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1

Solve the following initial value problems (IVP) using the Laplace transform method.

(i)

d2y

dy

-9 +3=0,

2

dt

dt

y(0) = 1 ,

dy

(0) = 1

dt

(equation 1)

[12 marks for this part]

(ii)

-2

dT

= 10 – 4T ,

dt

T (0) = 2

(equation 2)

[8 marks for this part]

[Total 20 marks for this question]

2

Your exam could be marked with a range of possible grades, simplified as on the following

state diagram:

Fail

0.5

0.1

0.6

Acceptible

Acceptable

Standard

0.5

0.1

Genius

0.8

0.1

To begin with the chances are that you will pass with a standard result. Each 45 minutes (the

time allocated for each answer) your grade could go up or down to a neighbouring grade or

stay put.

There is a chance that your Genius is so great that the examiner cannot appreciate what you

are saying and incorrectly marks it as nonsense.

What is your grade most likely to be after the 3 hour exam?

What (if any) are the steady state results for a group of 14 students?

[10 marks]

[10 marks]

[Total 20 marks for this question]

3

The Cannon Hill furniture Company produces tables and chairs. Each table takes five hours

of labor from the carpentry department and three hours of labor from the finishing

department. Each chair requires three hours of carpentry and two hours of finishing. During

the current week, 230 hours of carpentry time are available and 90 hours of finishing time.

Each table produced gives a profit of £85 and each chair a profit of £55. How many chairs

and tables should be made?

[Total 20 marks for this question]

4

A very new European Rapid Reaction Force for Fire has been created today and begins

operation between three Countries A, B and C. Its main resource is a super aircraft

Funderbird2 with a massive water cannon that even carries a small mini-submarine for

fighting fires at sea. Unfortunately, it can only be in one Country at a time.

A

0.1

0.1

0.4

B

0.3

C

What is the matrix representation of this problem?

[5 marks]

The probabilities of going to a fire in another Country, given that the force are in a given

Country to begin with, are shown on the sketch above (any resemblance to any particular

Country or Nation is purely coincidence and not intended). The probabilities were obtained

as a weekly average using statistics for fires over many recorded weeks.

When will the plane take up permanent residence in one Country?

[15 marks]

[Total 20 marks for this question]

5

0.1

A

0.5

0.1

0.3

B

C

0.05

Figure 3. State diagram for Question 3.

For the state diagram shown on Figure 3 the initial state is C and the time steps are days.

Find the following:

i)

Find the stochastic transitional probability matrix

[5 marks for this part]

ii)

Find the probability of being in state C after two time steps

[5 marks for this part]

iii)

What is the steady state probability of being in state A?

[8 marks for this part]

iv)

What is the steady state probability of being in either state B OR state

C?

[2 marks for this part]

[Total 20 marks for this question]

…

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