Questions - AQA D2 - A-Level Maths

Questions — AQA D2 (121 questions)

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State the vertex that represents the assembly hall.
Find the value of the cut shown on the diagram.
State the maximum flow along the routes $A B D$ and $G E D$.
1. Taking your answers to part (c) as the initial flow, use a labelling procedure on Figure 4 to find the maximum flow through the network.
2. State the value of the maximum flow and, on Figure 5, illustrate a possible flow along each edge corresponding to this maximum flow.
3. Verify that your flow is a maximum flow by finding a cut of the same value.
On a particular day, there is an obstruction allowing no more than 15 pupils per minute to pass through vertex $E$. State the maximum number of pupils that can move through the network per minute on this particular day.

5 A linear programming problem involving variables $x$ and $y$ is to be solved. The objective function to be maximised is $P = 4 x + 9 y$. The initial Simplex tableau is given below.

Write down the three inequalities in $x$ and $y$ represented by this tableau.
The Simplex method is to be used to solve this linear programming problem by initially choosing a value in the $x$-column as the pivot.
1. Explain why the initial pivot has value 1.
2. Perform two iterations using the Simplex method.
3. Comment on how you know that the optimum solution has been achieved and state your final values of $P , x$ and $y$.

6 Two people, Rowan and Colleen, play a zero-sum game. The game is represented by the following pay-off matrix for Rowan. Colleen

\multirow{4}{*}{Rowan}	Strategy	$\mathrm { C } _ { 1 }$	$\mathrm { C } _ { 2 }$	$\mathrm { C } _ { 3 }$
	$\mathrm { R } _ { 1 }$	-3	-4	1
	$\mathbf { R } _ { \mathbf { 2 } }$	1	5	-1
	$\mathbf { R } _ { \mathbf { 3 } }$	-2	-3	4

Explain the meaning of the term 'zero-sum game'.
Show that this game has no stable solution.
Explain why Rowan should never play strategy $R _ { 1 }$.
1. Find the optimal mixed strategy for Rowan.
2. Find the value of the game.
  Surname Other Names
  Centre Number Candidate Number
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  \section*{MATHEMATICS
  Unit Decision 2} \section*{Insert} Thursday 8 June 2006 9.00 am to 10.30 am Insert for use in Questions 1, 3 and 4.
  Fill in the boxes at the top of this page.
  Fasten this insert securely to your answer book.

1 [Figures 1 and 2, printed on the insert, are provided for use in this question.]
The following diagram shows an activity diagram for a building project. The time needed for each activity is given in days.
\includegraphics[max width=\textwidth, alt={}, center]{0c40b693-72d3-459c-bbb7-b9584a108b8e-02_698_1321_767_354}

Complete the precedence table for the project on Figure 1.
Find the earliest start times and latest finish times for each activity and insert their values on Figure 2.
Find the critical path and state the minimum time for completion of the project.
Find the activity with the greatest float time and state the value of its float time.

2 The daily costs, in pounds, for five managers A, B, C, D and E to travel to five different centres are recorded in the table below.

	A	B	C	D	E
Centre 1	10	11	8	12	5
Centre 2	11	5	11	6	7
Centre 3	12	8	7	11	4
Centre 4	10	9	14	10	6
Centre 5	9	9	7	8	9

Using the Hungarian algorithm, each of the five managers is to be allocated to a different centre so that the overall total travel cost is minimised.

By reducing the rows first and then the columns, show that the new table of values is
3 6 3 6 0
4 0 6 0 2
6 4 3 6 0
2 3 8 3 0
0 2 0 0 2
Show that the zeros in the table in part (a) can be covered with three lines and use adjustments to produce a table where five lines are required to cover the zeros.
Hence find the two possible ways of allocating the five managers to the five centres with the least possible total travel cost.
Find the value of this minimum daily total travel cost.

3 Two people, Rose and Callum, play a zero-sum game. The game is represented by the following pay-off matrix for Rose.

	Callum
\cline { 2 - 5 }		$\mathbf { C } _ { \mathbf { 1 } }$	$\mathbf { C } _ { \mathbf { 2 } }$	$\mathbf { C } _ { \mathbf { 3 } }$
\cline { 2 - 5 }	$\mathbf { R } _ { \mathbf { 1 } }$	5	2	- 1
\cline { 2 - 5 } Rose	$\mathbf { R } _ { \mathbf { 2 } }$	- 3	- 1	5
\cline { 2 - 5 }	$\mathbf { R } _ { \mathbf { 3 } }$	4	1	- 2
\cline { 2 - 5 }
\cline { 2 - 5 }

1. State the play-safe strategy for Rose and give a reason for your answer.
2. Show that there is no stable solution for this game.
Explain why Rose should never play strategy $\mathbf { R } _ { \mathbf { 3 } }$.
Rose adopts a mixed strategy, choosing $\mathbf { R } _ { \mathbf { 1 } }$ with probability $p$ and $\mathbf { R } _ { \mathbf { 2 } }$ with probability $1 - p$.
1. Find expressions for the expected gain for Rose when Callum chooses each of his three possible strategies. Simplify your expressions.
2. Illustrate graphically these expected gains for $0 \leqslant p \leqslant 1$.
3. Hence determine the optimal mixed strategy for Rose.
4. Find the value of the game.

4 A linear programming problem involving variables $x$ and $y$ is to be solved. The objective function to be maximised is $P = 3 x + 5 y$. The initial Simplex tableau is given below.

In addition to $x \geqslant 0 , y \geqslant 0$, write down three inequalities involving $x$ and $y$ for this problem.
1. By choosing the first pivot from the $\boldsymbol { y }$-column, perform one iteration of the Simplex method.
2. Explain how you know that the optimal value has not been reached.
1. Perform one further iteration.
2. Interpret the final tableau and state the values of the slack variables.

5 [Figure 3, printed on the insert, is provided for use in this question.]
A maker of exclusive furniture is planning to build three cabinets $A , B$ and $C$ at the rate of one per month. The order in which they are built is a matter of choice, but the costs will vary because of the materials available and suppliers' costs. The expected costs, in pounds, are given in the table.

\multirow[t]{2}{*}{Month}	\multirow[t]{2}{*}{Already built}	Cost
		$\boldsymbol { A }$	B	$\boldsymbol { C }$
1	-	500	440	475
2	A	-	440	490
	B	510	-	500
	$\boldsymbol { C }$	520	490	-
3	$\boldsymbol { A }$ and $\boldsymbol { B }$	-	-	520
	$\boldsymbol { A }$ and $\boldsymbol { C }$	-	500	-
	$\boldsymbol { B }$ and $\boldsymbol { C }$	510	-	-

Use dynamic programming, working backwards from month 3, to determine the order of manufacture that minimises the total cost. You may wish to use Figure 3 for your working.
It is discovered that the figures given were actually the profits, not the costs, for each item. Modify your solution to find the order of manufacture that maximises the total profit. You may wish to use the final column of Figure 3 for your working.

6 [Figures 4, 5 and 6, 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]{0c40b693-72d3-459c-bbb7-b9584a108b8e-07_713_1456_539_294}

1. Find the value of the cut $C$.
2. State what can be deduced about the maximum flow from $S$ to $T$.
Figure 4, printed on the insert, shows a partially completed diagram for a feasible flow of 20 litres per second from $S$ to $T$. Indicate, on Figure 4, the flows along the edges $M P , P N , Q R$ and $N R$.
1. Taking your answer from part (b) as an initial flow, indicate potential increases and decreases of the flow along each edge on Figure 5.
2. Use flow augmentation on Figure 5 to find the maximum flow from $S$ to $T$. You should indicate any flow augmenting paths in the table and modify the potential increases and decreases of the flow on the network.
3. Illustrate the maximum flow on Figure 6.

1 [Figures 1 and 2, printed on the insert, are provided for use in this question.]
The following diagram shows an activity network for a project. The time needed for each activity is given in days.
\includegraphics[max width=\textwidth, alt={}, center]{f98d4434-458a-4118-92ed-309510d7975a-02_940_1698_721_164}

Find the earliest start time and the latest finish time for each activity and insert their values on Figure 1.
Find the critical paths and state the minimum time for completion.
On Figure 2, draw a cascade diagram (Gantt chart) for the project, assuming each activity starts as early as possible.
Activity $C$ takes 5 days longer than first expected. Determine the effect on the earliest start time for other activities and the minimum completion time for the project.
(2 marks)

2 The following table shows the scores of five people, Alice, Baji, Cath, Dip and Ede, after playing five different computer games.

	Alice	Baji	Cath	Dip	Ede
Game 1	17	16	19	17	20
Game 2	20	13	15	16	18
Game 3	16	17	15	18	13
Game 4	13	14	18	15	17
Game 5	15	16	20	16	15

Each of the five games is to be assigned to one of the five people so that the total score is maximised. No person can be assigned to more than one game.

Explain why the Hungarian algorithm may be used if each number, $x$, in the table is replaced by $20 - x$.
Form a new table by subtracting each number in the table above from 20, and hence show that, by reducing columns first and then rows, the resulting table of values is as below.
3 1 1 1 0
0 4 5 2 2
4 0 5 0 7
5 1 0 1 1
5 1 0 2 5
Show that the zeros in the table in part (b) can be covered with one horizontal and three vertical lines. Hence use the Hungarian algorithm to reduce the table to a form where five lines are needed to cover the zeros.
Hence find the possible allocations of games to the five people so that the total score is maximised.
State the value of the maximum total score.

3 Two people, Roseanne and Collette, play a zero-sum game. The game is represented by the following pay-off matrix for Roseanne.

\multirow{2}{*}{}		Collette
	Strategy	$\mathrm { C } _ { 1 }$	$\mathbf { C } _ { \mathbf { 2 } }$	$\mathrm { C } _ { 3 }$
\multirow{2}{*}{Roseanne}	$\mathrm { R } _ { 1 }$	-3	2	3
	$\mathbf { R } _ { \mathbf { 2 } }$	2	-1	-4

1. Find the optimal mixed strategy for Roseanne.
2. Show that the value of the game is - 0.5 .
1. Collette plays strategy $\mathrm { C } _ { 1 }$ with probability $p$ and strategy $\mathrm { C } _ { 2 }$ with probability $q$. Write down, in terms of $p$ and $q$, the probability that she plays strategy $\mathrm { C } _ { 3 }$.
2. Hence, given that the value of the game is - 0.5 , find the optimal mixed strategy for Collette.

4 A linear programming problem consists of maximising an objective function $P$ involving three variables $x , y$ and $z$. Slack variables $s , t , u$ and $v$ are introduced and the Simplex method is used to solve the problem. Several iterations of the method lead to the following tableau.

$\boldsymbol { P }$	$x$	$y$	$\boldsymbol { Z }$	$\boldsymbol { s }$	$\boldsymbol { t }$	$\boldsymbol { u }$	$v$	value
1	0	-12	0	5	-3	0	0	37
0	1	-8	0	1	2	0	0	16
0	0	4	0	0	3	0	1	20
0	0	2	0	-3	2	1	0	14
0	0	1	1	2	5	0	0	8

1. The pivot for the next iteration is chosen from the $\boldsymbol { y }$-column. State which value should be chosen and explain the reason for your choice.
2. Perform the next iteration of the Simplex method.
Explain why your new tableau solves the original problem.
State the maximum value of $P$ and the values of $x , y$ and $z$ that produce this maximum value.
State the values of the slack variables at the optimum point. Hence determine how many of the original inequalities still have some slack when the optimum is reached.

5 [Figure 3, printed on the insert, is provided for use in this question.]
A small firm produces high quality cabinets.
It can produce up to 4 cabinets each month.
Whenever at least one cabinet is made during that month, the overhead costs for that month are $\pounds 300$. It is possible to hold in stock a maximum of 2 cabinets during any month.
The cost of storage is $\pounds 50$ per cabinet per month.
The orders for cabinets are shown in the table below. There is no stock at the beginning of January and the firm plans to clear all stock after completing the April orders.

Month	January	February	March	April
Number of cabinets required	3	3	5	2

Determine the total cost of storing 2 cabinets and producing 3 cabinets in a given month.
By completing the table of values on Figure 3, or otherwise, use dynamic programming, working backwards from April, to find the production schedule which minimises total costs.
Each cabinet is sold for $\pounds 2000$ but there is an additional cost of $\pounds 300$ for materials to make each cabinet and $\pounds 2000$ per month in wages. Determine the total profit for the four-month period.

6 [Figures 4, 5 and 6, 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]{f98d4434-458a-4118-92ed-309510d7975a-06_796_1337_518_338}

1. Find the value of the cut $C$.
2. Hence state what can be deduced about the maximum flow from $S$ to $T$.
Figure 4, printed on the insert, shows a partially completed diagram for a feasible flow of 32 litres per second from $S$ to $T$. Indicate, on Figure 4, the flows along the edges $P Q , U Q$ and $U T$.
1. Taking your feasible flow from part (b) as an initial flow, indicate potential increases and decreases of the flow along each edge on Figure 5.
2. Use flow augmentation on Figure 5 to find the maximum flow from $S$ to $T$. You should indicate any flow augmenting paths in the table and modify the potential increases and decreases of the flow on the network.
3. Illustrate the maximum flow on Figure 6.

1 [Figure 1, printed on the insert, is provided for use in this question.]
A decorating project is to be undertaken. The table shows the activities involved.

Activity	Immediate Predecessors	Duration (days)
A	-	5
B	-	3
C	-	2
D	A, $B$	4
E	$B , C$	1
$F$	D	2
G	E	9
H	$F , G$	1
I	$H$	6
$J$	$H$	5
$K$	$I , J$	2

Complete an activity network for the project on Figure 1.
On Figure 1, indicate:
1. the earliest start time for each activity;
2. the latest finish time for each activity.
State the minimum completion time for the decorating project and identify the critical path.
Activity $F$ takes 4 days longer than first expected.
1. Determine the new earliest start time for activities $H$ and $I$.
2. State the minimum delay in completing the project.

2 Two people, Rowena and Colin, play a zero-sum game.
The game is represented by the following pay-off matrix for Rowena.

\multirow{5}{*}{Rowena}	Colin
	Strategy	$\mathrm { C } _ { 1 }$	$\mathbf { C } _ { \mathbf { 2 } }$	$\mathrm { C } _ { 3 }$
	$\mathbf { R } _ { \mathbf { 1 } }$	-4	5	4
	$\mathbf { R } _ { \mathbf { 2 } }$	2	-3	-1
	$\mathbf { R } _ { \mathbf { 3 } }$	-5	4	3

Explain what is meant by the term 'zero-sum game'.
Determine the play-safe strategy for Colin, giving a reason for your answer.
Explain why Rowena should never play strategy $R _ { 3 }$.
Find the optimal mixed strategy for Rowena.

3 Five lecturers were given the following scores when matched against criteria for teaching five courses in a college.

	Course 1	Course 2	Course 3	Course 4	Course 5
Ron	13	13	9	10	13
Sam	13	14	12	17	15
Tom	16	10	8	14	14
Una	11	14	12	16	10
Viv	12	14	14	13	15

Each lecturer is to be allocated to exactly one of the courses so as to maximise the total score of the five lecturers.

Explain why the Hungarian algorithm may be used if each number, $x$, in the table is replaced by $17 - x$.
Form a new table by subtracting each number in the table above from 17. Hence show that, by reducing rows first and then columns, the resulting table of values is as below.
0 0 3 3 0
4 3 4 0 2
0 6 7 2 2
5 2 3 0 6
3 1 0 2 0
Show that the zeros in the table in part (b) can be covered with two horizontal and two vertical lines. Hence use the Hungarian algorithm to reduce the table to a form where five lines are needed to cover the zeros.
Hence find the possible allocations of courses to the five lecturers so that the total score is maximised.
State the value of the maximum total score.

4 A linear programming problem involving variables $x , y$ and $z$ is to be solved. The objective function to be maximised is $P = 4 x + y + k z$, where $k$ is a constant. The initial Simplex tableau is given below.

$\boldsymbol { P }$	$\boldsymbol { x }$	$\boldsymbol { y }$	$\boldsymbol { z }$	$s$	$\boldsymbol { t }$	value
1	-4	-1	$- k$	0	0	0
0	1	2	3	1	0	7
0	2	1	4	0	1	10

In addition to $x \geqslant 0 , y \geqslant 0$ and $z \geqslant 0$, write down two inequalities involving $x , y$ and $z$ for this problem.
1. The first pivot is chosen from the $\boldsymbol { x }$-column. Identify the pivot and perform one iteration of the Simplex method.
2. Given that the optimal value of $P$ has not been reached after this first iteration, find the possible values of $k$.
Given that $k = 10$ :
1. perform one further iteration of the Simplex method;
2. interpret the final tableau.

5 [Figure 2, printed on the insert, is provided for use in this question.]
A company has a number of stores. The following network shows the possible actions and profits over the next five years. The number on each edge is the expected profit, in millions of pounds. A negative number indicates a loss due to investment in new stores.
\includegraphics[max width=\textwidth, alt={}, center]{1bf0d8b7-9f91-437a-bc18-3bfe5ca12223-06_1006_1583_591_223}

Working backwards from $\boldsymbol { T }$, use dynamic programming to maximise the expected profits over the five years. You may wish to complete the table on Figure 2 as your solution.
State the maximum expected profit and the sequence of vertices from $S$ to $T$ in order to achieve this.
(2 marks)

6 [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]{1bf0d8b7-9f91-437a-bc18-3bfe5ca12223-07_849_1363_518_326}

Find the value of the cut $C$.
Figure 3, on the insert, shows a partially completed diagram for a feasible flow of 40 litres per second from $S$ to $T$. Indicate, on Figure 3, the flows along the edges $A E , E F$ and $F G$.
1. Taking your answer 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 $S$ to $T$. You should indicate any flow augmenting paths in the table and modify the potential increases and decreases of the flow on the network.
Illustrate the maximum flow on Figure 5.
Find a cut with value equal to that of the maximum flow.

1
Figure 1 below shows an activity diagram for a construction project. The time needed for each activity is given in days.

Find the earliest start time and the latest finish time for each activity and insert their values on Figure 1.
Find the critical paths and state the minimum time for completion of the project.
On Figure 2 opposite, draw a cascade diagram (Gantt chart) for the project, assuming that each activity starts as early as possible.
Activity $J$ takes longer than expected so that its duration is $x$ days, where $x \geqslant 3$. Given that the minimum time for completion of the project is unchanged, find a further inequality relating to the maximum value of $x$.
\begin{figure}[h]
\captionsetup{labelformat=empty} \caption{Figure 1} \includegraphics[alt={},max width=\textwidth]{d0902228-7041-4449-9ccb-770352ce6bef-02_910_1355_1414_411}
\end{figure}
Critical paths are $\_\_\_\_$
Minimum completion time is $\_\_\_\_$ days. \begin{figure}[h]
\captionsetup{labelformat=empty} \caption{Figure 2} \includegraphics[alt={},max width=\textwidth]{d0902228-7041-4449-9ccb-770352ce6bef-03_940_1160_390_520}
\end{figure}
$\_\_\_\_$

2 The times taken in minutes for five people, Ann, Baz, Cal, Di and Ez, to complete each of five different tasks are recorded in the table below. Neither Ann nor Di can do task 2, as indicated by the asterisks in the table.

Given that $k$ is a constant, complete the Simplex tableau below for the following linear programming problem. Maximise $$P = k x + 6 y + 5 z$$ subject to $$\begin{gathered} 2 x + y + 4 z \leqslant 11
x + 3 y + 6 z \leqslant 18
x \geqslant 0 , y \geqslant 0 , z \geqslant 0 \end{gathered}$$
Use the Simplex method to perform one iteration of your tableau for part (a), choosing a value in the $\boldsymbol { y }$-column as pivot.
1. In the case when $k = 1$, explain why the maximum value of $P$ has now been reached and write down this maximum value of $P$.
2. In the case when $k = 3$, perform one further iteration and interpret your new tableau. \section*{Answer space for question 3}
$\boldsymbol { P }$ $\boldsymbol { x }$ $\boldsymbol { y }$ $\boldsymbol { Z }$ $s$ $\boldsymbol { t }$ value
1 $- k$ -6 -5 0 0 0
0
0
$\boldsymbol { P }$ $\boldsymbol { x }$ $\boldsymbol { y }$ $\boldsymbol { Z }$ $\boldsymbol { s }$ $\boldsymbol { t }$ value
\section*{Answer space for question 3}
1. $\_\_\_\_$

Two people, Adam and Bill, play a zero-sum game. The game is represented by the following pay-off matrix for Adam. 4
Roza plays a different zero-sum game against a computer. The game is represented by the following pay-off matrix for Roza.

Activity	Immediate Predecessors	Duration (days)
A	-	5
B	-	3
C	-	2
D	A, \(B\)	4
E	\(B , C\)	1
\(F\)	D	2
G	E	9
H	\(F , G\)	1
I	\(H\)	6
\(J\)	\(H\)	5
\(K\)	\(I , J\)	2

\(\boldsymbol { P }\)	\(\boldsymbol { x }\)	\(\boldsymbol { y }\)	\(\boldsymbol { z }\)	\(s\)	\(\boldsymbol { t }\)	value
1	-4	-1	\(- k\)	0	0	0
0	1	2	3	1	0	7
0	2	1	4	0	1	10

\(\boldsymbol { P }\)	\(\boldsymbol { x }\)	\(\boldsymbol { y }\)	\(\boldsymbol { Z }\)	\(s\)	\(\boldsymbol { t }\)	value
1	\(- k\)	-6	-5	0	0	0
0
0

\multirow{4}{*}{Rowan}	Strategy	\(\mathrm { C } _ { 1 }\)	\(\mathrm { C } _ { 2 }\)	\(\mathrm { C } _ { 3 }\)
	\(\mathrm { R } _ { 1 }\)	-3	-4	1
	\(\mathbf { R } _ { \mathbf { 2 } }\)	1	5	-1
	\(\mathbf { R } _ { \mathbf { 3 } }\)	-2	-3	4

	Callum
\cline { 2 - 5 }		\(\mathbf { C } _ { \mathbf { 1 } }\)	\(\mathbf { C } _ { \mathbf { 2 } }\)	\(\mathbf { C } _ { \mathbf { 3 } }\)
\cline { 2 - 5 }	\(\mathbf { R } _ { \mathbf { 1 } }\)	5	2	- 1
\cline { 2 - 5 } Rose	\(\mathbf { R } _ { \mathbf { 2 } }\)	- 3	- 1	5
\cline { 2 - 5 }	\(\mathbf { R } _ { \mathbf { 3 } }\)	4	1	- 2
\cline { 2 - 5 }
\cline { 2 - 5 }

\(\boldsymbol { P }\)	\(x\)	\(y\)	\(\boldsymbol { Z }\)	\(\boldsymbol { s }\)	\(\boldsymbol { t }\)	\(\boldsymbol { u }\)	\(v\)	value
1	0	-12	0	5	-3	0	0	37
0	1	-8	0	1	2	0	0	16
0	0	4	0	0	3	0	1	20
0	0	2	0	-3	2	1	0	14
0	0	1	1	2	5	0	0	8

$\boldsymbol { P }$

$\boldsymbol { x }$

$\boldsymbol { y }$

$\boldsymbol { t }$