Questions - AQA D2 - A-Level Maths

AQA D2 2012 June Q5

5 Dave plans to renovate three houses, $A , B$ and $C$, at the rate of one per year. The order in which they are renovated is a matter of choice, but some costs vary over the three years. The expected costs, in thousands of pounds, are given in the table below. (b)

Year	Already renovated	House renovated	Calculation	Value
3	$A$ and $B$	C
	$A$ and $C$	B
	$B$ and $C$	A

2	A	B
		C

	B	A
		C

	C	A
		B

1

Optimum order $\_\_\_\_$

AQA D2 2012 June Q6

6

The network shows a flow from $S$ to $T$ along a system of pipes, with the capacity in litres per second indicated on each edge.
\includegraphics[max width=\textwidth, alt={}, center]{d0902228-7041-4449-9ccb-770352ce6bef-14_510_936_411_552}

Show that the value of the cut shown on the diagram is 36 .

The cut shown on the diagram can be represented as $\{ S , B \} , \{ A , C , T \}$. Complete the table below to give the value of each of the 8 possible cuts.

Cut		Value
$\{ S \}$	$\{ A , B , C , T \}$	30
$\{ S , A \}$	$\{ B , C , T \}$	29
$\{ S , B \}$	$\{ A , C , T \}$	36
$\{ S , C \}$	$\{ A , B , T \}$	33
$\{ S , A , B \}$	$\{ C , T \}$
$\{ S , A , C \}$	$\{ B , T \}$
$\{ S , B , C \}$	$\{ A , T \}$
$\{ S , A , B , C \}$	$\{ T \}$	30

State the value of the maximum flow through the network, giving a reason for your answer. Maximum flow is $\_\_\_\_$ because $\_\_\_\_$
Indicate on the diagram below a possible flow along each edge corresponding to this maximum flow.
\includegraphics[max width=\textwidth, alt={}, center]{d0902228-7041-4449-9ccb-770352ce6bef-15_469_933_406_550}

The capacities along $S C$ and along $A T$ are each increased by 4 litres per second.
1. Using your values from part (a)(iv) as the initial flow, indicate potential increases and decreases on the diagram below and use the labelling procedure to find the new maximum flow through the network. You should indicate any flow augmenting paths in the table and modify the potential increases and decreases of the flow on the diagram.
  \includegraphics[max width=\textwidth, alt={}, center]{d0902228-7041-4449-9ccb-770352ce6bef-15_470_935_1315_260}
  Path
  Additional
  Flow
2. Use your results from part (b)(i) to illustrate the flow along each edge that gives this new maximum flow, and state the value of the new maximum flow. New maximum flow is $\_\_\_\_$
  \includegraphics[max width=\textwidth, alt={}, center]{d0902228-7041-4449-9ccb-770352ce6bef-15_474_933_2078_550}

AQA D2 2014 June Q1

1 A major project has been divided into a number of tasks, as shown in the table. The minimum time required to complete each task is also shown. \section*{Answer space for question 1}

\includegraphics[max width=\textwidth, alt={}]{c2b62fee-d320-4701-a5bb-b2e4b8cc0952-03_424_410_296_685}

AQA D2 2014 June Q2

2 Alex and Roberto play a zero-sum game. The game is represented by the following pay-off matrix for Alex. \begin{table}[h]

\captionsetup{labelformat=empty} \caption{Roberto}

\multirow{5}{*}{Alex Strategy}	D	E	F	G
	A	5	- 4	- 1	1
	B	4	3	0	1
	C	- 3	0	- 5	- 2

\end{table}

Show that this game has a stable solution and state the play-safe strategy for each player.
List any saddle points.

AQA D2 2014 June Q3

3 The diagram below shows a network of pipes with source $A$ and $\operatorname { sink } J$. The capacity of each pipe is given by the number on each edge.
\includegraphics[max width=\textwidth, alt={}, center]{c2b62fee-d320-4701-a5bb-b2e4b8cc0952-08_816_1280_443_386}

Find the values of the cuts $\mathrm { C } _ { 1 }$ and $\mathrm { C } _ { 2 }$.
Find by inspection a flow of 60 units, with flows of 25,10 and 25 along $H J , G J$ and $I J$ respectively. Illustrate your answer on Figure 1.
1. On a certain day the section $E H$ is blocked, as shown on Figure 2. Find, by inspection or otherwise, the maximum flow on this day and illustrate your answer on Figure 2.
2. Show that the flow obtained in part (c)(i) is maximal. \begin{figure}[h]
  \captionsetup{labelformat=empty} \caption{Figure 1} \includegraphics[alt={},max width=\textwidth]{c2b62fee-d320-4701-a5bb-b2e4b8cc0952-09_595_1065_376_475}
  \end{figure}
\begin{figure}[h]
\captionsetup{labelformat=empty} \caption{Figure 2} \includegraphics[alt={},max width=\textwidth]{c2b62fee-d320-4701-a5bb-b2e4b8cc0952-09_617_1061_1142_477}
\end{figure} Maximum flow = $\_\_\_\_$

AQA D2 2014 June Q4

4

Display the following linear programming problem in a Simplex tableau. $$\begin{array} { l c } \text { Maximise } & P = 3 x + 6 y + 2 z
\text { subject to } & x + 3 y + 2 z \leqslant 11
& 3 x + 4 y + 2 z \leqslant 21
\text { and } & x \geqslant 0 , y \geqslant 0 , z \geqslant 0 . \end{array}$$
The first pivot to be chosen is from the $y$-column. Perform one iteration of the Simplex method.
Perform one further iteration.
Interpret the tableau obtained in part (c) and state the values of your slack variables.

AQA D2 2014 June Q5

7 marks

5 Mark and Owen play a zero-sum game. The game is represented by the following pay-off matrix for Mark.

	Owen
\cline { 2 - 5 }\cline { 2 - 5 }	Strategy	D	E	F
A	4	1	- 1
\cline { 2 - 5 } Mark	B	3	- 2	- 2
\cline { 2 - 5 }	C	- 2	0	3

Explain why Mark should never play strategy B.
It is given that the value of the game is 0.6 . Find the optimal strategy for Owen.
(You are not required to find the optimal mixed strategy for Mark.)
[0pt] [7 marks]

AQA D2 2014 June Q6

12 marks

6 The network below has 11 vertices and 16 edges connecting some pairs of vertices. The numbers on the edges are their weights. The weight of the edge $D G$ is given in terms of $x$. There are three routes from $A$ to $K$ that have the same minimum total weight.
\includegraphics[max width=\textwidth, alt={}, center]{c2b62fee-d320-4701-a5bb-b2e4b8cc0952-16_863_1444_552_299} Working backwards from $\boldsymbol { K }$, use dynamic programming, to find:

the minimum total weight from $A$ to $K$;
the value of $x$;
the three routes corresponding to the minimum total weight. You must complete the table opposite as your solution.
[0pt] [12 marks] \section*{Answer space for question 6}
Stage State From Calculation Value
1 I K
$J$ K

AQA D2 2014 June Q7

7 The table shows the times taken, in minutes, by four people, $A , B , C$ and $D$, to carry out the tasks $W , X , Y$ and $Z$. Some of the times are subject to the same delay of $x$ minutes, where $4 < x < 11$.

AQA D2 2014 June Q8

8 An activity diagram for a project is shown below. The duration of each activity is given in weeks. The earliest start time and the latest finish time for each activity are shown on the diagram.
\includegraphics[max width=\textwidth, alt={}, center]{c2b62fee-d320-4701-a5bb-b2e4b8cc0952-22_640_1626_475_209}

Find the values of $x , y$ and $z$.
State the critical path.
Some of the activities can be speeded up at an additional cost. The following table lists the activities that can be speeded up together with the minimum possible duration of these activities. The table also shows the additional cost of reducing the duration of each of these activities by one week.

AQA D2 2015 June Q1

2 marks

1 Figure 2, on the page opposite, shows an activity diagram for a project. Each activity requires one worker. The duration required for each activity is given in hours.

On Figure 1 below, complete the precedence table.
Find the earliest start time and the latest finish time for each activity and insert their values on Figure 2.
List the critical paths.
Find the float time of activity $E$.
Using Figure 3 opposite, draw a Gantt diagram to illustrate how the project can be completed in the minimum time, assuming that each activity is to start as early as possible.
Given that there is only one worker available for the project, find the minimum completion time for the project.
Given that there are two workers available for the project, find the minimum completion time for the project. Show a suitable allocation of tasks to the two workers.
[0pt] [2 marks] \begin{table}[h]
\captionsetup{labelformat=empty} \caption{Figure 1}
Activity Immediate predecessor(s)
A
B
C
D
E
$F$
G
$H$
I
J
\end{table} \begin{figure}[h]
\captionsetup{labelformat=empty} \caption{Figure 2} \includegraphics[alt={},max width=\textwidth]{b0f9523e-51dd-495f-99ec-4724243b5619-03_1071_1561_376_278}
\end{figure} \begin{figure}[h]
\captionsetup{labelformat=empty} \caption{Figure 3} \includegraphics[alt={},max width=\textwidth]{b0f9523e-51dd-495f-99ec-4724243b5619-03_801_1301_1644_420}
\end{figure}

AQA D2 2015 June Q2

2 Stan and Christine play a zero-sum game. The game is represented by the following pay-off matrix for Stan. \begin{table}[h]

\captionsetup{labelformat=empty} \caption{Christine}

\multirow{5}{*}{Stan}	Strategy	D	E	F
	A	3	- 3	- 1
	B	- 1	- 4	2
	C	1	0	- 3
\cline { 2 - 5 }			- 2

\end{table}

Find the play-safe strategy for each player.
Show that there is no stable solution.
Explain why a suitable pay-off matrix for Christine is given by

AQA D2 2015 June Q3

11 marks

3 In the London 2012 Olympics, the Jamaican $4 \times 100$ metres relay team set a world record time of 36.84 seconds. Athletes take different times to run each of the four legs.
The coach of a national athletics team has five athletes available for a major championship. The lowest times that the five athletes take to cover each of the four legs is given in the table below. The coach is to allocate a different athlete from the five available athletes, $A , B , C , D$ and $E$, to each of the four legs to produce the lowest total time.

	Leg 1	Leg 2	Leg 3	Leg 4
Athlete $\boldsymbol { A }$	9.84	8.91	8.98	8.70
Athlete $\boldsymbol { B }$	10.28	9.06	9.24	9.05
Athlete $\boldsymbol { C }$	10.31	9.11	9.22	9.18
Athlete $\boldsymbol { D }$	10.04	9.07	9.19	9.01
Athlete $\boldsymbol { E }$	9.91	8.95	9.09	8.74

Use the Hungarian algorithm, by reducing the columns first, to assign an athlete to each leg so that the total time of the four athletes is minimised. State the allocation of the athletes to the four legs and the total time.
[0pt] [11 marks]

\includegraphics[max width=\textwidth, alt={}]{b0f9523e-51dd-495f-99ec-4724243b5619-08_1200_1705_1507_155}

AQA D2 2015 June Q4

4

Display the following linear programming problem in a Simplex tableau. $$\begin{array} { l r } \text { Maximise } & P = 2 x + 3 y + 4 z
\text { subject to } & x + y + 2 z \leqslant 20
& 3 x + 2 y + z \leqslant 30
& 2 x + 3 y + z \leqslant 40
\text { and } & x \geqslant 0 , y \geqslant 0 , z \geqslant 0 \end{array}$$
1. The first pivot to be chosen is from the $z$-column. Identify the pivot and explain why this particular value is chosen.
2. Perform one iteration of the Simplex method.
1. Perform one further iteration.
2. Interpret your final tableau and state the values of your slack variables.

AQA D2 2015 June Q5

2 marks

5 Tom is going on a driving holiday and wishes to drive from $A$ to $K$.
The network below shows a system of roads. The number on each edge represents the maximum altitude of the road, in hundreds of metres above sea level. Tom wants to ensure that the maximum altitude of any road along the route from $A$ to $K$ is minimised.
\includegraphics[max width=\textwidth, alt={}, center]{b0f9523e-51dd-495f-99ec-4724243b5619-14_1522_1363_660_342}

Working backwards from $\boldsymbol { K }$, use dynamic programming to find the optimal route when driving from $A$ to $K$. You must complete the table opposite as your solution.
Tom finds that the road $C F$ is blocked. Find Tom's new optimal route and the maximum altitude of any road on this route.
[0pt] [2 marks] \section*{Answer space for question 5}
Stage State From Value
1 H $K$
I $K$
$J$ $K$
2
Optimal route is $\_\_\_\_$
Tom's route is $\_\_\_\_$
Maximum altitude is $\_\_\_\_$ Figure 4 below shows a network of pipes.
The capacity of each pipe is given by the number not circled on each edge. The numbers in circles represent an initial flow. \begin{figure}[h]
\captionsetup{labelformat=empty} \caption{Figure 4} \includegraphics[alt={},max width=\textwidth]{b0f9523e-51dd-495f-99ec-4724243b5619-18_1039_1623_561_191}
\end{figure}
Find the value of the initial flow.
1. Use the initial flow and the labelling procedure on Figure 5 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 6, 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.
On a particular day, there is a restriction at vertex $G$ which allows a maximum flow through $G$ of 30 . Find, by inspection, the maximum flow through the network on this day.
Initial flow $=$ $\_\_\_\_$
1. Figure 5
  \includegraphics[max width=\textwidth, alt={}, center]{b0f9523e-51dd-495f-99ec-4724243b5619-19_2158_1559_543_296}
  $7 \quad$ Arsene and Jose play a zero-sum game. The game is represented by the following pay-off matrix for Arsene, where $x$ is a constant. The value of the game is 2.5 .
  Jose
  \cline { 2 - 4 } Strategy C D
  \cline { 2 - 4 } Arsene A $x + 3$ 1
  \cline { 2 - 4 } B $x + 1$ 3
  \cline { 2 - 4 }
  \cline { 2 - 4 }
Find the optimal mixed strategy for Arsene.
Find the value of $x$.
\includegraphics[max width=\textwidth, alt={}]{b0f9523e-51dd-495f-99ec-4724243b5619-22_1636_1707_1071_153}

\includegraphics[max width=\textwidth, alt={}]{b0f9523e-51dd-495f-99ec-4724243b5619-24_2288_1705_221_155}

AQA D2 2016 June Q1

1
Figure 1 below shows an activity diagram for a project. Each activity requires one worker. The duration required for each activity is given in hours.

Find the earliest start time and the latest finish time for each activity and insert these values on Figure 1.
1. Find the critical path.
2. Find the float time of activity $F$.
Using Figure 2 on page 3, draw a resource histogram to illustrate how the project can be completed in the minimum time, assuming that each activity is to start as early as possible.
1. Given that there are two workers available for the project, find the minimum completion time for the project.
2. Write down an allocation of tasks to the two workers that corresponds to your answer in part (d)(i). \section*{Answer space for question 1} \begin{figure}[h]
  \captionsetup{labelformat=empty} \caption{Figure 1} \includegraphics[alt={},max width=\textwidth]{34de3f03-a275-44fb-88b2-b88038bcec97-02_687_1655_1941_189}
  \end{figure} \section*{Answer space for question 1} \begin{figure}[h]
  \captionsetup{labelformat=empty} \caption{Figure 2} \includegraphics[alt={},max width=\textwidth]{34de3f03-a275-44fb-88b2-b88038bcec97-03_1115_1575_434_283}
  \end{figure}
  \includegraphics[max width=\textwidth, alt={}]{34de3f03-a275-44fb-88b2-b88038bcec97-03_1024_1593_1683_267}

AQA D2 2016 June Q2

2 Alan, Beth, Callum, Diane and Ethan work for a restaurant chain. The costs, in pounds, for the five people to travel to each of five different restaurants are recorded in the table below. Alan cannot travel to restaurant 1 and Beth cannot travel to restaurants 3 and 5, as indicated by the asterisks in the table.

AQA D2 2016 June Q3

3 marks

3
Maximise $\quad P = 2 x - 3 y + 4 z$
subject to $\quad x + 2 y + z \leqslant 20$
$x - y + 3 z \leqslant 24$
$3 x - 2 y + 2 z \leqslant 30$
and $\quad x \geqslant 0 , y \geqslant 0 , z \geqslant 0$.

Display the linear programming problem in a Simplex tableau.
1. The first pivot to be chosen is from the $z$-column. Identify the pivot and explain why this particular value is chosen.
2. Perform one iteration of the Simplex method.
3. Perform one further iteration.
Interpret your final tableau and state the values of your slack variables.
[0pt] [3 marks]

AQA D2 2016 June Q4

4 Monica and Vladimir play a zero-sum game. The game is represented by the following pay-off matrix for Monica.

AQA D2 2016 June Q5

5 Robert is planning to renovate four houses, $A , B , C$ and $D$, at the rate of one per month. The houses can be renovated in any order but the costs will vary because some of the materials left over from renovating one house can be used for the next one. The expected profits, in hundreds of pounds, are given in the table below.

AQA D2 2016 June Q6

6 The network shows a system of pipes with lower and upper capacities for each pipe in litres per second.
\includegraphics[max width=\textwidth, alt={}, center]{34de3f03-a275-44fb-88b2-b88038bcec97-22_817_744_397_648}

1. Find the value of the cut $X$.
2. Hence state what can be deduced about the maximum flow from $A$ to $H$.
Figure 3 shows a partially completed diagram for a feasible flow of 28 litres per second from $A$ to $H$. Indicate, on Figure 3, the flows along the edges $B D , B E$ and $C D$.
1. Using your feasible flow from part (b) as an initial flow, indicate potential increases and decreases of the flow along each edge on Figure 4.
2. Use flow augmentation on Figure 4 to find the maximum flow from $A$ to $H$. You should indicate any flow augmenting paths in the table and modify the potential increases and decreases of the flow on the network.
3. State the maximum flow and indicate a maximum flow on Figure 5. \section*{Answer space for question 6} \begin{figure}[h]
  \captionsetup{labelformat=empty} \caption{Figure 3} \includegraphics[alt={},max width=\textwidth]{34de3f03-a275-44fb-88b2-b88038bcec97-23_682_689_312_397}
  \end{figure} \begin{figure}[h]
  \captionsetup{labelformat=empty} \caption{Figure 4} \includegraphics[alt={},max width=\textwidth]{34de3f03-a275-44fb-88b2-b88038bcec97-23_935_1477_1037_365}
  \end{figure} Figure 5
  \includegraphics[max width=\textwidth, alt={}]{34de3f03-a275-44fb-88b2-b88038bcec97-24_2032_1707_219_153}

Cut		Value
\(\{ S \}\)	\(\{ A , B , C , T \}\)	30
\(\{ S , A \}\)	\(\{ B , C , T \}\)	29
\(\{ S , B \}\)	\(\{ A , C , T \}\)	36
\(\{ S , C \}\)	\(\{ A , B , T \}\)	33
\(\{ S , A , B \}\)	\(\{ C , T \}\)
\(\{ S , A , C \}\)	\(\{ B , T \}\)
\(\{ S , B , C \}\)	\(\{ A , T \}\)
\(\{ S , A , B , C \}\)	\(\{ T \}\)	30

Questions — AQA D2 (121 questions)

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