CIE AS Maths: Pure 1

Topic Questions

1.7 Modelling with Functions

1
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2 marks

In football’s Premier League a team is awarded 3 points for each match they win, 1 point for each drawn match and no points for a loss.

(i)

Using Wfor the number of matches won, D for the number of matches drawn, and P for the total number of points a team has, write down a formula for P in terms of W and D.

(ii)
Briefly explain why there is no need to use L in the formula above for the number of matches lost.

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2a
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3 marks

Water is leaking from a pipe. The rate of the leak is directly proportional to the speed of water flowing through the pipe.

It was observed that the rate of the leak was 4 litres per second,1 space straight s to the power of negative 1 end exponent when the speed of water flowing through the pipe was 2 metres per second, straight m space straight s to the power of negative 1 end exponent.

Find a formula linking the rate of the leak, L space1 space straight s to the power of negative 1 end exponent   to the speed of water flowing through the pipe, straight s space straight m space straight s to the power of negative 1 end exponent.

2b
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3 marks
(i)

Find the rate of the leak if the speed of water is 2.5 straight m space straight s to the power of negative 1 end exponent.

(ii)
Find the speed of water if the rate of the leak is 31 space straight s to the power of negative 1 end exponent.

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3
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4 marks

A soft ball is thrown upwards from the top of a building.
The height, h m of the ball above the ground after t seconds is modelled by the function

h left parenthesis t right parenthesis equals 12 plus 20 t minus 5 t to the power of 2 space space space space space space space space space end exponent t greater or equal than 0

(i)

Evaluate h left parenthesis 0 right parenthesis and hence deduce the height of the building.

(ii)
Find the height of the ball after 4 seconds.

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4
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4 marks

The number of cases of an unknown virus are modelled by the formula

V equals 2700 over open parentheses d minus 30 close parentheses squared            0 less or equal than d less than 30

where d spaceis the number of days after the virus was first discovered and V is the total number of cases to date.

(i)

Find the number of cases when the virus was first discovered.

(ii)
Sketch a graph of V spaceagainst d spacelabelling the coordinates of the point where the graph intersects the V-axis.

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5a
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1 mark

The number of toys, P , produced by a machine in one hour is modelled by the function

P left parenthesis T right parenthesis equals left parenthesis 20 minus T right parenthesis left parenthesis T minus 40 right parenthesis space space space space space space space space space space space space space space space space space space space space 20 less or equal than T less or equal than 40

where T space degree C is the temperature of the machine.

Find the number of toys produced in one hour when the temperature of the machine is 25 space degree C.

5b
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6 marks
(i)

 Show that  P left parenthesis T right parenthesis equals 60 T minus T squared minus 800.

(ii)

By completing the square show further that P left parenthesis T right parenthesis equals 100 minus left parenthesis T minus 30 right parenthesis squared.

(iii)
Using your answer from (ii) or otherwise, find the temperature at which the machine is at its most productive, and how many toys it produces in one hour at this temperature.

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6
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6 marks

A patient takes a new medication at midday.  The amount of drug,D mg, remaining in their bloodstream  hours after midday is modelled by the formula

D left parenthesis h right parenthesis equals 1 plus 6 h minus h squared space space space space space space space space space space space space space space space 0 less or equal than h less or equal than 6

(i)

Evaluate D left parenthesis 0 right parenthesis and deduce the amount of drug in the patient’s bloodstream before they take any medication.

(ii)

Find the amount of drug in the patient’s bloodstream at 2 pm.

(iii)
Show that by 6pm the amount of drug in the patient’s bloodstream has returned to its starting amount.

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7a
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2 marks

A company selling books models the number of books sold per year,N, using the formula

N equals 10 space 000 minus 200 c

where c is the price per book in pounds sterling.

(i)

Find the number of books the company can expect to sell if they are priced at £18 each?

(ii)
Work out the total income the company will receive if they sell all books at £18 each.
7b
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2 marks

Find the number of books the company can expect to sell if they are priced at £16 each and work out the total income the company will receive if they sell all books at this price.

7c
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1 mark

What do your answers to part (a) and (b) suggest about the relationship between the price of a book, the number sold and the total income received?

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8a
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2 marks

A cricket ball is projected directly upwards from ground level.

The motion of the cricket ball is modelled by the function

h left parenthesis t right parenthesis equals 20 t minus 5 t squared space space space space space space space t greater than 0

where h metres is the height of the cricket ball above ground level after t seconds.

(i)

Factorise h left parenthesis t right parenthesis.

(ii)
Hence, find the time at which the cricket ball returns to ground level.
8b
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3 marks

The motion of a second cricket ball is modelled by a similar function

d left parenthesis t right parenthesis equals 16 t minus 5 t squared space space space space space space space space t greater than 0

where d metres is the height of the cricket ball above ground level after t seconds.

Find the times at which this cricket ball is exactly above the ground.

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9
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3 marks

The graph below shows a suggested model for estimating the value of a brand-new car costing £28 000. a spaceis the car’s age in years and £V  is the car’s value in thousands.

q9-2-12-modelling-with-functions-edexcel-a-level-pure-maths-easy

(i)

Use the model to predict a car’s value after 8 years.

(ii)

Use the model to predict how long it takes a brand-new car to halve in value.

(iii)
This model predicts that regardless of how old a car is, it’s value never
reaches £0. Suggest a reason to justify this property of the model.

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10
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4 marks

A skydiver jumps from a moving aircraft when it is directly above a fixed point, O, on the ground.  The trajectory of the skydiver is then modelled by the function

      h open parentheses x close parentheses space equals space 4900 space minus space x squared

where h space m is the height of the skydiver above the ground and x m is the horizontal distance along the ground from point O.

(i)
At what height does the skydiver jump from?
(ii)
How far above the ground is the skydiver when their horizontal distance from 0 is 50 space m?
(iii)
Explain why the model is not suitable for values of x larger than 70 space m.

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11a
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3 marks

An Aerobatic Display Team use smoke trails as part of their flying display.

At the start of the display, each aeroplane holds a tank containing 100 litres of smoke.

The amount of smoke remaining in a tank is inversely proportional to 10 more than the total time that the smoke has been used for.

Show that

      W equals fraction numerator 1000 over denominator t space plus space 10 end fraction

where W litres is the amount of smoke left in the tank t seconds after the smoke
first started being used.

11b
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3 marks

Find the amount of smoke left in the tank after 40 seconds of use.

11c
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1 mark

Explain why the model suggests the tank never empties.

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12a
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3 marks

The linear function T open parentheses h close parentheses space equals space 38 space minus space 1.2 h is used to model the temperature of a human body, T degree C comma space h hours after death.

(i)
According to this model, what would the expected temperature be of the body of

a human who died 6 hours ago?

(ii)
If the temperature of a body is 27.2 °C, how long ago did the person die?

 

12b
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2 marks

Police suspect a murder victim was killed at around 11 am.
The body was discovered at 3.30 pm the same day.
If the police’s suspicions are correct, what should the temperature of the body have been when it was discovered?

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13a
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3 marks

The revenue (money received before expenses, tax, etc.) Modx makes from selling maths t-shirts is calculated by multiplying the selling price by the number of items sold. Last year a company sold 8000 t-shirts, at a price of £12 each, thus giving them a revenue of £96 000 .

This year, in order to maximise their revenue, Modx intend to increase the t-shirt price by £x. It is expected, however, that each price increase of £1  will reduce the number of items sold by 400.

The expected revenue for this year is thus modelled by the equation

      R equals 96000 plus 3200 x minus 400 x squared

where £x  is the increase in price per item.

By factorising 3200 x space minus space 400 x squared and completing the square, show that the expected revenue can be rewritten as  R space equals space 102 space 400 space minus space 400 open parentheses x minus 4 close parentheses squared. 

13b
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3 marks
(i)
Write down the maximum revenue the company should expect this year.
(ii)
What price should the company sell the item at this year order to maximise 
        revenue?

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1a
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2 marks

It has often been said, to avoid relegation from football’s Premier League, teams should aim to score at least 40 points in a season.  Each team plays 38 games, a win is rewarded with 3 points and a draw with 1 point.  No points are awarded for a loss.

Using Wand D as the number of wins and draws respectively, write down two inequalities. One relating to the number of points a team needs to avoid relegation and one relating to the number of games played.

1b
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1 mark

Explain why W greater or equal than 0 and D greater or equal than 0 must also be conditions related to the problem.

1c
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2 marks

A team has won 4 games and drawn 4 after playing 17 games.
Write down two updated inequalities for the number of wins and draws required for the remainder of the season in order to avoid relegation.

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2a
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2 marks

The leakage rate of water from a pipe, L space 1 space straight s to the power of negative 1 end exponent (litres per second), is directly proportional to the square root of the flow rate,straight s space straight m space straight s to the power of negative 1 end exponent  (meters per second), which is the speed of the water flowing through the pipe.
It was observed that the leaking rate was 0.72 space straight l space straight s to the power of negative 1 end exponent when the flow rate was 0.64 space straight m space straight s to the power of negative 1 end exponent

Write down an equation connecting L spaceand S.

.

2b
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2 marks

Find the flow rate when the leakage rate is 0.49 space straight l space straight s to the power of negative 1 end exponent.

2c
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2 marks

An alternative model for the leakage rate is L equals 0.5 straight s.
Apart from when there is no leak find a flow rate and a leakage rate for when both models predict the same result.

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3a
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1 mark

A soft ball is thrown upwards from the top of a building.
The height, h m of the ball above the ground after t seconds is modelled by the function

h left parenthesis t right parenthesis equals 15 plus 8.4 t minus 4.9 t squared space space space space space space t greater than 0

What is the significance of the constant 15 in the function?

3b
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2 marks

At what time is the ball at the same height as when it was thrown?

3c
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3 marks

Find the time at which the ball is at its maximum height and what this maximum height is.

3d
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2 marks

How long does it take for the ball to first hit the ground?

3e
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2 marks

Given that the ball first hits the ground at a distance 20 m from the base of the building find the shortest distance between this point and where the ball was thrown from.

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4a
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2 marks

The number of cases of an unknown virus are modelled by the formula

V equals begin inline style 400 over open parentheses d minus 20 close parentheses squared end style          0 less or equal than d less than 20

where d spaceis the number of days after the first case was discovered and V is the total number of cases to date.

Find the number of cases after 10 days and after 15 days.

4b
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2 marks

Sketch a graph of V spaceagainst d, clearly marking the coordinate where the graph intersects the V-axis.

4c
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1 mark

Scientists suggest the model is not accurate beyond 15 days.
Suggest a reason why.

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5a
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1 mark

A machine produces toys at a rate dependent on the machine’s temperature. The more extreme the temperature of the machine, the less productive it is.

The productivity of the machine is measured by

P left parenthesis T right parenthesis equals 0.015 T left parenthesis 22 minus T right parenthesis left parenthesis T minus 75 right parenthesis space space space space space space space space space space space space space space space space space 22 less or equal than T less or equal than 75

where P is the number of toys produced per hour and T space degree C is the temperature of the machine.

Suggest a reason why the machine only operates between 22 space degree C and 75 space degree C.

5b
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3 marks

Productivity is at its peak when the machine is producing around 544 toys per hour.
Find the approximate temperature of the machine at this rate of production.

5c
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2 marks

The temperature of the machine rises by 7 space degree C space for every hour it is in constant use.
In order to prevent a breakdown the machine is switched off once the temperature exceeds 60.5 space degree C.

Assuming the machine is at 22 space degree C when it is switched on, find the number of hours it can run continuously for, before having to be being switched off.

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6a
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1 mark

A patient takes some medication at midday.  The amount of drug, D mg, remaining in their bloodstream h hours after midday is modelled by the formula

 D equals 0.03 plus 0.25 h minus 0.05 h squared space space space space space space space space space space 0 less or equal than h less or equal than 5

What amount of drug is already naturally occurring in the patient’s bloodstream before taking any medication?

6b
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2 marks

After what time does the amount of drug in the patient’s bloodstream return to its natural level?

6c
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2 marks

It is safe for the patient to take more medication once the amount of drug in their bloodstream falls below 0.23 mg. When is the earliest a patient can take a second dose of the medication?

6d
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1 mark

Explain why your answer to part (c) should not be 1pm despite this being a solution to the relevant equation?

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7a
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1 mark

Last year, a company sold 12 000 copies of a book, at a price of £15 each.

This year, the company wants to increase the price of the book and predicts that for every £2 increase in price, annual sales will drop by 400 books.

The formula N equals a plus b c is used to model the number of books sold annually. Where N is the number of books sold,c  is the selling price of the book and a and b are constants.

Using the company’s prediction regarding the expected impact of price increases, write down another equation involving N comma a comma b and  c

7b
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2 marks

Find the values of a spaceand b.

7c
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1 mark

Hence write down the model used for the number of books sold annually.

7d
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2 marks

The income, £ I, the company generates from sales of the book is given by

I equals c left parenthesis a plus b c right parenthesis

where a and b take the same values as in part (b).

Find the price the company should charge per book in order to maximise their income.

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8a
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2 marks

A slow-motion camera is used to record the motion of a cricket ball projected directly upwards from ground level.  The motion of the cricket ball is modelled by the function

h left parenthesis t right parenthesis equals 11 t minus 4.9 t squared space space space space space space space t greater than 0

where h metres is the height of the cricket ball above ground level after t seconds. The camera will capture the cricket balls’ motion whilst it remains at least 3 m above ground level.

Find the maximum height the cricket ball reaches and how long it takes to reach this point.

8b
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3 marks

Find the length of time for which the camera will capture the cricket ball’s motion.

8c
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1 mark

The slow-motion camera slows real-time down 200 times.  So, 1 second of real-time recorded footage would be 200 seconds of slow-motion footage.

How many seconds of slow-motion cricket ball footage will the camera capture?

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9a
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1 mark

The graphs below show two different suggested models for estimating the value of a brand-new car costing £10 000.  a  is the car’s age in years and £V is the car’s value in thousands

q9a-2-12-modelling-with-functions-edexcel-a-level-pure-maths-hard

Other than when brand new, at what age do the two models predict the same value for the car?

9b
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1 mark

At what age does Model 1 predict the car will become worthless?

9c
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1 mark

State a problem with using Model 1 for older cars.

9d
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1 mark

State a problem with using Model 1 for older cars.

9e
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2 marks

Compare the two models for estimating a car’s value at 8 years old and higher. Suggest which model you think is more realistic, justifying your answer.

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10a
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2 marks

A fountain is designed so that water is projected over a walkway. The path of the water is modelled by the formula

y equals x left parenthesis 5 minus x right parenthesis space space space space 0 less or equal than x less or equal than 5

where x is the horizontal distance in meters from the base of the fountain at ground level and y is the height of the water in metres.

Sketch the graph of the model, labelling any intersections with the coordinate axes.

10b
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3 marks

The average person is 1.7 m tall and needs a ground width of 1.2 m in order to walk comfortably.

Work out the maximum number of average sized people that can comfortably walk under the fountain side by side without getting wet.

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11a
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4 marks

A skydiver jumps from a moving aircraft at an altitude of 10 000 feet directly above a fixed point, O, on the ground.  The trajectory of the skydiver is then modelled by the function 

      h left parenthesis x right parenthesis equals 3048 minus 0.5 x squared

where h space m s the height of the skydiver above the ground and x space m is the horizontal distance along the ground from point O.

 

(i)
Explain why the value 10 000 does not appear in the model.
(ii)

Find the height of the skydiver at the point when they have covered a ground distance of 60 space m.

(iii)
How much ground has the skydiver covered when they land on the ground?
11b
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3 marks
(i)
Using a straight line between the start and end points of the skydive, find an approximation for the distance travelled by the skydiver.
(ii)

Given that the skydive took 84 seconds, find an approximation for the average speed of the skydiver.  State whether this is an underestimate or an overestimate.

 

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12a
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6 marks

The Blue Blades Aerobatic Display Team use white and blue smoke trails as part of their flying display.

At the start of the display, each aeroplane holds a tank of 360 litres of white smoke fluid and a separate tank of blue smoke fluid.

A function of the form

       w equals fraction numerator k over denominator t space plus space p end fraction minus q

is used to model the amount of white smoke fluid remaining in its tank,  wlitres, after time t seconds of use.  k,p and q are positive constants.

(i) Given that  q space equals space 90 and it takes 50 seconds for the amount of white smoke fluid in

its tank to halve, find the values of k and p.

(ii) Hence find the number of minutes of use that a tank of 360 litres of white smoke

fluid will last for.

12b
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3 marks

A similar model used for the blue smoke is as follows

      b space equals space fraction numerator 8400 space over denominator t space plus space 40 end fraction space minus space 30

Find the initial amount of blue smoke fluid in the tank, and determine how many minutes of use the blue smoke fluid will last for.

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13
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6 marks

The revenue (money received before expenses, tax, etc.) a company makes from selling a particular item is calculated by multiplying the selling price by the number of items sold.

Last year Toys2Go sold 120 000 doll’s houses at a price of £80 each.

Toys2Go wish to maximise their revenue from sales of the doll’s houses this year by increasing the price.  However a price increase will also lead to a reduction in the number of doll’s houses sold.

Toys2Go expect that 5000 fewer doll’s houses will be sold this year compared to last year for every £5 increase they make to the price.

Show that the expected revenue for this year, £R , would be

      R space equals space 10 space 000 space 000 space minus space 25 space 000 open parentheses x space minus space 4 close parentheses squared

where is the number of £5 increases Toys2Go make to the price of a doll’s house.

Hence find the price Toys2Go should sell the doll’s house for this year in order to maximise revenue. State the revenue expected.

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14a
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1 mark

A manufacturer claims their flask will keep a hot drink warm for up to 8 hours.

In this sense, warm is considered  to be 40 degree C or higher.

It is assumed a hot drink has an initial temperature of 80 degree space C.

A linear model of the temperature, T degree space C comma spaceinside the flask t hours from when a hot drink is first made is of the form

      T space equals space a space plus space b t

where a and b are constants.

write down the value of a.

14b
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2 marks

Assuming that a hot drink has a temperature of 40 degree space C after 8 hours, find the value of b.

14c
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1 mark

When does the model predit the temperature has decreased by 20 degree space C.

14d
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1 mark

Suggest  a problem if the model were to be used for values of t larger than 8.

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1a
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2 marks

It has often been said, to avoid relegation from football’s Premier League, teams should aim to score at least 40 points in a season.  Each team plays 38 games, a win is rewarded with 3 points and a draw with 1 point.  No points are awarded for a loss.

Using W and D spaceas the number of wins and draws respectively, write down an inequality describing the number of points a team should aim for to avoid relegation.

1b
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2 marks

Another condition on W spaceand D is W plus D less or equal than 38.

(i)

      Briefly explain why this second condition arises.

(ii)

Explain why W greater or equal than 0 spaceand D greater or equal than 0 must also be conditions.

1c
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2 marks

A team has won 3 games and drawn 5 after playing 19 games.
Write down an updated inequality for the number of points required during the remainder of the season in order to avoid relegation.

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2a
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2 marks

The leakage rate of water from a pipe, L space 1 space straight s to the power of negative 1 end exponent (litres per second), is directly proportional to flow rate, straight s space straight m space straight s to the power of negative 1 end exponent (meters per second), which is the speed of the water flowing through the pipe.
It was observed that the leakage rate was 0.31 space straight s to the power of negative 1 end exponent when the flow rate was 0.6 space straight m space straight s to the power of negative 1 end exponent.

Show that the constant of proportionality is 0.5 and hence write down an equation connecting L spaceand straight s.

2b
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2 marks

Find the leakage rate when the flow rate is 1.8 space straight m space straight s to the power of negative 1 end exponent.

2c
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2 marks

The flow rate is reduced should the leakage rate exceed 0.8 space l space s to the power of negative 1 end exponent.
Find the maximum possible flow rate before it is reduced.

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3a
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1 mark

A soft ball is thrown upwards from the top of a 10 m tall building.
The height, h m of the ball above the ground after t seconds is modelled by the function

h left parenthesis t right parenthesis equals H plus 7.8 t minus 4.9 t squared space space space space space space space space space space space t greater than 0

Write down the value of H.

3b
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2 marks

Find the height of the ball after 2 seconds.

3c
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2 marks

At what time is the ball at the same height as it was when thrown?

3d
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2 marks

How long does it take for the ball to first hit the ground?

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4a
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2 marks

The number of cases of an unknown virus are modelled by the formula

V equals 225 over open parentheses d minus 15 close parentheses squared space space space space space space space space space space space 0 less or equal than d less than 15

where d spaceis the number of days after the first case was discovered and V is the total number of cases to date.

Find the number of cases after 10 days.

4b
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2 marks

Sketch a graph of V against d, clearly marking the coordinate where the graph intersects the V-axis.

4c
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1 mark

A politician says that after 5 days there were 2.25 cases of the virus.
Comment on the politician’s statement.

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5a
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1 mark

A machine produces toys at a rate dependent on the machine’s temperature. The more extreme the temperature of the machine, the less productive it is.

The productivity of the machine is measured by

P left parenthesis T right parenthesis equals 0.02 T left parenthesis 5 minus T right parenthesis left parenthesis T minus 60 right parenthesis space space space space space space space space space space space space space space space space space space space space 5 less or equal than T less or equal than 60

where P is the number of toys produced per hour and T space degree C is the temperature of the machine.

Find the number of toys produced per hour when the temperature of the machine is 36 space degree C.

5b
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2 marks

Show that P left parenthesis T right parenthesis equals 1.3 T squared minus 0.02 T cubed minus 6 T.

5c
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2 marks

Find the temperatures at which productivity is 80 toys per hour.

5d
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1 mark

Productivity is at its peak when the temperature of the machine is 41 space degree C.
Find the number of toys produced per hour at this temperature.

5e
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1 mark

Suggest a reason why the machine cannot operate below 5 space degree C.

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6a
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1 mark

A patient takes a new medication at midday.  The amount of drug, D mg, remaining in their bloodstream h hours after midday is modelled by the formula

D equals 0.04 plus 0.16 h minus 0.04 h squared space space space space space space space space space space 0 less or equal than h less or equal than 4

What amount of drug is already naturally occurring in the patient’s bloodstream before taking any medication?

6b
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2 marks

Without doing any calculations, explain how you can tell that the time the drug reaches its highest level is after 2 hours, at 2 pm.

6c
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2 marks

It is safe for the patient to take more medication once the amount of drug in their bloodstream falls below 0.16 mg. When is the earliest time the patient can take a second dose of the medication?

6d
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1 mark

At what time does the amount of drug in the patient’s bloodstream return to its natural level?

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7a
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2 marks

Last year, a company sold 10 000 books, at a price of £20 each.

The previous year, the company sold 10 250 books at a price of £19 each.

The company wants to increase the price again this year and uses the formula N equals a plus b c to model the number of books sold annually. Where N is the number of books sold, c is the selling price of the book and a and b are constants.

Write down two equations for the sales from the past two years and hence find the constants a and b.

7b
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1 mark

Write down the model for the number of books sold annually.

7c
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2 marks

This year the company intends raising the price of the book by £2.

(i)

How many books should the company expect to sell this year?

(ii)
Calculate the income the company should expect from book sales this year.
7d
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2 marks

Work out the book sale income for last year and the year before.

7e
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2 marks

Briefly comment on the relationship between number of books sold and the income for this three-year period.

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8a
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2 marks

A cricket ball is projected directly upwards from ground level.  The motion of the cricket ball is modelled by the function

h left parenthesis t right parenthesis equals 13 t minus 4.9 t squared space space space space space space space space space space space space space t greater than 0

where h metres is the height of the cricket ball above ground level after t seconds.

Find the times at which the cricket ball is exactly 3 m above the ground.

8b
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1 mark

For how long is the cricket ball at least 3 m above the ground?

8c
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2 marks

A player catches the cricket ball (on its way down) at a height of 0.8 m above the ground.

How long was the cricket ball in the air for?

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9a
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1 mark

The graph below shows a suggested model for estimating the value of a brand new car costing £18 000. a  is the car’s age in years and £V is the car’s value in thousands.

q9a-2-12-modelling-with-functions-edexcel-a-level-pure-maths-medium

Use the model to predict a car’s value after 5 years.

9b
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1 mark

A car (of the same make and model) is seen advertised for sale at £3250.
How old would you expect the car to be?

9c
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1 mark

In terms of its value, what does the model suggest is a disadvantage of buying a brand new car?

9d
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1 mark

A 16 year old car was scrapped, and the owner received £200 for spare parts.
State a problem with using this model for very old cars.

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10a
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1 mark

A fountain is designed so that water is projected over a walk way. The path of the water is modelled by the formula

y equals x left parenthesis 4 minus x right parenthesis space space space space 0 less or equal than x less or equal than 4

where x is the horizontal distance in meters from the base of the fountain at ground level and y is the height of the water in metres.

Sketch a graph of the model, labelling any intersections with the coordinate axes.

10b
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2 marks

Find the height of the water at a ground width of 1.3 m.

10c
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2 marks

The average person is 1.7 m tall and needs a ground width of 1.2 m in order to walk comfortably.

Find the distance at ground level between the two points where the water height is 1.7 m.

10d
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1 mark

Use your answer to part (c) to work out the maximum number of average sized people that can comfortably walk under the fountain side by side without getting wet.

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11
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7 marks

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12a
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3 marks

The White Blades Aerobatic Display Team use white smoke trails as part of their flying display.

 At the start of the display, each aeroplane holds a tank containing 180 litres of white smoke fluid.

The amount of white smoke fluid remaining in an aeroplane’s tank is inversely proportional to the total time that white smoke has been produced plus 25 seconds.

Find an equation to model the relationship between the amount of white smoke fluid left in the tank, W, and the total time, t, that white smoke has been produced.

12b
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3 marks

12c
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1 mark

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13a
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3 marks

13b
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2 marks

13c
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3 marks

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14a
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3 marks

14b
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3 marks

14c
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2 marks

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15a
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1 mark

15b
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2 marks

15c
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2 marks

15d
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1 mark

15e
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1 mark

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1a
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2 marks

It has often been said, to avoid relegation from football’s Premier League, teams should aim to score at least 40 points in a season.  Each team plays 38 games, a win is rewarded with 3 points and a draw with 1 point.  No points are awarded for a loss.

Using Wand D spaceas the number of wins and draws respectively, write down two inequalities. One relating to the number of points a team needs to avoid relegation and one relating to the number of games played.

1b
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1 mark

Explain why W greater or equal than 0 and D greater or equal than 0 must also be conditions related to the problem.

1c
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2 marks

On the axes below, display all the inequalities from parts (a) and (b).

q1c-2-12-modelling-with-functions-edexcel-a-level-pure-maths-veryhard

1d
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1 mark

Using your graph, or otherwise, determine the minimum number of games a team can win and still avoid relegation. Justify your answer by showing how the team can still accumulate at least 40 points.

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2a
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4 marks

The leakage rate of water from a pipe, L space straight l space straight s to the power of negative 1 end exponent (litres per second), is directly proportional to the cube root of the flow rate, straight s space straight m space straight s to the power of negative 1 end exponent (meters per second), which is the speed of the water flowing through the pipe.
It was observed that the leakage rate was 0.63 space straight l space straight s to the power of negative 1 end exponent when the flow rate was 0.729 space straight m space straight s to the power of negative 1 end exponent.

Find the flow rate when the leakage rate is 0.21 space straight l space straight s to the power of negative 1 end exponent.

2b
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2 marks

An alternative model for the leakage rate is L equals 0.4 straight s.
Apart from when there is no leak find a flow rate and a leakage rate for when both models predict the same result.

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3a
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1 mark

A soft ball is thrown upwards from the top of a building.
The height, h m of the ball above the ground after t seconds is modelled by the function

h left parenthesis t right parenthesis equals H plus 9.8 t minus 4.9 t squared space space space space space space space space t greater than 0

What does the constant H indicate in the function?

3b
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2 marks

At what time is the ball at the same height as when it was thrown?

3c
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2 marks

Find in terms of H comma space how long it takes for the ball to first hit the ground.

3d
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2 marks

How much longer does a ball launched from a 25 m tall building stay in the air for compared to a ball launched from a 15 m tall building?

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4a
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2 marks

The number of cases of an unknown virus are modelled by the formula

V equals 625 over open parentheses d minus 25 close parentheses squared          0 less or equal than d less than 25

where d is the number of days after the first case was discovered and V is the total number of cases to date.

Sketch a graph of V spaceagainst d, clearly marking the coordinate where the graph intersects the V-axis.

4b
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1 mark

Explain why the model is not appropriate for d greater or equal than 25.

4c
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1 mark

Scientists suggest the model is not accurate after 18 days.
Suggest a reason why.

4d
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2 marks

The model is a graph transformation of the graph with equation y equals begin inline style a over x squared end style, where a equals 625. Describe this transformation.

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5a
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1 mark

A machine produces toys at a rate dependent on the machine’s temperature. The more extreme the temperature of the machine, the less productive it is.

The productivity of the machine is measured by

P left parenthesis T right parenthesis equals 0.01 T left parenthesis 22 minus T right parenthesis left parenthesis T minus 60 right parenthesis space space space space space space space space space space space space space space space 22 less or equal than T less or equal than 60

where P is the number of toys produced per hour and T space degree C is the temperature of the machine.

Suggest a reason for the temperature condition 22 less or equal than T less or equal than 60.

5b
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2 marks

Sketch the graph of the machine’s productivity for 22 less or equal than T less or equal than 60.

5c
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2 marks

Using your graph to estimate the temperature at which productivity is at its peak, calculate the number of toys produced at this temperature.

5d
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2 marks

The temperature of the machine rises by 6 °C for every hour it is in constant use.
In order to prevent a breakdown the machine is switched off once the temperature exceeds 52 °C.

(i)
Assuming the machine is at 22 °C when it is switched on, find the number of hours the machine can run continuously for before having to be switched off.

(ii)
Suggest a reason why it may be better to switch the machine off before it reaches this temperature?

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6a
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1 mark

A patient takes some medication at midday.  The amount of drug, D mg, remaining in their bloodstream h hours after midday is modelled by the formula

D equals 0.06 plus 0.21 h minus a h squared space space space space space space spacewhere a is a constant

What amount of drug is already naturally occurring in the patient’s bloodstream before taking any medication?

6b
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2 marks

After six hours the amount of drug in the patient’s bloodstream has returned to its natural level. Find the value of .

6c
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2 marks

It is particularly dangerous for the patient to take any other medication whilst the amount of this drug in their bloodstream remains at 0.3 mg or higher.
Find the times between which the patient should refrain from taking any other medication.

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7a
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2 marks

Last year, a company sold 15 000 copies of a book, at a price of £25 each.
This year, the company wants to increase the price of the book and predicts that for every £2.50 increase in price, annual sales will drop by 750 books.

The formula N equals a plus b c is used to model the number of books sold annually. Where N is the number of books sold, c is selling price of the book and a and b are constants.

Find the values of a spaceand b spaceand hence, write down the model used for the number of books sold annually.

7b
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1 mark

Write down an equation for I, where £ I is the annual income generated from sales of the book.

7c
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3 marks

Find the maximum amount of income the company should get from sales of the book this year, the price they should charge for each book and the number of books they should sell.

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8a
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2 marks

A slow-motion camera is used to record the motion of a cricket ball projected directly upwards from ground level.  The motion of the cricket ball is modelled by the function

h left parenthesis t right parenthesis equals 15 t minus 4.9 t squared space space space space space space space space space t greater than 0

where h metres is the height of the cricket ball above ground level after t seconds.
The camera will capture the cricket balls’ motion whilst it is between heights of 2 m and 4 m above the ground.

Find the maximum height the cricket ball reaches and how long it takes to reach this point.

8b
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2 marks

Find the times between which the camera will capture the cricket ball’s motion.

8c
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2 marks

The slow-motion camera slows real-time down 200 times.  So, 1 second of real-time recorded footage would be 200 seconds of slow-motion footage.

How many seconds of slow-motion cricket ball footage will the camera capture?

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9a
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1 mark

The graphs below show two different suggested models for estimating the value of a brand new car costing £12 000. a  is the car’s age in years and £V is the car’s value in thousands.

q9a-2-12-modelling-with-functions-edexcel-a-level-pure-maths-veryhard

Other than when brand new, at what age do the two models predict the same value for the car?

9b
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3 marks

The value of one particular car was tracked and recorded every two years as shown in the table below.

Age 2 4 6 8 10 12
Value £10 200 £7 600 £4 300 £2 000 £1 600 £1 200

The car was scrapped after 14 years with the value for parts given as £200.

Based on the data given above, compare the two models in terms of their suitability and comment on which you think is a more suitable model. Justify your choices.

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10a
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2 marks

A fountain is designed so that water is projected over a walk way. The path of the water is modelled by the formula

y equals x left parenthesis 6 minus x right parenthesis space space space space space space space space space space space space space 0 less or equal than x less or equal than 6

where x is the horizontal distance in meters from the base of the fountain at ground level and y is the height of the water in metres.

Sketch the graph of the model, labelling any intersections with the coordinate axes and the maximum point the fountain reaches.

10b
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2 marks

The average person is 1.7 m tall and needs a ground width of 1.2 m in order to walk comfortably.

Work out the maximum number of average sized people that can comfortably walk under the fountain side by side without getting wet.

10c
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5 marks

Using a model of the form y equals x left parenthesis A minus x right parenthesis, work out the minimum ground width of the fountain required in order for three average sized people to comfortably walk side by side under the fountain without getting wet.

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11
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6 marks

A skydiver jumps from a moving aircraft directly above a fixed point, O, on the ground.  The trajectory of the skydiver is then modelled by the function

      h left parenthesis x right parenthesis equals 4000 minus 0.1 x squared

where h space m is the height of the skydiver above the ground and  is the horizontal x space m distance along the ground from point O.

(i)
From what altitude does the skydiver jump from the aircraft?
(ii)
Find the height of the skydiver at the point when they have covered a ground
distance of 60 space m.
(iii)
How much ground has the skydiver covered when they land on the ground?
(iv)
The landing zone is in the shape of a circle with an area of 90 space 000 space m squared .
Assuming the skydiver lands exactly in the centre of this circle determine whether the point O lies within the landing zone or not.

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12a
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5 marks

The Red Blades Aerobatic Display Team use white and red smoke trails as part of their flying display.

 

At the start of the display, each aeroplane holds a tank containing 270 litres of white smoke fluid, and another tank containing 90 litres of red smoke fluid.

 

A function of the form

      w equals fraction numerator 30 space 375 over denominator t plus p end fraction minus q

is proposed to model the amount of white smoke fluid remaining in its tank, w litres, after time t seconds of its use. p and q are positive constants.

The white smoke fluid runs out after 6 minutes of use.
Find the values of p and q.

 

 are positive constants.

 

12b
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4 marks

A similar model for the red smoke fluid is also used

      r equals fraction numerator 36 space 000 over denominator t plus 240 end fraction minus 60

Find the time at which the white smoke fluid tank and the red smoke fluid tank have the same amount of fluid in them. Comment on your answer.

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13
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6 marks

The revenue (money received before expenses, tax, etc.) a company makes from selling a particular item is calculated by multiplying the selling price by the number of items sold.

 

Last year Toys-were-Us sold 3 million computer games at a price of £50 each.

 

Toys-were-Us wish to maximise their revenue from sales of computer games this year.
However a change in price could lead to a change in the number of computer games sold.

 

Toys-were-Us expect there will be a change of 500 000 computer games sold for every £2 change they make in the price.

So for every £2 increase to the price, sales drop by 500 000.

For every £2 decrease in price, sales increase by 500 000.

 

Find the price Toys-were-Us should sell computer games at this year in order to maximise revenue.  State the expected number of games they will sell and the expected revenue.

 

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