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OCR MEI D2 2013 June Q1
16 marks Easy -1.8
1
  1. 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.
  2. 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.
  3. 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
16 marks Standard +0.8
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.
  1. 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}goodmoderatepoorwindyproportion of forecasts
    good0.40.50.10.00.6
    not good0.150.250.350.250.4
  2. 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.
OCR MEI D2 2013 June Q3
20 marks Standard +0.3
3 Five towns, 1, 2, 3, 4 and 5, are connected by direct routes as shown. The arc weights represent distances. \includegraphics[max width=\textwidth, alt={}, center]{a09472cd-8f65-4cca-9683-c386053e66aa-4_632_540_312_744}
  1. The printed 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 5 to vertex 2.
    (C) Use the final distance table to produce a complete network with weights representing the shortest distances between vertices.
  2. Use the nearest neighbour algorithm, starting at vertex \(\mathbf { 4 }\), to produce a Hamilton cycle in the complete network. Give the length of your cycle.
  3. Interpret your Hamilton cycle from part (ii) in terms of towns actually visited.
  4. Find an improved Hamilton cycle by applying the nearest neighbour algorithm starting from one of the other vertices.
  5. Using the complete network of shortest distances (excluding loops), find a lower bound for the solution to the Travelling Salesperson Problem by deleting vertex 4 and its arcs, and by finding the length of a minimum connector for the remainder. (You may find the minimum connector by inspection.)
  6. Given that the sum of the road lengths in the original network is 43 , give a walk of minimum length which traverses every arc on the original network at least once, and which returns to the start. Show your methodology. Give the length of your walk.
OCR MEI D2 2013 June Q4
20 marks Standard +0.8
4 Colin has a hobby from which he makes a small income. He makes bowls, candle holders and key fobs.
The materials he uses include wood, metal parts, polish and sandpaper. They cost, on average, \(\pounds 15\) per bowl, \(\pounds 6\) per candle holder and \(\pounds 2\) per key fob. Colin has a monthly budget of \(\pounds 100\) for materials. Colin spends no more than 30 hours per month on manufacturing these objects. Each bowl takes 4 hours, each candle holder takes 2 hours and each key fob takes half an hour.
  1. Let \(b\) be the number of bowls Colin makes in a month, \(c\) the number of candle holders and \(f\) the number of key fobs. Write out, in terms of these variables, two constraints corresponding to the limit on monthly expenditure on materials, and to the limit on Colin's time. Colin sells the objects at craft fairs. He charges \(\pounds 30\) for a bowl, \(\pounds 15\) for a candle holder and \(\pounds 3\) for a key fob.
  2. Set up an initial simplex tableau for the problem of maximising Colin's monthly income subject to your constraints from part (i), assuming that he sells all that he produces.
  3. Use the simplex algorithm to solve your LP, and interpret the solution from the simplex algorithm. Over a spell of several months Colin finds it difficult to sell bowls so he stops making them.
  4. Modify and solve your LP, using simplex, to find how many candle holders and how many key fobs he should make, and interpret your solution. At the next craft fair Colin takes an order for 4 bowls. He promises to make exactly 4 bowls in the next month.
  5. Set up this modified problem either as an application of two-stage simplex, or as an application of the big-M method. You are not required to solve the problem. The solution now is for Colin to produce 4 bowls, \(6 \frac { 2 } { 3 }\) candle holders and no key fobs.
  6. What is Colin's best integer solution to the problem?
  7. Your answer to part (vi) is not necessarily the integer solution giving the maximum profit for Colin. Explain why.
OCR MEI D2 2014 June Q1
16 marks Moderate -0.8
1 Keith is wondering whether or not to insure the value of his house against destruction. His friend Georgia has told him that it is a waste of money. Georgia argues that the insurance company sets its premiums (how much it charges for insurance) to take account of the probability of destruction, plus an extra fee for its profit. Georgia argues that house-owners are, on average, simply paying fees to the insurance company. Keith's house is valued at \(\pounds 400000\). The annual premium for insuring its value against destruction is \(\pounds 100\). Past statistics show that the probability of destruction in any one year is 0.0002 .
  1. Draw a decision tree to model Keith's decision and the possible outcomes.
  2. Compute Keith's EMV and give the course of action which corresponds to that EMV.
  3. What would be the insurance premium if there were no fee for the insurance company? For the remainder of the question the insurance premium is still \(\pounds 100\).
    Suppose that, instead of EMV, Keith uses the utility function utility \(= ( \text { money } ) ^ { 0.5 }\).
  4. Compute Keith's utility and give his corresponding course of action. Keith suspects that it may be the case that he lives in an area in which the probability of destruction in a given year, \(p\), is not 0.0002 .
  5. Draw a decision tree, using the EMV criterion, to model Keith's decision in terms of \(p\), the probability of destruction in the area in which Keith lives.
  6. Find the value of \(p\) which would make it worthwhile for Keith to insure his house using the EMV criterion.
  7. Explain why Keith may wish to insure even if \(p\) is less than the value which you found in part (vi). [1]
    (a) A national Sunday newspaper runs a "You are the umpire" series, in which questions are posed about whether a batsman in cricket is given "out", and why, or "not out". One Sunday the readers were told that a ball had either hit the bat and then the pad, or had missed the bat and hit the pad; the umpire could not be sure which. The ball had then flown directly to a fielder, who had caught it. The LBW (leg before wicket) rule is complicated. The readers were told that this batsman should be given out (LBW) if the ball had not hit the bat. On the other hand, if the ball had hit the bat, then he should be given out (caught). Readers were asked what the decision should be. The answer given in the newspaper was that this batsman should be given not out because the umpire could not be sure that the batsman was out (LBW), and could not be sure that he was out (caught).
OCR MEI D2 2014 June Q3
20 marks Standard +0.3
3 Three products, A, B and C are to be made.
Three supplements are included in each product. Product A has 10 g per kg of supplement \(\mathrm { X } , 5 \mathrm {~g}\) per kg of supplement Y and 5 g per kg of supplement Z . Product B has 5 g per kg of supplement \(\mathrm { X } , 5 \mathrm {~g}\) per kg of supplement Y and 3 g per kg of supplement Z .
Product C has 12 g per kg of supplement \(\mathrm { X } , 7 \mathrm {~g}\) per kg of supplement Y and 5 g per kg of supplement Z .
There are 12 kg of supplement X available, 12 kg of supplement Y , and 9 kg of supplement Z .
Product A will sell at \(\pounds 7\) per kg and costs \(\pounds 3\) per kg to produce. Product B will sell at \(\pounds 5\) per kg and costs \(\pounds 2\) per kg to produce. Product C will sell at \(\pounds 4\) per kg and costs \(\pounds 3\) per kg to produce. The profit is to be maximised.
  1. Explain how the initial feasible tableau shown in Fig. 3 models this problem. \begin{table}[h]
    Pabcs 1s 2s 3RHS
    1- 4- 3- 10000
    01051210012000
    055701012000
    05350019000
    \captionsetup{labelformat=empty} \caption{Fig. 3}
    \end{table}
  2. Use the simplex algorithm to solve this problem, and interpret the solution.
  3. In the solution, one of the basic variables appears at a value of 0 . Explain what this means. There is a contractual requirement to provide at least 500 kg of product A .
  4. Show how to incorporate this constraint into the initial tableau ready for an application of the two-stage simplex method. Briefly describe how the method works. You are not required to perform the iterations.
OCR MEI D2 2015 June Q1
16 marks Moderate -0.5
1 A furniture manufacturer is planning a production run. He will be making wardrobes, drawer units and desks. All can be manufactured from the same wood. He has available \(200 \mathrm {~m} ^ { 2 }\) of wood for the production run. Allowing for wastage, a wardrobe requires \(5 \mathrm {~m} ^ { 2 }\), a drawer unit requires \(3 \mathrm {~m} ^ { 2 }\), and a desk requires \(2 \mathrm {~m} ^ { 2 }\). He has 200 hours available for the production run. A wardrobe requires 4.5 hours, a drawer unit requires 5.2 hours, and a desk requires 3.8 hours. The completed furniture will have to be stored at the factory for a short while before being shipped. The factory has \(50 \mathrm {~m} ^ { 3 }\) of storage space available. A wardrobe needs \(1 \mathrm {~m} ^ { 3 }\), a drawer unit needs \(0.75 \mathrm {~m} ^ { 3 }\), and a desk needs \(0.5 \mathrm {~m} ^ { 3 }\). The manufacturer needs to know what he should produce to maximise his income. He sells the wardrobes at \(\pounds 80\) each, the drawer units at \(\pounds 65\) each and the desks at \(\pounds 50\) each.
  1. Formulate the manufacturer's problem as an LP.
  2. Use the Simplex algorithm to solve the LP problem.
  3. Interpret the results.
  4. An extra \(25 \mathrm {~m} ^ { 2 }\) of wood is found and is to be used. The new optimal solution is to make 44 wardrobes, no drawer units and no desks. However, this leaves some of each resource (wood, hours and space) left over. Explain how this can be possible.
  5. Given that \(x\) and \(y\) are propositions, draw a 4-line truth table for \(x \Rightarrow y\), allowing \(x\) and \(y\) to take all combinations of truth values. If \(x\) is false and \(x \Rightarrow y\) is true, what can be deduced about the truth value of \(y\) ? A story has it that, in a lecture on logic, the philosopher Bertrand Russell (1872-1970) mentioned that a false proposition implies any proposition. A student challenged this, saying "In that case, given that \(1 = 0\), prove that you are the Pope."
    Russell immediately replied, "Add 1 to both sides of the equation: then we have \(2 = 1\). The set containing just me and the Pope has 2 members. But \(2 = 1\), so the set has only 1 member; therefore, I am the Pope." Russell's string of statements is an example of a deductive sequence. Let \(a\) represent " \(1 = 0\) ", \(b\) represent " \(2 = 1\) ", \(c\) represent "Russell and the Pope are 2" and \(d\) represent "Russell and the Pope are 1". Then Russell's deductive sequence can be written as \(( a \wedge ( a \Rightarrow b ) \wedge c ) \Rightarrow d\).
  6. Assuming that \(a\) is false, \(b\) is false, \(a \Rightarrow b\) is true, \(c\) is true, and that \(d\) can take either truth value, draw a 2-line truth table for \(( a \wedge ( a \Rightarrow b ) \wedge c ) \Rightarrow d\).
  7. What does the table tell you about \(d\) with respect to the false proposition \(a\) ?
  8. Explain why Russell introduced propositions \(b\) and \(c\) into his argument.
  9. Russell could correctly have started a deductive sequence: \(a \wedge [ a \Rightarrow ( ( 0.5 = - 0.5 ) \Rightarrow ( 0.25 = 0.25 ) ) ]\).
    Had he have done so could he correctly have continued it to end at \(d\) ?
    Justify your answer.
  10. Draw a combinatorial circuit to represent \(( a \wedge ( a \Rightarrow b ) \wedge c ) \Rightarrow d\). 3 Floyd's algorithm is applied to the incomplete network on 4 nodes drawn below. The weights on the arcs represent journey times. \includegraphics[max width=\textwidth, alt={}, center]{4b5bc097-1052-4e44-8623-a84ceaab0289-4_400_558_347_751} The final matrices are shown below. \begin{table}[h] \begin{center} \captionsetup{labelformat=empty} \caption{final time matrix} \begin{tabular}{ | l | r | r | r | r | } \cline { 2 - 5 } \multicolumn{1}{c|}{} & \multicolumn{1}{c|}{\(\mathbf { 1 }\)} & \(\mathbf { 2 }\) & \(\mathbf { 3 }\) & \multicolumn{1}{c|}{\(\mathbf { 4 }\)}
    \hline \(\mathbf { 1 }\) & 6 & 5 & 3 & 10
    \hline
OCR MEI D2 2015 June Q3
20 marks Moderate -0.5
3 Floyd's algorithm is applied to the incomplete network on 4 nodes drawn below. The weights on the arcs represent journey times. \includegraphics[max width=\textwidth, alt={}, center]{4b5bc097-1052-4e44-8623-a84ceaab0289-4_400_558_347_751} The final matrices are shown below. \begin{table}[h] \begin{center} \captionsetup{labelformat=empty} \caption{final time matrix} \begin{tabular}{ | l | r | r | r | r | } \cline { 2 - 5 } \multicolumn{1}{c|}{} & \multicolumn{1}{c|}{\(\mathbf { 1 }\)} & \(\mathbf { 2 }\) & \(\mathbf { 3 }\) & \multicolumn{1}{c|}{\(\mathbf { 4 }\)}
\hline \(\mathbf { 1 }\) & 6 & 5 & 3 & 10
\hline \(\mathbf { 2 }\) & 5 & 4 & 2 & 5
\hline \(\mathbf { 3 }\) & 3 & 2 & 4 & 7
\hline
OCR MEI D2 2015 June Q4
20 marks Standard +0.3
\(\mathbf { 4 }\) & 10 & 5 & 7 & 10
\hline \end{tabular} \end{center} \end{table} \begin{table}[h]
\captionsetup{labelformat=empty} \caption{final route matrix}
\cline { 2 - 5 } \multicolumn{1}{c|}{}\(\mathbf { 1 }\)\(\mathbf { 2 }\)\(\mathbf { 3 }\)\(\mathbf { 4 }\)
\(\mathbf { 1 }\)3333
\(\mathbf { 2 }\)3334
\(\mathbf { 3 }\)1222
\(\mathbf { 4 }\)2222
\end{table}
  1. Draw the complete network of shortest times.
  2. Explain how to use the final route matrix to find the quickest route from node \(\mathbf { 4 }\) to node \(\mathbf { 1 }\) in the original incomplete network. Give this quickest route. A new node, node 5, is added to the original incomplete network. The new journey times are shown in the table. \begin{center} \begin{tabular}{ | l | c | c | c | c | } \cline { 2 - 5 } \multicolumn{1}{c|}{} & \(\mathbf { 1 }\) & \(\mathbf { 2 }\) & \(\mathbf { 3 }\) & \(\mathbf { 4 }\) \hline
OCR MEI D2 2015 June Q5
Standard +0.3
\(\mathbf { 5 }\) & 4 & - & - & 2
\hline \end{tabular} \end{center} (iii) Draw the complete 5-node network of shortest times. (You are not required to use an algorithm to find the shortest times.)
(iv) Write down the final time matrix and the final route matrix for the complete 5 -node network. (You do not need to apply Floyd's algorithm.)
(v) (A) Apply the nearest neighbour algorithm to the complete 5-node network of shortest times, starting at node 1. Give the time for the cycle you produce.
(B) Starting at node 3, a possible cycle in the complete 5-node network of shortest times is \(\mathbf { 3 2 1 5 4 3 . }\) Give the actual cycle to which this corresponds in the incomplete 5-node network of journey times.
(vi) By deleting node 5 and its arcs from the complete 5 -node network of shortest times, and then using Kruskal's algorithm, produce a lower bound for the solution to the associated practical travelling salesperson problem. Show clearly your use of Kruskal's algorithm.
(vii) In the incomplete 5-node network of journey times, find a quickest route starting at node \(\mathbf { 5 }\) and using each of the 7 arcs at least once. Give the time of your route. 4 Helen has a meeting to go to in London. She has to travel from her home in G on the south coast to KC in London. She can drive from G to the west to A to catch a train, or she can drive to the east to W to catch a train on a different line. From both A and W she can travel to mainline terminuses V or LB in London. She will then travel by tube either from V to KC , or from LB to KC . The times for the various steps of her journey are shown in the tables. Both train services and tube services are subject to occasional delays, and these are modelled in the tables.
Driving timesto Ato W
From G20 min15 min
\multirow{2}{*}{Train journey}To VTo LB
normal timeprobability of delaydelaynormal timeprobability of delaydelay
From A1 hr 40 min0.0510 min1 hr 45 min0.0510 min
From W1 hr 30 min0.1020 min1 hr 35 min0.1020 min
\multirow{2}{*}{
Tube
journey
}		To KC
\cline { 2 - 4 }normal timeprobability of delaydelay
From V7 min0.202 min
From LB9 min0.102 min
Helen wants to choose the route which will give the shortest expected journey time.
(i) Draw a decision tree to model Helen's decisions and the possible outcomes.
(ii) Calculate Helen's shortest expected journey time and give the route which corresponds to that shortest expected journey time. As she gets into her car, Helen hears a travel bulletin on the radio warning of a broken escalator at V. This means that routes through V will take Helen 10 minutes longer.
(iii) Find the value of the radio information, explaining your calculation.
(iv) Why might the shortest expected journey time not be the best criterion for choosing a route for Helen?
OCR MEI D2 2016 June Q1
16 marks Moderate -0.5
1 Martin is considering paying for a vaccination against a disease. If he catches the disease he would not be able to work and would lose \(\pounds 900\) in income because he would have to stay at home recovering. The vaccination costs \(\pounds 20\). The vaccination would reduce his risk of catching the disease during the year from 0.02 to 0.001 .
  1. Draw a decision tree for Martin.
  2. Evaluate the EMV of Martin's loss at each node of your tree, and give the action that Martin should take to minimise the EMV of his loss. Martin can answer a medical questionnaire which will give an estimate of his susceptibility to the disease. If he is found to be susceptible, then his chance of catching the disease is 0.05 . Vaccination will reduce that to 0.0025 . If he is found not to be susceptible, then his chance of catching the disease is 0.01 and vaccination will reduce it to 0.0005 . Historically, \(25 \%\) of people are found to be susceptible.
  3. What is the EMV of this questionnaire? Martin decides not to answer the questionnaire. He also decides that there is more than just his EMV to be considered in deciding whether or not to have the vaccination. The vaccination itself is likely to have side effects, but catching the disease would be very unpleasant. Martin estimates that he would find the effects of the disease 1000 times more unpleasant than the effects of the vaccination.
  4. Analyse which course of action would minimise the unpleasantness for Martin.
OCR MEI D2 2016 June Q2
16 marks Easy -1.2
2
  1. Emelia: 'I won't go out for a walk if it's not dry or not warm.'
    Gemma: ‘It’s warm. Let’s go!’
    Will what Gemma has said convince Emelia, and if not, why not?
  2. If it is daytime and the car headlights are on, then it is raining. If the dashboard lights are dimmed then the car headlights are on.
    It is daytime.
    It is not raining.
    1. What can you deduce?
    2. Prove your deduction.
  3. In this part of the question the switch X is represented by \includegraphics[max width=\textwidth, alt={}, center]{d254fbd2-7443-4b6d-87ba-f0d71fce5e17-3_104_138_824_1226} The switch can be wired into a circuit so that current flows when
    the switch is up \includegraphics[max width=\textwidth, alt={}, center]{d254fbd2-7443-4b6d-87ba-f0d71fce5e17-3_103_177_1005_593}
    but does not flow when it is down \includegraphics[max width=\textwidth, alt={}, center]{d254fbd2-7443-4b6d-87ba-f0d71fce5e17-3_111_167_1000_1334} Or the switch can be wired so that current flows when
    the switch is down \includegraphics[max width=\textwidth, alt={}, center]{d254fbd2-7443-4b6d-87ba-f0d71fce5e17-3_109_172_1228_639}
    but does not flow when it is up \includegraphics[max width=\textwidth, alt={}, center]{d254fbd2-7443-4b6d-87ba-f0d71fce5e17-3_109_174_1228_1327}
    1. Explain how the following circuit models \(( \mathrm { A } \wedge \mathrm { B } ) \Rightarrow \mathrm { C }\). \includegraphics[max width=\textwidth, alt={}, center]{d254fbd2-7443-4b6d-87ba-f0d71fce5e17-3_365_682_1484_694} In the following circuit B1 and B2 represent 'ganged' switches. This means that the two switches are either both up or both down. \includegraphics[max width=\textwidth, alt={}, center]{d254fbd2-7443-4b6d-87ba-f0d71fce5e17-3_364_1278_2042_397}
    2. Given that A is down, C is up and current is flowing, what can you deduce?
OCR MEI D2 2016 June Q3
20 marks Standard +0.8
3 Neil is refurbishing a listed building. There are two types of paint that he can use for the inside walls. One costs \(\pounds 1.45\) per \(\mathrm { m } ^ { 2 }\) and the other costs \(\pounds 0.95\) per \(\mathrm { m } ^ { 2 }\). He must paint the lower half of each wall in the more expensive paint. He has \(350 \mathrm {~m} ^ { 2 }\) of wall to paint. He has a budget of \(\pounds 400\) for wall paint. The more expensive paint is easier to use, and so Neil wants to use as much of it as possible. Initially, the following LP is constructed to help Neil with his purchasing of paint.
Let \(x\) be the number of \(\mathrm { m } ^ { 2 }\) of wall painted with the expensive paint.
Let \(y\) be the number of \(\mathrm { m } ^ { 2 }\) of wall painted with the less expensive paint. $$\begin{array} { l l } \text { Maximise } & P = x + y \\ \text { subject to } & 1.45 x + 0.95 y \leqslant 400 \\ & y - x \leqslant 0 \\ & x \geqslant 0 \\ & y \geqslant 0 \end{array}$$
  1. Explain the purpose of the inequality \(y - x \leqslant 0\).
  2. The formulation does not include the inequality \(x + y \geqslant 350\). State what this constraint models and why it has been omitted from the formulation.
  3. Use the simplex algorithm to solve the LP. Pivot first on the "1" in the \(y\) column. Interpret your solution. The solution shows that Neil needs to buy more paint. He negotiates an increase in his budget to \(\pounds 450\).
  4. Find the solution to the LP given by changing \(1.45 x + 0.95 y \leqslant 400\) to \(1.45 x + 0.95 y \leqslant 450\), and interpret your solution. Neil realises that although he now has a solution, that solution is not the best for his requirements.
  5. Explain why the revised solution is not optimal for Neil. In order to move to an optimal solution Neil needs to change the objective of the LP and add another constraint to it.
  6. Write down the new LP and the initial tableau for using two-stage simplex to solve it. Give a brief description of how to use two-stage simplex to solve it. \includegraphics[max width=\textwidth, alt={}, center]{d254fbd2-7443-4b6d-87ba-f0d71fce5e17-5_497_558_269_751}
    (a) Solve the route inspection problem in the network above, showing the methodology you used to ensure that your solution is optimal. Show your route.
    (b) Floyd's algorithm is applied to the same network to find the complete network of shortest distances. After three iterations the distance and route matrices are as follows. \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 \(\mathbf { 1 }\) & 48 & 24 & 28 & 11 & 15
    \hline \(\mathbf { 2 }\) & 24 & 8 & 4 & 11 & 16
    \hline \(\mathbf { 3 }\) & 28 & 4 & 8 & 7 & 12
    \hline
OCR MEI D2 2016 June Q4
20 marks Standard +0.3
\(\mathbf { 4 }\) & 11 & 11 & 7 & 14 & 14
\hline
OCR MEI D2 2016 June Q5
Standard +0.8
\(\mathbf { 5 }\) & 15 & 16 & 12 & 14 & 24
\hline \end{tabular} \end{center}
\cline { 2 - 6 } \multicolumn{1}{c|}{}\(\mathbf { 1 }\)\(\mathbf { 2 }\)\(\mathbf { 3 }\)\(\mathbf { 4 }\)\(\mathbf { 5 }\)
\(\mathbf { 1 }\)22245
\(\mathbf { 2 }\)13333
\(\mathbf { 3 }\)22245
\(\mathbf { 4 }\)13335
\(\mathbf { 5 }\)13343
  1. Perform the fourth iteration of the algorithm, and show that there is no change to either matrix in the final iteration.
  2. Show how to use the matrices to give the shortest distance and the shortest route from vertex \(\mathbf { 1 }\) to vertex 2.
  3. Draw the complete network of shortest distances.
  4. 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.
  5. 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.
OCR MEI AS Paper 1 2018 June Q1
2 marks Easy -1.2
1 Write \(\frac { 8 } { 3 - \sqrt { 5 } }\) in the form \(a + b \sqrt { 5 }\), where \(a\) and \(b\) are integers to be found.
OCR MEI AS Paper 1 2018 June Q2
4 marks Easy -1.8
2 Find the binomial expansion of \(( 3 - 2 x ) ^ { 3 }\).
OCR MEI AS Paper 1 2018 June Q3
3 marks Moderate -0.8
3 A particle is in equilibrium under the action of three forces in newtons given by $$\mathbf { F } _ { 1 } = \binom { 8 } { 0 } , \quad \mathbf { F } _ { 2 } = \binom { 2 a } { - 3 a } \quad \text { and } \quad \mathbf { F } _ { 3 } = \binom { 0 } { b } .$$ Find the values of the constants \(a\) and \(b\).
OCR MEI AS Paper 1 2018 June Q4
4 marks Moderate -0.3
4 Fig. 4 shows a block of mass \(4 m \mathrm {~kg}\) and a particle of mass \(m \mathrm {~kg}\) connected by a light inextensible string passing over a smooth pulley. The block is on a horizontal table, and the particle hangs freely. The part of the string between the pulley and the block is horizontal. The block slides towards the pulley and the particle descends. In this motion, the friction force between the table and the block is \(\frac { 1 } { 2 } m g \mathrm {~N}\). \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{1513048a-d53b-4b85-82f4-c86e0d81f8f8-3_204_741_1151_662} \captionsetup{labelformat=empty} \caption{Fig. 4}
\end{figure} Find expressions for
  • the acceleration of the system,
  • the tension in the string.
OCR MEI AS Paper 1 2018 June Q5
7 marks Standard +0.8
5
  1. Sketch the graphs of \(y = 4 \cos x\) and \(y = 2 \sin x\) for \(0 ^ { \circ } \leqslant x \leqslant 180 ^ { \circ }\) on the same axes.
  2. Find the exact coordinates of the point of intersection of these graphs, giving your answer in the form (arctan \(a , k \sqrt { b }\) ), where \(a\) and \(b\) are integers and \(k\) is rational.
  3. A student argues that without the condition \(0 ^ { \circ } \leqslant x \leqslant 180 ^ { \circ }\) all the points of intersection of the graphs would occur at intervals of \(360 ^ { \circ }\) because both \(\sin x\) and \(\cos x\) are periodic functions with this period. Comment on the validity of the student's argument.
OCR MEI AS Paper 1 2018 June Q6
5 marks Moderate -0.8
6 In this question you must show detailed reasoning.
You are given that \(\mathrm { f } ( x ) = 4 x ^ { 3 } - 3 x + 1\).
  1. Use the factor theorem to show that \(( x + 1 )\) is a factor of \(\mathrm { f } ( x )\).
  2. Solve the equation \(\mathrm { f } ( x ) = 0\).
OCR MEI AS Paper 1 2018 June Q7
6 marks Standard +0.3
7 A toy boat of mass 1.5 kg is pushed across a pond, starting from rest, for 2.5 seconds. During this time, the boat has an acceleration of \(2 \mathrm {~m} \mathrm {~s} ^ { - 2 }\). Subsequently, when the only horizontal force acting on the boat is a constant resistance to motion, the boat travels 10 m before coming to rest. Calculate the magnitude of the resistance to motion.
OCR MEI AS Paper 1 2018 June Q9
9 marks Standard +0.3
9 The curve \(y = ( x - 1 ) ^ { 2 }\) maps onto the curve \(\mathrm { C } _ { 1 }\) following a stretch scale factor \(\frac { 1 } { 2 }\) in the \(x\)-direction.
  1. Show that the equation of \(\mathrm { C } _ { 1 }\) can be written as \(y = 4 x ^ { 2 } - 4 x + 1\). The curve \(\mathrm { C } _ { 2 }\) is a translation of \(y = 4.25 x - x ^ { 2 }\) by \(\binom { 0 } { - 3 }\).
  2. Show that the normal to the curve \(\mathrm { C } _ { 1 }\) at the point \(( 0,1 )\) is a tangent to the curve \(\mathrm { C } _ { 2 }\).
OCR MEI AS Paper 1 2018 June Q10
9 marks Standard +0.3
10 Rory runs a distance of 45 m in 12.5 s . He starts from rest and accelerates to a speed of \(4 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). He runs the remaining distance at \(4 \mathrm {~ms} ^ { - 1 }\). Rory proposes a model in which the acceleration is constant until time \(T\) seconds.
  1. Sketch the velocity-time graph for Rory's run using this model.
  2. Calculate \(T\).
  3. Find an expression for Rory's displacement at time \(t \mathrm {~s}\) for \(0 \leqslant t \leqslant T\).
  4. Use this model to find the time taken for Rory to run the first 4 m . Rory proposes a refined model in which the velocity during the acceleration phase is a quadratic function of \(t\). The graph of Rory's quadratic goes through \(( 0,0 )\) and has its maximum point at \(( S , 4 )\). In this model the acceleration phase lasts until time \(S\) seconds, after which the velocity is constant.
  5. Sketch a velocity-time graph that represents Rory's run using this refined model.
  6. State with a reason whether \(S\) is greater than \(T\) or less than \(T\). (You are not required to calculate the value of \(S\).)
OCR MEI AS Paper 1 2018 June Q11
13 marks Moderate -0.8
11 The intensity of the sun's radiation, \(y\) watts per square metre, and the average distance from the sun, \(x\) astronomical units, are shown in Fig. 11 for the planets Mercury and Jupiter. \begin{table}[h]
\(x\)\(y\)
Mercury0.307514400
Jupiter4.95055.8
\captionsetup{labelformat=empty} \caption{Fig. 11}
\end{table} The intensity \(y\) is proportional to a power of the distance \(x\).
  1. Write down an equation for \(y\) in terms of \(x\) and two constants.
  2. Show that the equation can be written in the form \(\ln y = a + b \ln x\).
  3. In the Printed Answer Booklet, complete the table for \(\ln x\) and \(\ln y\) correct to 4 significant figures.
  4. Use the values from part (iii) to find \(a\) and \(b\).
  5. Hence rewrite your equation from part (i) for \(y\) in terms of \(x\), using suitable numerical values for the constants.
  6. Sketch a graph of the equation found in part (v).
  7. Earth is 1 astronomical unit from the sun. Find the intensity of the sun's radiation for Earth.