Modelling with Logs & Exponentials (CIE A Level Maths: Pure 3)

Exam Questions

3 hours42 questions
1
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4 marks

State whether the following functions could represent exponential growth or exponential decay.

(i)

straight f open parentheses x close parentheses equals 5 straight e to the power of 2 x end exponent

(ii)

straight f open parentheses straight t close parentheses equals 100 straight e to the power of negative straight t end exponent

(iii)

straight f open parentheses straight a close parentheses equals 20 straight e to the power of negative ka end exponent space comma space space straight k greater than 0

(iv)
straight f open parentheses straight t close parentheses equals Ae to the power of kt space comma space space straight A comma straight k greater than 0

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2
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3 marks

Write the following in the form ,e to the power of k x end exponent where k is a constant and k greater than 0.

(i)

e to the power of 3 x end exponent cross times e to the power of 2 x end exponent

(ii)

5 to the power of x

(iii)
2 to the power of x

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3
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3 marks

Write the following in the form e to the power of negative k x end exponent, where k is a constant and k greater than 0.

(i)

e to the power of negative 2 x end exponent over e to the power of 4 x end exponent

(ii)

open parentheses 1 fifth close parentheses to the power of x

(iii)
open parentheses 1 half close parentheses to the power of x

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4
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3 marks

The diagram below shows a sketch of the graph of y equals e to the power of negative x end exponent.

On the diagram, add the graph of y equals e to the power of negative 2 x end exponent labelling the point at which the graph intersects the y-axis.

Write down the equation of any asymptotes on the graph.

q5-6-3-modelling-with-exponentials-and-logarithms-edexcel-a-level-pure-maths-easy

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5a
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3 marks

By taking logarithms (base e) of both sides show that the equation

y equals A e to the power of k x end exponent

can be written in the form ln space y equals k x plus ln space A

5b
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4 marks

Hence ...

(i)
write the equation y equals 2 e to the power of 0.01 x end exponent in the form ln space y equals k x plus ln space A.

(ii)
write the equation ln space y equals 0.3 x space plus space In space 5 in the form y equals A e to the power of k x end exponent.

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6a
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1 mark

In an effort to prevent extinction scientists released 24 rare birds into a newly constructed nature reserve.

The population of birds, within the reserve, is modelled by

B equals A e to the power of 0.4 t end exponent

B is the number of birds after t years of being released into the reserve.

A is a constant.

Write down the value of A.

6b
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2 marks

According to this model, how many birds will be in the reserve after 2 years?

6c
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2 marks

How many years after release will it take for the population of birds to double?

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7a
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1 mark

A simple model for the acceleration of a rocket, A space ms to the power of negative 2 end exponent, is given as 

A equals 10 e to the power of 0.1 t end exponent

where t is the time in seconds after lift-off.  

What is the meaning of the value 10 in the model?

7b
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2 marks

Find the acceleration of the rocket 15 seconds after lift-off.

7c
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3 marks

Find how long it takes for the acceleration to reach 100 space ms to the power of negative 2 end exponent.

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8a
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1 mark

An exponential growth model for the number of bacteria in an experiment is of the form N equals A e to the power of k t end exponent

N is the number of bacteria and t is the time in hours since the experiment began.

A are k constants.

A scientist records the number of bacteria every hour for 3 hours.

The results are shown in the table below.

 t,hours 0 1 2 3 4
 N, no. of bacteria 100 210 320 730 1580
 ln space N space (3SF) 4.61 5.35 5.77 6.59 7.37

Plot the observations on the graph below - plotting ln space N against t.

q8-6-3-modelling-with-exponentials-and-logarithms-edexcel-a-level-pure-maths-easy

8b
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2 marks

Using the points (0, 4.61) and (4, 7.37), find an equation for a line of best fit in the form ln space N equals m t plus ln space c, where m spaceand c are constants to be found.

8c
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2 marks

Hence estimate the values of A and k.

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1a
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1 mark

Write open parentheses begin inline style 1 third end style close parentheses to the power of x  in the form  e to the power of k x end exponent.

1b
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2 marks

Write open parentheses begin inline style 2 over 7 end style close parentheses to the power of t  in the form  e to the power of k t end exponent.
State whether this would represent exponential growth or exponential decay.

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2a
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1 mark

Write open parentheses begin inline style 7 over 10 end style close parentheses to the power of x  in the form  e to the power of negative k x end exponent.

2b
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2 marks

Sketch the graph of y equals open parentheses begin inline style 7 over 10 end style close parentheses to the power of x.
State the coordinates of the y-axis intercept.
Write down the equation of the asymptote.

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3a
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1 mark

By taking logarithms (base e) of both sides show that the equation

y equals 5 e to the power of 0.1 x end exponent

can be written as

ln space y equals 0.1 x plus ln space 5

3b
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2 marks

Given  y equals A e to the power of k x end exponent  and  ln space y equals 4.1 x plus ln space 8, find the values of A and k.

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4a
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1 mark

By taking logarithms (base 10) of both sides show that the equation

y equals 2 x to the power of 3.2 end exponent

can be written as

log space y equals 3.2 space log space x plus log space 2

4b
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2 marks

Given  y equals A x to the power of b and  log space y equals 1.8 space log space x plus log space 5, find the values of A and b.

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5a
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1 mark

By taking logarithms (base 2) of both sides show that the equation

y equals 3 cross times 2 to the power of 4 x end exponent

can be written as

log subscript 2 space y equals 4 x plus log subscript 2 space 3

5b
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2 marks

Given y equals A b to the power of k x end exponent  and log subscript 3 space y equals 5 x plus log subscript 3 space 7 , find the values of A comma b and k.

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6a
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1 mark

In an effort to prevent extinction scientists released some rare birds into a newly constructed nature reserve.

The population of birds, within the reserve, is modelled by

B equals 16 e to the power of 0.85 t end exponent

B is the number of birds after t years of being released into the reserve.

Write down the number of birds the scientists released into the nature reserve.

6b
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2 marks

According to this model, how many birds will be in the reserve after 3 years?

6c
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2 marks

How long will it take for the population of birds within the reserve to reach 500?

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7a
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1 mark

A simple model for the acceleration of a rocket,A space m s to the power of negative 2 end exponent , is given as

 A equals A subscript 0 e to the power of 0.2 t end exponent

where t is the time in seconds after lift-off.  A subscript 0 is a constant.

What does the constant A subscript 0 represent?

7b
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2 marks

After 10 seconds, the acceleration is 20 space m s to the power of negative 2 end exponent.
Find the value of A subscript 0.

7c
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2 marks

Find how long it takes for the acceleration of the rocket to reach 100 space ms to the power of negative 2 end exponent

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8a
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1 mark

Carbon-14 is a radioactive isotope of the element carbon.
Carbon-14 decays exponentially – as it decays it loses mass.
Carbon-14 is used in carbon dating to estimate the age of objects.

The time it takes the mass of carbon-14 to halve (called its half-life) is approximately 5700 years.

 A model for the mass of carbon-14, m g, in an object of age t years is

m equals m subscript 0 e to the power of negative k t end exponent

where m subscript 0 and k are constants.

For an object initially containing 100g of carbon-14, write down the value of m subscript 0.

8b
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2 marks

Briefly explain why, if m subscript 0 equals 100,m  will equal 50g  when t equals 5700 years.

8c
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2 marks

Using the values from part (b), show that the value of k spaceis 1.22 cross times 10 to the power of negative 4 end exponent to three significant figures.

8d
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2 marks

A different object currently contains 60g of carbon-14.
In 2000 years’ time how much carbon-14 will remain in the object?

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9a
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1 mark

An exponential growth model for the number of bacteria in an experiment is of the form N equals N subscript 0 space a to the power of k t end exponent. N is the number of bacteria and t is the time in hours since the experiment began.N subscript 0 comma a and k are constants.  A scientist records the number of bacteria at various points over a six-hour period.  The results are shown in the table below.

  t, hours 0 2 4 6

  N, no. of bacteria

100 180 340 620

  log subscript 3 space N (3SF)

4.19

4.73

5.31 5.85

Plot the observations on the graph below - plotting log subscript 3 space N against t.

q9a-6-3-modelling-with-exponentials-and-logarithms-edexcel-a-level-pure-maths-easy

9b
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2 marks

Using the points (0, 4.19) and (6, 5.85), find an equation for a line of best fit in the form log subscript 3 space N equals m t plus log subscript 3 space c, where m spaceand c are constants to be found.

9c
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2 marks

The equation N equals N subscript 0 space a to the power of k t end exponent  can be written in the form  log subscript a space N equals k t plus log subscript a space N subscript 0.
Use your answer to part (b) to estimate the values of , N subscript 0 comma a space, and k.

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10a
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2 marks

An exponential model of the form  D equals A e to the power of negative k t end exponent  is used to model the amount of a pain-relieving drug (D mg/ml) there is in a patient’s bloodstream, t hours after the drug was administered by injection. A  and k are constants.

The graph below shows values of In D plotted against t with a line of best fit drawn.

q10a-6-3-modelling-with-exponentials-and-logarithms-edexcel-a-level-pure-maths-medium

(i)        Use the graph and line of best fit to estimate ln space D at time t equals 0.

(ii)       Work out the gradient of the line of best fit.

10b
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1 mark

Use your answers to part (a) to write down an equation for the line of best fit in the form ln space D equals m t plus ln space c,  where m and c are constants.

10c
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1 mark

Show that D equals A e to the power of negative k t end exponent can be rearranged to give ln space D equals negative k t plus ln space A

10d
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2 marks

Hence find estimates for the constants A spaceand k.

10e
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2 marks

Find the time when the amount of the pain-relieving drug in the patient’s bloodstream is 1.5 mg/ml.

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11a
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2 marks

A small company makes a profit of £2500 in its first year of business and £3700 in the second year.  The company decides they will use the model

P equals P subscript 0 space y to the power of k

to predict future years’ profits.

£P is the profit in the y to the power of t h end exponent year of business.

P subscript 0 and k are constants.

Write down two equations connecting P subscript 0 and k.

11b
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2 marks

Find the values of P subscript 0 and k.

11c
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2 marks

Find the predicted profit for years 3 and 4.

11d
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2 marks

Show that

P equals P subscript 0 space y to the power of k

can be written in the form

log space P equals log space P subscript 0 plus k space log space y

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1a
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2 marks

Write  open parentheses 3 over 5 close parentheses to the power of x in the form e to the power of k x end exponent , giving the value of k to three significant figures.

1b
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2 marks

Write open parentheses begin inline style 4 over 7 end style close parentheses to the power of 3 t end exponent  in the form e to the power of k t end exponent , giving the value of k to three significant figures.
State, and justify, whether this would represent exponential growth or decay.

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2a
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2 marks

Write open parentheses 0.7 close parentheses to the power of x plus 1 end exponent  in the form  A e to the power of negative k x end exponent.

2b
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2 marks

Sketch the graph of y equals open parentheses 0.7 close parentheses to the power of x plus 1 end exponent minus 3.
State the coordinates of the y-axis intercept.
Write down the equation of the asymptote.

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3a
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2 marks

Show that the equation

x equals 7 e to the power of negative 0.2 t end exponent

can be written as

ln space x equals ln space 7 minus 0.2 straight t

3b
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2 marks

Rewrite the equation ln space y equals 4.1 x plus ln space 8 in the form  y equals A e to the power of k x end exponent.

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4a
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2 marks

Show that the equation

y equals 2 x to the power of begin inline style 3 over 4 end style end exponent

can be written as

log space y equals 0.75 space log space x plus log space 2

4b
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2 marks

Rewrite the equation log space y equals 4.7 space log space x plus log space 12 in the form  y equals A x to the power of b.

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5a
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2 marks

Show that the equation

y equals 0.1 cross times 2 to the power of 0.01 x end exponent

can be written as

log subscript 2 space y equals 0.01 x minus log subscript 2 space 10

5b
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2 marks

Rewrite the equation log subscript 3 space y equals 6.3 x plus log subscript 3 space 4 in the form  y equals A b to the power of k x end exponent.

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6a
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1 mark

Scientists introduced a small number of rare breed deer to a large wildlife sanctuary.

The population of deer, within the sanctuary, is modelled by

D equals 20 e to the power of 0.1 t end exponent

D is the number of deer after t years of first being introduced to the sanctuary.

Write down the number of deer the scientists introduced to the sanctuary.

6b
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2 marks

How many years does it take for the deer population to double?

6c
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1 mark

Give one criticism of the model for population growth.

6d
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2 marks

The scientists suggest that the population of deer are separated after either 25 years or when their population exceeds 400.
Find the earliest time the deer should be separated.

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7a
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2 marks

A simple model for the acceleration of a rocket,A space ms to the power of negative 2 end exponent , is given as

A equals 5 e to the power of k t end exponent

where t is the time in seconds after lift-off. k  is a constant.

After 4 seconds the acceleration of the rocket is 10 space ms to the power of negative 2 end exponent.
Find the value of k.

7b
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2 marks

Find the time at which the acceleration of the rocket has increased by 200%.

7c
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2 marks

Sketch the graph of the acceleration of the rocket, against time, stating the coordinates of the point that shows the initial acceleration of the rocket.

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8a
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2 marks

Carbon-14 is a radioactive isotope of the element carbon.
Carbon-14 decays exponentially – as it decays it loses mass.
Carbon-14 is used in carbon dating to estimate the age of objects.

The time it takes the mass of carbon-14 to halve (called its half-life) is approximately 5700 years

 A model for the mass of carbon-14, y g, in an object originally containing 100 g,
at time t years is

y equals 100 e to the power of negative k t end exponent

where k is a constant.

Find the value of k, giving your answer to three significant figures.

8b
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2 marks

The object is considered as having no radioactivity once the mass of carbon-14 it contains falls below 0.5 g. Find out how old the object would have to be, to be considered non-radioactive.

8c
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2 marks

A different object currently contains 25g of carbon-14.
In 500 years’ time how much carbon-14 will remain in the object?

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9a
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2 marks

An exponential growth model for the number of bacteria in an experiment is of the form

N equals N subscript 0 space a to the power of k t end exponent

N is the number of bacteria and t is the time in hours since the experiment began. N subscript 0 comma a and k are constants.

 A scientist records the number of bacteria at various points over a six-hour period.
The results are in the table below.

 t, hours 0 2 4 6
 N, no. of bacteria 200 350 600 1100

Plot the observations on the graph below - plotting log subscript 5 space N against t.
Draw a line of best fit.

q10a-6-3-modelling-with-exponentials-and-logarithms-edexcel-a-level-pure-maths-hard

9b
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2 marks

Find an equation for your line of best fit in the form log subscript 5 space N equals m t plus log subscript 5 c.

 

9c
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2 marks

Estimate the values of ,N subscript 0 ,a and k.

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10a
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2 marks

An exponential model of the form

D equals A e to the power of negative k t end exponent

is used to model the amount of a pain-relieving drug (D mg/ml) there is in a patient’s bloodstream, t hours after the drug was administered by injection.  A and k are constants.

The graph below shows values of ln space D plotted against t

q11a-6-3-modelling-with-exponentials-and-logarithms-edexcel-a-level-pure-maths-hard

Using the points marked P and Q, find an equation for the line of best fit, giving your answer in the form ln space D equals m t plus ln space c, where m and c are constants to be found.

10b
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2 marks

Hence find estimates for the constants A and space k

10c
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2 marks

The patient is allowed a second injection of the drug once the amount of drug in the bloodstream falls below 1% of the initial dose.
Find, to the nearest minute, how long until the patient is allowed a second injection of the drug.

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11a
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2 marks

The annual profits, in thousands of pounds, of a small company in the first 4 years of business are given in the table below.

  a comma spaceyears in business 1 2 3 4
  P,annual profit  £3100 £4384 £5369 £6200

Using this data the company uses the model

P equals P subscript 1 a to the power of k

to predict future years’ profits. P subscript 1  and k are constants.

Use data from the table to find the values of P subscript 1 and k.

11b
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2 marks

Show that log space P equals k space log space a plus log space P subscript 1, where P subscript 1 and k take the values found in part (a).

11c
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1 mark

State a potential problem with using the model to predict the profit in the company’s 12th year of business.

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1a
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1 mark

Write  open parentheses 0.8 close parentheses to the power of x in the form e to the power of k x end exponent , giving the value of k to three significant figures.

1b
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2 marks
(i)

Write  open parentheses begin inline style 2 over 3 end style close parentheses to the power of 4 t plus 1 end exponent  in the form  A e to the power of k t end exponent, giving the values of A and k to three significant
figures where necessary.

(ii)

State, and justify, whether this would represent exponential growth or decay.

(iii)
Write down the initial value of A e to the power of k t end exponent.

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2
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4 marks

Sketch the graph of  y equals open parentheses 3 over 5 close parentheses to the power of 2 x plus 1 end exponent minus 4.

State the coordinates of any points where the graph intercepts the coordinate axes.

Write down the equations of any asymptotes.

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3a
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2 marks

Rewrite the equation ln space x equals 2 t plus ln space 6 in the form  x equals A e to the power of k t end exponent.

3b
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2 marks

Sketch the graph of ln space equals 2 t plus ln space 6 by plotting  ln space x  against t.

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4a
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2 marks

Rewrite the equation  y equals 3.6 x to the power of negative 0.4 end exponent  in the form log space y equals log space A minus b log space x

4b
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2 marks

Sketch the graph of log space y against  log space x.

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5a
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3 marks

Rewrite the equation y equals 2 over 3 cross times 5 to the power of negative 0.2 x end exponent  in the form log subscript b space y equals log subscript b space p minus q x  where b is an integer and p and q are rational numbers.

5b
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2 marks

Sketch the graph of log subscript b space y against x.

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6a
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2 marks

Scientists introduced a small number of apes into a previously unpopulated forest.

The population of apes in the forest is modelled by

A equals 16 e to the power of k m end exponent

where A is the number of apes after m months of first being introduced to the forest.

State, with a reason, whether you would expect the value of k to be positive or negative.

6b
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2 marks

After 8 months, the number of apes in the forest has increased by 50%.
Find the value of k.

6c
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2 marks

Scientists believe the forest cannot sustain a population of apes greater than 3000.
What length of time is the model for the population of the apes reliable for?

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7a
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4 marks

A manufacturer claims their flask will keep a hot drink warm for up to 7 hours.

In this sense, warm is considered to be 50 degree straight C or higher.

Assuming a hot drink is made at 85 degree straight C and its temperature inside the flask is 50 degree straight C after exactly 7 hours, find:

(i)
a linear model for the temperature of the drink inside the flask of the form T equals a plus b t, and

(ii)
an exponential model for the temperature of the drink inside the flask of the form T equals A e to the power of negative k t end exponent

where T degree straight C is the temperature of the drink in the flask after t hours and a comma b comma A spaceand k are constants.

7b
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2 marks

Compare the rate of change of the temperature of the drink inside the flask of both models after 3 hours.

7c
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1 mark

A user of the flask suggests that hot drinks are only kept warm for 5 hours.
Suggest a reason why the user’s experience may not be up to the claims of the manufacturer.

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8a
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2 marks

A simple model for the acceleration of a rocket, A space ms to the power of negative 1 end exponent, is given as

A equals R e to the power of k t end exponent

where t is the time in seconds after lift-off.  R and k are constants.

Negative time is often used in rocket launches as a way of counting down until lift off. Despite this the model above is still not suitable for use with negative  t spacevalues
Briefly explain why.

8b
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3 marks

After 5 seconds the acceleration of the rocket is 12 space ms to the power of negative 2 end exponent and after 20 seconds its acceleration is 50 space ms to the power of negative 2 end exponent.  Find the values of R spaceand k.

8c
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1 mark

A space enthusiast suggests that a linear model (of the form A equals R plus c t) would be more suitable.
Using the figures in (b), explain why the enthusiast’s model is unrealistic.

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9a
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1 mark

Carbon-14 is a radioactive isotope of the element carbon.
Carbon-14 decays exponentially – as it decays it loses mass.
Carbon-14 is used in carbon dating to estimate the age of objects.

The time it takes carbon-14 to halve (called its half-life) is approximately 5700 years.

A model for the mass of carbon-14, m g, in an object, at time t years is

m equals M subscript 0 space e to the power of negative k t end exponent

where M subscript 0 space end subscriptand k are constants.

Briefly explain the meaning of the constant M subscript 0.

9b
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3 marks

Find the value of k, giving your answer in the form fraction numerator ln space a over denominator b end fraction, where a spaceand b are integers to be found.

9c
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2 marks

An object currently contains 200 g of carbon-14. In 20 000 years’ time, how much carbon-14, to the nearest gram, remains in the object?

9d
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3 marks

The half-life of carbon-14 is believed to only be accurate to ±40 years.
A fossilised bone currently contains 3 cross times 10 to the power of negative 6 end exponent g of carbon-14.
It is estimated the bone would have originally contained 1 cross times 10 to the power of negative 2 end exponent g of carbon-14.

Find upper and lower estimates for the age of the bone, giving your answers to two significant figures.

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10a
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3 marks

An exponential growth model for the number of bacteria in an experiment is of the form

N equals N subscript 0 a to the power of k t end exponent

N is the number of bacteria and t is the time in hours since the experiment began. N subscript 0 comma a and k  are constants.

A scientist records the number of bacteria at various points over a six-hour period.
The results are in the table below.

  t, hours

0 1.5 3

4.5

6

  N, no. of bacteria

120 190 360 680 1230

By plotting log subscript 2 space N against t, drawing a line of best fit and finding its equation, estimate the values of N subscript 0,a , and k.

q10a-6-3-modelling-with-exponentials-and-logarithms-edexcel-a-level-pure-maths-veryhard

10b
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2 marks

What does the model predict for the value of N after twelve hours?
Comment on the reliability of this prediction.

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11a
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3 marks

An exponential model of the form

D equals A e to the power of negative k t end exponent

is used to model the concentration of a pain-relieving drug (D mg/ml) in a patient’s bloodstream t hours after the drug was administered by injection.  A and k are constants.

The graph below shows values of ln space D plotted against t

q11a-6-3-modelling-with-exponentials-and-logarithms-edexcel-a-level-pure-maths-veryhard

Find estimates for the constants A spaceand k.

11b
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2 marks

Find the time, to the nearest minute, at which the rate of decrease of the concentration of the drug in the patient’s bloodstream is 12 mg/ml/hour.

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12a
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3 marks

The annual profits, in thousands of pounds, of a small company in the first 4 years of business are given in the table below.

  a, years in business 1 2 3 4
  log space P£ P spaceis annual profit) 3.74 3.86 3.94 4.01

Using this data the company uses the model

P equals P subscript 1 a to the power of k

to predict future years’ profits. P subscript 1  and k are constants.

Use the results in the table to estimate the values of P subscript 1 spaceand k.

12b
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1 mark

Many new companies make a loss in their first year of business.
Briefly explain why, in such circumstances, a model of the form used above would not be suitable.

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