Questions - OCR MEI D2 - A-Level Maths

1. Describe in words the conditions under which the foglights will come on. Fig. 1.2 shows a combinatorial circuit. \begin{figure}[h]
  \includegraphics[alt={},max width=\textwidth]{ab28be76-9329-41c8-90fe-ff1bdb28f788-2_367_1235_1183_431} \captionsetup{labelformat=empty} \caption{Fig. 1.2}
  \end{figure}
2. Write the output in terms of a Boolean expression involving $s , h$ and $f$.
3. Use a truth table to prove that $\mathrm { s } \wedge \mathrm { h } \wedge \mathrm { f } = \sim ( \sim \mathrm { s } \vee \sim \mathrm { h } ) \wedge \mathrm { f }$.
A car's first gear can be engaged ( g ) if either both the road speed is low ( r ) and the clutch is depressed ( d ), or if both the road speed is low ( r ) and the engine speed is the correct multiple of the road speed (m).
1. Draw a switching circuit to represent the conditions under which first gear can be engaged. Use two ganged switches to represent $r$, and single switches to represent each of $\mathrm { d } , \mathrm { m }$ and g .
2. Draw a combinatorial circuit to represent the Boolean expression $\mathrm { r } \wedge ( \mathrm { d } \vee \mathrm { m } ) \wedge \mathrm { g }$.
3. Use Boolean algebra to prove that $\mathrm { r } \wedge ( \mathrm { d } \vee \mathrm { m } ) \wedge \mathrm { g } = ( ( \mathrm { r } \wedge \mathrm { d } ) \vee ( \mathrm { r } \wedge \mathrm { m } ) ) \wedge \mathrm { g }$.
4. Draw another switching circuit to represent the conditions under which first gear can be selected, but without using a ganged switch.

OCR MEI D2 2005 June Q2

2 Karl is considering investing in a villa in Greece. It will cost him 56000 euros ( € 56000 ). His alternative is to invest his money, $\pounds 35000$, in the United Kingdom. He is concerned with what will happen over the next 5 years. He estimates that there is a $60 \%$ chance that a house currently worth $€ 56000$ will appreciate to be worth $€ 75000$ in that time, but that there is a $40 \%$ chance that it will be worth only $€ 55000$. If he invests in the United Kingdom then there is a $50 \%$ chance that there will be $20 \%$ growth over the 5 years, and a $50 \%$ chance that there will be $10 \%$ growth.

Given that $\pounds 1$ is worth $€ 1.60$, draw a decision tree for Karl, and advise him what to do, using the EMV of his investment (in thousands of euros) as his criterion. In fact the $\pounds / €$ exchange rate is not fixed. It is estimated that at the end of 5 years, if there has been $20 \%$ growth in the UK then there is a $70 \%$ chance that the exchange rate will stand at 1.70 euros per pound, and a $30 \%$ chance that it will be 1.50 . If growth has been $10 \%$ then there is a $40 \%$ chance that the exchange rate will stand at 1.70 and a $60 \%$ chance that it will be 1.50 .
Produce a revised decision tree incorporating this information, and give appropriate advice. A financial analyst asks Karl a number of questions to determine his utility function. He estimates that for $x$ in cash (in thousands of euros) Karl's utility is $x ^ { 0.8 }$, and that for $y$ in property (in thousands of euros), Karl's utility is $y ^ { 0.75 }$.
Repeat your computations from part (ii) using utility instead of the EMV of his investment. Does this change your advice?
Using EMVs, find the exchange rate (number of euros per pound) which will make Karl indifferent between investing in the UK and investing in a villa in Greece.
Show that, using Karl's utility function, the exchange rate would have to drop to 1.277 euros per pound to make Karl indifferent between investing in the UK and investing in a villa in Greece.

OCR MEI D2 2005 June Q3

3 The distance and route matrices shown in Fig. 3.1 are the result of applying Floyd's algorithm to the incomplete network on 4 vertices shown in Fig. 3.2. Distance Matrix \begin{center} \begin{tabular}{ | c | c | c | c | c | } \multicolumn{1}{l}{} & $\mathbf { 1 }$ & $\mathbf { 2 }$ & $\mathbf { 3 }$ & $\mathbf { 4 }$
\hline $\mathbf { 1 }$ & 4 & 2 & 3 & 9
\hline $\mathbf { 2 }$ & 2 & 2 & $\mathbf { 1 }$ & 7
\hline $\mathbf { 3 }$ & 3 & $\mathbf { 1 }$ & 2 & 6
\hline

OCR MEI D2 2005 June Q4

$\mathbf { 4 }$ & 9 & 7 & 6 & 12
\hline \end{tabular} \end{center} Route Matrix

	$\mathbf { 1 }$	$\mathbf { 2 }$	$\mathbf { 3 }$	$\mathbf { 4 }$
$\mathbf { 1 }$	2	2	2	2
$\mathbf { 2 }$	$\mathbf { 1 }$	3	3	3
$\mathbf { 3 }$	2	2	2	4
$\mathbf { 4 }$	3	3	3	3

Fig. 3.1 \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{ab28be76-9329-41c8-90fe-ff1bdb28f788-4_296_310_918_904} \captionsetup{labelformat=empty} \caption{Fig. 3.2}

\end{figure}

Draw the complete network of shortest distances.
Explain how to use the route matrix to find the shortest route from vertex 4 to vertex 1 in the original incomplete network. A new vertex, vertex 5, is added to the original network. Its distances from vertices to which it is connected are shown in Fig. 3.3. \begin{table}[h]
\cline { 2 - 5 } \multicolumn{1}{c|}{} 1 2 3 4
5 - 3 - 1
\captionsetup{labelformat=empty} \caption{Fig. 3.3}
\end{table}
Draw the extended network and the complete 5 -node network of shortest distances. (You are not required to use an algorithm to find the shortest distances.)
Produce the shortest distance matrix and the route matrix for the extended 5-node network.
Apply the nearest neighbour algorithm to your $5 \times 5$ distance matrix, starting at vertex 1. Give the length of the cycle produced, together with the actual cycle in the original 5-node network.
By deleting vertex 1 and its arcs, and by using Prim's algorithm on the reduced distance matrix, produce a lower bound for the solution to the practical travelling salesperson problem in the original 5-node network. Show clearly your use of the matrix form of Prim's algorithm.
In the original 5-node network find a shortest route starting at vertex $\mathbf { 1 }$ and using each of the 6 arcs at least once. Give the length of your route. 4 Kassi and Theo are discussing how much oil and how much vinegar to use to dress their salad. They agree to use between 5 and 10 ml of oil and between 3 and 6 ml of vinegar and that the amount of oil should not exceed twice the amount of vinegar. Theo prefers to have more oil than vinegar. He formulates the following problem to maximise the proportion of oil: $$\begin{array} { l c } \text { Maximise } & \frac { x } { x + y }
\text { subject to } & 0 \leqslant x \leqslant 10 ,
& 0 \leqslant y \leqslant 6 ,
& x - 2 y \leqslant 0 . \end{array}$$
Explain why this problem is not an LP.
Use the simplex method to solve the following LP. $$\begin{array} { l c } \text { Maximise } & x - y
\text { subject to } & 0 \leqslant x \leqslant 10
& 0 \leqslant y \leqslant 6
& x - 2 y \leqslant 0 \end{array}$$
Kassi prefers to have more vinegar than oil. She formulates the following LP. $$\begin{array} { l l } \text { Maximise } & y - x
\text { subject to } & 5 \leqslant x \leqslant 10 ,
& 3 \leqslant y \leqslant 6 ,
& x - 2 y \leqslant 0 . \end{array}$$ Draw separate graphs to show the feasible regions for this problem and for the problem in part (ii).
Explain why the formulation in part (ii) produced a solution for Theo's problem, and why it is more difficult to use the simplex method to solve Kassi's problem in part (iii).
Produce an initial tableau for using the two-stage simplex method to solve Kassi's problem. Explain briefly how to proceed.

OCR MEI D2 2006 June Q1

Use a truth table to prove $\sim ( \sim \mathrm { T } \Rightarrow \sim \mathrm { S } ) \Leftrightarrow ( \sim \mathrm { T } \wedge \mathrm { S } )$.
Prove that $( \mathrm { A } \Rightarrow \mathrm { B } ) \Leftrightarrow ( \sim \mathrm { A } \vee \mathrm { B } )$ and hence use Boolean algebra to prove that $$\sim ( \sim \mathrm { T } \Rightarrow \sim \mathrm {~S} ) \Leftrightarrow ( \sim \mathrm { T } \wedge \mathrm {~S} ) .$$
A teacher wrote on a report "It is not the case that if Joanna doesn't try then she won't succeed." He meant to say that if Joanna were to try then she would have a chance of success. By letting T be "Joanna will try" and S be "Joanna will succeed", find the real meaning of what the teacher wrote.

OCR MEI D2 2006 June Q3

3 Emma has won a holiday worth $\pounds 1000$. She is wondering whether or not to take out an insurance policy which will pay out $\pounds 1000$ if she should fall ill and be unable to go on the holiday. The insurance company tells her that this happens to 1 in 200 people. The insurance policy costs $\pounds 10$. Thus Emma's monetary value if she buys the insurance and does not fall ill is $\pounds 990$.

Draw a decision tree for Emma's problem. Use the EMV criterion in your calculations.
Interpret your tree and say what the maximum cost of the insurance would have to be for Emma to consider buying it if she uses the EMV criterion. Suppose that Emma's utility function is given by utility $= \sqrt [ 3 ] { \text { monetary value } }$.
Using expected utility as the criterion, should Emma purchase the insurance? Under this criterion what is the cost at which she will be indifferent to buying or not buying it? Emma could pay for a blood pressure check to help her to make her decision. Statistics show that $75 \%$ of checks are positive, and that when a check is positive the chance of missing a holiday through ill heath is 0.001 . However, when a check is negative the chance of cancellation through ill health is 0.017.
Draw a decision tree to help Emma decide whether or not to pay for the check. Use EMV, not expected utility, in your calculations and assume that the insurance policy costs $\pounds 10$. What is the maximum amount that she should pay for the blood pressure check?

OCR MEI D2 2006 June Q4

Explain how this formulation models the problem, and say why the analyst has not simplified the last inequality to $a + b + c \leqslant 15$.
The final constraint is different from the others in that the resource is integer valued. Explain why that does not impose an additional difficulty for this problem.
Solve this problem using the simplex algorithm. Interpret your solution and say what resources are left over. On a particular day an order is received for two Cuddly Cats, and the extra constraint $c \geqslant 2$ is added to the formulation.
Set up an initial simplex tableau to deal with the modified problem using either the big-M approach or two-phase simplex. Do not perform any iterations on your tableau.
Show that the solution given by $a = 8 , b = 2$ and $c = 5$ uses all of the resources, but that $a = 6 , b = 6$ and $c = 2$ gives more profit. What resources are left over from the latter solution?

OCR MEI D2 2007 June Q1

A joke has it that army recruits used to be instructed: "If it moves, salute it. If it doesn't move, paint it." Assume that this instruction has been carried out completely in the local universe, so that everything that doesn't move has been painted.
1. A recruit encounters something which is not painted. What should he do, and why?
2. A recruit encounters something which is painted. Do we know what he or she should do? Justify your answer.
Use a truth table to prove $( ( ( m \Rightarrow s ) \wedge ( \sim m \Rightarrow p ) ) \wedge \sim p ) \Rightarrow s$.
You are given the following two rules. $$\begin{aligned} & 1 \quad ( a \Rightarrow b ) \Leftrightarrow ( \sim b \Rightarrow \sim a )
& 2 \quad ( x \wedge ( x \Rightarrow y ) ) \Rightarrow y \end{aligned}$$ Use Boolean algebra to prove that $( ( ( m \Rightarrow s ) \wedge ( \sim m \Rightarrow p ) ) \wedge \sim p ) \Rightarrow s$.

OCR MEI D2 2007 June Q2

2 Bill is at a horse race meeting. He has $\pounds 2$ left with two races to go. He only ever bets $\pounds 1$ at a time. For each race he chooses a horse and then decides whether or not to bet on it. In both races Bill's horse is offered at "evens". This means that, if Bill bets $\pounds 1$ and the horse wins, then Bill will receive back his $\pounds 1$ plus $\pounds 1$ winnings. If Bill's horse does not win then Bill will lose his $\pounds 1$.

Draw a decision tree to model this situation. Show Bill's payoffs on your tree, i.e. how much money Bill finishes with under each possible outcome. Assume that in each race the probability of Bill's horse winning is the same, and that it has value $p$.
Find Bill's EMV when
(A) $p = 0.6$,
(B) $p = 0.4$. Give his best course of action in each case.
Suppose that Bill uses the utility function utility $= ( \text { money } ) ^ { x }$, to decide whether or not to bet $\pounds 1$ on one race. Show that, with $p = 0.4$, Bill will not bet if $x = 0.5$, but will bet if $x = 1.5$.

OCR MEI D2 2007 June Q3

3 Floyd's algorithm is applied to the following network:
\includegraphics[max width=\textwidth, alt={}, center]{483a681e-011a-464f-b7cb-007d894d1516-3_398_394_331_833} At the end of the third iteration of the algorithm the distance and route 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 $\mathbf { 1 }$ & 6 & 3 & 10 & 5
\hline $\mathbf { 2 }$ & 3 & 6 & 7 & 2
\hline $\mathbf { 3 }$ & 10 & 7 & 14 & 1
\hline

OCR MEI D2 2007 June Q4

$\mathbf { 4 }$ & 5 & 2 & 1 & 2
\hline \end{tabular} \end{center}

\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

	\(\mathbf { 1 }\)	\(\mathbf { 2 }\)	\(\mathbf { 3 }\)	\(\mathbf { 4 }\)
\(\mathbf { 1 }\)	2	2	2	2
\(\mathbf { 2 }\)	\(\mathbf { 1 }\)	3	3	3
\(\mathbf { 3 }\)	2	2	2	4
\(\mathbf { 4 }\)	3	3	3	3

\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

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