Differential Equations (CIE A Level Maths: Pure 3)

Exam Questions

4 hours32 questions
1a
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2 marks

Find the general solution to the differential equation

fraction numerator straight d x over denominator straight d t end fraction equals 9 t squared plus 4 t.

1b
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3 marks

Find the particular solution to the differential equation

fraction numerator straight d S over denominator straight d x end fraction equals 4 e to the power of 2 x end exponent

given that the graph of S against x passes through the point (0 , 5).

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2a
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2 marks

By separating the variables, show that the solution to the differential equation

fraction numerator straight d y over denominator straight d x end fraction equals 2 x y comma space space y greater than 0

can be found by solving

integral 1 over y space space straight d y equals integral 2 x space space d x.

2b
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3 marks

Show that the general solution to the differential equation in part (a) is

y equals e to the power of x squared plus c end exponent

where c is a constant.

2c
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2 marks

By letting  A equals e to the power of c, show that the general solution to the differential equation in part (a) can be written in the form

y equals A e to the power of x squared end exponent.

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3a
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1 mark

The velocity of a particle is given by the differential equation

fraction numerator d s over denominator d t end fraction equals 8 t plus 1

where s is the displacement of the particle in metres from a fixed point O at time t seconds. At time t equals 0 the particle is located at point O.

Write down the initial velocity of the particle.

3b
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2 marks

Use the method of separating variables to find an expression for the displacement of the particle from O after t seconds.

3c
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2 marks

Find  fraction numerator d squared s over denominator d t squared end fraction  and hence explain why the acceleration of the particle is constant.

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4a
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1 mark

The differential equation

fraction numerator straight d V over denominator straight d t space end fraction equals negative k V comma

is used to model the rate at which water is leaking from a container.

V is the volume of water in the container at time t seconds.
k is a constant.

Explain the use of a negative sign on the right-hand side of the differential equation, and the impact this has on the value of the constant k.

4b
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3 marks

Show that

integral 1 over V space d V equals k integral space straight d t

and hence find the general solution to the differential equation.

4c
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2 marks

Given that k equals 0.02 and the initial volume of the container is 300 litres, find the particular solution to the differential equation, giving your answer in the form

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

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5a
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3 marks

Given that y greater than 1, find the general solution to the differential equation

fraction numerator 1 over denominator y minus 1 end fraction space fraction numerator d y over denominator d x end fraction equals 6 x squared

giving your answer in the form

y equals A e to the power of straight f left parenthesis x right parenthesis end exponent plus 1.

5b
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3 marks

Given that y greater than negative 2, find the general solution to the differential equation

fraction numerator d y over denominator d x end fraction equals 9 left parenthesis y plus 2 right parenthesis x to the power of begin inline style 1 half end style end exponent

giving your answer in the form

y equals A e to the power of straight f open parentheses x close parentheses end exponent minus 2.

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6a
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4 marks

Use the given boundary condition to find the particular solution to the following differential equations:

fraction numerator d y over denominator d x end fraction equals sec squared x space           space space x equals straight pi over 3,    space y equals 2 square root of 3,

6b
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5 marks

sec space x space fraction numerator d y over denominator d x end fraction equals cosec space y space space space space   space space x equals pi over 2,     y equals 0,

giving your answer in the form  cos space y equals straight f left parenthesis x right parenthesis .

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7a
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3 marks

A large weather balloon is being inflated.

The rate of change of its volume, fraction numerator d V over denominator d t end fraction, where V m3 is the volume of the balloon t minutes after inflation commenced, is inversely proportional to its volume.

(i)
Form a differential equation to describe the relationship between V and t as the weather balloon is inflated.

(ii)
The rate of inflation of the balloon is 10 m3 min-1 when its volume is 20 m3.
Use this information to find the constant of proportionality.
7b
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3 marks

Show that the general solution to the differential equation found in part (a) is

V squared equals 400 t plus c

where c is a constant.

7c
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3 marks
(i)
When not in use, the weather balloon is stored flat and so can initially be considered to have a volume of 0 m3.
Use this information to find the particular solution to the differential equation.

(ii)
What is the volume of the balloon after 25 minutes?

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8a
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2 marks

A tree disease is spreading throughout a large forested area.

The differential equation

e to the power of negative k t end exponent space fraction numerator straight d N over denominator straight d t end fraction equals 2 over 5

where k is a positive constant, is used to model the number of infected trees, N, at a time t days after the disease was first discovered.

Show that

integral 5 space space straight d N equals integral 2 e to the power of k t end exponent space space straight d t.

8b
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2 marks

Hence show that

N equals fraction numerator 2 e to the power of k t end exponent over denominator 5 k end fraction plus c

where c is a constant.

8c
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4 marks

Given that k equals 0.1 and that when the disease was first discovered, four trees were infected.

Find the particular solution of the differential equation and use it to estimate the number of infected trees after 30 days.

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1a
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2 marks

Find the general solution to the differential equation

9 t squared minus 4 plus fraction numerator d x over denominator d t end fraction equals 0

1b
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3 marks

Find the particular solution to the differential equation

fraction numerator d V over denominator d x end fraction minus 4 equals 2 e to the power of x

given that the graph of V against x passes through the point (0, 3).

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2a
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3 marks

By separating the variables, show that the general solution to the differential equation

fraction numerator d y over denominator d x end fraction equals 4 x y comma space space space space space space space space space space space space space y greater than 0

can be written as

y equals e to the power of 2 x squared plus c end exponent

where c is the constant of integration.

2b
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2 marks
(i)
By renaming the constant ec as A, show that the general solution from part (a) can be written in the form

y equals A e to the power of 2 x squared end exponent

(ii)
Explain the significance of the value of A in that form of the general solution, and suggest what it might represent if the equation were being used to model a real- life problem.

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3
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4 marks

The velocity of a particle is given by the differential equation

fraction numerator d s over denominator d t end fraction equals 6 t squared minus 2 t plus 5

where s is the displacement of the particle in metres from a fixed point O, and t is the time in seconds. At time space t equals 0 spacethe particle is located at point O.

(i)
Write down the initial velocity of the particle.

(ii)
Find an expression for the displacement, s m, of the particle after t seconds.

(iii)
Find an expression for the acceleration, a m s-2, of the particle after t seconds.

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4a
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2 marks

A large container of water is leaking at a rate directly proportional to the volume of water in the container.

Using the variables V, for the volume of water in the container, and t, for time, write down a differential equation involving the term fraction numerator d V over denominator d t end fraction, for the volume of water in the container.

4b
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2 marks

The general solution to the differential equation in part (a) can be written in the form

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

where k is a positive constant.

(i)
State, in the context of the question, the significance of the constant A.
(ii)
Briefly explain where the negative sign in the solution comes from in the context of the question.

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5a
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4 marks

Given that y greater than 2, find the general solution to the differential equation

fraction numerator d y over denominator d x end fraction equals x squared left parenthesis y minus 2 right parenthesis

5b
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3 marks

Find the general solution to the differential equation

fraction numerator d y over denominator d x end fraction equals sin to the power of 2 space end exponent 2 y

giving your answer in the form x equals straight f left parenthesis y right parenthesis.

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6a
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6 marks

Find particular solutions to the following differential equations, using the given boundary conditions.

sin squared x fraction numerator d y over denominator d x end fraction equals cos squared y                space x equals straight pi over 4 comma space space space y equals 0 space

6b
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6 marks

e to the power of negative 3 x end exponent fraction numerator d y over denominator d x end fraction equals 2 e to the power of y space end exponent space space space space space space space space space space space space space space space space space space space x equals 0 comma space space space y equals 0

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7a
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3 marks

A large weather balloon is being inflated at a rate that is inversely proportional to its volume.

(i)
Using the variables V m3 for the volume of the balloon and t seconds for the time since inflation began, write down a differential equation to describe the relationship between V and t as the weather balloon is inflated.

(ii)
The rate of inflation of the balloon is 5 m3 s-1 when its volume is 48 m3.
Use this information to find the constant of proportionality.
7b
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4 marks

Find the general solution to the differential equation found in part (a).

7c
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3 marks
(i)
When not in use, the weather balloon is stored flat and so can initially be considered to have a volume of 0 m3.  Use this information to find the particular solution to the differential equation found in part (a).

(ii)
What is the volume of the balloon after 50 minutes?

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8a
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7 marks

A tree disease is spreading throughout a large forested area.

When the disease was first discovered, three trees were infected.

Ten days later ten trees were infected.

The differential equation

1 over t space fraction numerator d N over denominator d t end fraction equals k N

where k is a positive constant, is used to model the number of infected trees, N, at a time t days after the disease was first discovered.

Find the particular solution to the differential equation.

8b
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4 marks

Scientists believe the majority of the forest can be saved from infection if action is taken before 30 trees are infected.

Measured from the time when the disease was first discovered, how many days does the model predict the scientists have to take action in order to save the majority of the forest from infection?

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1a
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2 marks

Find the general solution to the differential equation

5 minus sin space 2 t space plus fraction numerator d x over denominator d t end fraction equals 0

1b
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3 marks

Find the particular solution to the differential equation

3 e to the power of 4 x end exponent minus fraction numerator d V over denominator d x end fraction equals 2

where the graph of V against x passes through the point (0, -4).

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2a
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5 marks

Show that the general solution to the differential equation

fraction numerator d y over denominator d x end fraction equals 3 x squared y comma space space space space space space space space space space space space space space space space space y not equal to 0


is

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


where A is a constant.

2b
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3 marks

On the same set of axes sketch the graphs of the solution for the instances where

(i)
the constant A is greater than 0
(ii)
the constant A is less than 0

In each case be sure to state where the graph intercepts the y-axis.

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3
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4 marks

The acceleration of a particle is given by the differential equation

fraction numerator d squared s over denominator d t squared end fraction equals 18 t minus 4

where s is the displacement of the particle in metres from a fixed reference point O, and t is the time in seconds.  

The particle starts its journey at point O at t equals 0 comma and has an initial velocity of 1 m s-1.

Find an expression for the displacement of the particle in terms of t.

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4a
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2 marks

A large container of water is leaking at a rate directly proportional to the volume of water in the container.

Defining any variables, write down a differential equation that describes how the volume of water in the container varies with time.

4b
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3 marks

By separating the variables, find the general solution to your differential equation from part (a).

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5a
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4 marks

Find the general solution to the differential equation

fraction numerator 2 y minus 1 over denominator 3 end fraction space fraction numerator d y over denominator d x end fraction equals x squared y squared minus x squared y comma space space space space space space space space space y greater than 1

5b
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4 marks

Find the general solution to the differential equation


3 fraction numerator d y over denominator d x end fraction equals fraction numerator cosec space y cubed over denominator space y to the power of space space 2 end exponent end fraction


giving your answer in the form x equals straight f left parenthesis y right parenthesis.

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6a
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5 marks

Show that the general solution to the differential equation

y cot space x space space fraction numerator d y over denominator d x end fraction equals y squared plus 3

can be written in the form

y squared plus 3 equals A space sec to the power of 2 space end exponent x

where A is a constant.

6b
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6 marks

Find the particular solution to the following differential equation, using the given boundary condition

e to the power of x squared end exponent space fraction numerator d y over denominator d x end fraction equals 2 x space cosec space 3 y space           space x equals 0 comma space space space y equals pi over 3

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7a
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2 marks

A large weather balloon is being inflated at a rate that is inversely proportional to the square of its volume.

Defining variables for the volume of the balloon (m3) and time (seconds) write down a differential equation to describe the relationship between volume and time as the weather balloon is inflated.

7b
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6 marks

Given that initially the balloon may be considered to have a volume of zero, and that after 400 seconds of inflating its volume is 600 m3, find the particular solution to your differential equation.

7c
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2 marks

Although it can be inflated further, the balloon is considered ready for release when its volume reaches 1250 m3.  If the balloon needs to be ready for a midday release, what is the latest time that it can start being inflated?

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8a
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7 marks

A bar of soap in the shape of a cuboid is placed in a bowl of warm water and its volume is recorded at regular intervals.  The water is maintained at a constant temperature.

Before being placed in the water the soap measures 3 cm by 6 cm by 10 cm.

Two minutes later the bar of soap measures 2.85 cm by 5.7 cm by 9.5 cm.

The rate of decrease in volume of the bar of soap is modelled as being directly proportional to its volume.

Defining any variables you use, find and solve a differential equation linking the volume of the bar of soap and time.

8b
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2 marks

What happens to the volume of the bar of soap for large values of t?
Briefly explain why this could be considered a criticism of the model.

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1a
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2 marks

Find the general solution to the differential equation

1 half sec squared space 3 t plus 2 fraction numerator d x over denominator d t end fraction equals 0

1b
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6 marks

Find the particular solution to the differential equation

2 x e to the power of 4 x end exponent minus 3 fraction numerator d V over denominator d x end fraction equals 1

where the graph of V against x passes through the point (0,2).

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2a
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5 marks

Show that the general solution to the differential equation

2 x fraction numerator d y over denominator d x end fraction equals 3 k x cubed y comma space space space space space space space space space space space space space space space y not equal to 0 space

is

y equals A e to the power of begin inline style 1 half end style k x cubed end exponent

where A and k are constants.

2b
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3 marks

On separate diagrams sketch a graph of the solution for x greater or equal than 0 in the instances when

(i)
the constant k is greater than 0,
(ii)
the constant k is less than 0.


On both diagrams state where the graph intercepts the y-axis.
You may assume A greater than 0 in both cases.

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3
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4 marks

The acceleration of a particle moving in a straight line is given by the differential equation

fraction numerator straight d squared s over denominator straight d t squared end fraction equals 18 sin space 3 t space minus 2

where s m is the displacement of the particle relative to a fixed point O, and t is the elapsed time in seconds.  The particle starts its journey with a velocity of  -6 m s-1, from a point 50 m in the positive direction from the point O.

Find an expression for the displacement, s, of the particle in terms of t.

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4
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8 marks

A large container of water is leaking at a rate directly proportional to the square of the volume of water in the container.

(i)
Given that the initial volume of water in the container is 4000 litres and that after 10 minutes the volume of water in the container has dropped by 30%, write down and solve a differential equation connecting the volume, V, of water in the container to the time, t.

 

(ii)
What does your solution predict will happen to the volume of water in the container after a very long time?

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5a
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6 marks

Newton’s Law of Cooling states that the rate of cooling of an object is directly proportional to the difference between the object’s temperature and the ambient temperature (temperature of the object’s surroundings).

By setting up and solving an appropriate differential equation, show that

T equals T subscript a m b end subscript plus A e to the power of negative k t end exponent

where T °C is the temperature of the object, Tamb °C is the ambient temperature, t is time, and k greater than 0 and A are both constants.  You may assume in working out your solution that the ambient temperature is constant, and that the temperature of the object is greater than the ambient temperature.

5b
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4 marks

A meat processing factory must store its products at a temperature below -1 °C.

Due to the production process, products, before cooling, typically have a temperature between 5 °C and 10 °C.  

The company therefore has a policy that any products failing to cool to below -1 °C within 6 minutes of being processed must be discarded.

The factory stores its products in a freezer with a constant ambient temperature of -4 °C.

A product that has just finished being processed has a temperature of 7 °C and is immediately placed in the freezer.  One minute later its temperature has dropped to 4.7 °C.  Determine whether or not this product will need to be discarded.

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6
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5 marks

Find the general solution to the differential equation

fraction numerator d y over denominator d x end fraction equals 2 x y plus 2 x minus y minus 1 comma space space y greater than negative 1

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7a
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4 marks

Using the standard integral result

integral s e c space k theta space space d theta equals 1 over k ln open vertical bar space tan space open parentheses fraction numerator k theta over denominator 2 end fraction plus pi over 4 close parentheses space close vertical bar plus c

(where k is a constant, and c is a constant of integration), show that the solution to the differential equation

cos space x space fraction numerator d y over denominator d x end fraction equals cos space y space

with boundary conditionspace x equals 0y equals pi,  may be written in the form

open vertical bar tan space open parentheses y over 2 plus pi over 4 close parentheses space close vertical bar equals open vertical bar tan space open parentheses x over 2 plus pi over 4 close parentheses close vertical bar

7b
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6 marks
(i)
Show that the relationship between x and y in  open vertical bar tan space open parentheses y over 2 plus pi over 4 close parentheses close vertical bar space equals open vertical bar tan open parentheses space x over 2 plus pi over 4 close parentheses close vertical bar may also be expressed by the set of all equations of the forms

y equals x plus 2 n pi space space space space space space o r space space space space space space y equals negative x plus open parentheses 2 n minus 1 close parentheses pi

where n is an integer.

(ii)
Hence deduce that the particular solution to the differential equation in part (a), with the given boundary condition, is  y equals pi minus x .

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8a
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6 marks

A tree disease is spreading throughout a large forested area.

The rate of increase in the number of infected trees is modelled by the differential equation

fraction numerator straight d N over denominator straight d t end fraction equals k N open parentheses N minus 1 close parentheses comma space space space N greater than 1

where N is the number of infected trees, t is the time in days since the disease was first identified and k is a positive constant.

Solve the differential equation above, and show that the general solution can be written in the form

N equals fraction numerator 1 over denominator 1 minus A e to the power of k t end exponent end fraction

where A is a positive constant.

8b
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3 marks

Initially two trees were identified as diseased.
A fortnight later, 4 trees were infected.
Using this information, find the values of the constants A and k.

8c
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3 marks

By considering the solution to the differential equation along with the values of A and k found in part (b), suggest a range of values of  t  for which the model might be considered reliable.

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