Questions D2 - A-Level Maths

\cline { 2 - 5 } \multicolumn{1}{c\|}{}	$\mathbf { 1 }$	$\mathbf { 2 }$	$\mathbf { 3 }$	$\mathbf { 4 }$
$\mathbf { 1 }$	2	2	2	2
$\mathbf { 2 }$	1	1	3	4
$\mathbf { 3 }$	2	2	2	4
$\mathbf { 4 }$	2	2	3	3

Perform the fourth (final) iteration of the algorithm.
Explain how to use the final matrices to find the shortest distance and the shortest route from vertex $\mathbf { 1 }$ to vertex $\mathbf { 3 }$, and give the distance and route.
Draw the complete network of shortest distances.
Apply the nearest neighbour algorithm, starting at vertex $\mathbf { 1 }$, to your complete network of shortest distances. Give the Hamilton cycle it produces, its length, and the corresponding route through the original network.
By considering vertex 2 and its arcs, construct a lower bound for the length of the solution to the travelling salesperson problem in the original network.
Explain what you can deduce from your answers to parts (iv) and (v). 4 Noel is designing a hotel patio. It will consist of decking and paving.
Decking costs $\pounds 4$ per $\mathrm { m } ^ { 2 }$ and paving costs $\pounds 2$ per $\mathrm { m } ^ { 2 }$. He has a budget of $\pounds 2500$.
Noel prefers paving to decking, and he wants the area given to paving to be at least twice that given to decking. He wants to have as large a patio as possible.
Noel's problem is formulated as the following LP.
Let $x$ be the number of $\mathrm { m } ^ { 2 }$ of decking.
Let $y$ be the number of $\mathrm { m } ^ { 2 }$ of paving. $$\begin{aligned} \text { Maximise } \quad P & = x + y
\text { subject to } 2 x + y & \leqslant 1250
2 x - y & \leqslant 0
x & \geqslant 0
y & \geqslant 0 \end{aligned}$$
Use the simplex algorithm to solve this LP. Pivot first on the positive element in the $y$ column. Noel would like to have at least $200 \mathrm {~m} ^ { 2 }$ of decking.
Add a line corresponding to this constraint to your solution tableau from part (i), and modify the resulting table either for two-stage simplex or the big-M method. Hence solve the problem. Noel finally decides that he will minimise the annual cost of maintenance, which is given by $\pounds ( 0.75 x + 1.25 y )$, subject to the additional constraint that there is at least $1000 \mathrm {~m} ^ { 2 }$ of patio.
Starting from your solution to part (ii), use simplex to solve this problem.

OCR MEI D2 2008 June Q1

The Plain English Society presents an annual "Foot in Mouth" award for "a truly baffling comment". In 2004 it was presented to Boris Johnson MP for a comment on the $12 ^ { \text {th } }$ December 2003 edition of "Have I Got News For You":
"I could not fail to disagree with you less."
1. Explain why this can be rewritten as:
  "I could succeed in agreeing with you more."
2. Rewrite the comment more simply in your own words without changing its meaning.
Two switches are to be wired between a mains electricity supply and a light so that when the state of either switch is changed the state of the light changes (i.e. from off to on, or from on to off). Draw a switching circuit to achieve this. The switches are both 2-way switches, thus:
\includegraphics[max width=\textwidth, alt={}, center]{88acde67-e22b-478a-9145-48abe931beff-2_127_220_895_1054}
Construct a truth table to show the following. $$[ ( a \wedge b ) \vee ( ( \sim a ) \wedge ( \sim b ) ) ] \Leftrightarrow [ ( ( \sim a ) \vee b ) \wedge ( a \vee ( \sim b ) ) ]$$

OCR MEI D2 2008 June Q2

2 Jane has a house on a Mediterranean island. She spends eight weeks a year there, either visiting twice for four weeks each trip or four times for two weeks each trip. Jane is wondering whether it is best for her to fly out and rent a car, or to drive out.
Flights cost $\pounds 500$ return and car rental costs $\pounds 150$ per week.
Driving out costs $\pounds 900$ for ferries, road tolls, fuel and overnight expenses.

Draw a decision tree to model this situation. Advise Jane on the cheapest option. As an alternative Jane considers buying a car to keep at the house. This is a long-term alternative, and she decides to cost it over 10 years. She has to cost the purchase of the car and her flights, and compare this with the other two options. In her costing exercise she decides that she will not be tied to two trips per year nor to four trips per year, but to model this as a random process in which she is equally likely to do either.
Draw a decision tree to model this situation. Advise Jane on how much she could spend on a car using the EMV criterion.
Explain what is meant by "the EMV criterion" and state an alternative approach.

OCR MEI D2 2008 June Q3

3 The weights on the network represent distances.
\includegraphics[max width=\textwidth, alt={}, center]{88acde67-e22b-478a-9145-48abe931beff-3_401_702_315_683}

1. Apply Floyd's algorithm to the network to find the complete network of shortest distances, showing that the final matrices are as follows. \begin{center} \begin{tabular}{ | c | c | c | c | c | } \cline { 2 - 5 } \multicolumn{1}{c|}{} & $\mathbf { 1 }$ & $\mathbf { 2 }$ & $\mathbf { 3 }$ & $\mathbf { 4 }$
  \hline $\mathbf { 1 }$ & 22 & 14 & 11 & 23
  \hline $\mathbf { 2 }$ & 14 & 28 & 15 & 27
  \hline $\mathbf { 3 }$ & 11 & 15 & 22 & 12
  \hline

OCR MEI D2 2008 June Q4

$\mathbf { 4 }$ & 23 & 27 & 12 & 24
\hline \end{tabular} \end{center}

\cline { 2 - 5 } \multicolumn{1}{c\|}{}	$\mathbf { 1 }$	$\mathbf { 2 }$	$\mathbf { 3 }$	$\mathbf { 4 }$
$\mathbf { 1 }$	3	2	3	3
$\mathbf { 2 }$	1	1	3	3
$\mathbf { 3 }$	1	2	1	4
$\mathbf { 4 }$	3	3	3	3

Draw the complete network of shortest distances.
(ii) Starting at vertex 1, apply the nearest neighbour algorithm to the complete network of shortest distances to find a Hamilton cycle. Give the length of your cycle and interpret it in the original network.
(iii) By temporarily deleting vertex $\mathbf { 1 }$ and its connecting arcs from the complete network of shortest distances, find a lower bound for the solution to the Travelling Salesperson's Problem in that network. Say what this implies in the original network.
(b) Solve the route inspection problem in the original network, and say how you can be sure that your solution is optimal. 4 A factory's output includes three products. To manufacture a tonne of product $\mathrm { A } , 3$ tonnes of water are needed. Product B needs 2 tonnes of water per tonne produced, and product C needs 5 tonnes of water per tonne produced. Product A uses 5 hours of labour per tonne produced, product B uses 6 hours and product C uses 2 hours. There are 60 tonnes of water and 50 hours of labour available for tomorrow's production.
(i) Formulate a linear programming problem to maximise production within the given constraints.
(ii) Use the simplex algorithm to solve your LP, pivoting first on your column relating to product C.
(iii) An extra constraint is imposed by a contract to supply at least 8 tonnes of A . Use either two stage simplex or the big M method to solve this revised problem.

OCR MEI D2 2009 June Q1

The following was said in a charity appeal on Radio 4 in October 2006.
"It is hard to underestimate the effect that your contribution will make."
Rewrite the comment more simply in your own words without changing its meaning.
A machine has three components, A, B and C, each of which is either active or inactive.
- The machine is active if A and B are both active.
- The machine is active if A is inactive and C is active.
- The machine is active if B is inactive and C is active.
- Otherwise the machine is inactive.
The states (active or inactive) of the components and the machine are to be modelled by a combinatorial circuit in which "active" is represented by "true" and "inactive" is represented by "false". Draw such a circuit.
Construct a truth table to show the following. $$[ ( ( \mathrm { a } \wedge \mathrm {~b} ) \vee ( ( \sim \mathrm { a } ) \wedge \mathrm { c } ) ) \vee ( ( \sim \mathrm { b } ) \wedge \mathrm { c } ) ] \Leftrightarrow \sim [ ( ( \sim \mathrm { a } ) \wedge ( \sim \mathrm { c } ) ) \vee ( ( \sim \mathrm { b } ) \wedge ( \sim \mathrm { c } ) ) ]$$

OCR MEI D2 2009 June Q2

2 Zoe is preparing for a Decision Maths test on two topics, Decision Analysis (D) and Simplex (S). She has to decide whether to devote her final revision session to D or to S . There will be two questions in the test, one on D and one on S . One will be worth 60 marks and the other will be worth 40 marks. Historically there is a 50\% chance of each possibility. Zoe is better at $D$ than at $S$. If her final revision session is on $D$ then she would expect to score $80 \%$ of the $D$ marks and $50 \%$ of the $S$ marks. If her final session is on $S$ then she would expect to score $70 \%$ of the S marks and $60 \%$ of the D marks.

Compute Zoe's expected mark under each of the four possible circumstances, i.e. Zoe revising $D$ and the D question being worth 60 marks, etc.
Draw a decision tree for Zoe. Michael claims some expertise in forecasting which question will be worth 60 marks. When he forecasts that it will be the D question which is worth 60 , then there is a $70 \%$ chance that the D question will be worth 60 . Similarly, when he forecasts that it will be the S question which is worth 60 , then there is a $70 \%$ chance that the S question will be worth 60 . He is equally likely to forecast that the D or the S question will be worth 60.
Draw a decision tree to find the worth to Zoe of Michael's advice.

OCR MEI D2 2009 June Q3

3 A farmer has 40 acres of land. Four crops, A, B, C and D are available.
Crop A will return a profit of $\pounds 50$ per acre. Crop B will return a profit of $\pounds 40$ per acre.
Crop C will return a profit of $\pounds 40$ per acre. Crop D will return a profit of $\pounds 30$ per acre.
The total number of acres used for crops A and B must not be greater than the total number used for crops C and D. The farmer formulates this problem as:
Maximise $\quad 50 a + 40 b + 40 c + 30 d$,
subject to $\quad a + b \leqslant 20$,
$a + b + c + d \leqslant 40$.

Explain what the variables $a , b , c$ and $d$ represent. Explain how the first inequality was obtained.
Explain why expressing the constraint on the total area of land as an inequality will lead to a solution in which all of the land is used.
Solve the problem using the simplex algorithm. Suppose now that the farmer had formulated the problem as:
Maximise $\quad 50 a + 40 b + 40 c + 30 d$,
subject to $\quad a + b \leqslant 20$,
$a + b + c + d = 40$.
Show how to adapt this problem for solution either by the two-stage simplex method or the big-M method. In either case you should show the initial tableau and describe what has to be done next. You should not attempt to solve the problem.

OCR MEI D2 2009 June Q4

4 The diagram shows routes connecting five cities. Lengths are in km .
\includegraphics[max width=\textwidth, alt={}, center]{ae8dc840-0c61-4ba5-b082-1f5f3e6c2cef-4_442_995_319_534}

Produce the initial matrices for an application of Floyd's algorithm to find the complete network of shortest distances between the five cities. The following are the distance and route matrices after the third iteration of Floyd's algorithm. \begin{center} \begin{tabular}{ | c | c | c | c | c | c | } \cline { 2 - 6 } \multicolumn{1}{c|}{} & $\mathbf { 1 }$ & $\mathbf { 2 }$ & $\mathbf { 3 }$ & $\mathbf { 4 }$ & $\mathbf { 5 }$
\hline $\mathbf { 1 }$ & 44 & 22 & 42 & 15 & 15
\hline $\mathbf { 2 }$ & 22 & 44 & 20 & 5 & 23
\hline $\mathbf { 3 }$ & 42 & 20 & 40 & 25 & 43
\hline $\mathbf { 4 }$ & 15 & 5 & 25 & 10 & 16
\hline

OCR MEI D2 2009 June Q5

$\mathbf { 5 }$ & 15 & 23 & 43 & 16 & 30
\hline \end{tabular} \end{center}

\cline { 2 - 6 } \multicolumn{1}{c\|}{}	$\mathbf { 1 }$	$\mathbf { 2 }$	$\mathbf { 3 }$	$\mathbf { 4 }$	$\mathbf { 5 }$
$\mathbf { 1 }$	2	2	2	4	5
$\mathbf { 2 }$	1	1	3	4	5
$\mathbf { 3 }$	2	2	2	2	2
$\mathbf { 4 }$	1	2	2	2	5
$\mathbf { 5 }$	1	2	2	4	1

(ii) Perform the fourth iteration. There are no changes on the fifth iteration, so your answer to part (ii) should give the complete network of shortest distances.
(iii) Use your matrices to find the shortest distance and route from vertex $\mathbf { 3 }$ to vertex $\mathbf { 1 }$, and explain how you do it.
(iv) Draw the complete network of shortest distances, not including the loops.
(v) Apply the nearest neighbour algorithm to your network in part (iv), starting at vertex 2. Give the length of the Hamilton cycle that is produced. Interpret the Hamilton cycle in terms of the original diagram and state what the algorithm has achieved. OCR is committed to seeking permission to reproduce all third-party content that it uses in its assessment materials. OCR has attempted to identify and contact all copyright holders whose work is used in this paper. To avoid the issue of disclosure of answer-related information to candidates, all copyright acknowledgements are reproduced in the OCR Copyright Acknowledgements Booklet. This is produced for each series of examinations, is given to all schools that receive assessment material and is freely available to download from our public website (\href{http://www.ocr.org.uk}{www.ocr.org.uk}) after the live examination series.
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OCR MEI D2 2010 June Q1

Mickey ate the last of the cheese. Minnie was put out at this. Mickey's defence was "There wasn't enough left not to eat it all". Let "c" represent "there is enough cheese for two" and "e" represent "one person can eat all of the cheese".
1. Which of the following best captures Mickey's argument? $$\mathrm { c } \Rightarrow \mathrm { e } \quad \mathrm { c } \Rightarrow \sim \mathrm { e } \quad \sim _ { \mathrm { c } } \Rightarrow \mathrm { e } \quad \sim _ { \mathrm { c } } \Rightarrow \sim \mathrm { e }$$ In the ensuing argument Minnie concedes that if there's a lot left then one should not eat it all, but argues that this is not an excuse for Mickey having eaten it all when there was not a lot left.
2. Prove that Minnie is right by writing down a line of a truth table which shows that $$( c \Rightarrow \sim e ) \Leftrightarrow ( \sim c \Rightarrow e )$$ is false.
  (You may produce the whole table if you wish, but you need to indicate a specific line of the table.)
1. Show that the following combinatorial circuit is modelling an implication statement. Say what that statement is, and prove that the circuit and the statement are equivalent.
  \includegraphics[max width=\textwidth, alt={}, center]{c3a528e4-b5fe-4bff-a77e-e3199bb225a1-2_188_533_1272_767}
2. Express the following combinatorial circuit as an equivalent implication statement.
  \includegraphics[max width=\textwidth, alt={}, center]{c3a528e4-b5fe-4bff-a77e-e3199bb225a1-2_314_835_1599_616}
3. Explain why $( \sim \mathrm { p } \wedge \sim \mathrm { q } ) \Rightarrow \mathrm { r }$, together with $\sim \mathrm { r }$ and $\sim \mathrm { q }$, give p .

OCR MEI D2 2010 June Q2

2 The network is a representation of a show garden. The weights on the arcs give the times in minutes to walk between the six features represented by the vertices, where paths exist.
\includegraphics[max width=\textwidth, alt={}, center]{c3a528e4-b5fe-4bff-a77e-e3199bb225a1-3_483_985_342_539}

Why might it be that the time taken to walk from vertex $\mathbf { 2 }$ to vertex $\mathbf { 3 }$ via vertex $\mathbf { 4 }$ is less than the time taken by the direct route, i.e. the route from $\mathbf { 2 }$ to $\mathbf { 3 }$ which does not pass through any other vertices? The matrices shown below are the results of the first iteration of Floyd's algorithm when applied to the network. \begin{center} \begin{tabular}{ | c | c | c | c | c | c | c | } \cline { 2 - 7 } \multicolumn{1}{c|}{} & $\mathbf { 1 }$ & $\mathbf { 2 }$ & $\mathbf { 3 }$ & $\mathbf { 4 }$ & $\mathbf { 5 }$ & $\mathbf { 6 }$
\hline $\mathbf { 1 }$ & $\infty$ & 15 & $\infty$ & $\infty$ & 7 & 8
\hline $\mathbf { 2 }$ & 15 & 30 & 6 & 2 & 6 & 23
\hline

OCR MEI D2 2010 June Q3

$\mathbf { 3 }$ & $\infty$ & 6 & $\infty$ & 3 & $\infty$ & $\infty$
\hline

OCR MEI D2 2010 June Q4

$\mathbf { 4 }$ & $\infty$ & 2 & 3 & $\infty$ & 10 & 17
\hline

OCR MEI D2 2010 June Q5

$\mathbf { 5 }$ & 7 & 6 & $\infty$ & 10 & 14 & 8
\hline

OCR MEI D2 2011 June Q1

Heard in Parliament: "Will the minister not now discontinue her proposal to ban the protest?"
The minister replied "Yes I will."
To what had the minister committed herself logically, and why might that not have been her intention?
In a cricket tournament an umpire might be required to decide whether or not a batsman is out 'lbw', ie 'leg before wicket'. The lbw law for the tournament refers to parts of the cricket pitch as shown in the diagram (assuming a right-handed batsman):
\includegraphics[max width=\textwidth, alt={}, center]{52b8153f-e655-4852-a0f8-6f1c1e9c9170-2_254_1045_717_507} The umpire has to make a number of judgements:
A Would the ball have hit the wicket?
B Did the ball hit the batsman, or part of his equipment other than the bat, without hitting the bat?
C Did the ball hit the batsman, or part of his equipment other than the bat, before hitting the bat?
D Was the part of the batsman or his equipment which was hit by the ball, between the wickets when it was hit? E Was the part of the batsman or his equipment which was hit by the ball, outside of the wicket on the off side when it was hit? F Was the batsman attempting to play a stroke?
The law can be interpreted as saying that the batsman is out lbw if $[ ( \mathrm { A } \wedge \mathrm { B } ) \vee ( \mathrm { A } \wedge \mathrm { C } ) ] \wedge [ \mathrm { D } \vee ( \mathrm { E } \wedge \sim \mathrm { F } ) ]$.
The tournament's umpiring manual, in attempting to simplify the law, states that the batsman is out lbw if $\mathrm { A } \wedge ( \mathrm { B } \vee \mathrm { C } ) \wedge ( \mathrm { D } \vee \mathrm { E } ) \wedge ( \mathrm { D } \vee \sim \mathrm { F } )$. For an lbw decision this requires 4 conditions each to be true.
1. Use the rules of Boolean algebra to show that the manual's rule is logically equivalent to the law as stated above, naming the rules used at each step. A trainee umpire, using the manual, considers each condition in turn and judges that the following are true: A; B; E; D.
2. What is her decision and why?
3. What is odd about her judgement, and does this make the logic invalid?

OCR MEI D2 2011 June Q2

2 A government has just created a new ministry, the Ministry of Administrative Affairs. The ministry is to have four departments:
the Administration
the Bureaucracy
the Certification Service
the Duplication Section.
Each of these departments is to be established in a separate office on one of four existing sites. The diagram shows the direct journey times in minutes between these four sites.
\includegraphics[max width=\textwidth, alt={}, center]{52b8153f-e655-4852-a0f8-6f1c1e9c9170-3_342_403_721_829}

Use Floyd's algorithm to find the shortest journey times between the four office sites.
Draw a network showing your shortest times.
Use appropriate algorithms to find upper and lower bounds for the optimum solution to the Travelling Salesperson Problem in the original network, briefly explaining the steps taken.
A van is to be organised to deliver bundles of paperwork between the departments. Why might the optimum solution to the TSP be useful in planning this, and why might it not be?
Journeys to locations 2 and 3, from locations 1 and 4, are subject to a congestion charge which is equivalent in costing terms to 15 minutes of journey time. What sort of network would be needed to model this?

OCR MEI D2 2011 June Q3

3 Magnus has been researching career possibilities. He has just completed his GCSEs, and could leave school and get a good job. He estimates, discounted at today's values and given a 49 year working life, that there is a $50 \%$ chance of such a job giving him lifetime earnings of $\pounds 1.5 \mathrm {~m}$, a $30 \%$ chance of $\pounds 1.75 \mathrm {~m}$, and a $20 \%$ chance of $\pounds 2 \mathrm {~m}$. Alternatively Magnus can stay on at school and take A levels. He estimates that, if he does so, there is a 75\% chance that he will achieve good results. If he does not achieve good results then he will still be able to take the same job as earlier, but he will have lost two years of his lifetime earnings. This will give a $50 \%$ chance of lifetime earnings of $\pounds 1.42 \mathrm {~m}$, a $30 \%$ chance of $\pounds 1.67 \mathrm {~m}$ and a $20 \%$ chance of $\pounds 1.92 \mathrm {~m}$. If Magnus achieves good A level results then he could take a better job, which should give him discounted lifetime earnings of $\pounds 1.6 \mathrm {~m}$ with $50 \%$ probability or $\pounds 2 \mathrm {~m}$ with $50 \%$ probability. Alternatively he could go to university. This would cost Magnus another 3 years of lifetime earnings and would not guarantee him a well-paid career, since graduates sometimes choose to follow less well-paid vocations. His research shows him that graduates can expect discounted lifetime earnings of $\pounds 1 \mathrm {~m}$ with $20 \%$ probability, $\pounds 1.5 \mathrm {~m}$ with $30 \%$ probability, $\pounds 2 \mathrm {~m}$ with $30 \%$ probability, and $\pounds 3 \mathrm {~m}$ with $20 \%$ probability.

Draw up a decision tree showing Magnus's options.
Using the EMV criterion, find Magnus's best course of action, and give its value. Magnus has read that money isn't everything, and that one way to reflect this is to use a utility function and then compare expected utilities. He decides to investigate the outcome of using a function in which utility is defined to be the square root of value.
Using the expected utility criterion, find Magnus's best course of action, and give its utility.
The possibility of high earnings ( $\pounds 3 \mathrm {~m}$ ) swings Magnus's decision towards a university education. Find what value instead of $\pounds 3 \mathrm {~m}$ would make him indifferent to choosing a university education under the EMV criterion. (Do not change the probabilities.)

OCR MEI D2 2011 June Q4

4 A small alpine hotel is planned. Permission has been obtained for no more than 60 beds, and these can be accommodated in rooms containing one, two or four beds. The total floor areas needed are $15 \mathrm {~m} ^ { 2 }$ for a one-bed room, $25 \mathrm {~m} ^ { 2 }$ for a two-bed room and $40 \mathrm {~m} ^ { 2 }$ for a four-bed room. The total floor area of the bedrooms must not exceed $700 \mathrm {~m} ^ { 2 }$. Marginal profit contributions per annum, in thousands of euros, are estimated to be 5 for a one-bed room, 9 for a two-bed room and 15 for a four-bed room.

Formulate a linear programming problem to find the mix of rooms which will maximise the profit contribution within the two constraints.
Use the simplex algorithm to solve the problem, and interpret your solution. It is decided that, for marketing reasons, at least 5 one-bed rooms must be provided.
Solve this modified problem using either the two-stage simplex method or the big-M method. You may wish to adapt your final tableau from part (ii) to produce an initial tableau, but you are not required to do so.
The simplex solution to the revised problem is to provide 5 one-bed rooms, 15 two-bed rooms and 6.25 four-bed rooms, giving a profit contribution of $€ 253750$. Interpret this solution in terms of the real world problem.
Compare the following solution to your answer to part (iv): 8 one-bed rooms, 12 two-bed rooms and 7 four-bed rooms. Explain your findings.

OCR MEI D2 2012 June Q1

When marking coursework, a teacher has to complete a form which includes the following:
□
In your opinion is this the original work of the pupil? (tick as appropriate)
I have no reason to believe that it is not □
I cannot confirm that it is □
1. The teacher suspects that a pupil has copied work from the internet. For each box, state whether the teacher should tick the box or not.
2. The teacher has no suspicions about the work of another pupil, and has no information about how the work was produced. Which boxes should she tick?
3. Explain why the teacher must always tick at least one box.
Angus, the ski instructor, says that the class will have to have lunch in Italy tomorrow if it is foggy or if the top ski lift is not working. On the next morning Chloe, one of Angus's students, says that it is not foggy, so they can have lunch in Switzerland. Produce a line of a truth table which shows that Chloe's deduction is incorrect. You may produce a complete truth table if you wish, but you must indicate a row which shows that Chloe's deduction is incorrect.
You are given that the following two statements are true. $$\begin{aligned} & ( \mathrm { X } \vee \sim \mathrm { Y } ) \Rightarrow \mathrm { Z }
& \sim \mathrm { Z } \end{aligned}$$ Use Boolean algebra to show that Y is true.

OCR MEI D2 2012 June Q2

2 Adrian is considering selling his house and renting a flat.
Adrian still owes $\pounds 150000$ on his house. He has a mortgage for this, for which he has to pay $\pounds 4800$ annual interest. If he sells he will pay off the $\pounds 150000$ and invest the remainder of the proceeds at an interest rate of $2.5 \%$ per annum. He will use the interest to help to pay his rent. His estate agent estimates that there is a $30 \%$ chance that the house will sell for $\pounds 225000$, a $50 \%$ chance that it will sell for $\pounds 250000$, and a $20 \%$ chance that it will sell for $\pounds 275000$. A flat will cost him $\pounds 7500$ per annum to rent.

Draw a decision tree to help Adrian to decide whether to keep his house, or to sell it and rent a flat. Compare the EMVs of Adrian's annual outgoings, and ignore the costs of selling.

Would the analysis point to a different course of action if Adrian were to use a square root utility function, instead of EMVs? Adrian's circumstances change so that he has to decide now whether to sell or not in one year's time. Economic conditions might then be less favourable for the housing market, the same, or more favourable, these occurring with probabilities $0.3,0.3$ and 0.4 respectively. The possible selling prices and their probabilities are shown in the table.

Economic conditions and probabilities		Selling prices ( £) and probabilities
less favourable	0.3	200000	0.2	225000	0.3	250000	0.5
unchanged	0.3	225000	0.3	250000	0.5	275000	0.2
more favourable	0.4	250000	0.3	300000	0.5	350000	0.2

Draw a decision tree to help Adrian to decide what to do. Compare the EMVs of Adrian's annual outgoings. Assume that he will still owe $\pounds 150000$ in one year's time, and that the cost of renting and interest rates do not change.

OCR MEI D2 2012 June Q3

3 The weights on the network represent distances.
\includegraphics[max width=\textwidth, alt={}, center]{eb4e9c34-7d8f-4118-b7ec-edcd9567077f-4_451_544_324_740}

The answer book shows the initial tables and the results of iterations $1,2,3$ and 5 when Floyd's algorithm is applied to the network.
(A) Complete the two tables for iteration 4.
(B) Use the final route table to give the shortest route from vertex $\mathbf { 3 }$ to vertex $\mathbf { 5 }$.
(C) Use the final distance table to produce a complete network with weights representing the shortest distances between vertices.
Using the complete network of shortest distances, find a lower bound for the solution to the Travelling Salesperson Problem by deleting vertex 5 and its arcs, and by finding the length of a minimum connector for the remainder. (You may find the minimum connector by inspection.)
Use the nearest neighbour algorithm, starting at vertex $\mathbf { 1 }$, to produce a Hamilton cycle in the complete network. Give the length of your cycle.
Interpret your Hamilton cycle in part (iii) in terms of the original network.
Give a walk of minimum length which traverses every arc on the original network at least once, and which returns to the start. Give the length of your walk.

OCR MEI D2 2012 June Q4

4 A publisher is considering producing three books over the next week: a mathematics book, a novel and a biography. The mathematics book will sell at $\pounds 10$ and costs $\pounds 4$ to produce. The novel will sell at $\pounds 5$ and costs $\pounds 2$ to produce. The biography will sell at $\pounds 12$ and costs $\pounds 5$ to produce. The publisher wants to maximise profit, and is confident that all books will be sold. There are constraints on production. Each copy of the mathematics book needs 2 minutes of printing time, 1 minute of packing time, and $300 \mathrm {~cm} ^ { 3 }$ of temporary storage space. Each copy of the novel needs 1.5 minutes of printing time, 0.5 minutes of packing time, and $200 \mathrm {~cm} ^ { 3 }$ of temporary storage space. Each copy of the biography needs 2.5 minutes of printing time, 1.5 minutes of packing time, and $400 \mathrm {~cm} ^ { 3 }$ of temporary storage space. There are 10000 minutes of printing time available on several printing presses, 7500 minutes of packing time, and $2 \mathrm {~m} ^ { 3 }$ of temporary storage space.

Explain how the following initial feasible tableau models this problem.
P $x$ $y$ $z$ $s 1$ $s 2$ $s 3$ RHS
1 - 6 - 3 - 7 0 0 0 0
0 2 1.5 2.5 1 0 0 10000
0 1 0.5 1.5 0 1 0 7500
0 300 200 400 0 0 1 2000000
Use the simplex algorithm to solve your LP, and interpret your solution.
The optimal solution involves producing just one of the three books. By how much would the price of each of the other books have to be increased to make them worth producing? There is a marketing requirement to provide at least 1000 copies of the novel.
Show how to incorporate this constraint into the initial tableau ready for an application of the two-stage simplex method. Briefly describe how to use the modified tableau to solve the problem. You are NOT required to perform the iterations.

OCR MEI D2 2013 June Q1

A graph is simple if it contains neither loops nor multiple arcs, ie none of the following:
\includegraphics[max width=\textwidth, alt={}, center]{a09472cd-8f65-4cca-9683-c386053e66aa-2_81_134_219_1683}
or
\includegraphics[max width=\textwidth, alt={}, center]{a09472cd-8f65-4cca-9683-c386053e66aa-2_79_589_301_328} In an examination question, students were asked to describe in words when a graph is simple. Mark the following responses as right or wrong, giving reasons for your decisions if you mark them wrong.
1. A graph is simple if there are no loops and if two nodes are connected by a single arc.
2. A graph is simple if there are no loops and no two nodes are connected by more than one arc.
3. A graph is simple if there are no loops and two arcs do not have the same ends.
4. A graph is simple if there are no loops and there is at most one route from one node to another.
The following picture represents a two-way switch
\includegraphics[max width=\textwidth, alt={}, center]{a09472cd-8f65-4cca-9683-c386053e66aa-2_149_138_932_1119} It can either be in the up state
\includegraphics[max width=\textwidth, alt={}, center]{a09472cd-8f65-4cca-9683-c386053e66aa-2_104_104_1128_790}
or in the down state • or in the down state . Two switches can be used to construct a circuit in which changing the state of either switch changes the state of a lamp.
\includegraphics[max width=\textwidth, alt={}, center]{a09472cd-8f65-4cca-9683-c386053e66aa-2_309_543_1448_762} Georgios tries to connect together three two-way switches so that changing the state of any switch changes the state of the lamp. His circuit is shown below. The switches have been labelled 1,2 and 3.
\includegraphics[max width=\textwidth, alt={}, center]{a09472cd-8f65-4cca-9683-c386053e66aa-2_496_547_1946_760}
1. List the possible combination of switch states and determine whether the lamp is on or off for each of them.
2. Say whether or not Georgios has achieved his objective, justifying your answer.
Use a truth table to show that $( \mathrm { A } \wedge ( \mathrm { B } \vee \mathrm { C } ) ) \vee \sim ( \sim \mathrm { A } \vee ( \mathrm { B } \wedge \mathrm { C } ) ) \Leftrightarrow \mathrm { A }$.

OCR MEI D2 2013 June Q2

2 Graham skis each year in an Italian resort which shares a ski area with a Swiss resort. He can buy an Italian lift pass, or an international lift pass which gives him access to Switzerland as well as to Italy. For his 6-day holiday the Italian pass costs $€ 200$ and the international pass costs $€ 250$. If he buys an Italian pass then he can still visit Switzerland by purchasing day supplements at $€ 30$ per day. If the weather is good during his holiday, then Graham visits Switzerland three times. If the weather is moderate he goes twice. If poor he goes once. If the weather is windy then the lifts are closed, and he is not able to go at all. In his years of skiing at the resort he has had good weather on $30 \%$ of his visits, moderate weather on $40 \%$, poor weather on $20 \%$ and windy weather on $10 \%$ of his visits.

Draw a decision tree to help Graham decide whether to buy an Italian lift pass or an international lift pass. Give the action he should take to minimize the EMV of his costs. When he arrives at the resort, and before he buys his lift pass, he finds that he has internet access to a local weather forecast, and to records of the past performance of the forecast. The 6-day forecast is limited to "good"/"not good", and the records show the actual weather proportions following those forecasts. It also shows that $60 \%$ of historical forecasts have been "good" and $40 \%$ "not good".
\backslashbox{Forecast}{Actual} good moderate poor windy proportion of forecasts
good 0.4 0.5 0.1 0.0 0.6
not good 0.15 0.25 0.35 0.25 0.4
Draw a decision tree to help Graham decide the worth of consulting the forecast before buying his lift pass. Give the actions he should take to minimize the EMV of his costs.

\cline { 2 - 5 } \multicolumn{1}{c\|}{}	\(\mathbf { 1 }\)	\(\mathbf { 2 }\)	\(\mathbf { 3 }\)	\(\mathbf { 4 }\)
\(\mathbf { 1 }\)	2	2	2	2
\(\mathbf { 2 }\)	1	1	3	4
\(\mathbf { 3 }\)	2	2	2	4
\(\mathbf { 4 }\)	2	2	3	3

\cline { 2 - 5 } \multicolumn{1}{c\|}{}	\(\mathbf { 1 }\)	\(\mathbf { 2 }\)	\(\mathbf { 3 }\)	\(\mathbf { 4 }\)
\(\mathbf { 1 }\)	3	2	3	3
\(\mathbf { 2 }\)	1	1	3	3
\(\mathbf { 3 }\)	1	2	1	4
\(\mathbf { 4 }\)	3	3	3	3

\cline { 2 - 6 } \multicolumn{1}{c\|}{}	\(\mathbf { 1 }\)	\(\mathbf { 2 }\)	\(\mathbf { 3 }\)	\(\mathbf { 4 }\)	\(\mathbf { 5 }\)
\(\mathbf { 1 }\)	2	2	2	4	5
\(\mathbf { 2 }\)	1	1	3	4	5
\(\mathbf { 3 }\)	2	2	2	2	2
\(\mathbf { 4 }\)	1	2	2	2	5
\(\mathbf { 5 }\)	1	2	2	4	1

P	\(x\)	\(y\)	\(z\)	\(s 1\)	\(s 2\)	\(s 3\)	RHS
1	- 6	- 3	- 7	0	0	0	0
0	2	1.5	2.5	1	0	0	10000
0	1	0.5	1.5	0	1	0	7500
0	300	200	400	0	0	1	2000000

\backslashbox{Forecast}{Actual}	good	moderate	poor	windy	proportion of forecasts
good	0.4	0.5	0.1	0.0	0.6
not good	0.15	0.25	0.35	0.25	0.4

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