Questions D2 - A-Level Maths

Draw a complete network showing the shortest distances between the six areas.
(3 marks)
Obtain a minimum spanning tree for the complete network and hence find an upper bound for the length of the driver's route.
(5 marks)
Improve this upper bound to find an upper bound of less than 55 miles.
By deleting $A$, find a lower bound for the total length of the route.

Edexcel D2 Q7

7. Mrs. Hartley organises the tennis fixtures for her school. On one day she has to send a team of 10 players to a match against school $A$ and a team of 6 players to a match against school $B$. She has to select the two teams from a squad that includes 7 players who live in village $C$, 5 players who live in village $D$ and 8 players who live in village $E$. Having a small budget, Mrs. Hartley wishes to minimise the total amount spent on travel. The table below shows the cost, in pounds, for one player to travel from each village to each of the schools they are competing against.

\cline { 2 - 3 } \multicolumn{1}{c\|}{}	$A$	$B$
$C$	2	3
$D$	2	5
$E$	7	6

Use the north-west corner rule to find an initial solution to this problem.
Obtain improvement indices for this initial solution.
Use the stepping-stone method to obtain an optimal solution and state the pattern of transportation that this represents. \section*{Please hand this sheet in for marking}
Stage State Action
\multirow[t]{2}{*}{1} G GI
H HI
\multirow[t]{3}{*}{2} D
DG
DH
E
EG
$E H$
F
FG
FH
\multirow[t]{3}{*}{3} A
AD
$A E$
$A F$
B
BD
BE
$B F$
C
CD
CE
CF
4 Home
Home-A
Home-B
Home-C
\section*{Please hand this sheet in for marking}
\includegraphics[max width=\textwidth, alt={}, center]{4e50371b-0c1c-4b4e-b21d-60858ae160df-8_662_1025_529_440}
Sheet for answering question 6 (cont.)

Edexcel D2 Q1

This question should be answered on the sheet provided.

\begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{e892e87c-1c2d-4f97-ac23-41e38663d0f0-02_485_995_285_477} \captionsetup{labelformat=empty} \caption{Fig. 1}

\end{figure} The network in Figure 1 shows the distances, in miles, between the five villages in which Sarah is planning to enquire about holiday work, with village $A$ being Sarah's home village.

Illustrate this situation as a complete network showing the shortest distances.
(2 marks)
Use the nearest neighbour algorithm, starting with $A$, to find an upper bound to the length of a tour beginning and ending at $A$.
(2 marks)
Interpret the tour found in part (b) in terms of the original network.
(2 marks)

Edexcel D2 Q2

2. The payoff matrix for player $A$ in a two-person zero-sum game with value $V$ is shown below.

\cline { 3 - 5 } \multicolumn{2}{c\|}{}	$B$
\cline { 2 - 5 } \multicolumn{2}{c\|}{}	I	II	III
\multirow{3}{*}{$A$}	I	- 1	4	- 3
\cline { 2 - 5 }	II	- 3	7	1
\cline { 2 - 5 }	III	5	- 2	- 1

Formulate this information as a linear programming problem, the solution to which will give the optimal strategy for player $B$.

Rewrite the matrix as necessary and state the new value of the game, $v$, in terms of $V$.
Define your decision variables.
Write down the objective function in terms of your decision variables.
Write down the constraints.

Edexcel D2 Q3

3. This question should be answered on the sheet provided. Arthur is planning a bus journey from town $A$ to town $L$. There are various routes he can take but he will have to change buses three times - at $B , C$ or $D$, at $E , F , G$ or $H$ and at $I , J$ or $K$. \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{e892e87c-1c2d-4f97-ac23-41e38663d0f0-03_764_1410_477_315} \captionsetup{labelformat=empty} \caption{Fig. 2}

\end{figure} Figure 2 shows the bus routes that Arthur can use. The number on each arc shows the average waiting time, in minutes, for a bus to come on that route. As the forecast is for rain, Arthur wishes to plan his journey so that the maximum waiting time at any one stop is as small as possible. Use dynamic programming to find the route that Arthur should use.
(9 marks)

Edexcel D2 Q4

4. A furniture manufacturer has three workshops, $W _ { 1 } , W _ { 2 }$ and $W _ { 3 }$. Orders for rolls of fabric are to be placed with three suppliers, $S _ { 1 } , S _ { 2 }$ and $S _ { 3 }$. The supply, demand and cost per roll in pounds, according to which supplier each workshop uses, are given in the table below.

	$W _ { 1 }$	$W _ { 2 }$	$W _ { 3 }$	Available
$S _ { 1 }$	12	11	17	30
$S _ { 2 }$	7	5	10	25
$S _ { 3 }$	5	6	8	10
Required	20	15	30

Starting with the north-west corner method of finding an initial solution, find an optimal transportation pattern which minimises the total cost. State the final solution and its total cost.
(11 marks)

Edexcel D2 Q5

5. A travel company offers a touring holiday which stops at four locations, $A , B , C$ and $D$. The tour may be taken with the locations appearing in any order, but the number of days spent in each location is dependent on its position in the tour, as shown in the table below.

\multirow{2}{*}{}	Stage
	1	2	3	4
A	7	8	5	6
$B$	6	9	6	5
C	9	8	5	7
D	7	7	6	6

Showing the state of the table at each stage, use the Hungarian algorithm to find the order in which to complete the tour so as to maximise the total number of days. State the maximum total number of days that can be spent in the four locations.
(11 marks)

Edexcel D2 Q6

6. The payoff matrix for player $A$ in a two-person zero-sum game is shown below.

\cline { 3 - 5 } \multicolumn{2}{c\|}{}	$B$
\cline { 2 - 5 } \multicolumn{2}{c\|}{}	I	II	III
\multirow{3}{*}{$A$}	I	3	5	- 2
\cline { 2 - 5 }	II	7	- 4	- 1
\cline { 2 - 5 }	III	9	- 4	8

Applying the dominance rule, explain, with justification, which strategy can be ignored by
1. player $A$,
2. player $B$.
For the reduced table, find the optimal strategy for
1. player $A$,
2. player $B$.
Find the value of the game. Turn over

Edexcel D2 Q7

7. This question should be answered on the sheet provided. A tinned food producer delivers goods to six supermarket warehouses, $B , C , D , E , F$ and $G$, from its base, $A$. The distances, in kilometres, between each location are given in the table below. \section*{Please hand this sheet in for marking}

Edexcel D2 Q1

This question should be answered on the sheet provided.

The table below shows the distances in miles between five villages. Jane lives in village $A$ and is about to take her daughter's friends home to villages $B , C , D$ and $E$. She will begin and end her journey at $A$ and wishes to travel the minimum distance possible.

	$A$	$B$	$C$	$D$	$E$
$A$	-	4	7	8	2
$B$	4	-	1	5	6
$C$	7	1	-	2	7
$D$	8	5	2	-	3
$E$	2	6	7	3	-

Obtain a minimum spanning tree for the network and hence find an upper bound for the length of Jane's journey.
Using a shortcut, improve this upper bound to find an upper bound of less than 15 miles.
(2 marks)

Edexcel D2 Q2

2. The payoff matrix for player $A$ in a two-person zero-sum game with value $V$ is shown below.

\cline { 3 - 5 } \multicolumn{2}{c\|}{}	$B$
\cline { 2 - 5 } \multicolumn{2}{c\|}{}	I	II	III
\multirow{3}{*}{$A$}	I	6	- 4	- 1
\cline { 2 - 5 }	II	${ } ^ { - } 2$	5	3
\cline { 2 - 5 }	III	5	1	- 3

Formulate this information as a linear programming problem, the solution to which will give the optimal strategy for player $B$.

Rewrite the matrix as necessary and state the new value of the game, $v$, in terms of $V$.
Define your decision variables.
Write down the objective function in terms of your decision variables.
Write down the constraints.

Edexcel D2 Q3

3. This question should be answered on the sheet provided. The table below gives distances, in miles, for a network relating to a travelling salesman problem. Use dynamic programming to find the route which satisfies the wish of the organisers. State the length of the shortest stage on this route.

Edexcel D2 Q5

5. Four athletes are put forward for selection for a mixed stage relay race at a local competition. They may each be selected for a maximum of one stage and only one athlete can be entered for each stage. The average time, in seconds, for each athlete to complete each stage is given below, based on past performances.

\multirow{2}{*}{}	Stage
	1	2	3
Alex	19	69	168
Darren	22	64	157
Leroy	20	72	166
Suraj	23	66	171

Use the Hungarian algorithm to find an optimal allocation which will minimise the team's total time. Your answer should show clearly how you have applied the algorithm.

Edexcel D2 Q6

6. The payoff matrix for player $X$ in a two-person zero-sum game is shown below.

\cline { 3 - 4 } \multicolumn{2}{c\|}{}	$Y$
\cline { 3 - 4 } \multicolumn{2}{c\|}{}	$Y _ { 1 }$	$Y _ { 2 }$
\multirow{2}{*}{$X$}	$X _ { 1 }$	${ } ^ { - } 2$	4
\cline { 2 - 4 }	$X _ { 2 }$	6	${ } ^ { - } 1$

Explain why the game does not have a saddle point.
Find the optimal strategy for
1. player $X$,
2. player $Y$.
Find the value of the game.

Edexcel D2 Q7

7. A transportation problem has costs, in pounds, and supply and demand, in appropriate units, as given in the transportation tableau below. (c) \section*{Please hand this sheet in for marking}

Stage	State	Action
\multirow[t]{3}{*}{1}	I	IL
	J	JL
	K	KL
\multirow[t]{3}{*}{2}	$F$	FI FJ FK
	G	GI GJ GK
	H	HI HJ HK
\multirow[t]{4}{*}{3}	B	$B F B G B H$
	C	CF CH
	D	DF DH
	E	EF $E G E H$
4	A	$A B A C$ AD $A E$

Edexcel D2 Q1

A glazing company runs a promotion for a special type of window. As a result of this the company receives orders for 30 of these windows from business $B _ { 1 } , 18$ from business $B _ { 2 }$ and 22 from business $B _ { 3 }$. The company has stocks of 20 of these windows at factory $F _ { 1 } , 35$ at factory $F _ { 2 }$ and 15 at factory $F _ { 3 }$. The table below shows the profit, in pounds, that the company will make for each window it sells according to which factory supplies each business.

\cline { 2 - 4 } \multicolumn{1}{c\|}{}	$B _ { 1 }$	$B _ { 2 }$	$B _ { 3 }$
$F _ { 1 }$	20	14	17
$F _ { 2 }$	18	19	19
$F _ { 3 }$	15	17	23

The glazing company wishes to supply the windows so that the total profit is a maximum.
Formulate this information as a linear programming problem.

State your decision variables.
Write down the objective function in terms of your decision variables.
Write down the constraints and state what each one represents.

Edexcel D2 Q2

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

\includegraphics[alt={},max width=\textwidth]{726bca96-7f98-4ed5-b642-f5007a958c8b-03_492_862_301_502} \captionsetup{labelformat=empty} \caption{Fig. 1}

\end{figure} Figure 1 shows a network in which the nodes represent five major rides in a theme park and the arcs represent paths between these rides. The numbers on the arcs give the length, in metres, of the paths.

By inspection, add additional arcs to make a complete network showing the shortest distances between the rides.
(2 marks)
Use the nearest neighbour algorithm, starting at $A$, and your complete network to find an upper bound to the length of a tour visiting each ride exactly once.
Interpret the tour found in part (b) in terms of the original network.

Edexcel D2 Q3

3. Whilst Clive is in hospital, four of his friends decide to redecorate his lounge as a welcomehome surprise. They divide the work to be done into four jobs which must be completed in the following order:

strip the wallpaper,
paint the woodwork and ceiling,
hang the new wallpaper,
replace the fittings and tidy up.

The table below shows the time, in hours, that each of the friends is likely to take to complete each job.

	Alice	Bhavin	Carl	Dieter
Strip wallpaper	5	3	5	4
Paint	7	5	6	4
Hang wallpaper	8	4	7	6
Replace fittings	5	3	2	3

As they do not know how long Clive will be in hospital his friends wish to complete the redecoration in the shortest possible total time.

Use the Hungarian method to obtain the optimal allocation of the jobs, showing the state of the table after each stage in the algorithm.
(6 marks)
Hence, find the minimum time in which the friends can redecorate the lounge.
(1 mark)

Edexcel D2 Q4

4. This question should be answered on the sheet provided. The owner of a small plane is planning a journey from her local airport, $A$ to the airport nearest her parents, $K$. On the journey she will make three refuelling stops, the first at $B , C$ or $D$, the second at $E , F$ or $G$ and the third at $H , I$ or $J$. \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{726bca96-7f98-4ed5-b642-f5007a958c8b-05_727_1303_523_356} \captionsetup{labelformat=empty} \caption{Fig. 2}

\end{figure} Figure 2 shows all the possible flights that can be made on the journey with the number by each arc indicating the distance of each flight in hundreds of miles. As her plane does not have a large fuel tank, the owner wishes to choose a route that minimises the maximum distance of any one flight. Find the route that she should use and state the maximum distance of any one stage on this route.

Edexcel D2 Q5

5. A car-hire firm has six branches in a region. Three of the branches, $A , B$ and $C$, have spare cars, whereas the other three, $D , E$ and $F$, require cars. The total number of cars required is equal to the number of cars available. The table below shows the cost in pounds of sending one car from each branch with spares to each branch needing more cars and the number of cars available or required by each branch.

\backslashbox{Branches with spare cars}{Branches needing cars}	$D$	$E$	$F$	Available
$A$	6	4	7	7
B	8	5	3	8
C	4	4	2	5
Required	5	9	6

Use the north-west corner method to obtain a possible pattern of moving cars and find its cost. The firm wishes to minimise the cost of redistributing the cars.
Calculate shadow costs for the pattern found in part (a) and improvement indices for each unoccupied cell.
State, with a reason, whether or not the pattern found in part (a) is optimal.

Edexcel D2 Q6

6. This question should be answered on the sheet provided. A furniture company in Leeds is considering opening outlets in six other cities.
The table below shows the distances, in miles, between all seven cities.

	Leeds	Liverpool	Manchester	Newcastle	Nottingham	Oxford	York
Leeds	-	71	40	96	71	165	28
Liverpool	71	-	31	155	92	155	93
Manchester	40	31	-	136	62	141	67
Newcastle	96	155	136	-	156	250	78
Nottingham	71	92	62	156	-	94	78
Oxford	165	155	141	250	94	-	172
York	28	93	67	78	78	172	-

Starting with Leeds, obtain and draw a minimum spanning tree for this network of cities showing your method clearly.
(4 marks)
A representative of the company is to visit each of the areas being considered. He wishes to plan a journey of minimum length starting and ending in Leeds and visiting each of the other cities in the table once. Assuming that the network satisfies the triangle inequality,
find an initial upper bound for the length of his journey,
improve this upper bound to find an upper bound of less than 575 miles.
By deleting Leeds, find a lower bound for his journey. Turn over

Edexcel D2 Q7

7. The payoff matrix for player $A$ in a two-person zero-sum game is shown below.

Edexcel D2 Q1

A team of gardeners is called in to attend to the grounds of a stately home. The three gardeners will each be assigned to one of three areas, the lawns, the hedgerows and the flower beds. The table below shows the estimated time, in hours, it will take each gardener to do each job.

\cline { 2 - 4 } \multicolumn{1}{c\|}{}	Lawns	Hedgerows	Flower Beds
Alan	4	4.5	6
Beth	3	4	5
Colin	3.5	5	6

The team wishes to complete the tasks in the least total time.
Formulate this information as a linear programming problem.

State your decision variables.
Write down the objective function in terms of your decision variables.
Write down the constraints and explain what each one represents.

Edexcel D2 Q2

2. This question should be answered on the sheet provided. A pool player is to play in four tournaments. Some of the tournaments take place simultaneously and the player has to choose one of each of the following: $$\begin{array} { l l } 1 ^ { \text {st } } \text { tournament: } & A , B \text { or } C ,
2 ^ { \text {nd } } \text { tournament: } & D , E \text { or } F ,
3 ^ { \text {rd } } \text { tournament: } & G , H \text { or } I ,
4 ^ { \text {th } } \text { tournament: } & J , K \text { or } L \end{array}$$ Each tournament has six rounds and the player estimates how well he will do in each tournament based on which tournament he plays before it. The table below shows his expectations with each number indicating the round he expects to reach and a "7" indicating he expects to win the tournament.

\multirow{2}{*}{}		Expected performance in tournament
		A	B	C	$D$	E	$F$	$G$	$H$	I	J	$K$	$L$
\multirow{10}{*}{Previous tournament}	None	5	3	3
	A				6	3	7
	B				5	5	4
	C				7	5	5
	D							5	3	3
	E							3	5	6
	$F$							3	6	5
	G										2	4	1
	H										3	2	2
	$I$										2	5	3

He wishes to choose the tournaments such that his worst performance is as good as possible. Use dynamic programming to find which tournaments he should play.
(10 marks) Turn over

Edexcel D2 Q3

3. Four people are contributing to the entertainment section of an email magazine. For one issue reviews are required for a film, a musical, a ballet and a concert such that each person reviews one show. The people in charge of the magazine will pay each person's expenses and the cost, in pounds, for each reviewer to attend each show are given below.

	Film	Musical	Ballet	Concert
Andrew	5	20	12	18
Betty	6	18	15	16
Carlos	4	21	9	15
Davina	5	16	11	13

Use the Hungarian algorithm to find an optimal assignment which minimises the total cost. State the total cost of this allocation.
(10 marks)

\cline { 2 - 3 } \multicolumn{1}{c\|}{}	\(A\)	\(B\)
\(C\)	2	3
\(D\)	2	5
\(E\)	7	6

	\(A\)	\(B\)	\(C\)	\(D\)	\(E\)
\(A\)	-	4	7	8	2
\(B\)	4	-	1	5	6
\(C\)	7	1	-	2	7
\(D\)	8	5	2	-	3
\(E\)	2	6	7	3	-

\cline { 3 - 4 } \multicolumn{2}{c\|}{}	\(Y\)
\cline { 3 - 4 } \multicolumn{2}{c\|}{}	\(Y _ { 1 }\)	\(Y _ { 2 }\)
\multirow{2}{*}{\(X\)}	\(X _ { 1 }\)	\({ } ^ { - } 2\)	4
\cline { 2 - 4 }	\(X _ { 2 }\)	6	\({ } ^ { - } 1\)

\backslashbox{Branches with spare cars}{Branches needing cars}	\(D\)	\(E\)	\(F\)	Available
\(A\)	6	4	7	7
B	8	5	3	8
C	4	4	2	5
Required	5	9	6

\multirow{2}{*}{}		Expected performance in tournament
		A	B	C	\(D\)	E	\(F\)	\(G\)	\(H\)	I	J	\(K\)	\(L\)
\multirow{10}{*}{Previous tournament}	None	5	3	3
	A				6	3	7
	B				5	5	4
	C				7	5	5
	D							5	3	3
	E							3	5	6
	\(F\)							3	6	5
	G										2	4	1
	H										3	2	2
	\(I\)										2	5	3

	\(W _ { 1 }\)	\(W _ { 2 }\)	\(W _ { 3 }\)	Available
\(S _ { 1 }\)	12	11	17	30
\(S _ { 2 }\)	7	5	10	25
\(S _ { 3 }\)	5	6	8	10
Required	20	15	30

\cline { 2 - 4 } \multicolumn{1}{c\|}{}	\(B _ { 1 }\)	\(B _ { 2 }\)	\(B _ { 3 }\)
\(F _ { 1 }\)	20	14	17
\(F _ { 2 }\)	18	19	19
\(F _ { 3 }\)	15	17	23

Questions D2 (547 questions)

Browse by module

Browse by board

Edexcel D2 Q6

Edexcel D2 Q7

Edexcel D2 Q1

Edexcel D2 Q2

Edexcel D2 Q3

Edexcel D2 Q4

Edexcel D2 Q5

Edexcel D2 Q6

Edexcel D2 Q7

Edexcel D2 Q1

Edexcel D2 Q2

Edexcel D2 Q3

Edexcel D2 Q5

Edexcel D2 Q6

Edexcel D2 Q7

Edexcel D2 Q1

Edexcel D2 Q2

Edexcel D2 Q3

Edexcel D2 Q4

Edexcel D2 Q5

Edexcel D2 Q6

Edexcel D2 Q7

Edexcel D2 Q1

Edexcel D2 Q2

Edexcel D2 Q3