a), to find the time when R first equals 1.

Give your answer to 3 significant figures.

2c
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3 marks

(i) Write down the minimum R value the model predicts.

(ii) Approximately, how often (in months) does this minimum infection rate occur?

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3
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4 marks

Following an oil spill into the ocean, experts recorded the rate of oil leaking at regular intervals whilst engineers worked to clear up the spillage.

The rate of oil leaking (measured in barrels per week) was recorded every fortnight for eight weeks. The results are shown in the table below.

Time

0

2

4

6

8

Rate of leak

0

14 000

9600

3400

1800

The trapezium rule is used to estimate the total amount of oil leaked during the eight weeks.  All the data in the table is to be used.

(i) Write down the width, h, of a single trapezium.

(ii) Show that the trapezium rule estimates that the total amount of oil spilled in the eight weeks is 56000 barrels, to 2 significant figures.

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4
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4 marks

The game of Tanball is played on a flat table.

A player rolls a ball from a fixed point, at any angle, with the aim of it coming to rest in the winning zone.

A particular player decides to roll the ball at an angle of  pi over 4 radians. This is illustrated by the graph below with the ball being rolled from the origin and the shaded area being the winning zone.

q4-10-2-modelling-involving-numerical-methods-easy-a-level-maths-pure-screenshots

The boundary of the winning zone is given by part of the curve with equation

y equals 1 minus tan open parentheses space square root of 3 left parenthesis x plus 1 right parenthesis end root space close parentheses

(i) Using the iteration formula 

x subscript n plus 1 end subscript equals 1 minus tan space open parentheses square root of 3 left parenthesis x subscript n plus 1 right parenthesis end root close parentheses

with the initial starting value x subscript 0 equals 1.5, find space x subscript 1 comma space x subscript 2and x subscript 3, writing each answer to 5 decimal places.

(ii) Use repeated iteration to find the value of x correct to 3 significant figures. 

(iii) Write down the y-coordinate of the point where this player’s ball should cross the winning boundary.

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5a
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2 marks

The profile of a vertical drop along a rollercoaster track is modelled using part of the function 

straight f open parentheses x close parentheses equals 10 minus left parenthesis x cubed minus 3 x squared minus x right parenthesis

The graph of h equals straight f open parentheses x close parentheses is shown below.

q6-10-2-modelling-involving-numerical-methods-easy-a-level-maths-pure-screenshots
  • h is the height, in metres, above the ground of the front carriage of the rollercoaster

  • x is the horizontal distance from the origin, in metres, of the front carriage of the rollercoaster

(i) Show that the equation straight f open parentheses x close parentheses equals 15 can be rewritten as  x cubed minus 3 x squared minus x plus 5 equals 0.

(ii) Find an expression for straight f apostrophe open parentheses x close parentheses

5b
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4 marks

The rollercoaster briefly stops at the point labelled S on the graph, where straight f open parentheses x close parentheses equals 15.

Apply the Newton-Raphson method with the initial value x subscript 0 equals 2.6 to find the x-coordinate of point S.

Give your answer to 3 significant figures.

5c
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1 mark

State a possible problem if  x subscript 0 equals 1.7 is used instead of the initial value in part (b).

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6a
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1 mark

According to legend, unicorn tears can heal an injury almost instantly.

If a unicorn tear is applied to a burn of initial size B mm2  on human skin it will heal according to the model

straight b open parentheses t close parentheses equals B minus t cubed plus square root of t space space space space space space space space space space space space space space space space space space space space t greater or equal than 0

where b is the area of the burn, in square millimetres, at time t seconds after the unicorn tear has been applied.

Show that the equation straight b open parentheses t close parentheses equals 0 can be written as

t equals cube root of B plus square root of t end root

6b
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3 marks

Use the iteration formula 

space t subscript n plus 1 end subscript equals cube root of 40 plus square root of t subscript n space end root

with an initial value ofspace t subscript 0 equals 3 to find how many seconds it takes a burn of size 40 mm2 to heal once a unicorn tear is applied.

Give your final answer to 3 significant figures.

6c
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3 marks

An alternative iteration formula is 

t subscript n plus 1 end subscript equals open parentheses t subscript n superscript 3 minus B close parentheses squared

(i) Using t subscript 0 equals 3, find t subscript 1 comma space t subscript 2 and t subscript 3, for the same initial burn size as part (b), giving each answer to 3 significant figures.

(ii) Explain how you can deduce whether this sequence of estimates is converging or diverging.

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7a
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2 marks

Following an explosion on the Longwater drilling rig, oil began to leak into the ocean from a damaged underwater pipe.

It took several months for experts to seal the pipe and stop further oil from leaking.

For the first fifteen weeks of the oil spill, the rate of oil leaking from the pipe (measured in barrels per week) was recorded every three weeks.

The results are shown in the table below.

Time

0

3

6

9

12

15

Rate of leak

0

2500

8000

15 000

25 000

37 500

The trapezium rule will be used to estimate the total amount of oil leaked during the first fifteen weeks of the spillage. All the data in the table will be used.

(i) State the number of trapezia that will be used.

(ii) State the width of each trapezium that will be used.

7b
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3 marks

Use the trapezium rule to estimate the amount of oil spilled in the first fifteen weeks.

Give your answer to 3 significant figures.

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8a
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3 marks

Traffic is monitored by three average speed cameras, along a stretch of the road where the speed limit is 30 mph.

A car passes the first camera at time zero. The car’s speed and the time it passes each camera, are recorded.

The results are shown in the table below. 

Camera

1

2

3

Time (hours)

0

0.25

0.5

Speed (mph)

32

38

27

Use the trapezium rule with the data in the table to find an estimate for the distance between the first and last camera.

Give your estimate to 3 significant figures.

8b
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2 marks

Determine whether or not the car is driving within the speed limit.

You must show your reasoning clearly.

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1a
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5 marks

Scientists are analysing the infection rate, R, at which a virus, Pi-flu, spreads amongst a population.  They model R using the function

  R open parentheses t close parentheses equals 4 over 5 plus 2 over 3 sin space t space space space space space space space space space space space space space space space space space space space space space space space left parenthesis t greater or equal than 0 right parenthesis

where t is the number of months since the first case of the virus was discovered.

The diagram below shows the graphs of y equals R left parenthesis t right parenthesis and y equals 1.

0ynXVuWJ_q2a-10-2-modelling-involving-numerical-methods-easy-a-level-maths-pure-screenshots

(i) Show that the equation R open parentheses t close parentheses equals 1 can be written in the form 

2 over 3 space sin space t space minus 1 fifth equals 0

(ii) Apply the Newton-Raphson method with t subscript 0 equals 0.5, along with a suitable function from part (i), to find the number of months after Pi-flu was discovered that the value of R first equals 1.

Give your answer to 3 significant figures.

1b
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3 marks

(i) Write down the maximum infection rate value that the model predicts.

(ii) Approximately, how often does this maximum infection rate occur? Give your answer to the nearest whole number of months.

(iii) Suggest a possible reason, in context, for your answer in part (ii).

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2a
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4 marks

The energy company PowerX operate a wind turbine. Engineers from PowerX model the power output, P kW (kiloWatts), of the wind turbine over a twelve-hour period according to the function

space space P open parentheses t close parentheses equals 50 plus 10 sin square root of 10 t end root space space space space space space space space space space space space space space space space space space space space space space 0 less or equal than t less or equal than 12

where t is time measured in hours. The graph of P against t is shown below.

q3-10-2-modelling-involving-numerical-methods-easy-a-level-maths-pure-screenshots

Using the trapezium rule, with six trapezia each of width 1, find an estimate for the total power generated by the wind turbine in the first six hours.

You may use the table below to help.

Time (t)

0

1

2

3

4

5

6

Power (P)

 

 

 

 

 

 

 

2b
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1 mark

Explain why it may be difficult for the engineers from PowerX to determine whether the total amount of power found using the method in part (a) is an overestimate or an underestimate.

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3a
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2 marks

The village of Greendale lies on a straight road, as modelled by the line y equals x on the graph below.  To ease rush hour congestion, a bypass is to be built around Greendale. The path of the bypass is modelled by the equatio

y equals 6 square root of 1 minus fraction numerator 1 over denominator x plus 1 end fraction end root

q1-10-2-modelling-involving-numerical-methods-medium-a-level-maths-pure-screenshots

The bypass runs from the origin to the point P open parentheses p comma space p close parentheses.

On the graph above, draw a staircase diagram to show how using the iteration formula 

x subscript n plus 1 end subscript equals 6 square root of 1 minus fraction numerator 1 over denominator x subscript n plus 1 end fraction end root

where 0 less than x subscript 0 less than p leads to convergence at the point P.

3b
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2 marks

Use the iteration formula 

space x subscript n plus 1 end subscript equals 6 square root of 1 minus fraction numerator 1 over denominator x subscript n plus 1 end fraction end root

with x subscript 0 equals 5 to find the value of p, correct to 2 decimal places.

3c
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2 marks

By considering a suitable interval and the function

straight f open parentheses x close parentheses equals x minus 6 square root of 1 minus fraction numerator 1 over denominator x plus 1 end fraction end root

show that your answer to part (b) is correct to 2 decimal places.

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4a
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2 marks

The game of Curveball is played on a flat table.

A player rolls a ball from a fixed point, at any angle, with the aim of it coming to rest in the winning zone.

A particular player decides to roll the ball at an angle of 45°.
This is illustrated by the graph below with the ball being rolled from the origin and the shaded area being the winning zone.

q5-10-2-modelling-involving-numerical-methods-medium-a-level-maths-pure-screenshots

The boundary of the winning zone is given by part of the curve with equation

y equals 1 half sin squared open parentheses straight e to the power of x close parentheses

Use the iteration formula

x subscript n plus 1 end subscript equals 1 half sin squared open parentheses straight e to the power of x subscript n end exponent close parentheses

with a starting value of x subscript 0 equals 0.5 to find the x-coordinate of the point where this player’s ball should cross the winning zone boundary.

Give your answer to 3 significant figures.

4b
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2 marks

Use your answer to part (a) to find the minimum distance the ball should travel for this player to win Curveball.

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5a
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3 marks

Coronation Street has a speed limit of 40 mph and traffic is monitored along one stretch of the road by four average speed cameras.

Vera is driving her car along Coronation Street and passes the first camera at time zero. Vera’s speed, measured in miles per hour, and the time she passes each camera, are recorded.

The results are shown in the table below. 

Camera

1

2

3

4

Time (hours)

0

0.1

0.2

0.3

Speed (mph)

36

40

38

35

Use a suitable method with all the data in the table to estimate the distance between the first and last camera.

Give your estimate to 3 significant figures.

5b
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2 marks

Vera’s car consumes fuel at a rate of 52.6 miles per gallon.

Estimate the amount of fuel, in gallons, Vera’s car uses between the first and last camera.

Give your answer to 1 significant figure.

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6a
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1 mark

The profile of the Wibbley-Wobbley-Slide is modelled using part of the function

straight f open parentheses x close parentheses equals 20 minus x plus sin space x

The graph of space h equals straight f open parentheses x close parentheses, where x greater or equal than 0, is shown below.

q7-10-2-modelling-involving-numerical-methods-medium-a-level-maths-pure-screenshots
  • h is the height, in metres, above the ground of a person using the slide.

  • x is the horizontal distance, in metres, of a person on the slide, measured from the point directly underneath the top of the slide (the origin).

Find the height above the ground of a person on the slide when they have travelled 10 m horizontally.

6b
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4 marks

Apply the Newton-Raphson method with a starting approximation of x subscript 0 equals 20 to find the total horizontal distance travelled by a person using the slide.

Give your answer to 3 significant figures.

6c
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2 marks

Using a suitable interval, show that your answer to part (b) is correct to 3 significant figures.

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7a
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1 mark

The air pressure inside a ThetaAir aeroplane is kept between 10 and 12 psi (pounds per square inch). A computer automatically adjusts the air pressure according to the model

  P open parentheses t close parentheses equals 11 plus 0.1 t sin space t space space space space space space space space space space space space space space space space space space space space t greater or equal than 0

where P is the pressure (psi) and t is the time, in hours, since take off.

The graph of the model is shown below.

q8-10-2-modelling-involving-numerical-methods-medium-a-level-maths-pure-screenshots

Explain why this model is only suitable for flights roughly less than 10 hours long.

7b
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3 marks

(i) Show that when  P equals 10, t satisfies the equation

1 plus 0.1 t sin space t equals 0

(ii) Find an expression for fraction numerator d P over denominator d t end fraction

7c
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3 marks

Apply the Newton-Raphson method with a starting value of t subscript 0 equals 10 to find the time that the air pressure on a ThetaAir aeroplane first goes outside of the range 10 to 12 psi.

Give your answer to three significant figures.

7d
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2 marks

The air pressure increases back to 10 psi after 11.5 hours. 

Estimate the length of time, to the nearest minute, that the air pressure was below 10 psi.

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8a
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2 marks

According to legend, a unicorn can heal an injury almost instantly by touching it with its horn.

When a unicorn touches a cut in human skin of length L mm, it will heal according to the model

straight f open parentheses t close parentheses equals L straight e to the power of negative t end exponent minus t 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 space space t greater or equal than 0

where straight f is the length of the cut in millimetres, at time t seconds after the unicorn has touched the injury with its horn.

(i) Write down the value of straight f at the point when the cut is completely healed.

(ii) Show that, for a cut in human skin of length 5 mm the equationspace straight f open parentheses t close parentheses equals 0 spacecan be rearranged into the form

t equals negative ln open parentheses t over 5 close parentheses

8b
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4 marks

The iteration formula

t subscript n plus 1 end subscript equals negative ln open parentheses fraction numerator space t subscript n over denominator 5 end fraction close parentheses space

with the initial value t subscript 0 equals 1 is used to find how many seconds it takes a cut of size 5 mm to heal once a unicorn has touched it with its horn.

(i) Write down the values of the estimatesspace t subscript 1 comma space t subscript 2 space and t subscript 3 to 4 decimal places.

(ii) Give your final answer to 2 significant figures.

(iii) State the number of iterations required, n, for convergence to 2 significant figures.

8c
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1 mark

Explain why the model should restrict the range of straight f left parenthesis t right parenthesis to be greater than or equal to zero.

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1a
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3 marks

The speed of a small jet aircraft was measured every 5 seconds, starting from the time it turned onto a runway, until the time when it left the ground.

The results are given in the table below with the time in seconds and the speed in m s-1.

Time (s)

0

5

10

15

20

25

Speed (m s-1)

2

5

10

18

28

42

Using all of this information, estimate the length of runway used by the jet to take off.

1b1 mark

Given that the jet accelerated smoothly in these 25 seconds, explain whether your answer to part (a) is an underestimate or an overestimate of the length of runway used by the jet to take off.

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2a
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1 mark

The village of Crinkley Bottom lies on a straight road, as modelled by the line y equals x on the graph below.  Rush hour traffic causes much air pollution in the village so to improve the air quality around Crinkley Bottom a bypass is to be built.

The path of the bypass is modelled by part of the equation y equals x squared sin space x

q1-10-2-modelling-involving-numerical-methods-hard-a-level-maths-pure-screenshots

The bypass is to be built with a roundabout south of the village at the origin and a northern roundabout which re-joins the road through Crinkley Bottom at the point P open parentheses p comma space p close parentheses.

On the graph, draw a staircase diagram to show how using the iteration formula 

x subscript n plus 1 end subscript equals x subscript n superscript 2 sin space x subscript n space

with 0 less than x subscript 0 less than p leads to convergence at the southern roundabout.

2b
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2 marks

A second iteration formula is given by

x subscript n plus 1 end subscript equals square root of fraction numerator x subscript n over denominator sin space x subscript n end fraction end root

Use repeated iteration, with x subscript 0 equals 1, to find the position of the roundabout at P, to 4 significant figures.

2c
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2 marks

Using a suitable interval and suitable function that should be stated, show that your answer to part (b) is correct to 4 significant figures.

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3a
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4 marks

Scientists are analysing the infection rate, R, at which a virus, Newrap-20, spreads amongst a population. R is modelled using the function

space space R open parentheses t close parentheses equals 0.6 plus 1 over t sin space open parentheses 2 t close parentheses space minus straight e to the power of negative t space space space space space space end exponent space space space space space space space space space space space space space space space space space space space space left parenthesis t greater than 0 right parenthesis

where t is the number of months since the first case of the virus was discovered.

To control the spread of Newrap-20, scientists need to keep the value of R below 1.

The diagram below shows the graphs of y equals R left parenthesis t right parenthesis and y equals 1.

q2-10-2-modelling-involving-numerical-methods-hard-a-level-maths-pure-screenshots

Apply the Newton-Raphson method with a starting approximation of t subscript 0 equals 1 to find the number of months after Newrap-20 was discovered, that the value of R first drops below 1.

Give your answer to 3 significant figures.

3b
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1 mark

Given that the first two months both have 31 days each, find the number of days (to the nearest day) for which the value of R first drops below 1.

3c
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1 mark

In the long term, the scientists expect the value of R to settle to a constant value.

Use the model to write down this constant value of R.

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4a
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4 marks

The energy company PowerY own and maintain a wind farm. Engineers from PowerY model the power output, P kW (kiloWatts), of a single wind turbine over a twelve-hour period according to the function

 space P open parentheses t close parentheses equals 50 plus 30 straight e to the power of negative 0.1 end exponent to the power of t sin space t space space space space space space space space space space space space space space space space space space space space space space space space 0 less or equal than t less or equal than 12

where t is time measured in hours.  The graph of P against t is shown below.

q3-10-2-modelling-involving-numerical-methods-hard-a-level-maths-pure-screenshots

Use the trapezium rule with a step size of 1 to estimate the total power generated by the wind turbine in the last four hours of the twelve-hour period.

Give your answer to the nearest whole number of kiloWatts.

4b
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2 marks

Estimate the average amount of power per minute produced by the wind turbine during these last four hours.

Give your answer in kiloWatt min-1 to 1 decimal place.

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5
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3 marks

The drilling rig AlphaBeta began leaking oil into the North Sea following a technical fault. It took 14 hours for engineers to trace and repair the fault.

During that time the rate of oil leaking, measured in tonnes per hour, was recorded every 2 hours.  The results are shown in the table below.

Time

0

2

4

6

8

10

12

14

Rate of leak

0

8

12

18

26

38

18

0

For safety reasons oil rigs are required to shut down and stop all operations until an inspection is carried out should the total amount of oil leaked during any incident exceed 250 tonnes.

Use all the data in the table to determine whether or not the AlphaBeta rig should be shut down.

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6a
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3 marks

The game of Logball is played on a flat table.

A player rolls a ball from a fixed point, at any angle, with the aim of it coming to rest within a winning zone.

A particular player decides to roll the ball at an angle of 45°, as illustrated in the graph below, with the ball being rolled from the origin and the shaded area being the winning zone.

The lower boundary of the winning zone has equation 

y equals 3 minus ln space open parentheses x plus 1 close parentheses squared space space space space space space space space space space space space space space x comma space y greater or equal than 0

The upper boundary of the winning zone has equation 

y equals 4 minus ln space open parentheses x plus 1 close parentheses squared space space space space space space space space space space space space space x comma space y greater or equal than 0

q5-10-2-modelling-involving-numerical-methods-hard-a-level-maths-pure-screenshots

Use a suitable iteration formula with an initial value of x subscript 0 equals 1.8 to find the minimum distance this player’s ball needs to travel to stop within the winning zone.

Give your answer to 2 significant figures.

6b
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3 marks

Use a suitable iteration formula with an initial value of x subscript 0 equals 2.5 to find the maximum distance this player’s ball can travel yet remain within the winning zone.

Give your answer to 2 significant figures.

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7a
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3 marks

A stretch of road along Equality Street has a speed limit of 70 mph and traffic is monitored by six average speed cameras.

Ricky is driving his car along Equality Street and passes the first camera at time zero.  Ricky’s speed, measured in miles per hour, and the time he passes each camera, are recorded in the table below. 

Camera

1

2

3

4

5

6

Time (hours)

0

0.05

0.1

0.15

0.2

0.25

Speed (mph)

68

72

69

71

70

70

Use all the results above to estimate the distance between the first and last camera.

7b
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2 marks

A driver will receive a speeding ticket if their average speed between the first and last camera exceeds the speed limit.

Based on this information and the answer in part (a), determine whether or not Ricky should receive a speeding ticket. 

You must give a reason for your answer.

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8a
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7 marks

The profile of the first part of the Big Delta rollercoaster is modelled using the function 

straight f open parentheses x close parentheses equals x sin space open parentheses 1 half x close parentheses space

The graph of h equals straight f open parentheses x close parentheses is shown below.

q7-10-2-modelling-involving-numerical-methods-hard-a-level-maths-pure-screenshots
  • h is the height, in metres of a person sat on the rollercoaster. h = 0 is the height at which passengers board the rollercoaster.

  • x is the horizontal distance travelled, in metres, measured from the starting point.

Apply the Newton-Raphson method with a starting approximation of x subscript 0 equals 14 to find the horizontal distance travelled by the first time the Big Delta reaches a height of 10 metres above its starting point.

Give your answer to three significant figures.

8b
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2 marks

Using a suitable interval and a suitable function, show that your answer to part (a) is correct to 3 significant figures.

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9a
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6 marks

The air pressure inside a Flappyjet aeroplane is kept between 10.5 and 11.5 psi (pounds per square inch). A computer makes automatic adjustments according to the model

space space P equals 11 plus 0.07 t space cos space 2 t space space space space space space space space space space space space space space space space space space space space space space t greater or equal than 0

where P is the pressure (psi) and t is the time of flight, in hours, since take off.

The graph of the model is shown below.

q8-10-2-modelling-involving-numerical-methods-hard-a-level-maths-pure-screenshots

The Newton-Raphson method is used to find the time at which the air pressure first drops below 10.5 psi.

(i) Show that the formula is

t subscript n plus 1 end subscript equals t subscript n minus fraction numerator k plus 0.07 t subscript n cos space 2 t subscript n over denominator 0.07 left parenthesis cos space 2 t subscript n minus 2 t subscript n sin space 2 t subscript n space right parenthesis end fraction space

where k is a constant to be found.

(ii) Apply the Newton-Raphson method with t subscript 0 equals 7 to find the time at which the air pressure first drops below 10.5 psi, to 3 significant figures.

 

9b
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2 marks

Human beings can cope with the air pressure being below 10.5 psi as long as it is for no longer than 30 minutes.

Point P has coordinates (9.1 , 11.5).

Given that the air pressure increases to 10.5 psi at t equals 8.10, determine whether or not the model above can be used for flights up to 9 hours long.

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10a
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3 marks

According to legend, unicorn tears have magical healing powers.

When a unicorn tear is applied to a bruise of size A mm2 it will heal according to the model

straight f open parentheses t close parentheses equals A straight e to the power of negative 0.15 t end exponent minus 0.2 t space space space space space space space space space space space space space space space space space space space space space space space space t greater or equal than 0

where straight f is the area of the bruise, in square millimetres, at time t seconds after the unicorn tear is applied.

Show that, for a bruise of initial size 20 mm2, the equation straight f open parentheses t close parentheses equals 0 can be rearranged into the form

t equals 20 over 3 ln space open parentheses 100 over t close parentheses

10b
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2 marks

Using the equation from part (a) as an iteration formula with an initial value ofspace t subscript 0 equals 12, find how many seconds it takes a bruise of size 20 mm2 to heal once a unicorn tear is applied.

Give your answer to 3 significant figures.

10c
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2 marks

It is rumoured that a unicorn tear can heal bruises one hundred thousand times faster than they would heal naturally.

Estimate how many days it would take a bruise of initial size 20 mm2 to heal without a unicorn tear.

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1a
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1 mark
Graph showing velocity (v) against time (t). The curve rises from origin, peaks, then falls back to the axis at time, T.
Figure 2

A car stops at two sets of traffic lights.

Figure 2 shows a graph of the speed of the car, v space ms to the power of negative 1 end exponent, as it travels between the two sets of traffic lights.

The car takes T seconds to travel between the two sets of traffic lights.

The speed of the car is modelled by the equation

v equals left parenthesis 10 minus 0.4 t right parenthesis ln left parenthesis t plus 1 right parenthesis space space space space space space space space space space 0 less or equal than t less or equal than T

where t seconds is the time after the car leaves the first set of traffic lights.

According to the model, find the value of T

1b
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4 marks

Show that the maximum speed of the car occurs when

t equals fraction numerator 26 over denominator 1 plus ln open parentheses t plus 1 close parentheses end fraction minus 1

1c
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3 marks

Using the iteration formula

t subscript n plus 1 end subscript equals fraction numerator 26 over denominator 1 plus ln open parentheses t subscript n plus 1 close parentheses end fraction minus 1

with t subscript 1 equals 7

(i) find the value of t subscript 3 to 3 decimal places,

(ii) find, by repeated iteration, the time taken for the car to reach maximum speed.

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2a
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1 mark

The village of Camberwick Green lies on a straight road, as modelled by the line y equals x on the graph below, measured in miles. Rush hour traffic through the village causes both congestion and air pollution.To ease congestion and improve the air quality around Camberwick Green a bypass ,modelled by part of the equation

y equals 1 plus 2 ln space left parenthesis x squared right parenthesis

is to be built.

q1-10-2-modelling-involving-numerical-methods-veryhard-a-level-maths-pure-screenshots

The bypass is to be built with a roundabout south of the village at point S and a northern roundabout which re-joins the road through Camberwick Green at the point P open parentheses p comma space p close parentheses.

Write down the coordinates of the southern roundabout at point S.

2b
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3 marks

(i) Draw a staircase diagram on the graph in part (a) to how an iteration formula with a starting value of x subscript 0 in the interval open square brackets 1 comma space p close square brackets will converge to the northern roundabout at point P.

(ii) Use a suitable iteration formula with an initial value of x subscript 0 equals 10 to find the value of p to 5 significant figures.

2c
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2 marks

Find the length of road through Camberwick Green that will benefit from the construction of the bypass.

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3a
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7 marks

Scientists are analysing the infection rate, R, at which a virus Divid-e20 spreads amongst a population.They model R using the function

  R open parentheses t close parentheses equals 0.5 plus straight e to the power of negative 0.3 t end exponent sin space open parentheses t close parentheses space space space space space space space space space space space space space space space space space space space space space left parenthesis t greater than 0 right parenthesis

where t is the number of months since the first case of the virus was discovered.

To control the spread of Divid-e20 scientists need to keep the value of R below 1.

The diagram below shows the graphs of y equals R left parenthesis t right parenthesis and R equals 1.

q2-10-2-modelling-involving-numerical-methods-veryhard-a-level-maths-pure-screenshots

Use the Newton-Raphson method, with an initial value of t subscript 0 equals 0.5, to find the number of months after Divid-e20 was discovered that R first equals 1.

Give your answer to 3 decimal places.

3b
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2 marks

Given that the infection rate for Divid-e20 first went above 1 on 1st July, and drops back below 1 roughly 1.997 months (to three decimal places) after Divid-e20 was first discovered, calculate the date when the infection rate drops back below 1.

3c
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1 mark

In the long term, scientists expect the value of R to remain roughly constant around 0.5.

Explain how to deduce this value from the model, R open parentheses t close parentheses.

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4a
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4 marks

The energy company PowerZ own and maintain a wind farm.
Engineers from PowerZ model the power output, P kW (kiloWatts), of a single wind turbine over a twelve-hour period according to the function

space space P open parentheses t close parentheses equals 60 plus 5 straight e to the power of 0.7 square root of t end exponent sin space t 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 0 less or equal than t less or equal than 12

where t is the number of hours after 6pm. The graph of P against t is shown below.

q3-10-2-modelling-involving-numerical-methods-veryhard-a-level-maths-pure-screenshots

Estimate the total power generated by this wind turbine between midnight and 4am.

4b
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2 marks

Once every twelve hours, PowerZ turn each wind turbine off for half an hour for maintenance and safety checks.

Use the model to suggest, with a reason, at what time between 6pm and 6am the maintenance and safety checks should be carried out on the wind turbine.

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5a
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4 marks

The XnValdez container ship was carrying 4000 tonnes of crude oil when it ran aground and oil started leaking from its hull into the ocean.

It took 36 days for experts to stop the oil leak.

During that time the leakage rate of the oil, measured in tonnes per day, was recorded every 4 days.  The results are shown in the table below.

Time

0

4

8

12

16

20

24

28

32

36

Rate of leak

0

10

20

40

70

110

160

230

150

0

An environmental disaster is declared if the total amount of crude oil that leaks into the ocean exceeds 2500 tonnes.

Use all the data in the table to determine whether or not the environmentalists were right to declare the XnValdez incident an environmental disaster.

5b
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3 marks

By replacing trapezia with suitable rectangles, use the data in the table to find a lower bound and an upper bound estimate to the total amount of crude oil leaked by the XnValdez.

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6
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6 marks

The game of Funcball is played on a flat table. A player rolls a ball from a fixed point, at any angle, with the aim of it coming to rest within a winning zone.

The winning zone is modelled by the function

straight f open parentheses x close parentheses equals ln space open parentheses 3 x plus 4 close parentheses minus 0.25 x squared.

The lower boundary of the winning zone has equationspace y equals straight f left parenthesis x right parenthesis comma space space space x comma space y greater or equal than 0

The upper boundary of the winning zone has equation y equals straight f open parentheses x close parentheses plus 1 comma space space x comma space y greater or equal than 0

A particular player decides to roll the ball at an angle of 45°, as illustrated by the graph below, with the ball being rolled from the origin and the shaded area being the winning zone.

q5-10-2-modelling-involving-numerical-methods-veryhard-a-level-maths-pure-screenshots

Using suitable iteration formulas with the initial values x subscript 0 equals 1.5 and x subscript 0 equals 2.1 where appropriate, find the distances between which the ball must stop for this player to win Funcball.

Give your answers to 2 significant figures.

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7a
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2 marks

A stretch of road along Baker Street has a speed limit of 60 mph and traffic is monitored by eight average speed cameras.

Sherlock is driving along Baker Street and passes the first camera at time zero.  Sherlock’s speed, measured in miles per hour, and the time he passes each camera, are recorded in the table below. 

Camera

1

2

3

4

5

6

7

8

Time (minutes)

0

4

8

12

16

20

54

n/a

Speed (mph)

64

59

61

62

57

58

60

n/a

(i) Suggest a reason as to why the last camera did not record any data for Sherlock’s journey.

(ii) Suggest a reason as to why there was a longer time gap between the sixth and seventh camera.

7b
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3 marks

Use the results from the table to estimate the distance between the first and sixth camera.

7c
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2 marks

Explain why it would not be suitable to apply the trapezium rule formula for estimating the distance between the first and seventh camera.

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8
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6 marks

The top of the slope at the No-Sno Dry Ski Centre is at a height of 25 metres above sea level. The profile of the slope, shown in the graph below, is modelled by the function

h open parentheses x close parentheses equals 25 minus square root of x minus 1 over 16 x sin space open parentheses x over 20 close parentheses space space space           0 less or equal than x less or equal than 160

where x is the horizontal distance travelled, in metres, by a skier using the slope.

q7-10-2-modelling-involving-numerical-methods-veryhard-a-level-maths-pure-screenshots

Apply the Newton-Raphson method, with an initial value of x subscript 0 equals 70, to find the horizontal distance travelled by a skier when they are at a height of 18 m above sea level and travelling uphill.

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9a
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2 marks

The air pressure inside a PiPlane aircraft is kept between 10 and 12 psi (pounds per square inch). A computer adjusts the air pressure automatically, according to the model

  P equals 12 minus 0.3 t space sin squared t space space space space space space space space space space space space space space space space space space space space space space space t greater or equal than 0

where P is the pressure (psi) and t is the time in hours since take off, as shown in the graph below.

q8-10-2-modelling-involving-numerical-methods-veryhard-a-level-maths-pure-screenshots

Explain why this model will never lead to an air pressure above 12 psi.

9b
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4 marks

Apply the Newton-Raphson method with a starting approximation of t subscript 0 equals 7.5 to find the time at which the air pressure first drops below 10 psi

Give your answer to 3 significant figures.

9c
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3 marks

Human beings can tolerate air pressure below 10 psi for up to an hour, after which there can be an adverse effect on their comfort and behaviour. In the confines of an aircraft this is undesirable for all PiPlane passengers and crew.

Apply the Newton-Raphson method a second time with the starting approximation of t subscript 0 equals 8.5 to find the next time the air pressure is 10 psi.

Hence show that the model is suitable for flights of at least 10 hours.

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10a
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1 mark

According to legend, unicorn tears have magical healing powers.

Without unicorn tears, a bruise of initial size A mm2 will heal according to the model

straight f open parentheses t close parentheses equals A straight e to the power of negative 0.25 t end exponent minus 0.1 t space space space space space space space space space space space space space space space space space space space space t greater or equal than 0

where straight f is the area of the bruise, in square millimetres, at time t days since the bruise first appeared.

For one particular value of A, the iteration formula

t subscript n plus 1 end subscript equals 4 ln space open parentheses 120 over t subscript n close parentheses

withspace t subscript 0 equals 10 space gives how many days it takes a bruise to heal without unicorn tears.

Use repeated iteration to find this value, to 3 significant figures.

10b
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3 marks

With unicorn tears, a bruise of the same initial size will heal according to the model

u open parentheses T close parentheses equals A T straight e to the power of negative T end exponent minus 0.1 T space space space space space space space space space space space space space T greater or equal than 0

where time T is measured in seconds.

(i) Find the particular value of Athat was used to create the iteration formula in part (a).

(ii) Find how many seconds it takes the same size bruise to heal using unicorn tears.

10c
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1 mark

Using your answers from parts (a) and (b), calculate approximately how many times quicker a bruise heals when unicorn tears are used.

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