Questions - OCR Further Discrete AS

Use a graphical method to find the optimal value of $P$, and the corresponding values of $x$ and $y$. An additional constraint is introduced.
This constraint means that the value of $y$ must be at least $k$ times the value of $x$, where $k$ is a positive constant.
1. Determine the set of values of $k$ for which the optimal value of $P$ found in part (a) is unchanged.
2. Determine, in terms of $k$, the values of $x , y$ and $P$ in the cases when the optimal solution is different from that found in part (a).

OCR Further Discrete AS 2024 June Q1

1 There are six non-isomorphic trees with exactly six vertices.
A student has drawn the diagram below showing six trees each with exactly six vertices. However, two of the trees that the student has drawn are isomorphic. \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{6f64abca-108c-4b81-8ccf-124dfd9cc2f6-2_222_243_388_246} \captionsetup{labelformat=empty} \caption{A}

\end{figure} \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{6f64abca-108c-4b81-8ccf-124dfd9cc2f6-2_214_240_392_635} \captionsetup{labelformat=empty} \caption{B}

\end{figure} \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{6f64abca-108c-4b81-8ccf-124dfd9cc2f6-2_214_241_392_1023} \captionsetup{labelformat=empty} \caption{C}

\end{figure} \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{6f64abca-108c-4b81-8ccf-124dfd9cc2f6-2_218_246_758_246} \captionsetup{labelformat=empty} \caption{D}

\end{figure} \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{6f64abca-108c-4b81-8ccf-124dfd9cc2f6-2_216_243_760_632} \captionsetup{labelformat=empty} \caption{E}

\end{figure} \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{6f64abca-108c-4b81-8ccf-124dfd9cc2f6-2_214_241_762_1023} \captionsetup{labelformat=empty} \caption{F}

\end{figure}

Identify which two of these trees are isomorphic.
Draw an example of the missing tree. Graph G has exactly six vertices.
The degree sequence of G is $1,1,1,3,3,3$.
Without using a sketch, calculate the number of edges in graph G.
Explain how the result from part (c) shows that graph G is not a tree. In a simple graph with six vertices each vertex degree must be one of the values $0,1,2,3,4$ or 5 .
Use the pigeonhole principle to show that in a simple graph with six vertices there must be at least two vertices with the same vertex degree.

OCR Further Discrete AS 2024 June Q2

2 In a game two players are each dealt five cards from a set of ten different cards.
Player 1 is dealt cards A, B, F, G and J.
Player 2 is dealt cards C, D, E, H and I. Each player chooses a card to play.
The players reveal their choices simultaneously. The pay-off matrix below shows the points scored by player 1 for each combination of cards. Pay-off for player 1 \begin{table}[h]

\captionsetup{labelformat=empty} \caption{Player 2}

		C
\cline { 3 - 7 } \multirow{4}{*}{Alayer 1}	D	E	H	I
\cline { 3 - 7 }	A	4	1	3	2	2
\cline { 3 - 7 }	B	0	2	1	2	1
\cline { 3 - 7 }	F	0	1	1	2	3
\cline { 2 - 7 }	G	2	0	3	3	3
\cline { 3 - 7 }	J	1	2	3	0	2
\cline { 3 - 7 }
\cline { 3 - 7 }

\end{table}

Determine the play-safe strategy for player 1, ignoring any effect on player 2. The pay-off matrix below shows the points scored by player 2 for each combination of cards.
Pay-off for player 2 Player 1 \begin{table}[h]
\captionsetup{labelformat=empty} \caption{Player 2}
C D E H
\cline { 2 - 6 } A 2 2 0 1 I
\cline { 2 - 6 } B 3 1 2 1 2
\cline { 2 - 6 } F 3 2 2 1 0
\cline { 2 - 6 } G 1 3 0 0 0
\cline { 2 - 6 } J 2 1 0 3 1
\cline { 2 - 6 }
\cline { 2 - 6 }
\end{table}
Use a dominance argument to delete two columns from the pay-off matrix. You must show all relevant comparisons.
Explain, with reference to specific combinations of cards, why the game cannot be converted to a zero-sum game.

OCR Further Discrete AS 2024 June Q3

3 Heidi has a pack of cards.
Each card has a single digit on one side and is blank on the other side.
Each of the digits from 1 to 9 appears on exactly four cards.
Apart from the numerical values of the digits, the cards are indistinguishable from each other.
Heidi draws four cards from the pack, at random and without replacement. She places the four cards in a row to make a four-digit number. Determine how many different four-digit numbers Heidi could have made in each of the following cases.

The four digits are all different.
Two of the digits are the same and the other two digits are different.
There is no restriction on whether any of the digits are the same or not.

OCR Further Discrete AS 2024 June Q4

4 A project is represented by the activity network below. The activity durations are given in minutes.
\includegraphics[max width=\textwidth, alt={}, center]{6f64abca-108c-4b81-8ccf-124dfd9cc2f6-5_447_1020_392_246}

Give the reason for the dummy activity from event (3) to event (4).
Complete a forward pass to determine the minimum project completion time.
By completing a backward pass, calculate the float for each activity.
Determine the effect on the minimum project completion time if the duration of activity A changes from 2 minutes to 3 minutes. The duration of activity C changes to $m$ minutes, where $m$ need not be an integer. This reduces the minimum project completion time.
By considering the range of possible values of $m$, determine the minimum project completion time, in terms of $m$ where necessary.

OCR Further Discrete AS 2024 June Q5

5 A garden centre sells mixed packs of flower bulbs.
Each pack contains bulbs which produce flowers of two different colours. The cost, in $\pounds$, of a pack of each colour combination is shown in the table below.

Pack type	A	B	C	D	E
Colours	Red, orange	Red, yellow	Orange, yellow	Red, pink	Orange, pink
Cost $( \pounds )$	1.50	3.00	4.00	5.25	6.50

Represent the information in the table as a network in which the vertices are the colours and the arcs are the packs available, weighted using the costs.
Construct a minimum spanning tree for the network. Taylor wants to buy at most four different packs of bulbs and to ensure that the packs include bulbs capable of producing all four flower colours. Taylor wants to minimise the total cost of these packs.
Determine whether or not buying the packs represented by the solution to part (b) solves Taylor's problem.
Represent Taylor's problem as an LP formulation, in which the variables are the number of packs of each type.

OCR Further Discrete AS 2024 June Q6

6 Beth wants to buy some tokens for use in a game.
Each token is either a silver token or a gold token.
Silver tokens and gold tokens have different points values in the game.
Silver tokens have a value of 1.5 points each.
Gold tokens have a value of 4 points each.
Beth already has 2 silver tokens and 1 gold token.
She also has $\pounds 10$ that can be spent on buying more tokens.
Silver tokens can be bought for $\pounds 2$ each.
Gold tokens can be bought for $\pounds 6$ each.
After buying some tokens, Beth has $x$ silver tokens and $y$ gold tokens.
She now has a total of at least 5 tokens and no more than 8 tokens.

Set up an LP formulation in $x$ and $y$ for the problem of maximising the points value of tokens that she finishes with.
Use a graphical method to determine how many tokens of each type Beth should buy to maximise the points value of her tokens.

OCR Further Discrete AS 2024 June Q7

7 Six items need to be packed into bins. Each bin has size $k$, where $k$ is an integer. The sizes of the items are
$\begin{array} { l l l l l l } 8 & 5 & 4 & 7 & 3 & 3 \end{array}$

Show a way to pack the items into two bins of size $k = 18$.
If exactly three bins are needed, determine the possible values of $k$.
Use the first-fit method to pack the items into bins of size 12.
Explain why, in general, the first-fit algorithm does not necessarily find the most efficient solution to a packing problem. A possible way to calculate the run-time of the first-fit method is to count for each item the number of bins, excluding full bins, that are considered before the bin where each item is packed. The total count for all packed items is then used to calculate the run-time of the method.
Determine the total count for the packing used in part (c). A student says that when $k = 13$ the total count is 0 because the first bin fills up before the second bin is considered.
Explain why the student's argument is not correct.
Determine the least possible value of $k$ for which the total count is 0 . \section*{END OF QUESTION PAPER}

OCR Further Discrete AS 2020 November Q1

1 Algorithm X is given below: STEP 1: $\quad$ Let $A = 0$
STEP 2: Input the value of $B \quad [ B$ must be a positive integer]
STEP 3: $\quad C = \operatorname { INT } ( B \div 3 )$
STEP 4: $D = B + 3 C$
STEP 5: $\quad$ Replace $A$ with $( D - A )$
STEP 6: Decrease $B$ by 1
STEP 7: If $B > 0$ go to STEP 3
STEP 8: Display the value of $\operatorname { ABS } ( A )$ and STOP

Trace through algorithm X when the input is $B = 5$. Only record changes to the values of the variables, $A , B , C$ and $D$, using a new column of the table in the Printed Answer Booklet each time one of these variables changes.
Explain carefully how you know that algorithm X is finite for all positive integer values of $B$.
Explain what it means to say that an algorithm is deterministic. The run-time of algorithm X is $k \times$ (number of times that STEP 6 is used), for some positive constant $k$.
Explain, with reference to your answer to part (b), why the order of algorithm X is $\mathrm { O } ( n )$, where $n$ is the input value of $B$. For each input $B$, algorithm Y achieves the same output as algorithm X , but the run-time of algorithm Y is $m \times$ (the number of digits in $B$ ), where $m$ is a positive constant.
Show that algorithm Y is more efficient than algorithm X .

OCR Further Discrete AS 2020 November Q2

2 Jameela needs to store ten packages in boxes. She has a list showing the size of each package. The boxes are all the same size and Jameela can use up to six of these boxes to store all the packages.

Which of the following is a question that Jameela could ask which leads to a construction problem? Justify your choice.
- In how many different ways can I fit the packages in the boxes?
- How can I fit the packages in the boxes?
The total volume of the packages is $1 \mathrm {~m} ^ { 3 }$. The volume of each of the six boxes is $0.25 \mathrm {~m} ^ { 3 }$.
Explain why a solution to the problem of storing all the packages in six boxes may not exist. The volume of each package is given below.
Package A B C D E F G H I J
Volume $\left( \mathrm { m } ^ { 3 } \right)$ 0.20 0.05 0.15 0.25 0.04 0.03 0.02 0.02 0.12 0.12
By considering the five largest packages (A, C, D, I and J) first, explain what happens if Jameela tries to pack the 10 packages using only four boxes. You may now assume that the packages will always fit in the boxes if there is enough volume.
Use first-fit to find a way of storing the packages in the boxes. Show the letters of the packages in each box, in the order that they are packed into that box. The order of the packages within a box does not matter and the order of the boxes does not matter. So, for example, having A and E in box 1 is the same as having E and A in box 2 , but different from having A in one box and E in a different box.
Suppose that packages A and B are not in the same box. In this case the following are true:
- there are 8 different ways to put any or none of $\mathrm { E } , \mathrm { F } , \mathrm { G }$ and H with A
- 3 include F with A
- 3 include G with A
- 1 includes both F and G with A
Use the inclusion-exclusion principle to determine how many of the 8 ways include neither package F nor package G.

OCR Further Discrete AS 2020 November Q3

3
\includegraphics[max width=\textwidth, alt={}, center]{c2deec7d-0617-4eb0-a47e-5b42ba55b753-4_399_1331_255_367}

By listing the vertices passed through, in the order they are used, give an example for the graph above of:
1. a walk from A to H that is not a trail,
2. a trail from A to H that is not a path,
3. a path from A to H that is not a cycle. The arcs are weighted to form a network. The arc weights are shown in the table, apart from the weight of arc AG. The arcs model a system of roads and the weights represent distances in km.
  A B C D E F G H
  A - 3 - 8 - - $x$ -
  B 3 - 5 - - - - -
  C - 5 - 4 3 - - -
  D 8 - 4 - 5 - - 9
  E - - 3 5 - 7 6 -
  F - - - - 7 - - -
  G $x$ - - - 6 - - 2
  H - - - 9 - - 2 -
  The road modelled by arc AG is temporarily closed.
Use the matrix form of Prim's algorithm to find a minimum spanning tree, of total weight 30 km , that does not include arc AG. The road modelled by arc AG is now reopened.
1. For what set of values of $x$ could AG be included in a minimum spanning tree for the full network?
2. Which arc from the minimum spanning tree in part (b) is not used when arc AG is included?

OCR Further Discrete AS 2020 November Q4

4 Bob is extending his attic with the help of some friends, including his architect friend Archie. The activities involved, their durations (in days) and Bob's notes are given below.

Activity		Duration (days)	Notes
A	Archie takes measurements	1
B	Archie draws up plans	3	Must come after A
C	Plans are approved	21	Must come after B
D	Bob orders materials	2	Must come after B
E	Materials delivered	10	Must come after D
F	Work area cleared	5	Must come after A
G	Plumbing and electrics	3	Must come after C, E and F
H	Floors, walls and ceilings	24	Must come after G
I	Staircase	2	Must come after H
J	Windows	1	Must come after H
K	Decorating	6	Must come after I and J

Archie has started to construct an activity network to represent the project.
\includegraphics[max width=\textwidth, alt={}, center]{c2deec7d-0617-4eb0-a47e-5b42ba55b753-5_401_1253_1475_406}

Complete the activity network in the Printed Answer Booklet and use it to determine
- the minimum completion time
- the critical activities
  for the project.
- Give two different reasons why the project might take longer than the minimum completion time.
- For the activity with the longest float
- calculate this float, in days
- interpret this float in context.

OCR Further Discrete AS 2020 November Q5

5 The number of points won by player 1 in a zero-sum game is shown in the pay-off matrix below, where $k$ is a constant. \begin{table}[h]

\captionsetup{labelformat=empty} \caption{Player 2}

	Strategy E	Strategy F	Strategy G	Strategy H
Strategy A	$2 k$	-2	$1 - k$	4
Strategy B	-3	3	4	-5
Strategy C	1	4	-4	2
Strategy D	4	-2	-5	6

\end{table}

In one game, player 2 chooses strategy H. Write down the greatest number of points that player 2 could win. You are given that strategy A is a play-safe strategy for player 1.
Determine the range of possible values for $k$.
Determine the column minimax value.

OCR Further Discrete AS 2020 November Q6

6 Tamsin is planning how to spend a day off. She will divide her time between walking the coast path, visiting a bird sanctuary and visiting the garden centre. Tamsin has given a value to each hour spent doing each activity. She wants to decide how much time to spend on each activity to maximise the total value of the activities.

Activity	Walking coast path	Visiting bird sanctuary	Visiting garden centre
Value	5 points per hour	3 points per hour	2 points per hour

Tamsin's requirements are that she will spend:

a total of exactly 6 hours on the three activities
at most 3.5 hours walking the coast path
at least as long at the bird sanctuary as at the garden centre
at least 1 hour at the garden centre.
1. If Tamsin spends $x$ hours walking the coast path and $y$ hours visiting the bird sanctuary, how many hours does she spend visiting the garden centre?

Tamsin models her problem as the linear programming formulation Maximise $P = 3 x + y$
Subject to $$\begin{aligned} & x \leqslant 3.5
& x + 2 y \geqslant 6
& x + y \leqslant 5
& x \geqslant 0 \end{aligned}$$ and

Explain why the maximum total value of the activities done is achieved when $3 x + y$ is maximised.
Show how the requirement that she spends at least as long at the bird sanctuary as at the garden centre leads to the constraint $x + 2 y \geqslant 6$.
Explain why there is no need to require that $y \geqslant 0$.

Represent the constraints graphically and hence find a solution to Tamsin's problem.

OCR Further Discrete AS Specimen Q1

1 Hussain wants to travel by train from Edinburgh to Southampton, leaving Edinburgh after 9 am and arriving in Southampton by 4 pm . He wants to leave Edinburgh as late as possible.
Hussain rings the train company to find out about the train times. Write down a question he might ask that leads to
(A) an existence problem,
(B) an optimisation problem.

OCR Further Discrete AS Specimen Q2

2 Some of the activities that may be involved in making a cup of tea are listed below. A: Boil water.
B: Put teabag in teapot, pour on boiled water and let tea brew.
C: Get cup from cupboard.
D: Pour tea into cup.
E: Add milk to cup.
F: Add sugar to cup. Activity A must happen before activity B.
Activities B and C must happen before activity D .
Activities E and F cannot happen until after activity C.
Other than that, the activities can happen in any order.

Lisa does not take milk or sugar in her tea, so she only needs to use activities $\mathrm { A } , \mathrm { B } , \mathrm { C }$ and D . In how many different orders can activities $\mathrm { A } , \mathrm { B } , \mathrm { C }$ and D be arranged, subject to the restrictions above?
Mick takes milk but no sugar, so he needs to use activities $\mathrm { A } , \mathrm { B } , \mathrm { C } , \mathrm { D }$ and E . Explain carefully why there are exactly nine different orders for these activities, subject to the restrictions above.
Find the number of different orders for all six activities, subject to the restrictions above. Explain your reasoning carefully.

OCR Further Discrete AS Specimen Q3

3 A zero-sum game is being played between two players, $X$ and $Y$. The pay-off matrix for $X$ is given below. \section*{Player X}

	Player $\boldsymbol { Y }$
	Strategy $\boldsymbol { R }$	Strategy $\boldsymbol { S }$
Strategy $\boldsymbol { P }$	4	- 2
Strategy $\boldsymbol { Q }$	- 3	1

Find an optimal mixed strategy for player $X$.
Give one assumption that must be made about the behaviour of $Y$ in order to make the mixed strategy of Player $X$ valid.

OCR Further Discrete AS Specimen Q4

4 Two graphs are shown below. Each has exactly five vertices with vertex orders 2, 3, 3, 4, 4 . \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{a6800c9f-583b-493a-906c-015df63b842f-3_605_616_360_278} \captionsetup{labelformat=empty} \caption{Graph 1}

\end{figure} \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{a6800c9f-583b-493a-906c-015df63b842f-3_420_501_497_1169} \captionsetup{labelformat=empty} \caption{Graph 2}

\end{figure}

Write down a semi-Eulerian route for graph 1 .
Explain how the vertex orders show that graph 2 is also semi-Eulerian.
By referring to specific vertices, explain how you know that these graphs are not simple.
By referring to specific vertices, explain how you know that these graphs are not isomorphic.

OCR Further Discrete AS Specimen Q5

5 There are three non-isomorphic trees on five vertices.

Draw an example of each of these trees.
State three properties that must be satisfied by the vertex orders of a tree on six vertices.
List the five different sets of possible vertex orders for trees on six vertices.
Draw an example of each type listed in part (iii).

OCR Further Discrete AS Specimen Q6

6 The following masses, in kg, are to be packed into bins. $$\begin{array} { l l l l l l l l l l } 8 & 5 & 9 & 7 & 7 & 9 & 1 & 3 & 3 & 8 \end{array}$$

Chloe says that first-fit decreasing gives a packing that requires 4 bins, but first-fit only requires 3 bins. Find the maximum capacity of the bins. First-fit requires one pass through the list and the time taken may be regarded as being proportional to the length of the list. Suppose that shuttle sort was used to sort the list into decreasing order.
What can be deduced, in this case, about the order of the time complexity, $\mathrm { T } ( n )$, for first-fit decreasing?

OCR Further Discrete AS Specimen Q7

7 A complete graph on five vertices is weighted to form a network, as given in the weighted matrix below.

	A	B	C	D	E
\cline { 2 - 6 } A	-	9	5	4	2
\cline { 2 - 6 } B	9	-	7	5	7
\cline { 2 - 6 } C	5	7	-	6	8
\cline { 2 - 6 } D	4	5	6	-	5
\cline { 2 - 6 } E	2	7	8	5	-
\cline { 2 - 6 }
\cline { 2 - 6 }

Apply Prim's algorithm to the copy of this weighted matrix in the Printed Answer Booklet to construct a minimum spanning tree for the five vertices.
Draw your minimum spanning tree, stating the order in which you built the tree and giving its total weight.
(a) Using only the arcs in the minimum spanning tree, which vertex should be chosen to find the smallest total of the weights of the paths from that vertex to each of the other vertices?
(b) State the minimum total for this vertex.
Show that the total number of comparisons needed to find a minimum spanning tree for a $5 \times 5$ matrix is 16 .
If a computer takes 4 seconds to find a minimum spanning tree for a network with 100 vertices, how long would it take to find a minimum spanning tree for a network with 500 vertices?

OCR Further Discrete AS Specimen Q8

8 A sweet shop sells three different types of boxes of chocolate truffles. The cost of each type of box and the number of truffles of each variety in each type of box are given in the table below.

Type	Cost (£)	Milk chocolate	Plain chocolate	White chocolate	Nutty chocolate
Assorted	2.00	5	5	5	5
No Nuts	1.00	5	8	7	0
Speciality	2.50	5	4	9	2

Narendra wants to buy some boxes of truffles so that in total he has at least 20 milk chocolate, 10 plain chocolate, 16 white chocolate and 12 nutty chocolate truffles.

Explain why Narendra needs to buy at least four boxes of truffles.
Narendra decides that he will buy exactly four boxes. Determine the minimum number of Assorted boxes that Narendra must buy.
For your answer in part (ii),
- list all the feasible solutions and
- find the cheapest solution.
Narendra finds that the sweet shop has sold out of Assorted boxes, but he then spots that it also sells small boxes of milk chocolate truffles and small boxes of nutty chocolate truffles. Each small box contains 4 truffles (all of one variety) and costs $\pounds 0.50$. He decides to buy $x$ boxes of No Nuts and $y$ boxes of Speciality, where $x + y < 4$, so that he has at least 10 plain chocolate and 16 white chocolate truffles. He will then buy as many small boxes as he needs to give a total of at least 20 milk chocolate and 12 nutty chocolate truffles.
(a) Set up constraints on the values of $x$ and $y$.
(b) Represent the feasible region graphically.
(c) Hence determine the cheapest cost for Narendra. {www.ocr.org.uk}) after the live examination series. If OCR has unwittingly failed to correctly acknowledge or clear any third-party content in this assessment material, OCR will be happy to correct its mistake at the earliest possible opportunity. For queries or further information please contact the Copyright Team, First Floor, 9 Hills Road, Cambridge CB2 1GE.
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OCR Further Discrete AS 2021 November Q1

1 A set consists of five distinct non-integer values, $\mathrm { A } , \mathrm { B } , \mathrm { C } , \mathrm { D }$ and E . The set is partitioned into non-empty subsets and there are at least two subsets in each partition.

Show that there are 15 different partitions into two subsets.
Show that there are 25 different partitions into three subsets.
Calculate the total number of different partitions. The numbers 12, 24, 36, 48, 60, 72, 84 and 96 are marked on a number line. The number line is then cut into pieces by making cuts at $\mathrm { A } , \mathrm { B } , \mathrm { C } , \mathrm { D }$ and E , where $0 < \mathrm { A } < \mathrm { B } < \mathrm { C } < \mathrm { D } < \mathrm { E } < 100$.
Explain why there must be at least one piece with two or more of the numbers 12, 24, 36, 48, 60, 72, 84 and 96.

OCR Further Discrete AS 2021 November Q2

2 Seven items need to be packed into bins. Each bin has capacity 30 kg . The sizes of the items, in kg, in the order that they are received, are as follows.
12
23
15
18
8
7
5

Find the packing that results using each of these algorithms.
1. The next-fit method
2. The first-fit method
3. The first-fit decreasing method
A student claims that all three methods from part (a) can be used for both 'online' and 'offline' lists. Explain why the student is wrong. The bins of capacity 30 kg are replaced with bins of capacity $M \mathrm {~kg}$, where $M$ is an integer.
The item of size 23 kg can be split into two items, of sizes $x \mathrm {~kg}$ and $( 23 - x ) \mathrm { kg }$, where $x$ may be any integer value you choose from 1 to 11. No other item can be split.
Determine the smallest value of $M$ for which four bins are needed to pack these eight items. Explain your reasoning carefully.

OCR Further Discrete AS 2021 November Q3

3 The diagram shows a simplified map of the main streets in a small town.
\includegraphics[max width=\textwidth, alt={}, center]{4db3fbc5-e664-4803-a5f2-05af376d2591-3_531_1127_301_246} Some of the junctions have traffic lights, these junctions are labelled A to F .
There are no traffic lights at junctions X and Y .
The numbers show distances, in km, between junctions. Alex needs to check that the traffic lights at junctions A to F are working correctly.

Find a route from A to E that has length 2.8 km . Alex has started to construct a table of shortest distances between junctions A to F.
\includegraphics[max width=\textwidth, alt={}, center]{4db3fbc5-e664-4803-a5f2-05af376d2591-3_543_1173_1420_246} For example, the shortest route from C to B has length 1.7 km , the shortest route from C to D has length 2.5 km and the shortest route from C to E has length 1.8 km .
Complete the copy of the table in the Printed Answer Booklet.
Use your table from part (b) to construct a minimum spanning tree for the complete graph on the six vertices A to F .
- Write down the total length of the minimum spanning tree.
- List which arcs of the original network correspond to the arcs used in your minimum spanning tree.
Beth starts from junction B and travels through every junction, including X and Y . Her route has length 5.1 km .
Write down the junctions in the order that Beth visited them. Do not draw on your answer from part (c).

	Player \(\boldsymbol { Y }\)
	Strategy \(\boldsymbol { R }\)	Strategy \(\boldsymbol { S }\)
Strategy \(\boldsymbol { P }\)	4	- 2
Strategy \(\boldsymbol { Q }\)	- 3	1

		C	D	E	H
\cline { 2 - 6 } A	2	2	0	1	I
\cline { 2 - 6 } B	3	1	2	1	2
\cline { 2 - 6 } F	3	2	2	1	0
\cline { 2 - 6 } G	1	3	0	0	0
\cline { 2 - 6 } J	2	1	0	3	1
\cline { 2 - 6 }
\cline { 2 - 6 }

Package	A	B	C	D	E	F	G	H	I	J
Volume \(\left( \mathrm { m } ^ { 3 } \right)\)	0.20	0.05	0.15	0.25	0.04	0.03	0.02	0.02	0.12	0.12

	A	B	C	D	E	F	G	H
A	-	3	-	8	-	-	\(x\)	-
B	3	-	5	-	-	-	-	-
C	-	5	-	4	3	-	-	-
D	8	-	4	-	5	-	-	9
E	-	-	3	5	-	7	6	-
F	-	-	-	-	7	-	-	-
G	\(x\)	-	-	-	6	-	-	2
H	-	-	-	9	-	-	2	-

Questions — OCR Further Discrete AS (52 questions)

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