Questions D2 - A-Level Maths

\cline { 3 - 4 } \multicolumn{2}{c\|}{}	$B$
\cline { 3 - 4 }	I	II
\multirow{2}{*}{$A$}	I	4	${ } ^ { - } 8$
\cline { 2 - 4 }	II	2	${ } ^ { - } 4$
\cline { 2 - 4 }	III	${ } ^ { - } 8$	2

Explain why the game does not have a saddle point.
Using a graphical method, find the optimal strategy for player $B$.
Find the optimal strategy for player $A$.
Find the value of the game.

Edexcel D2 Q5

5. A carpet manufacturer has two warehouses, $W _ { 1 }$ and $W _ { 2 }$, which supply carpets for three sales outlets, $S _ { 1 } , S _ { 2 }$ and $S _ { 3 }$. At one point $S _ { 1 }$ requires 40 rolls of carpet, $S _ { 2 }$ requires 23 rolls of carpet and $S _ { 3 }$ requires 37 rolls of carpet. At this point $W _ { 1 }$ has 45 rolls in stock and $W _ { 2 }$ has 40 rolls in stock. The following table shows the cost, in pounds, of transporting one roll from each warehouse to each sales outlet:

\cline { 2 - 4 } \multicolumn{1}{c\|}{}	$S _ { 1 }$	$S _ { 2 }$	$S _ { 3 }$
$W _ { 1 }$	8	7	11
$W _ { 2 }$	9	10	11

The company's manager wishes to supply the 85 rolls that are in stock such that transportation costs are kept to a minimum.

Use the north-west corner rule to obtain an initial solution to the problem.
Calculate improvement indices for the unused routes.
Use the stepping-stone method to obtain an optimal solution. Turn over

Edexcel D2 Q6

6. This question should be answered on the sheet provided. \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{073926c5-03cc-41d4-82bf-315740ead663-6_672_984_322_431} \captionsetup{labelformat=empty} \caption{Fig. 1}

\end{figure} A band is going on tour to play gigs in six towns, including their home town, $A$. The network in Figure 1 shows the distances, in miles, between the various towns. The band must begin and end their tour at $A$ and visit each of the other towns once, and they wish to keep the total distance travelled as small as possible.

By inspection, draw a complete network showing the shortest distances between the towns.
Use your complete network and the nearest neighbour algorithm, starting at $A$, to find an upper bound for the total distance travelled.
1. Use your complete network to obtain and draw a minimum spanning tree and hence obtain another upper bound for the total distance travelled.
2. Improve this upper bound using two shortcuts to find an upper bound below 225 miles.
By deleting $A$, find a lower bound for the total distance travelled.
State an interval of as small a width as possible within which $d$, the minimum distance travelled, in miles, must lie. \section*{Please hand this sheet in for marking}
Stage Previous tournament Current tournament
\multirow[t]{3}{*}{1} G
J
K
L
$H$
J
K
L
I
J
K
L
\multirow[t]{3}{*}{2} D
G
H
I
$E$
G
H
I
$F$
G
H
I
\multirow[t]{3}{*}{3} A
D
E
F
$B$
D
E
F
C
D
E
F
4 None
A
B
C
\section*{Please hand this sheet in for marking}
\includegraphics[max width=\textwidth, alt={}, center]{073926c5-03cc-41d4-82bf-315740ead663-8_684_992_461_427}
\section*{Sheet for answering question 6 (cont.)}
2. $\_\_\_\_$
$\_\_\_\_$

OCR MEI D2 2014 June Q2

Rachel thinks that the answer given in the newspaper article is not sensible. Give a verbal argument why Rachel might think that the batsman should be given out. Rachel tries to formalise her argument. She defines four simple propositions.
o: "The batsman is given out."
lb: "The batsman is given out (LBW)."
c: "The batsman is given out (caught)."
b: "The ball hit the bat."
An implication of the batsman not being out (LBW) is that the ball has hit the bat. Write this down in terms of Rachel's propositions.
Similarly, write down the implication of the batsman not being out (caught).
Using your answers to parts (ii) and (iii) write down the implication of a batsman being not out, in terms of $b$ and $\sim b$.
[0pt] [You may assume that if $\mathrm { w } \Rightarrow \mathrm { y }$ and $\mathrm { x } \Rightarrow \mathrm { z }$, then $( \mathrm { w } \wedge \mathrm { x } ) \Rightarrow ( \mathrm { y } \wedge \mathrm { z } )$. ]
By writing down the contrapositive of your implication from part (iv), produce an implication which supports Rachel's argument.
(b) A classroom rule has been broken by either Anja, Bobby, Catherine or Dimitria, or by a subset of those four. The teacher knows that Dimitria could not have done it on her own. Let $a$ be the proposition "Anja is guilty", and similarly for $b , c$ and $d$.
Express the teacher's knowledge as a compound proposition. Evidence emerges that Bobby and Catherine were elsewhere at the time, so they cannot be guilty. This can be expressed as the compound proposition $\sim ( b \vee c )$.
Construct a truth table to show the truth values of the compound proposition given by the conjunction of the two compound propositions, one from part (i) and one given above.
What does your truth table tell you about who is guilty? 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.
Explain how the initial feasible tableau shown in Fig. 3 models this problem. \begin{table}[h]
1(v)
1(vi)
1
\begin{center} \begin{tabular}{|l|l|} \hline 2(a)(i) &
\hline &
\hline &
\hline &
\hline &
\hline &
\hline &
\hline &
\hline &
\hline &
\hline

OCR D2 2007 January Q6

6 Answer this question on the insert provided. The table shows a partially completed dynamic programming tabulation for solving a maximin problem.

Stage	State	Action	Working	Maximin
\multirow{2}{*}{1}	0	0	4	4
	1	0	3	3
\multirow{6}{*}{2}	0	0	$\min ( 6,4 ) = 4$	\multirow{2}{*}{}
		1	$\min ( 2,3 ) = 2$
	\multirow{2}{*}{1}	0	$\min ( 2,4 ) =$	\multirow{2}{*}{}
		1	$\min ( 4,3 ) =$
	\multirow{2}{*}{2}	0	min(2,	\multirow{2}{*}{}
		1	min(3,
\multirow{3}{*}{3}	\multirow{3}{*}{0}	0	min(5,	\multirow{3}{*}{}
		1	$\min ( 5$,
		2	$\min ( 2$,

Complete the last two columns of the table in the insert.
State the maximin value and write down the maximin route.

OCR D2 2006 January Q1

1 Answer this question on the insert provided. Mrs Price has bought six T shirts for her children. Each child is to have two shirts.
Amanda would like the green shirt, the pink shirt or the red shirt.
Ben would like the green shirt, the turquoise shirt, the white shirt or the yellow shirt.
Carrie would like the pink shirt, the white shirt or the yellow shirt.

On the first diagram in the insert, draw a bipartite graph to show which child would like which shirt. The children are represented as $A 1 , A 2 , B 1 , B 2 , C 1$ and $C 2$ and the shirts as $G , P , R , T , W$ and $Y$. Initially, Mrs Price puts aside the green shirt and the pink shirt for Amanda, the turquoise shirt and the white shirt for Ben and the yellow shirt for Carrie.
Show this incomplete matching on the second diagram in the insert.
Write down an alternating path consisting of three arcs to enable the matching to be improved. Use your alternating path to match the children to the shirts.
Amanda decides that she does not like the green shirt after all. Which shirts should each child have now?

OCR D2 2006 January Q2

2 Answer this question on the insert provided. The diagram shows a directed network of paths with vertices labelled with (stage; state) labels. The weights on the arcs represent distances in km . The shortest route from $( 3 ; 0 )$ to $( 0 ; 0 )$ is required. Complete the dynamic programming tabulation on the insert, working backwards from stage 1 , to find the shortest route through the network. Give the length of this shortest route. \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{9c9b1a42-8d16-446a-85a1-4c08e5e368be-2_501_1018_1741_575} \captionsetup{labelformat=empty} \caption{Stage 3 Stage 2 Stage 1}

\end{figure}

OCR D2 2006 January Q5

5 Answer this question on the insert provided. The diagram shows an activity network for a project. The table lists the durations of the activities (in days).
\includegraphics[max width=\textwidth, alt={}, center]{9c9b1a42-8d16-446a-85a1-4c08e5e368be-4_652_867_429_393}

Activity	Duration
$A$	5
$B$	3
$C$	4
$D$	2
$E$	1
$F$	3
$G$	5
$H$	2
$I$	4
$J$	3

Explain why each of the dummy activities is needed.
Complete the blank column of the table in the insert to show the immediate predecessors for each activity.
Carry out a forward pass to find the early start times for the events. Record these at the eight vertices on the copy of the network on the insert. Also calculate the late start times for the events and record these at the vertices. Find the minimum completion time for the project and list the critical activities.
By how much would the duration of activity $C$ need to increase for $C$ to become a critical activity? Assume that each activity requires one worker and that each worker is able to do any of the activities. The activities may not be split. The duration of $C$ is 4 days.
Draw a resource histogram, assuming that each activity starts at its earliest possible time. How many workers are needed with this schedule?
Describe how, by delaying the start of activity $E$ (and other activities, to be determined), the project can be completed in the minimum time by just three workers.

OCR D2 2008 January Q5

5 Answer this question on the insert provided. The diagram shows an activity network for a project. The figures in brackets show the durations of the activities in days.
\includegraphics[max width=\textwidth, alt={}, center]{95fbb09b-0301-4fc1-b694-838b8d0b64a6-06_956_921_495_612}

Complete the table in the insert to show the precedences for the activities.
Use the boxes on the diagram in the insert to carry out a forward pass and a backward pass. Find the minimum project duration and list the critical activities. The number of people required for each activity is shown in the table below. The workers are all equally skilled at all of the activities.
Activity $A$ $B$ $C$ $D$ $E$ $F$ $G$ $H$ $I$ $J$
Number of workers 4 1 2 2 3 2 3 3 1 2
On graph paper, draw a resource histogram for the project with each activity starting at its earliest possible time.
Describe how the project can be completed in 21 days using just six workers.

OCR D2 2009 January Q1

1 Answer this question on the insert provided. The table shows a partially completed dynamic programming tabulation for solving a maximin problem.

Stage	State	Action	Working	Maximin
\multirow{4}{*}{1}	0	0	10
	1	0	11
	2	0	14
	3	0	15
\multirow{10}{*}{2}	\multirow{2}{*}{0}	0	(12, ) =	\multirow{2}{*}{}
		2	$( 10 , \quad ) =$
	\multirow{3}{*}{1}	0	$( 13 , \quad ) =$	\multirow{3}{*}{}
		1	$( 10 , \quad ) =$
		2	(11, ) =
	\multirow{3}{*}{2}	1	( 9, ) =	\multirow{3}{*}{}
		2	(10, ) =
		3	( 7, ) =
	\multirow{2}{*}{3}	1	( 8, ) =	\multirow{2}{*}{}
		3	(12, ) =
\multirow{4}{*}{3}	\multirow{4}{*}{0}	0	$( 15 , \quad ) =$	\multirow{4}{*}{}
		1	$( 14 , \quad ) =$
		2	(16, ) =
		3	(13, ) =

Complete the last two columns of the table in the insert.
State the maximin value and write down the maximin route.

OCR D2 2009 January Q2

2 Answer this question on the insert provided. The diagram shows an activity network for a project. The figures in brackets show the durations of the activities in days.
\includegraphics[max width=\textwidth, alt={}, center]{c5bfbe78-64c4-4254-ad83-0c90f4a54b18-3_497_1230_493_459}

Complete the table in the insert to show the precedences for the activities.
Use the boxes on the diagram in the insert to carry out a forward pass and a backward pass. Show that the minimum project completion time is 28 days and list the critical activities. The resource histogram below shows the number of workers required each day when the activities each begin at their earliest possible start time. Once an activity has been started it runs for its duration without a break.
\includegraphics[max width=\textwidth, alt={}, center]{c5bfbe78-64c4-4254-ad83-0c90f4a54b18-3_457_1543_1503_299}
By considering which activities are happening each day, complete the table in the insert to show the number of workers required for each activity. You are advised to start at day 28 and work back through the days towards day 1 . Only five workers are actually available, but they are all equally skilled at each of the activities. The project can still be completed in 28 days by delaying the start of activity $E$.
Find the minimum possible delay and the maximum possible delay on activity $E$ in this case.

OCR D2 2009 January Q3

3 Answer this question on the insert provided. \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{c5bfbe78-64c4-4254-ad83-0c90f4a54b18-4_625_1100_358_520} \captionsetup{labelformat=empty} \caption{Fig. 1}

\end{figure} Fig. 1 represents a system of pipes through which fluid can flow from a source, $S$, to a sink, $T$. It also shows a cut $\alpha$. The weights on the arcs show the lower and upper capacities of the pipes in litres per second.

Calculate the capacity of the cut $\alpha$.
By considering vertex $B$, explain why arc $S B$ must be at its lower capacity. Then by considering vertex $E$, explain why arc $C E$ must be at its upper capacity, and hence explain why arc $H T$ must be at its lower capacity.
On the diagram in the insert, show a flow through the network of 15 litres per second. Write down one flow augmenting route that allows another 1 litre per second to flow through the network. Show that the maximum flow is 16 litres per second by finding a cut of 16 litres per second. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{c5bfbe78-64c4-4254-ad83-0c90f4a54b18-4_602_1086_1809_568} \captionsetup{labelformat=empty} \caption{Fig. 2}
\end{figure} Fig. 2 represents the same system, but with pipe $E B$ installed the wrong way round.
Explain why there can be no feasible flow through this network.

OCR D2 2011 January Q6

6 Answer this question on the insert provided. Four friends have decided to sponsor four birds at a bird sanctuary. They want to construct a route through the bird sanctuary, starting and ending at the entrance/exit, that enables them to visit the four birds in the shortest possible time. The table below shows the times, in minutes, that it takes to get between the different birds and the entrance/exit. The friends will spend the same amount of time with each bird, so this does not need to be included in the calculation.

	Entrance/exit	Kite	Lark	Moorhen	Nightjar
Entrance/exit	-	10	14	12	17
Kite	10	-	3	2	6
Lark	14	3	-	2	4
Moorhen	12	2	2	-	3
Nightjar	17	6	4	3	-

Let the stages be $0,1,2,3,4,5$. Stage 0 represents arriving at the sanctuary entrance. Stage 1 represents visiting the first bird, stage 2 the second bird, and so on, with stage 5 representing leaving the sanctuary. Let the states be $0,1,2,3,4$ representing the entrance/exit, kite, lark, moorhen and nightjar respectively.

Calculate how many minutes it takes to travel the route $$( 0 ; 0 ) - ( 1 ; 1 ) - ( 2 ; 2 ) - ( 3 ; 3 ) - ( 4 ; 4 ) - ( 5 ; 0 ) .$$ The friends then realise that if they try to find the quickest route using dynamic programming with this (stage; state) formulation, they will get the route $( 0 ; 0 ) - ( 1 ; 1 ) - ( 2 ; 2 ) - ( 3 ; 3 ) - ( 4 ; 1 ) - ( 5 ; 0 )$, or this in reverse, taking 27 minutes.
Explain why the route $( 0 ; 0 ) - ( 1 ; 1 ) - ( 2 ; 2 ) - ( 3 ; 3 ) - ( 4 ; 1 ) - ( 5 ; 0 )$ is not a solution to the friends' problem. Instead, the friends set up a dynamic programming tabulation with stages and states as described above, except that now the states also show, in brackets, any birds that have already been visited. So, for example, state $1 ( 234 )$ means that they are currently visiting the kite and have already visited the other three birds in some order. The partially completed dynamic programming tabulation is shown opposite.
For the last completed row, i.e. stage 2, state 1(3), action 4(13), explain where the value 18 and the value 6 in the working column come from.

Complete the table in the insert and hence find the order in which the birds should be visited to give a quickest route and find the corresponding minimum journey time.

$21 + 3 = 24 18 + 6 = 24$

OCR D2 2007 June Q4

4 Answer this question on the insert provided. The table shows a partially completed dynamic programming tabulation for solving a minimax problem.

Stage	State	A ction	Working	M inimax
\multirow{3}{*}{1}	0	0	4	4
	1	0	3	3
	2	0	2	2
\multirow{9}{*}{2}	\multirow{3}{*}{0}	0	$\max ( 6,4 ) = 6$	\multirow{3}{*}{3}
		1	$\max ( 2,3 ) = 3$
		2	$\max ( 3,2 ) = 3$
	\multirow{3}{*}{1}	0	$\max ( 2,4 ) =$	\multirow{3}{*}{}
		1	$\max ( 4,3 ) =$
		2	$\max ( 5,2 ) =$
	\multirow{3}{*}{2}	0	max(2,	\multirow{3}{*}{}
		1	max(3,
		2	max(4,
\multirow{3}{*}{3}	\multirow{3}{*}{0}	0	max(5,	\multirow{3}{*}{}
		1	max(5,
		2	max(2,

On the insert, complete the last two columns of the table.
State the minimax value and write down the minimax route.
Complete the diagram on the insert to show the network that is represented by the table.

OCR D2 2007 June Q5

5 Answer this question on the insert provided. The network represents a system of pipes through which fluid can flow from a source, S , to a sink, T .
\includegraphics[max width=\textwidth, alt={}, center]{09d4aacd-026b-4d81-a826-3d3f29f9c105-5_1310_1301_447_424} The arrows are labelled to show excess capacities and potential backflows (how much more and how much less could flow in each pipe). The excess capacities and potential backflows are measured in litres per second. Currently the flow is 6 litres per second, all flowing along a single route through the system.

Write down the route of the 6 litres per second that is flowing from $S$ to $T$.
What is the capacity of the pipe AG and in which direction can fluid flow along this pipe?
Calculate the capacity of the $\operatorname { cut } \mathrm { X } = \{ \mathrm { S } , \mathrm { A } , \mathrm { B } , \mathrm { C } , \mathrm { D } , \mathrm { E } \} , \mathrm { Y } = \{ \mathrm { F } , \mathrm { G } , \mathrm { H } , \mathrm { I } , \mathrm { T } \}$.
Describe how a further 7 litres per second can flow from S to T and update the labels on the arrows to show your flow. Explain how you know that this is the maximum flow. \href{http://physicsandmathstutor.com}{physicsandmathstutor.com}

OCR MEI D2 2006 June Q2

2 Answer this question on the insert provided. Fig. 2 shows a network in which the weights on the arcs represent distances. \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{9716cf3f-afa5-44a4-a8cd-f7511449d06b-2_405_497_1046_776} \captionsetup{labelformat=empty} \caption{Fig. 2}

\end{figure}

Apply Floyd's algorithm on the insert provided to find the complete network of shortest distances.
Show how to use your final matrices to find the shortest route from vertex $\mathbf { 1 }$ to vertex 3, together with the length of that route.
Use the nearest neighbour algorithm, starting at vertex 1, to find a Hamilton cycle in the complete network of shortest distances. Give the corresponding cycle in the original network, together with its length.

AQA D2 Q4

4 [Figures 3, 4 and 5, printed on the insert, are provided for use in this question.]
The network shows a system of pipes, with the lower and upper capacities for each pipe in litres per second.
\includegraphics[max width=\textwidth, alt={}, center]{c18db720-6fe8-4e6c-bd0c-dc51cc341b47-005_547_1214_555_404}

Figure 3, on the insert, shows a partially completed diagram for a feasible flow of 10 litres per second from $S$ to $T$. Indicate, on Figure 3, the flows along the edges $M N , P Q , N P$ and $N T$.
1. Taking your answer from part (a) as an initial flow, use flow augmentation on Figure 4 to find the maximum flow from $S$ to $T$.
2. State the value of the maximum flow and illustrate this flow on Figure 5.
Find a cut with capacity equal to that of the maximum flow.

AQA D2 Q7

7 The network below shows a system of one-way roads. The number on each edge represents the number of bags for recycling that can be collected by driving along that road. A collector is to drive from $A$ to $I$.
\includegraphics[max width=\textwidth, alt={}, center]{c18db720-6fe8-4e6c-bd0c-dc51cc341b47-144_867_1644_552_191}

Working backwards from $\boldsymbol { I }$, use dynamic programming to find the maximum number of bags that can be collected when driving from $A$ to $I$. You must complete the table opposite as your solution.
State the route that the collector should take in order to collect the maximum number of bags.
Stage State From Value
1 G I
H I
2

AQA D2 Q8

8 The network below represents a system of pipes. The capacity of each pipe, in litres per second, is indicated on the corresponding edge.
\includegraphics[max width=\textwidth, alt={}, center]{c18db720-6fe8-4e6c-bd0c-dc51cc341b47-146_743_977_404_536}

Find the maximum flow along each of the routes $A B E H , A C F H$ and $A D G H$ and enter their values in the table on Figure 4 opposite.
1. Taking your answers to part (a) as the initial flow, use the labelling procedure on Figure 4 to find the maximum flow through the network. You should indicate any flow-augmenting routes in the table and modify the potential increases and decreases of the flow on the network.
2. State the value of the maximum flow and, on Figure 5 opposite, illustrate a possible flow along each edge corresponding to this maximum flow.
Confirm that you have a maximum flow by finding a cut of the same value. List the edges of your cut. \begin{table}[h]
\captionsetup{labelformat=empty} \caption{Figure 4}
Route Flow
$A B E H$
$A C F H$
$A D G H$
\end{table} \begin{figure}[h]
\captionsetup{labelformat=empty} \caption{Figure 4} \includegraphics[alt={},max width=\textwidth]{c18db720-6fe8-4e6c-bd0c-dc51cc341b47-147_746_972_397_845}
\end{figure} \begin{figure}[h]
\captionsetup{labelformat=empty} \caption{Figure 5} \includegraphics[alt={},max width=\textwidth]{c18db720-6fe8-4e6c-bd0c-dc51cc341b47-147_739_971_1311_539}
\end{figure}

AQA D2 2006 January Q1

1 Five trainers, Ali, Bo, Chas, Dee and Eve, held an initial training session with each of four teams over an assault course. The completion times in minutes are recorded below.

	Ali	Bo	Chas	Dee	Eve
Team 1	16	19	18	25	24
Team 2	22	21	20	26	25
Team 3	21	22	23	21	24
Team 4	20	21	21	23	20

Each of the four teams is to be allocated a trainer and the overall time for the four teams is to be minimised. No trainer can train more than one team.

Modify the table of values by adding an extra row of values so that the Hungarian algorithm can be applied.
Use the Hungarian algorithm, reducing columns first then rows, to decide which four trainers should be allocated to which team. State the minimum total training time for the four teams using this matching.

AQA D2 2006 January Q2

2 A manufacturing company is planning to build three new machines, $A , B$ and $C$, at the rate of one per month. The order in which they are built is a matter of choice, but the profits will vary according to the number of workers available and the suppliers' costs. The expected profits in thousands of pounds are given in the table.

\multirow[b]{2}{*}{Month}	\multirow[b]{2}{*}{Already built}	Profit (in units of £1000)
		$\boldsymbol { A }$	$\boldsymbol { B }$	$\boldsymbol { C }$
1	-	52	47	48
\multirow[t]{3}{*}{2}	A	-	58	54
	B	70	-	54
	$\boldsymbol { C }$	68	63	-
\multirow[t]{3}{*}{3}	$\boldsymbol { A }$ and $\boldsymbol { B }$	-	-	64
	$\boldsymbol { A }$ and $\boldsymbol { C }$	-	67	-
	$\boldsymbol { B }$ and $\boldsymbol { C }$	69	-	-

Draw a labelled network such that the most profitable order of manufacture corresponds to the longest path within that network.
Use dynamic programming to determine the order of manufacture that maximises the total profit, and state this maximum profit.

AQA D2 2006 January Q3

3 [Figures 1 and 2, printed on the insert, are provided for use in this question.] A building project is to be undertaken. The table shows the activities involved.

Activity	Immediate Predecessors	Duration (days)	Number of Workers Required
A	-	2	3
B	A	4	2
C	A	6	1
D	$B , C$	8	3
E	C	3	2
F	D	2	2
G	D, E	4	2
H	D, E	6	1
I	$F , G , H$	2	3

Complete the activity network for the project on Figure 1.
Find the earliest start time for each activity.
Find the latest finish time for each activity.
Find the critical path and state the minimum time for completion.
State the float time for each non-critical activity.
Given that each activity starts as early as possible, draw a resource histogram for the project on Figure 2.
There are only 3 workers available at any time. Use resource levelling to explain why the project will overrun and state the minimum extra time required.

AQA D2 2006 January Q4

4 [Figures 3, 4 and 5, printed on the insert, are provided for use in this question.]
The network shows a system of pipes, with the lower and upper capacities for each pipe in litres per second.
\includegraphics[max width=\textwidth, alt={}, center]{30a88efe-fe9e-4384-a3e3-da2a05326797-04_547_1214_555_404}

Figure 3, on the insert, shows a partially completed diagram for a feasible flow of 10 litres per second from $S$ to $T$. Indicate, on Figure 3, the flows along the edges $M N , P Q , N P$ and $N T$.
1. Taking your answer from part (a) as an initial flow, use flow augmentation on Figure 4 to find the maximum flow from $S$ to $T$.
2. State the value of the maximum flow and illustrate this flow on Figure 5.
Find a cut with capacity equal to that of the maximum flow.

AQA D2 2006 January Q5

Display the following linear programming problem in a Simplex tableau. $$\begin{array} { l c } \text { Maximise } & P = 3 x + 2 y + 4 z
\text { subject to } & x + 4 y + 2 z \leqslant 8
& 2 x + 7 y + 3 z \leqslant 21
& x \geqslant 0 , y \geqslant 0 , z \geqslant 0 \end{array}$$
Use the Simplex method to perform one iteration of your tableau for part (a), choosing a value in the $z$-column as pivot.
1. Perform one further iteration.
2. State whether or not this is the optimal solution, and give a reason for your answer.

AQA D2 2006 January Q6

6 Sam is playing a computer game in which he is trying to drive a car in different road conditions. He chooses a car and the computer decides the road conditions. The points scored by Sam are shown in the table.

		Road Conditions
\cline { 2 - 5 }		$\boldsymbol { C } _ { \mathbf { 1 } }$	$\boldsymbol { C } _ { \mathbf { 2 } }$	$\boldsymbol { C } _ { \mathbf { 3 } }$
\cline { 2 - 5 }	$\boldsymbol { S } _ { \mathbf { 1 } }$	- 2	2	4
\cline { 2 - 5 } Sam's Car	$\boldsymbol { S } _ { \mathbf { 2 } }$	2	4	5
\cline { 2 - 5 }	$\boldsymbol { S } _ { \mathbf { 3 } }$	5	1	2
\cline { 2 - 5 }
\cline { 2 - 5 }

Sam is trying to maximise his total points and the computer is trying to stop him.

Explain why Sam should never choose $S _ { 1 }$ and why the computer should not choose $C _ { 3 }$.
Find the play-safe strategies for the reduced 2 by 2 game for Sam and the computer, and hence show that this game does not have a stable solution.
Sam uses random numbers to choose $S _ { 2 }$ with probability $p$ and $S _ { 3 }$ with probability $1 - p$.
1. Find expressions for the expected gain for Sam when the computer chooses each of its two remaining strategies.
2. Calculate the value of $p$ for Sam to maximise his total points.
3. Hence find the expected points gain for Sam.
  Surname Other Names
  Centre Number Candidate Number
  Candidate Signature
  \section*{General Certificate of Education January 2006
  Advanced Level Examination} \section*{MATHEMATICS
  Unit Decision 2} MD02 \section*{Insert} Wednesday 18 January 20061.30 pm to 3.00 pm Insert for use in Questions 3 and 4.
  Fill in the boxes at the top of this page.
  Fasten this insert securely to your answer book.

\cline { 2 - 4 } \multicolumn{1}{c\|}{}	\(S _ { 1 }\)	\(S _ { 2 }\)	\(S _ { 3 }\)
\(W _ { 1 }\)	8	7	11
\(W _ { 2 }\)	9	10	11

Activity	Duration
\(A\)	5
\(B\)	3
\(C\)	4
\(D\)	2
\(E\)	1
\(F\)	3
\(G\)	5
\(H\)	2
\(I\)	4
\(J\)	3

Activity	\(A\)	\(B\)	\(C\)	\(D\)	\(E\)	\(F\)	\(G\)	\(H\)	\(I\)	\(J\)
Number of workers	4	1	2	2	3	2	3	3	1	2

Route	Flow
\(A B E H\)
\(A C F H\)
\(A D G H\)

		Road Conditions
\cline { 2 - 5 }		\(\boldsymbol { C } _ { \mathbf { 1 } }\)	\(\boldsymbol { C } _ { \mathbf { 2 } }\)	\(\boldsymbol { C } _ { \mathbf { 3 } }\)
\cline { 2 - 5 }	\(\boldsymbol { S } _ { \mathbf { 1 } }\)	- 2	2	4
\cline { 2 - 5 } Sam's Car	\(\boldsymbol { S } _ { \mathbf { 2 } }\)	2	4	5
\cline { 2 - 5 }	\(\boldsymbol { S } _ { \mathbf { 3 } }\)	5	1	2
\cline { 2 - 5 }
\cline { 2 - 5 }

\cline { 3 - 4 } \multicolumn{2}{c\|}{}	\(B\)
\cline { 3 - 4 }	I	II
\multirow{2}{*}{\(A\)}	I	4	\({ } ^ { - } 8\)
\cline { 2 - 4 }	II	2	\({ } ^ { - } 4\)
\cline { 2 - 4 }	III	\({ } ^ { - } 8\)	2

1(v)
1(vi)





1

Stage	State	From	Value
1	G	I
	H	I

2

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