Modelling with Exponentials & Logarithms (Edexcel A Level Maths: Pure): Exam Questions

5 hours57 questions

12 marks

A new smartphone was released by a company.

The company monitored the total number of phones sold, $n$ , at time $t$ days after the phone was released.

The company observed that, during this time, the rate of increase of $n$ was proportional to $n$ .

Use this information to write down a suitable equation for $n$ in terms of $t$ .

(You do not need to evaluate any unknown constants in your equation.)

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4 marks

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

(i) $f (x) = 5 e^{2 x}$

(ii) $f (t) = 100 e^{- t}$

(iii) $f (a) = 20 e^{- ka}, k > 0$

(iv) $f (t) = {Ae}^{kt}, A, k > 0$

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3 marks

Write the following in the form , $e^{k x}$ where $k$ is a constant and $k > 0$ .

(i) $e^{3 x} \times e^{2 x}$

(ii) $5^{x}$

(iii) $2^{x}$

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3 marks

Write the following in the form $e^{- k x}$ , where $k$ is a constant and $k > 0$ .

(i) $\frac{e^{- 2 x}}{e^{4 x}}$

(ii) ${(\frac{1}{5})}^{x}$

(iii) ${(\frac{1}{2})}^{x}$

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3 marks

The diagram below shows a sketch of the graph of $y = e^{- x}$ .

On the diagram, add the graph of $y = e^{- 2 x}$ 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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3 marks

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

$y = A e^{k x}$

can be written in the form $\ln y = k x + \ln A$

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4 marks

Hence ...

(i) write the equation $y = 2 e^{0.01 x}$ in the form $\ln y = k x + \ln A$ .

(ii) write the equation $\ln y = 0.3 x + In 5$ in the form $y = A e^{k x}$ .

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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

$B = A e^{0.4 t}$

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

$A$ is a constant.

Write down the value of $A$ .

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2 marks

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

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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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1 mark

A simple model for the acceleration of a rocket, $A {ms}^{- 2}$ , is given as

$A = 10 e^{0.1 t}$

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

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

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2 marks

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

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3 marks

Find how long it takes for the acceleration to reach $100 {ms}^{- 2}$ .

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1 mark

An exponential growth model for the number of bacteria in an experiment is of the form $N = A e^{k t}$

$N$ 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
$\ln N$ (3SF)	4.61	5.35	5.77	6.59	7.37

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

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

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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 $\ln N = m t + \ln c$ , where $m$ and $c$ are constants to be found.

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2 marks

Hence estimate the values of $A$ and $k$ .

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1a1 mark

Coffee is poured into a cup.

The temperature of the coffee, $H$ ℃, $t$ minutes after being poured into the cup is modelled by the equation

$H = A e^{- B t} + 30$

where $A$ and $B$ are constants.

Initially, the temperature of the coffee was 85 ℃.

State the value of $A$ .

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3 marks

Initially, the coffee was cooling at a rate of 7.5 ℃ per minute.

Find a complete equation linking $H$ and $t$ , giving the value of $B$ to 3 decimal places.

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4 marks

In a simple model, the value, $£ V$ , of a car depends on its age, $t$ , in years.

The following information is available for car $A$

its value when new is $£ 20 000$
its value after one year is $£ 16 000$

Use an exponential model to form, for car $A$ , a possible equation linking $V$ with $t$ .

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2 marks

The value of car $A$ is monitored over a 10-year period.

Its value after 10 years is $£ 2 000$

Evaluate the reliability of your model in light of this information.

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2c1 mark

The following information is available for car $B$

it has the same value, when new, as car $A$
its value depreciates more slowly than that of car $A$

Explain how you would adapt the equation found in (a) so that it could be used t model the value of car $B$ .

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3a1 mark

In this question you must show all your stages of working.

Solutions relying entirely on calculator technology are not acceptable.

The air pressure, $P$ kg/cm², inside a car tyre, $t$ minutes from the instant when the tyre developed a puncture is given by the equation

$\begin{matrix} P = k + 1.4 e^{- 0.5 t} & t \in ℝ & t \geq 0 \end{matrix}$

where $k$ is a constant.

Given that the initial air pressure inside the tyre was $2.2$ kg/cm² state the value of $k$ .

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3 marks

From the instant when the tyre developed the puncture, find the time taken for the air pressure to fall to $1$ kg/cm²

Give your answer in minutes to one decimal place.

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2 marks

Find the rate at which the air pressure in the tyre is decreasing exactly $2$ minutes from the instant when the tyre developed the puncture.

Give your answer in kg/cm² per minute to 3 significant figures.

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4a2 marks

The height, $h$ metres, of a plant, $t$ years after it was first measured, is modelled by the equation

$h = 2.3 - 1.7 e^{- 0.2 t} t \in ℝ t \geq 0$

Using the model, find the height of the plant when it was first measured,

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3 marks

Show that, exactly 4 years after it was first measured, the plant was growing at approximately 15.3 cm per year.

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4c1 mark

According to the model, there is a limit to the height to which this plant can grow.

Deduce the value of this limit.

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5a1 mark

The owners of a nature reserve decided to increase the area of the reserve covered by trees.

Tree planting started on 1st January 2005.

The area of the nature reserve covered by trees, $A$ km², is modelled by the equation

$A = 80 - 45 e^{c t}$

where $c$ is a constant and $t$ is the number of years after 1st January 2005.

Using the model, find the area of the nature reserve that was covered by trees just before tree planting started.

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4 marks

On 1st January 2019 an area of 60 km²of the nature reserve was covered by trees.

Use this information to find a complete equation for the model, giving your value of $c$ to 3 significant figures.

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5c1 mark

On 1st January 2020, the owners of the nature reserve announced a long-term plan to have 100 km² of the nature reserve covered by trees.

State a reason why the model is not appropriate for this plan.

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1 mark

Write ${(\frac{1}{3})}^{x}$ in the form $e^{k x}$ .

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2 marks

Write ${(\frac{2}{7})}^{t}$ in the form $e^{k t}$ .
State whether this would represent exponential growth or exponential decay.

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1 mark

Write ${(\frac{7}{10})}^{x}$ in the form $e^{- k x}$ .

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2 marks

Sketch the graph of $y = {(\frac{7}{10})}^{x}$ .
State the coordinates of the $y$ -axis intercept.
Write down the equation of the asymptote.

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1 mark

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

$y = 5 e^{0.1 x}$

can be written as

$\ln y = 0.1 x + \ln 5$

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2 marks

Given $y = A e^{k x}$ and $\ln y = 4.1 x + \ln 8$ , find the values of $A$ and $k$ .

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1 mark

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

$y = 2 x^{3.2}$

can be written as

$\log y = 3.2 \log x + \log 2$

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2 marks

Given $y = A x^{b}$ and $\log y = 1.8 \log x + \log 5$ , find the values of $A$ and $b$ .

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10a

1 mark

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

$y = 3 \times 2^{4 x}$

can be written as

$\log_{2} y = 4 x + \log_{2} 3$

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10b

2 marks

Given $y = A b^{k x}$ and $\log_{3} y = 5 x + \log_{3} 7$ , find the values of $A, b$ and $k$ .

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11a

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

$B = 16 e^{0.85 t}$

$B$ 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.

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11b

2 marks

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

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11c

2 marks

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

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12a

1 mark

A simple model for the acceleration of a rocket, $A m s^{- 2}$ , is given as

$A = A_{0} e^{0.2 t}$

where $t$ is the time in seconds after lift-off. $A_{0}$ is a constant.

What does the constant $A_{0}$ represent?

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12b

2 marks

After 10 seconds, the acceleration is $20 m s^{- 2}$ . Find the value of $A_{0}$ .

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12c

2 marks

Find how long it takes for the acceleration of the rocket to reach $100 {ms}^{- 2}$

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13a

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

$m = m_{0} e^{- k t}$

where $m_{0}$ and $k$ are constants.

For an object initially containing 100g of carbon-14, write down the value of $m_{0}$ .

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13b

2 marks

Briefly explain why, if $m_{0} = 100$ , $m$ will equal $50$ g when $t = 5700$ years.

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13c

2 marks

Using the values from part (b), show that the value of $k$ is $1.22 \times 10^{- 4}$ to three significant figures.

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13d

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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14a

1 mark

An exponential growth model for the number of bacteria in an experiment is of the form $N = N_{0} a^{k t}$ . $N$ is the number of bacteria and $t$ is the time in hours since the experiment began. $N_{0},$ $a$ 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
$\log_{3} N$ (3SF)	4.19	4.73	5.31	5.85

Plot the observations on the graph below - plotting $\log_{3} N$ against $t$ .

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

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14b

2 marks

Using the points (0, 4.19) and (6, 5.85), find an equation for a line of best fit in the form $\log_{3} N = m t + \log_{3} c$ , where $m$ and $c$ are constants to be found.

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14c

2 marks

The equation $N = N_{0} a^{k t}$ can be written in the form $\log_{a} N = k t + \log_{a} N_{0}$ .
Use your answer to part (b) to estimate the values of , $N_{0}, a$ , and $k$ .

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15a

2 marks

An exponential model of the form $D = A e^{- k t}$ 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 $\ln D$ at time $t = 0$ .

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

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15b

1 mark

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

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15c

1 mark

Show that $D = A e^{- k t}$ can be rearranged to give $\ln D = - k t + \ln A$

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15d

2 marks

Hence find estimates for the constants $A$ and $k$ .

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15e

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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16a

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

$P = P_{0} y^{k}$

to predict future years’ profits.

$£$ $P$ is the profit in the $y^{t h}$ year of business.

$P_{0}$ and $k$ are constants.

Write down two equations connecting $P_{0}$ and $k$ .

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16b

2 marks

Find the values of $P_{0}$ and $k$ .

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16c

2 marks

Find the predicted profit for years 3 and 4.

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16d

2 marks

Show that

$P = P_{0} y^{k}$

can be written in the form

$\log P = \log P_{0} + k \log y$

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4 marks

The mass, $A$ kg, of algae in a small pond, is modelled by the equation

$A = p q^{t}$

where $p$ and $q$ are constants and $t$ is the number of weeks after the mass of algae was first recorded.

Data recorded indicates that there is a linear relationship between $t$ and $\log_{10} A$ given by the equation

$\log_{10} A = 0.03 t + 0.5$

Use this relationship to find a complete equation for the model in the form

$A = p q^{t}$

giving the value of $p$ and the value of $q$ each to 4 significant figures.

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1b2 marks

With reference to the model, interpret

(i) the value of the constant $p$ ,

(ii) the value of the constant $q$ .

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3 marks

Find, according to the model,

(i) the mass of algae in the pond when $t = 8$ , giving your answer to the nearest $0.5$ kg,

(ii) the number of weeks it takes for the mass of algae in the pond to reach $4$ kg.

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1d1 mark

State one reason why this may not be a realistic model in the long term.

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4 marks

A scientist is studying the growth of two different populations of bacteria.

The number of bacteria, $N$ , in the first population is modelled by the equation

$N = A e^{k t} t \geq 0$

where $A$ and $k$ are positive constants and $t$ is the time in hours from the start of the study.

Given that

there were 1000 bacteria in this population at the start of the study
it took exactly 5 hours from the start of the study for this population to double

find a complete equation for the model.

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2 marks

Hence find the rate of increase in the number of bacteria in this population exactly 8 hours from the start of the study. Give your answer to 2 significant figures.

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3 marks

The number of bacteria, $M$ , in the second population is modelled by the equation

$M = 500 e^{1.4 k t} t \geq 0$

where $k$ has the value found in part (a) and $t$ is the time in hours from the start of the study.

Given that $T$ hours after the start of the study, the number of bacteria in the two different populations was the same, find the value of $T$ .

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4 marks

A quantity of ethanol was heated until it reached boiling point.

The temperature of the ethanol, $θ$ °C, at time $t$ seconds after heating began, is modelled by the equation

$θ = A - B e^{- 0.07 t}$

where $A$ and $B$ are positive constants.

Given that

the initial temperature of the ethanol was 18°C
after 10 seconds the temperature of the ethanol was 44°C

find a complete equation for the model, giving the values of $A$ and $B$ to 3 significant figures.

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3b2 marks

Ethanol has a boiling point of approximately 78°C

Use this information to evaluate the model.

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4a2 marks

The time, $T$ seconds, that a pendulum takes to complete one swing is modelled by the formula

$T = a l^{b}$

where $l$ metres is the length of the pendulum and $a$ and $b$ are constants.

Show that this relationship can be written in the form

$\log_{10} T = b \log_{10} l + \log_{10} a$

How did you do?

3 marks

Graph with x-axis labelled log₁₀ I and y-axis labelled log₁₀ T. Points at (-0.7, 0) and (0.21, 0.45) are connected by a straight line. — **Figure 3**

A student carried out an experiment to find the values of the constants $a$ and $b$ .

The student recorded the value of $T$ for different values of $l$ .

Figure 3 shows the linear relationship between $\log_{10} l$ and $\log_{10} T$ for the student's data. The straight line passes through the points $(- 0.7, 0)$ and $(0.21, 0.45)$

Using this information, find a complete equation for the model in the form

$T = a l^{b}$

giving the value of $a$ and the value of $b$ , each to 3 significant figures.

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4c1 mark

With reference to the model, interpret the value of the constant $a$ .

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2 marks

Write ${(\frac{3}{5})}^{x}$ in the form $e^{k x}$ , giving the value of $k$ to three significant figures.

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2 marks

Write ${(\frac{4}{7})}^{3 t}$ in the form $e^{k t}$ , giving the value of $k$ to three significant figures.
State, and justify, whether this would represent exponential growth or decay.

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2 marks

Write ${(0.7)}^{x + 1}$ in the form $A e^{- k x}$ .

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2 marks

Sketch the graph of $y = {(0.7)}^{x + 1} - 3$ .
State the coordinates of the $y$ -axis intercept.
Write down the equation of the asymptote.

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2 marks

Show that the equation

$x = 7 e^{- 0.2 t}$

can be written as

$\ln x = \ln 7 - 0.2 t$

How did you do?

2 marks

Rewrite the equation $\ln y = 4.1 x + \ln 8$ in the form $y = A e^{k x}$ .

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2 marks

Show that the equation

$y = 2 x^{\frac{3}{4}}$

can be written as

$\log y = 0.75 \log x + \log 2$

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2 marks

Rewrite the equation $\log y = 4.7 \log x + \log 12$ in the form $y = A x^{b}$ .

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2 marks

Show that the equation

$y = 0.1 \times 2^{0.01 x}$

can be written as

$\log_{2} y = 0.01 x - \log_{2} 10$

How did you do?

2 marks

Rewrite the equation $\log_{3} y = 6.3 x + \log_{3} 4$ in the form $y = A b^{k x}$ .

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10a

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

$D = 20 e^{0.1 t}$

$D$ 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.

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10b

2 marks

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

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10c

1 mark

Give one criticism of the model for population growth.

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10d

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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11a

2 marks

A simple model for the acceleration of a rocket, $A {ms}^{- 2}$ , is given as

$A = 5 e^{k t}$

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

After 4 seconds the acceleration of the rocket is $10 {ms}^{- 2}$ .
Find the value of $k$ .

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11b

2 marks

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

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11c

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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12a

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

$y = 100 e^{- k t}$

where $k$ is a constant.

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

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12b

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.

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12c

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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13a

2 marks

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

$N = N_{0} a^{k t}$

$N$ is the number of bacteria and $t$ is the time in hours since the experiment began. $N_{0}, a$ 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 $\log_{5} N$ against $t$ .
Draw a line of best fit.

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

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13b

2 marks

Find an equation for your line of best fit in the form $\log_{5} N = m t + \log_{5} c$ .

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13c

2 marks

Find the value of $N_{0}$ and estimate the value of $a^{k}$ .

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14a

2 marks

An exponential model of the form

$D = A e^{- k t}$

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 $\ln D$ 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 $\ln D = m t + \ln c$ , where $m$ and $c$ are constants to be found.

How did you do?

14b

2 marks

Hence find estimates for the constants $A$ and $k$

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14c

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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15a

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.

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

Using this data the company uses the model

$P = P_{1} a^{k}$

to predict future years’ profits. $P_{1}$ and $k$ are constants.

Use data from the table to find the values of $P_{1}$ and $k$ .

How did you do?

15b

2 marks

Show that $\log P = k \log a + \log P_{1}$ , where $P_{1}$ and $k$ take the values found in part (a).

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15c

1 mark

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

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1a1 mark

A scientist is studying the number of bees and the number of wasps on an island.

The number of bees, measured in thousands, $N_{b}$ , is modelled by the equation

$N_{b} = 45 + 220 e^{0.05 t}$

where $t$ is the number of years from the start of the study.

According to the model, find the number of bees at the start of the study.

How did you do?

3 marks

According to the model, show that, exactly $10$ years after the start of the study, the number of bees was increasing at a rate of approximately $18$ thousand per year.

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4 marks

The number of wasps, measured in thousands, $N_{w}$ , is modelled by the equation

$N_{w} = 10 + 800 e^{- 0.05 t}$

where $t$ is the number of years from the start of the study.

When $t = T$ , according to the models, there are an equal number of bees and wasps.

Find the value of $T$ to $2$ decimal places.

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3 marks

A research engineer is testing the effectiveness of the braking system of a car when it is driven in wet conditions.

The engineer measures and records the braking distance, $d$ metres, when the brakes are applied from a speed of $V$ km h^-1.

Graphs of $d$ against $V$ and $\log_{10} d$ against $\log_{10} V$ were plotted.

The results are shown below together with a data point from each graph.

Graph with an upward curve showing a point labelled (30, 20). Axes are labelled "d" (vertical) and "O V" (horizontal). — **Figure 5**

Graph with axes labelled log base 10 of d and V. A line intersects the vertical axis at (0, -1.77), indicating a logarithmic relationship. — **Figure 6**

Explain how Figure 6 would lead the engineer to believe that the braking distance should be modelled by the formula

$d = k V^{n}$ where $k$ and $n$ are constants

with $k \approx 0.017$ .

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3 marks

Using the information given in Figure 5, with $k = 0.017$ , find a complete equation for the model giving the value of $n$ to 3 significant figures.

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3 marks

Sean is driving this car at 60 km h^-1 in wet conditions when he notices a large puddle in the road 100 m ahead. It takes him 0.8 seconds to react before applying the brakes.

Use your formula to find out if Sean will be able to stop before reaching the puddle.

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3a2 marks

Graph showing a line segment with points at (0, 3) and (10, 2.79). Axes are labelled t and log₁₀V , with arrows indicating direction. — **Figure 2**

The value, $V$ pounds, of a mobile phone, $t$ months after it was bought, is modelled by

$V = a b^{t}$

where $a$ and $b$ are constants.

Figure 2 shows the linear relationship between $\log_{10} V$ and $t$ .

The line passes through the points $(0, 3)$ and $(10, 2.79)$ as shown.

Using these points find the initial value of the phone.

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3 marks

Find a complete equation for $V$ in terms of $t$ , giving the exact value of $a$ and giving the value of $b$ to $3$ significant figures.

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2 marks

Exactly $2$ years after it was bought, the value of the phone was $£ 320$

Use this information to evaluate the reliability of the model.

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4 marks

The value, £ $V$ , of a vintage car $t$ years after it was first valued on 1st January 2001, is modelled by the equation

$V = A p^{t}$ where $A$ and $p$ are constants

Given that the value of the car was £32 000 on 1st January 2005 and £50 000 on 1st January 2012

(i) find $p$ to 4 decimal places,

(ii) show that $A$ is approximately 24 800.

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4b2 marks

With reference to the model, interpret

(i) the value of the constant $A$ ,

(ii) the value of the constant $p$ .

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4 marks

Using the model, find the year during which the value of the car first exceeds £100 000.

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3 marks

Graph with axes labelled "log_10(h)" vertically and "log_10(m)" horizontally. A downward sloping line starts at 2.25 on the vertical axis and goes to the right. — **Figure 2**

The resting heart rate, $h$ , of a mammal, measured in beats per minute, is modelled by the equation

$h = p m^{q}$

where $p$ and $q$ are constants and $m$ is the mass of the mammal measured in kg.

Figure 2 illustrates the linear relationship between $\log_{10} h$ and $\log_{10} m$

The line meets the vertical $\log_{10} h$ axis at $2.25$ and has a gradient of $- 0.235$

Find, to 3 significant figures, the value of $p$ and the value of $q$ .

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3 marks

A particular mammal has a mass of 5 kg and a resting heart rate of 119 beats per minute.

Comment on the suitability of the model for this mammal.

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5c1 mark

With reference to the model, interpret the value of the constant $p$ .

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1 mark

Write ${(0.8)}^{x}$ in the form $e^{k x}$ , giving the value of $k$ to three significant figures.

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2 marks

(i) Write ${(\frac{2}{3})}^{4 t + 1}$ in the form $A e^{k t}$ , 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 $A e^{k t}$ .

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4 marks

Sketch the graph of $y = {(\frac{3}{5})}^{2 x + 1} - 4$ .

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

Write down the equations of any asymptotes.

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2 marks

Rewrite the equation $\ln x = 2 t + \ln 6$ in the form $x = A e^{k t}$ .

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2 marks

Sketch the graph of $\ln = 2 t + \ln 6$ by plotting $\ln x$ against $t$ .

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2 marks

Rewrite the equation $y = 3.6 x^{- 0.4}$ in the form $\log y = \log A - b \log x$

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2 marks

Sketch the graph of $\log y$ against $\log x$ .

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10a

3 marks

Rewrite the equation $y = \frac{2}{3} \times 5^{- 0.2 x}$ in the form $\log_{b} y = \log_{b} p - q x$ where $b$ is an integer and $p$ and $q$ are rational numbers.

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10b

2 marks

Sketch the graph of $\log_{b} y$ against $x$ .

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11a

2 marks

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

The population of apes in the forest is modelled by

$A = 16 e^{k m}$

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.

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11b

2 marks

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

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11c

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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12a

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 $50 ° C$ or higher.

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

(i) a linear model for the temperature of the drink inside the flask of the form $T = a + b t$ , and

(ii) an exponential model for the temperature of the drink inside the flask of the form $T = A e^{- k t}$

where $T ° C$ is the temperature of the drink in the flask after $t$ hours and $a, b, A$ and $k$ are constants.

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12b

2 marks

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

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12c

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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13a

2 marks

A simple model for the acceleration of a rocket, $A {ms}^{- 1}$ , is given as

$A = R e^{k t}$

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 $t$ values Briefly explain why.

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13b

3 marks

After 5 seconds the acceleration of the rocket is $12 {ms}^{- 2}$ and after 20 seconds its acceleration is $50 {ms}^{- 2}$ . Find the values of $R$ and $k$ .

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13c

1 mark

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

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14a

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

$m = M_{0} e^{- k t}$

where $M_{0}$ and $k$ are constants.

Briefly explain the meaning of the constant $M_{0}$ .

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14b

3 marks

Find the value of $k$ , giving your answer in the form $\frac{\ln a}{b}$ , where $a$ and $b$ are integers to be found.

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14c

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?

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14d

3 marks

The half-life of carbon-14 is believed to only be accurate to ±40 years.
A fossilised bone currently contains $3 \times 10^{- 6}$ g of carbon-14.
It is estimated the bone would have originally contained $1 \times 10^{- 2}$ 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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15a

3 marks

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

$N = N_{0} a^{k t}$

$N$ is the number of bacteria and $t$ is the time in hours since the experiment began. $N_{0}, a$ 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 $\log_{2} N$ against $t$ , drawing a line of best fit and finding its equation, estimate the values of $N_{0}$ , $a$ , and $k$ .

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

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15b

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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16a

3 marks

An exponential model of the form

$D = A e^{- k t}$

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 $\ln D$ plotted against $t$

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

Find estimates for the constants $A$ and $k$ .

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16b

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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17a

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
$\log P$ ( $£ P$ is annual profit)	3.74	3.86	3.94	4.01

Using this data the company uses the model

$P = P_{1} a^{k}$

to predict future years’ profits. $P_{1}$ and $k$ are constants.

Use the results in the table to estimate the values of $P_{1}$ and $k$ .

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17b

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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Modelling with Exponentials & Logarithms (Edexcel A Level Maths: Pure): Exam Questions

Proof

Proof

Proof by Contradiction

Algebra & Functions

Laws of Indices & Surds

Quadratics

Simultaneous Equations

Inequalities

Polynomials

Rational Expressions

Graphs of Functions

Functions

Transformations of Functions

Combinations of Transformations

Partial Fractions

Modelling with Functions

Further Modelling with Functions

Coordinate Geometry

Equation of a Straight Line

Circles

Sequences & Series

Binomial Expansion

General Binomial Expansion

Arithmetic Sequences & Series

Geometric Sequences & Series

General Sequences & Series

Modelling with Sequences & Series

Trigonometry

Basic Trigonometry

Trigonometric Functions

Trigonometric Equations

Radian Measure

Reciprocal & Inverse Trigonometric Functions

Compound & Double Angle Formulae

Further Trigonometric Equations

Trigonometric Proof

Modelling with Trigonometric Functions

Exponentials & Logarithms

Exponential & Logarithms

Laws of Logarithms

Modelling with Exponentials & Logarithms

Differentiation

Differentiation

Applications of Differentiation

Further Differentiation

Further Applications of Differentiation

Implicit Differentiation

Integration

Integration

Further Integration

Differential Equations

Parametric Equations

Parametric Equations

Calculus & Modelling with Parametric Equations

Numerical Methods

Numerical Methods

Modelling involving Numerical Methods

Vectors

Vectors in 2D

Vectors in 3D