Questions D2 - A-Level Maths

By deleting C, find another lower bound for the length of the route. State which is the best lower bound of the three, giving a reason for your answer.
By inspection complete the table of least distances. \begin{figure}[h]
\captionsetup{labelformat=empty} \caption{(8)
(8)
(Total 13 marks)} \includegraphics[alt={},max width=\textwidth]{a5d69a77-c196-483c-a550-1a55363555af-3_780_889_1069_1078}
\end{figure} (4) The table can now be taken to represent a complete network. The nearest neighbour algorithm was used to obtain upper bounds for the length of the route: Starting at D, an upper bound for the length of the route was found to be 838 km .
Starting at F, an upper bound for the length of the route was found to be 707 km .
Starting at C , use the nearest neighbour algorithm to obtain another upper bound for the length of the route. State which is the best upper bound of the
A B C D E F G H
A - 84 85 138 173 149 52
B 84 - 130 77 126 213 222 136
C 85 130 - 53 88 83 92
D 138 77 53 - 49 190
E 173 126 88 49 - 100 180 215
F 213 83 100 - 163 115
G 149 222 92 180 163 - 97
H 52 136 190 215 115 97 -
three, giving a reason for your answer.
(4) (Total 13 marks)

Edexcel D2 2006 January Q7

Define the terms
1. cut,
2. minimum cut, as applied to a directed network flow.
  \includegraphics[max width=\textwidth, alt={}, center]{a5d69a77-c196-483c-a550-1a55363555af-4_844_1465_338_299} The figure above shows a capacitated directed network and two cuts $C _ { 1 }$ and $C _ { 2 }$. The number on each arc is its capacity.
State the values of the cuts $C _ { 1 }$ and $C _ { 2 }$. Given that one of these two cuts is a minimum cut,
find a maximum flow pattern by inspection, and show it on the diagram below.
\includegraphics[max width=\textwidth, alt={}, center]{a5d69a77-c196-483c-a550-1a55363555af-4_597_1470_1656_296}
Find a second minimum cut for this network. In order to increase the flow through the network it is decided to add an arc of capacity 100 joining $D$ either to $E$ or to $G$.
State, with a reason, which of these arcs should be added, and the value of the increased flow.

Edexcel D2 2002 June Q1

1. \begin{figure}[h]

\captionsetup{labelformat=empty} \caption{Figure 1} \includegraphics[alt={},max width=\textwidth]{c4c64221-0373-4be9-abe3-5ff281922cdb-01_675_1052_378_485}

\end{figure} Figure 1 shows a network of roads connecting six villages $A , B , C , D , E$ and $F$. The lengths of the roads are given in km .

Complete the table in the answer booklet, in which the entries are the shortest distances between pairs of villages. You should do this by inspection. The table can now be taken to represent a complete network.
Use the nearest-neighbour algorithm, starting at $A$, on your completed table in part (a). Obtain an upper bound to the length of a tour in this complete network, which starts and finishes at $A$ and visits every village exactly once.
(3)
Interpret your answer in part (b) in terms of the original network of roads connecting the six villages.
(1)
By choosing a different vertex as your starting point, use the nearest-neighbour algorithm to obtain a shorter tour than that found in part (b). State the tour and its length.
(2)

Edexcel D2 2002 June Q2

2. A two-person zero-sum game is represented by the following pay-off matrix for player $A$.

		$B$
		I	II	III	IV
\multirow{3}{*}{$A$}	I	- 4	- 5	- 2	4
	II	- 1	1	- 1	2
	III	0	5	- 2	- 4
	IV	- 1	3	- 1	1

Determine the play-safe strategy for each player.
Verify that there is a stable solution and determine the saddle points.
State the value of the game to $B$.

Edexcel D2 2002 June Q3

3. \begin{figure}[h]

\captionsetup{labelformat=empty} \caption{Figure 2} \includegraphics[alt={},max width=\textwidth]{c4c64221-0373-4be9-abe3-5ff281922cdb-03_764_1514_283_141}

\end{figure} The network in Fig. 2 shows possible routes that an aircraft can take from $S$ to $T$. The numbers on the directed arcs give the amount of fuel used on that part of the route, in appropriate units. The airline wishes to choose the route for which the maximum amount of fuel used on any part of the route is as small as possible. This is the rninimax route.

Complete the table in the answer booklet.
(8)
Hence obtain the minimax route from $S$ to $T$ and state the maximum amount of fuel used on any part of this route.
(2)

Edexcel D2 2002 June Q4

4. Andrew ( $A$ ) and Barbara ( $B$ ) play a zero-sum game. This game is represented by the following payoff matrix for Andrew. $$A \left( \begin{array} { c c c } & B &
3 & 5 & 4
1 & 4 & 2
6 & 3 & 7 \end{array} \right)$$

Explain why this matrix may be reduced to $$\left( \begin{array} { l l } 3 & 5
6 & 3 \end{array} \right)$$
Hence find the best strategy for each player and the value of the game.
(8)

Edexcel D2 2002 June Q5

5. An engineering company has 4 machines available and 4 jobs to be completed. Each machine is to be assigned to one job. The time, in hours, required by each machine to complete each job is shown in the table below.

\cline { 2 - 5 } \multicolumn{1}{c\|}{}	Job 1	Job 2	Job 3	Job 4
Machine 1	14	5	8	7
Machine 2	2	12	6	5
Machine 3	7	8	3	9
Machine 4	2	4	6	10

Use the Hungarian algorithm, reducing rows first, to obtain the allocation of machines to jobs which minimises the total time required. State this minimum time.

Edexcel D2 2002 June Q6

6. The table below shows the distances, in km, between six towns $A , B , C , D , E$ and $F$.

\cline { 2 - 7 } \multicolumn{1}{c\|}{}	$A$	$B$	$C$	$D$	$E$	$F$
$A$	-	85	110	175	108	100
$B$	85	-	38	175	160	93
$C$	110	38	-	148	156	73
$D$	175	175	148	-	110	84
$E$	108	160	156	110	-	92
$F$	100	93	73	84	92	-

Starting from $A$, use Prim's algorithm to find a minimum connector and draw the minimum spanning tree. You must make your method clear by stating the order in which the arcs are selected.
(4)
1. Using your answer to part (a) obtain an initial upper hound for the solution of the travelling salesman problem.
2. Use a short cut to reduce the upper bound to a value less than 680 .
  (4)
Starting by deleting $F$, find a lower bound for the solution of the travelling salesman problem.
(4)

Edexcel D2 2002 June Q7

7. A steel manufacturer has 3 factories $F _ { 1 } , F _ { 2 }$ and $F _ { 3 }$ which can produce 35,25 and 15 kilotonnes of steel per year, respectively. Three businesses $B _ { 1 } , B _ { 2 }$ and $B _ { 3 }$ have annual requirements of 20,25 and 30 kilotonnes respectively. The table below shows the cost $C _ { i j }$ in appropriate units, of transporting one kilotonne of steel from factory $F _ { i }$ to business $B _ { j }$.

\cline { 3 - 5 } \multicolumn{2}{c\|}{}	Business
\cline { 3 - 5 } \multicolumn{2}{c\|}{}	$B _ { 1 }$	$B _ { 2 }$	$B _ { 3 }$
\multirow{3}{*}{Factory}	$F _ { 1 }$	10	4	11
\cline { 2 - 5 }	$F _ { 2 }$	12	5	8
\cline { 2 - 5 }	$F _ { 3 }$	9	6	7

The manufacturer wishes to transport the steel to the businesses at minimum total cost.

Write down the transportation pattern obtained by using the North-West corner rule.
Calculate all of the improvement indices $I _ { i j }$, and hence show that this pattern is not optimal.
Use the stepping-stone method to obtain an improved solution.
Show that the transportation pattern obtained in part (c) is optimal and find its cost.

Edexcel D2 2002 June Q8

8. \begin{figure}[h]

\captionsetup{labelformat=empty} \caption{Figure 4} \includegraphics[alt={},max width=\textwidth]{c4c64221-0373-4be9-abe3-5ff281922cdb-07_521_1404_285_343}

\end{figure} The network in Fig. 4 models a drainage system. The number on each arc indicates the capacity of that arc, in litres per second.

Write down the source vertices. \begin{figure}[h]
\captionsetup{labelformat=empty} \caption{Figure 5} \includegraphics[alt={},max width=\textwidth]{c4c64221-0373-4be9-abe3-5ff281922cdb-07_521_1402_1170_343}
\end{figure} Figure 5 shows a feasible flow through the same network.
State the value of the feasible flow shown in Fig. 5. Taking the flow in Fig. 5 as your initial flow pattern,
use the labelling procedure on Diagram 1 to find a maximum flow through this network. You should list each flow-augmenting route you use, together with its flow.
Show the maximal flow on Diagram 2 and state its value.
Prove that your flow is maximal.

Edexcel D2 2002 June Q9

9. T42 Co. Ltd produces three different blends of tea, Morning, Afternoon and Evening. The teas must be processed, blended and then packed for distribution. The table below shows the time taken, in hours, for each stage of the production of a tonne of tea. It also shows the profit, in hundreds of pounds, on each tonne.

\cline { 2 - 5 } \multicolumn{1}{c\|}{}	Processing	Blending	Packing	Profit ( $\pounds 100$ )
Morning blend	3	1	2	4
Afternoon blend	2	3	4	5
Evening blend	4	2	3	3

The total times available each week for processing, blending and packing are 35, 20 and 24 hours respectively. T42 Co. Ltd wishes to maximise the weekly profit. Let $x , y$ and $z$ be the number of tonnes of Morning, Afternoon and Evening blend produced each week.

Formulate the above situation as a linear programming problem, listing clearly the objective function, and the constraints as inequalities.
(4) An initial Simplex tableau for the above situation is
Basic
variable
$x$ $y$ $z$ $r$ $s$ $t$ Value
$r$ 3 2 4 1 0 0 35
$s$ 1 3 2 0 1 0 20
$t$ 2 4 3 0 0 1 24
$P$ - 4 - 5 - 3 0 0 0 0
Solve this linear programming problem using the Simplex algorithm. Take the most negative number in the profit row to indicate the pivot column at each stage. T42 Co. Ltd wishes to increase its profit further and is prepared to increase the time available for processing or blending or packing or any two of these three.
Use your answer to part (b) to advise the company as to which stage(s) it should increase the time available.

Edexcel D2 2002 June Q10

10. While solving a maximizing linear programming problem, the following tableau was obtained.

Basic variable	$x$	$y$	$z$	$r$	$s$	$t$	Value
$r$	0	0	$1 \frac { 2 } { 3 }$	1	0	$- \frac { 1 } { 6 }$	$\frac { 2 } { 3 }$
$y$	0	1	$3 \frac { 1 } { 3 }$	0	1	$- \frac { 1 } { 3 }$	$\frac { 1 } { 3 }$
$x$	1	0	-3	0	-1	$\frac { 1 } { 2 }$	1
P	0	0	1	0	1	1	11

Explain why this is an optimal tableau.
Write down the optimal solution of this problem, stating the value of every variable.
Write down the profit equation from the tableau. Use it to explain why changing the value of any of the non-basic variables will decrease the value of $P$.

Edexcel D2 2002 June Q11

11. A company wishes to transport its products from 3 factories $F _ { 1 } , F _ { 2 }$ and $F _ { 3 }$ to a single retail outlet $R$. The capacities of the possible routes, in van loads per day, are shown in Fig. 5. \begin{figure}[h]

\captionsetup{labelformat=empty} \caption{Figure 5} \includegraphics[alt={},max width=\textwidth]{c4c64221-0373-4be9-abe3-5ff281922cdb-10_723_1172_476_337}

\end{figure}

On Diagram 1 in the answer booklet add a supersource $S$ to obtain a capacitated network with a single source and a single sink. State the minimum capacity of each arc you have added.
1. State the maximum flow along $S F _ { 1 } A B R$ and $S F _ { 3 } C R$.
2. Show these maximum flows on Diagram 2 in the answer booklet, using numbers in circles. Taking your answer to part (b)(ii) as the initial flow pattern,
1. use the labelling procedure to find a maximum flow from $S$ to $R$. Your working should be shown on Diagram 3. List each flow-augmenting route you find together with its flow.
2. Prove that your final flow is maximal.

Edexcel D2 2002 June Q12

12. \begin{figure}[h]

\captionsetup{labelformat=empty} \caption{Figure 2} \includegraphics[alt={},max width=\textwidth]{c4c64221-0373-4be9-abe3-5ff281922cdb-11_618_1211_253_253}

\end{figure} A company has 3 warehouses $W _ { 1 } , W _ { 2 }$, and $W _ { 3 }$. It needs to transport the goods stored there to 2 retail outlets $R _ { 1 }$ and $R _ { 2 }$. The capacities of the possible routes, in van loads per day, are shown in Fig 2. Warehouses $W _ { 1 } , W _ { 2 }$ and $W _ { 3 }$ have 14, 12 and 14 van loads respectively available per day and retail outlets $R _ { 1 }$ and $R _ { 2 }$ can accept 6 and 25 van loads respectively per day.

On Diagram 1 on the answer sheet add a supersource $W$, a supersink $R$ and the appropriate directed arcs to obtain a single-source, single-sink capacitated network. State the minimum capacity of each arc you have added.
State the maximum flow along
1. $W \quad W _ { 1 } \quad A \quad R _ { 1 } \quad R$,
2. $W W _ { 3 } \quad C \quad R _ { 2 } \quad R$.
Taking your answers to part (b) as the initial flow pattern, use the labelling procedure to obtain a maximum flow through the network from $W$ to $R$. Show your working on Diagram 2. List each flowaugmenting route you use, together with its flow.
From your final flow pattern, determine the number of van loads passing through $B$ each day.

Edexcel D2 2003 June Q1

A two person zero-sum game is represented by the following pay-off matrix for player $A$.

\cline { 2 - 4 } \multicolumn{1}{c\|}{}	$B$ plays I	$B$ plays II	$B$ plays III
$A$ plays I	- 3	2	5
$A$ plays II	4	- 1	- 4

Write down the pay off matrix for player $B$.
Formulate the game as a linear programming problem for player $B$, writing the constraints as equalities and stating your variables clearly.

Edexcel D2 2003 June Q2

2. (a) Explain the difference between the classical and practical travelling salesman problems.
\includegraphics[max width=\textwidth, alt={}, center]{eabe577b-80d9-45f8-a8e8-0c3b139b96a8-1_691_1297_1014_383} The network in the diagram above shows the distances, in kilometres, between eight McBurger restaurants. An inspector from head office wishes to visit each restaurant. His route should start and finish at $A$, visit each restaurant at least once and cover a minimum distance.
(b) Obtain a minimum spanning tree for the network using Kruskal's algorithm. You should draw your tree and state the order in which the arcs were added.
(c) Use your answer to part (b) to determine an initial upper bound for the length of the route.
(d) Starting from your initial upper bound and using an appropriate method, find an upper bound which is less than 135 km . State your tour.

Edexcel D2 2003 June Q3

3. Talkalot College holds an induction meeting for new students. The meeting consists of four talks: I (Welcome), II (Options and Facilities), III (Study Tips) and IV (Planning for Success). The four department heads, Clive, Julie, Nicky and Steve, deliver one of these talks each. The talks are delivered consecutively and there are no breaks between talks. The meeting starts at 10 a.m. and ends when all four talks have been delivered. The time, in minutes, each department head takes to deliver each talk is given in the table below.

\cline { 2 - 5 } \multicolumn{1}{c\|}{}	Talk I	Talk II	Talk III	Talk IV
Clive	12	34	28	16
Julie	13	32	36	12
Nicky	15	32	32	14
Steve	11	33	36	10

Use the Hungarian algorithm to find the earliest time that the meeting could end. You must make your method clear and show
1. the state of the table after each stage in the algorithm,
2. the final allocation.
Modify the table so it could be used to find the latest time that the meeting could end.

Edexcel D2 2003 June Q4

4. A two person zero-sum game is represented by the following pay-off matrix for player $A$.

\cline { 2 - 4 } \multicolumn{1}{c\|}{}	$B$ plays I	$B$ plays II	$B$ plays III
$A$ plays I	2	- 1	3
$A$ plays II	1	3	0
$A$ plays III	0	1	- 3

Identify the play safe strategies for each player.
Verify that there is no stable solution to this game.
Explain why the pay-off matrix above may be reduced to
\cline { 2 - 4 } \multicolumn{1}{c|}{} $B$ plays I $B$ plays II $B$ plays III
$A$ plays I 2 - 1 3
$A$ plays II 1 3 0
Find the best strategy for player $A$, and the value of the game.

Edexcel D2 2003 June Q5

5. The manager of a car hire firm has to arrange to move cars from three garages $A , B$ and $C$ to three airports $D , E$ and $F$ so that customers can collect them. The table below shows the transportation cost of moving one car from each garage to each airport. It also shows the number of cars available in each garage and the number of cars required at each airport. The total number of cars available is equal to the total number required.

	Airport $D$	Airport $E$	Airport $F$	Cars available
Garage $A$	£20	£40	£10	6
Garage $B$	£20	£30	£40	5
Garage C	£10	£20	£30	8
Cars required	6	9	4

Use the North-West corner rule to obtain a possible pattern of distribution and find its cost.
(3)
Calculate shadow costs for this pattern and hence obtain improvement indices for each route.
Use the stepping-stone method to obtain an optimal solution and state its cost.

Edexcel D2 2003 June Q6

6. Kris produces custom made racing cycles. She can produce up to four cycles each month, but if she wishes to produce more than three in any one month she has to hire additional help at a cost of $\pounds 350$ for that month. In any month when cycles are produced, the overhead costs are $\pounds 200$. A maximum of 3 cycles can be held in stock in any one month, at a cost of $\pounds 40$ per cycle per month. Cycles must be delivered at the end of the month. The order book for cycles is

Month	August	September	October	November
Number of cycles required	3	3	5	2

Disregarding the cost of parts and Kris’ time,

determine the total cost of storing 2 cycles and producing 4 cycles in a given month, making your calculations clear. There is no stock at the beginning of August and Kris plans to have no stock after the November delivery.

Use dynamic programming to determine the production schedule which minimises the costs, showing your working in the table below.

Stage	Demand	State	Action	Destination	Value
\multirow[t]{3}{*}{1 (Nov)}	\multirow[t]{3}{*}{2}	0 (in stock)	(make) 2	0	200
		1 (in stock)	(make) 1	0	240
		2 (in stock)	(make) 0	0	80
\multirow[t]{2}{*}{2 (Oct)}	\multirow[t]{2}{*}{5}	1	4	0	$590 + 200 = 790$
		2	3	0

The fixed cost of parts is $\pounds 600$ per cycle and of Kris’ time is $\pounds 500$ per month. She sells the cycles for $\pounds 2000$ each.

Determine her total profit for the four month period.
(3)
(Total 18 marks)

Edexcel D2 2003 June Q7

7. \begin{figure}[h]

\captionsetup{labelformat=empty} \caption{Figure 1} \includegraphics[alt={},max width=\textwidth]{eabe577b-80d9-45f8-a8e8-0c3b139b96a8-4_759_1529_715_267}

\end{figure} Figure 1 shows a capacitated, directed network. The unbracketed number on each arc indicates the capacity of that arc, and the numbers in brackets show a feasible flow of value 68 through the network.

Add a supersource and a supersink, and arcs of appropriate capacity, to Diagram 1 below. \section*{Diagram 1} \includegraphics[max width=\textwidth, alt={}, center]{eabe577b-80d9-45f8-a8e8-0c3b139b96a8-4_684_1531_1816_267}
Find the values of $x$ and $y$, explaining your method briefly.
Find the value of cuts $C _ { 1 }$ and $C _ { 2 }$. Starting with the given feasible flow of 68,
use the labelling procedure on Diagram 2 to find a maximal flow through this network. List each flow-augmenting route you use, together with its flow. \section*{Diagram 2} \includegraphics[max width=\textwidth, alt={}, center]{eabe577b-80d9-45f8-a8e8-0c3b139b96a8-5_647_1506_612_283}
Show your maximal flow on Diagram 3 and state its value. \section*{Diagram 3} \includegraphics[max width=\textwidth, alt={}, center]{eabe577b-80d9-45f8-a8e8-0c3b139b96a8-5_654_1511_1567_278}
Prove that your flow is maximal.

Edexcel D2 2003 June Q8

8. The tableau below is the initial tableau for a maximising linear programming problem.

For this problem $x \geq 0 , y \geq 0 , z \geq 0$. Write down the other two inequalities and the objective function.
Solve this linear programming problem. You may not need to use all of these tableaux.
b.v. $x$ $y$ $z$ $r$ $s$ Value
$P$

b.v. $x$ $y$ $z$ $r$ $s$ Value
$P$

b.v. $x$ $y$ $z$ $r$ $s$ Value
$P$

b.v. $x$ $y$ $z$ $r$ $s$ Value
$P$
State the final value of $P$, the objective function, and of each of the variables.

Edexcel D2 2004 June Q1

In game theory explain what is meant by
1. zero-sum game,
2. saddle point.
  (Total 4 marks)
3. In a quiz there are four individual rounds, Art, Literature, Music and Science. A team consists of four people, Donna, Hannah, Kerwin and Thomas. Each of four rounds must be answered by a different team member. The table shows the number of points that each team member is likely to get on each individual round.
\cline { 2 - 5 } \multicolumn{1}{c|}{} Art Literature Music Science
Donna 31 24 32 35
Hannah 16 10 19 22
Kerwin 19 14 20 21
Thomas 18 15 21 23
Use the Hungarian algorithm, reducing rows first, to obtain an allocation which maximises the total points likely to be scored in the four rounds. You must make your method clear and show the table after each stage.
(Total 9 marks)

Edexcel D2 2004 June Q3

3. The table shows the least distances, in km, between five towns, $A , B , C , D$ and $E$. Nassim wishes to find an interval which contains the solution to the travelling salesman problem for this network.

Making your method clear, find an initial upper bound starting at $A$ and using
1. the minimum spanning tree method,
2. the nearest neighbour algorithm.
  $A$ $B$ $C$ $D$ $E$
  $A$ - 153 98 124 115
  $B$ 153 - 74 131 149
  $C$ 98 74 - 82 103
  $D$ 124 131 82 - 134
  $E$ 115 149 103 134 -
By deleting $E$, find a lower bound.
Using your answers to parts (a) and (b), state the smallest interval that Nassim could correctly write down.
(Total 12 marks)

Edexcel D2 2004 June Q4

4. Emma and Freddie play a zero-sum game. This game is represented by the following pay-off matrix for Emma. $\left( \begin{array} { r r r } - 4 & - 1 & 3
2 & 1 & - 2 \end{array} \right)$

Show that there is no stable solution.
Find the best strategy for Emma and the value of the game to her.
Write down the value of the game to Freddie and his pay-off matrix.

\cline { 3 - 5 } \multicolumn{2}{c\|}{}	Business
\cline { 3 - 5 } \multicolumn{2}{c\|}{}	\(B _ { 1 }\)	\(B _ { 2 }\)	\(B _ { 3 }\)
\multirow{3}{*}{Factory}	\(F _ { 1 }\)	10	4	11
\cline { 2 - 5 }	\(F _ { 2 }\)	12	5	8
\cline { 2 - 5 }	\(F _ { 3 }\)	9	6	7

Basic variable	\(x\)	\(y\)	\(z\)	\(r\)	\(s\)	\(t\)	Value
\(r\)	0	0	\(1 \frac { 2 } { 3 }\)	1	0	\(- \frac { 1 } { 6 }\)	\(\frac { 2 } { 3 }\)
\(y\)	0	1	\(3 \frac { 1 } { 3 }\)	0	1	\(- \frac { 1 } { 3 }\)	\(\frac { 1 } { 3 }\)
\(x\)	1	0	-3	0	-1	\(\frac { 1 } { 2 }\)	1
P	0	0	1	0	1	1	11

\cline { 2 - 4 } \multicolumn{1}{c\|}{}	\(B\) plays I	\(B\) plays II	\(B\) plays III
\(A\) plays I	- 3	2	5
\(A\) plays II	4	- 1	- 4

\cline { 2 - 4 } \multicolumn{1}{c\|}{}	\(B\) plays I	\(B\) plays II	\(B\) plays III
\(A\) plays I	2	- 1	3
\(A\) plays II	1	3	0
\(A\) plays III	0	1	- 3

\cline { 2 - 4 } \multicolumn{1}{c\|}{}	\(B\) plays I	\(B\) plays II	\(B\) plays III
\(A\) plays I	2	- 1	3
\(A\) plays II	1	3	0

	A	B	C	D	E	F	G	H
A	-	84	85	138	173		149	52
B	84	-	130	77	126	213	222	136
C	85	130	-	53	88	83	92
D	138	77	53	-	49			190
E	173	126	88	49	-	100	180	215
F		213	83		100	-	163	115
G	149	222	92		180	163	-	97
H	52	136		190	215	115	97	-

		\(B\)
		I	II	III	IV
\multirow{3}{*}{\(A\)}	I	- 4	- 5	- 2	4
	II	- 1	1	- 1	2
	III	0	5	- 2	- 4
	IV	- 1	3	- 1	1

\cline { 2 - 7 } \multicolumn{1}{c\|}{}	\(A\)	\(B\)	\(C\)	\(D\)	\(E\)	\(F\)
\(A\)	-	85	110	175	108	100
\(B\)	85	-	38	175	160	93
\(C\)	110	38	-	148	156	73
\(D\)	175	175	148	-	110	84
\(E\)	108	160	156	110	-	92
\(F\)	100	93	73	84	92	-

	Airport \(D\)	Airport \(E\)	Airport \(F\)	Cars available
Garage \(A\)	£20	£40	£10	6
Garage \(B\)	£20	£30	£40	5
Garage C	£10	£20	£30	8
Cars required	6	9	4

\cline { 2 - 5 } \multicolumn{1}{c\|}{}	Art	Literature	Music	Science
Donna	31	24	32	35
Hannah	16	10	19	22
Kerwin	19	14	20	21
Thomas	18	15	21	23

	A	B	C	D	E	F	G	H
A	-	84	85	138	173		149	52
B	84	-	130	77	126	213	222	136
C	85	130	-	53	88	83	92
D	138	77	53	-	49			190
E	173	126	88	49	-	100	180	215
F		213	83		100	-	163	115
G	149	222	92		180	163	-	97
H	52	136		190	215	115	97	-

Questions D2 (547 questions)

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Edexcel D2 2006 January Q6

Edexcel D2 2006 January Q7

Edexcel D2 2002 June Q1

Edexcel D2 2002 June Q2

Edexcel D2 2002 June Q3

Edexcel D2 2002 June Q4

Edexcel D2 2002 June Q5

Edexcel D2 2002 June Q6

Edexcel D2 2002 June Q7

Edexcel D2 2002 June Q8

Edexcel D2 2002 June Q9

Edexcel D2 2002 June Q10

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Edexcel D2 2003 June Q1

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Edexcel D2 2003 June Q3

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Edexcel D2 2003 June Q5

Edexcel D2 2003 June Q6

Edexcel D2 2003 June Q7

Edexcel D2 2003 June Q8

Edexcel D2 2004 June Q1

Edexcel D2 2004 June Q3

Edexcel D2 2004 June Q4

	A	B	C	D	E	F	G	H
A	-	84	85	138	173		149	52
B	84	-	130	77	126	213	222	136
C	85	130	-	53	88	83	92
D	138	77	53	-	49			190
E	173	126	88	49	-	100	180	215
F		213	83		100	-	163	115
G	149	222	92		180	163	-	97
H	52	136		190	215	115	97	-