Questions D1 - A-Level Maths

AQA D1 2015 June Q5

5 The network shows the paths mown through a wildflower meadow so that visitors can use these paths to enjoy the flowers. The lengths of the paths are shown in metres.
\includegraphics[max width=\textwidth, alt={}, center]{f5890e58-38c3-413c-8762-6f80ce6dcec7-10_1097_1603_413_214} The total length of all the paths is 1400 m .
The mower is kept in a shed at $A$. The groundskeeper must mow all the paths and return the mower to its shed.

Find the length of an optimal Chinese postman route starting and finishing at $A$.
State the number of times that the mower, following the optimal route, will pass through:
1. $C$;
2. $D$.

AQA D1 2015 June Q6

1 marks

6 The network shows the roads linking a warehouse at $A$ and five shops, $B , C , D , E$ and $F$. The numbers on the edges show the lengths, in miles, of the roads. A delivery van leaves the warehouse, delivers to each of the shops and returns to the warehouse.
\includegraphics[max width=\textwidth, alt={}, center]{f5890e58-38c3-413c-8762-6f80ce6dcec7-12_1241_1239_484_402}

Complete the table, on the page opposite, showing the shortest distances between the vertices.
1. Find the total distance travelled if the van follows the cycle $A E F B C D A$.
2. Explain why your answer to part (b)(i) provides an upper bound for the minimum journey length.
Use the nearest neighbour algorithm starting from $D$ to find a second upper bound.
By deleting $A$, find a lower bound for the minimum journey length.
Given that the minimum journey length is $T$, write down the best inequality for $T$ that can be obtained from your answers to parts (b), (c) and (d).
[0pt] [1 mark] \section*{Answer space for question 6} REFERENCE

	$\boldsymbol { A }$	$\boldsymbol { B }$	$\boldsymbol { C }$	$\boldsymbol { D }$	$\boldsymbol { E }$	$\boldsymbol { F }$
$\boldsymbol { A }$	-	7
$\boldsymbol { B }$	7	-	5
$\boldsymbol { C }$		5	-	4
$\boldsymbol { D }$			4	-	6
$\boldsymbol { E }$				6	-	10
$\boldsymbol { F }$					10	-

AQA D1 2015 June Q7

2 marks

7

A simple connected graph has 4 edges and $m$ vertices. State the possible values of $m$.
A simple connected graph has $n$ edges and 4 vertices. State the possible values of $n$.
A simple connected graph, $G$, has 5 vertices and is Eulerian but not Hamiltonian. Draw a possible graph $G$.
[0pt] [2 marks]

AQA D1 2015 June Q8

8 A student is tracing the following algorithm.
\includegraphics[max width=\textwidth, alt={}, center]{f5890e58-38c3-413c-8762-6f80ce6dcec7-18_1431_955_372_539}

Trace the algorithm illustrated in the flowchart for the case where the input value of $N$ is 5 .
Explain the role of $N$ in the algorithm.

AQA D1 2015 June Q9

10 marks

9 A company producing chicken food makes three products, Basic, Premium and Supreme, from wheat, maize and barley. A tonne $( 1000 \mathrm {~kg} )$ of Basic uses 400 kg of wheat, 200 kg of maize and 400 kg of barley.
A tonne of Premium uses 400 kg of wheat, 500 kg of maize and 100 kg of barley.
A tonne of Supreme uses 600 kg of wheat, 200 kg of maize and 200 kg of barley.
The company has 130 tonnes of wheat, 70 tonnes of maize and 72 tonnes of barley available. The company must make at least 75 tonnes of Supreme.
The company makes $\pounds 50$ profit per tonne of Basic, $\pounds 100$ per tonne of Premium and $\pounds 150$ per tonne of Supreme. They plan to make $x$ tonnes of Basic, $y$ tonnes of Premium and $z$ tonnes of Supreme.

Write down four inequalities representing the constraints (in addition to $x , y \geqslant 0$ ).
[0pt] [4 marks]
The company want exactly half the production to be Supreme. Show that the constraints in part (a) become $$\begin{aligned} x + y & \leqslant 130
4 x + 7 y & \leqslant 700
2 x + y & \leqslant 240
x + y & \geqslant 75
x & \geqslant 0
y & \geqslant 0 \end{aligned}$$
On the grid opposite, illustrate all the constraints and label the feasible region.
Write an expression for $P$, the profit for the whole production, in terms of $x$ and $y$ only.
[0pt] [2 marks]
1. By drawing an objective line on your graph, or otherwise, find the values of $x$ and $y$ which give the maximum profit.
  [0pt] [2 marks]
2. State the maximum profit and the amount of each product that must be made.
  [0pt] [2 marks] \section*{Answer space for question 9}
  \includegraphics[max width=\textwidth, alt={}]{f5890e58-38c3-413c-8762-6f80ce6dcec7-21_1349_1728_310_148}
  QUESTION
  PART
  REFERENCE
  \includegraphics[max width=\textwidth, alt={}, center]{f5890e58-38c3-413c-8762-6f80ce6dcec7-24_2488_1728_219_141}

AQA D1 2016 June Q1

3 marks

1 Alfred has bought six different chocolate bars. He wants to give a chocolate bar to each of his six friends. The table gives the names of the friends and indicates which of Alfred's six chocolate bars they like.

AQA D1 2016 June Q2

4 marks

2

Use a shuttle sort to rearrange into alphabetical order the following list of names:
Rob, Eve, Meg, lan, Xavi
Show the list at the end of each pass.
A list of ten numbers is sorted into ascending order, using a shuttle sort.
1. How many passes are needed?
2. Give the maximum number of comparisons needed in the sixth pass.
3. Given that the list is initially in descending order, find the total number of swaps needed.
  [0pt] [4 marks]

AQA D1 2016 June Q3

3 The network below shows vertices $A , B , C , D$ and $E$. The number on each edge shows the distance between vertices.
\includegraphics[max width=\textwidth, alt={}, center]{fb95068f-f76d-492a-b385-bce17b26ae30-06_563_736_402_651}

1. In the case where $x = 8$, use Kruskal's algorithm to find a minimum spanning tree for the network. Write down the order in which you add edges to your minimum spanning tree.
2. Draw your minimum spanning tree.
3. Write down the length of your minimum spanning tree.
Alice draws the same network but changes the value of $x$. She correctly uses Kruskal's algorithm and edge $C D$ is included in her minimum spanning tree.
1. Explain why $x$ cannot be equal to 7 .
2. Write down an inequality for $x$.

AQA D1 2016 June Q4

4 Amal delivers free advertiser magazines to all the houses in his village. The network shows the roads in his village. The number on each road shows the time, in minutes, that Amal takes to walk along that road.
\includegraphics[max width=\textwidth, alt={}, center]{fb95068f-f76d-492a-b385-bce17b26ae30-08_846_1264_445_388}

Amal starts his delivery round from his house, at vertex $A$. He must walk along each road at least once.
1. Find the length of an optimal Chinese postman route around the village, starting and finishing at Amal's house.
2. State the number of times that Amal passes his friend Dipak's house, at vertex $D$.
Dipak offers to deliver the magazines while Amal is away on holiday. Dipak must walk along each road at least once. Assume that Dipak takes the same length of time as Amal to walk along each road.
1. Dipak can start his journey at any vertex and finish his journey at any vertex. Find the length of time for an optimal route for Dipak.
2. State the vertices at which Dipak could finish, in order to achieve his optimal route.
1. Find the length of time for an optimal route for Dipak, if, instead, he wants to finish at his house, at vertex $D$, and can start his journey at any other vertex.
2. State the start vertex.

AQA D1 2016 June Q5

5 A fair comes to town one year and sets up its rides in two large fields that are separated by a river. The diagram shows the ten main rides, at $A , B , C , \ldots , J$. The numbers on the edges represent the times, in minutes, it takes to walk between pairs of rides. A footbridge connects the rides at $D$ and $F$.

1. Use Dijkstra's algorithm on the diagram below to find the minimum time to walk from $A$ to each of the other main rides.
2. Write down the route corresponding to the minimum time to walk from $A$ to $G$.
The following year, the fair returns. In addition to the information shown on the diagram, another footbridge has been built to connect the rides at $E$ and $G$. This reduces the time taken to travel from $A$ to $G$, but the time taken to travel from $A$ to $J$ is not reduced. The time to walk across the footbridge from $E$ to $G$ is $x$ minutes, where $x$ is an integer. Find two inequalities for $x$ and hence state the value of $x$. \section*{Answer space for question 5}
1. \includegraphics[max width=\textwidth, alt={}, center]{fb95068f-f76d-492a-b385-bce17b26ae30-12_659_1591_1692_223}

AQA D1 2016 June Q6

6 A connected graph is semi-Eulerian if exactly two of its vertices are of odd degree.

A graph is drawn with 4 vertices and 7 edges. What is the sum of the degrees of the vertices?
Draw a simple semi-Eulerian graph with exactly 5 vertices and 5 edges, in which exactly one of the vertices has degree 4 .
Draw a simple semi-Eulerian graph with exactly 5 vertices that is also a tree.
A simple graph has 6 vertices. The graph has two vertices of degree 5 . Explain why the graph can have no vertex of degree 1.

AQA D1 2016 June Q7

7 A company operates a steam railway between six stations. The minimum cost (in euros) of travelling between pairs of stations is shown in the table below.

On Figure 1 below, use Prim's algorithm, starting from $P$, to find a minimum spanning tree for the graph connecting $P , Q , R , S , T$ and $U$. State clearly the order in which you select the vertices and draw your minimum spanning tree. \section*{Question 7 continues on page 20} \begin{table}[h]

\captionsetup{labelformat=empty} \caption{Figure 1}

	$\boldsymbol { P }$	$\boldsymbol { Q }$	$\boldsymbol { R }$	$\boldsymbol { S }$	$\boldsymbol { T }$	$\boldsymbol { U }$
$\boldsymbol { P }$	-	14	7	11	6	12
$\boldsymbol { Q }$	14	-	8	10	9	10
$\boldsymbol { R }$	7	8	-	12	13	15
$\boldsymbol { S }$	11	10	12	-	5	11
$\boldsymbol { T }$	6	9	13	5	-	10
$\boldsymbol { U }$	12	10	15	11	10	-

\end{table}

Another station, $V$, is opened. The minimum costs (in euros) of travelling to and from $V$ to each of the other stations are added to the table in part (a), as shown in Figure 2(i) below. Further copies of this table are shown in Figure 2(ii). Arjen is on holiday and he plans to visit each station. He intends to board a train at $V$ and visit all the other stations, once only, before returning to $V$.

By first removing $V$, obtain a lower bound for the minimum travelling cost of Arjen's tour. (You may use Figure 2(i) for your working.)
Use the nearest neighbour algorithm twice, starting each time from $V$, to find two different upper bounds for the minimum cost of Arjen's tour. State, with a reason, which of your two answers gives the better upper bound. (You may use Figure 2(ii) for your working.)

Hence find an optimal tour of the seven stations. Explain how you know that it is optimal. Answer space for question 7(b) \begin{table}[h]

\captionsetup{labelformat=empty} \caption{Figure 2(ii)}

	$\boldsymbol { P }$	$Q$	$\boldsymbol { R }$	$\boldsymbol { S }$	$T$	$\boldsymbol { U }$	$V$
$\boldsymbol { P }$	-	14	7	11	6	12	15
$Q$	14	-	8	10	9	10	18
$\boldsymbol { R }$	7	8	-	12	13	15	14
$\boldsymbol { S }$	11	10	12	-	5	11	14
$T$	6	9	13	5	-	10	17
$\boldsymbol { U }$	12	10	15	11	10	-	12
$V$	15	18	14	14	17	12	-

\end{table}

AQA D1 2016 June Q8

8 Nerys runs a cake shop. In November and December she sells Christmas hampers. She makes up the hampers herself, in two sizes: Luxury and Special. Each day, Nerys prepares $x$ Luxury hampers and $y$ Special hampers.
It takes Nerys 10 minutes to prepare a Luxury hamper and 15 minutes to prepare a Special hamper. She has 6 hours available, each day, for preparing hampers. From past experience, Nerys knows that each day:

she will need to prepare at least 5 hampers of each size
she will prepare at most a total of 32 hampers
she will prepare at least twice as many Luxury hampers as Special hampers.

Each Luxury hamper that Nerys prepares makes her a profit of $\pounds 15$; each Special hamper makes a profit of $\pounds 20$. Nerys wishes to maximise her daily profit, $\pounds P$.

Show that $x$ and $y$ must satisfy the inequality $2 x + 3 y \leqslant 72$.
In addition to $x \geqslant 5$ and $y \geqslant 5$, write down two more inequalities that model the constraints above.
On the grid opposite draw a suitable diagram to enable this problem to be solved graphically. Indicate a feasible region and the direction of an objective line.
1. Use your diagram to find the number of each type of hamper that Nerys should prepare each day to achieve a maximum profit.
2. Calculate this profit.
  \includegraphics[max width=\textwidth, alt={}]{fb95068f-f76d-492a-b385-bce17b26ae30-27_2490_1719_217_150}
  \section*{DO NOT WRITE ON THIS PAGE ANSWER IN THE SPACES PROVIDED}

OCR MEI D1 2007 January Q1

1 Each of the following symbols consists of boundaries enclosing regions.
(0) 1

OCR MEI D1 2007 January Q3

3
$\triangle$
5
(5) 7
(8)
(O) The symbol representing zero has three regions, the outside, that between the two boundaries and the inside. To classify the symbols a graph is produced for each one. The graph has a vertex for each region, with arcs connecting regions which share a boundary. Thus the graph for
\includegraphics[max width=\textwidth, alt={}, center]{a56f79f4-3b71-4ae3-a88b-5a88b0197433-2_110_81_900_767}
is
\includegraphics[max width=\textwidth, alt={}, center]{a56f79f4-3b71-4ae3-a88b-5a88b0197433-2_49_350_959_959}

Produce the graph for the symbol
Give two symbols each having the graph
\includegraphics[max width=\textwidth, alt={}, center]{a56f79f4-3b71-4ae3-a88b-5a88b0197433-2_52_199_1187_1030}
Produce the graph for the symbol .
Produce a single graph for the composite symbol
\includegraphics[max width=\textwidth, alt={}, center]{a56f79f4-3b71-4ae3-a88b-5a88b0197433-2_112_158_1398_1165}
Give the name of a connected graph with $n$ nodes and $n - 1 \operatorname { arcs }$. 2 The following algorithm is a version of bubble sort.
Step 1 Store the values to be sorted in locations $\mathrm { L } ( 1 ) , \mathrm { L } ( 2 ) , \ldots , \mathrm { L } ( \mathrm { n } )$ and set i to be the number, n , of values to be sorted. Step $2 \quad$ Set $\mathrm { j } = 1$.
Step 3 Compare the values in locations $\mathrm { L } ( \mathrm { j } )$ and $\mathrm { L } ( \mathrm { j } + 1 )$ and swap them if that $\mathrm { in } \mathrm { L } ( \mathrm { j } )$ is larger than that in $\mathrm { L } ( \mathrm { j } + 1 )$. Step 4 Add 1 to j.
Step 5 If j is less than i then go to step 3.
Step 5 Write out the current list, $\mathrm { L } ( 1 ) , \mathrm { L } ( 2 ) , \ldots , \mathrm { L } ( \mathrm { n } )$.
Step 6 Subtract 1 from i .
Step 7 If i is larger than 1 then go to step 2.
Step 8 Stop.
Apply this algorithm to sort the following list. $$\begin{array} { l l l l l } 109 & 32 & 3 & 523 & 58 \end{array} .$$ Count the number of comparisons and the number of swaps which you make in applying the algorithm.
Put the five values into the order which maximises the number of swaps made in applying the algorithm, and give that number.
Bubble sort has quadratic complexity. Using bubble sort it takes a computer 1.5 seconds to sort a list of 1000 values. Approximately how long would it take to sort a list of 100000 values? (Give your answer in hours and minutes.) 3 A bag contains five pieces of paper labelled A, B, C, D and E. One piece is drawn at random from the bag. If the piece is labelled with a vowel (A or E) then the process stops. Otherwise the piece of paper is replaced, the bag is shaken, and the process is repeated. You are to simulate this process to estimate the mean number of draws needed to get a vowel.
Show how to use single digit random numbers to simulate the process efficiently. You need to describe exactly how your simulation will work.
Use the random numbers in your answer book to run your simulation 5 times, recording your results.
From your results compute an estimate of the mean number of draws needed to get a vowel.
State how you could produce a more accurate estimate. Section B (48 marks)

OCR MEI D1 2007 January Q5

5
(5) 7
(8)
(O) The symbol representing zero has three regions, the outside, that between the two boundaries and the inside. To classify the symbols a graph is produced for each one. The graph has a vertex for each region, with arcs connecting regions which share a boundary. Thus the graph for
\includegraphics[max width=\textwidth, alt={}, center]{a56f79f4-3b71-4ae3-a88b-5a88b0197433-2_110_81_900_767}
is
\includegraphics[max width=\textwidth, alt={}, center]{a56f79f4-3b71-4ae3-a88b-5a88b0197433-2_49_350_959_959}

Produce the graph for the symbol
Give two symbols each having the graph
\includegraphics[max width=\textwidth, alt={}, center]{a56f79f4-3b71-4ae3-a88b-5a88b0197433-2_52_199_1187_1030}
Produce the graph for the symbol .
Produce a single graph for the composite symbol
\includegraphics[max width=\textwidth, alt={}, center]{a56f79f4-3b71-4ae3-a88b-5a88b0197433-2_112_158_1398_1165}
Give the name of a connected graph with $n$ nodes and $n - 1 \operatorname { arcs }$. 2 The following algorithm is a version of bubble sort.
Step 1 Store the values to be sorted in locations $\mathrm { L } ( 1 ) , \mathrm { L } ( 2 ) , \ldots , \mathrm { L } ( \mathrm { n } )$ and set i to be the number, n , of values to be sorted. Step $2 \quad$ Set $\mathrm { j } = 1$.
Step 3 Compare the values in locations $\mathrm { L } ( \mathrm { j } )$ and $\mathrm { L } ( \mathrm { j } + 1 )$ and swap them if that $\mathrm { in } \mathrm { L } ( \mathrm { j } )$ is larger than that in $\mathrm { L } ( \mathrm { j } + 1 )$. Step 4 Add 1 to j.
Step 5 If j is less than i then go to step 3.
Step 5 Write out the current list, $\mathrm { L } ( 1 ) , \mathrm { L } ( 2 ) , \ldots , \mathrm { L } ( \mathrm { n } )$.
Step 6 Subtract 1 from i .
Step 7 If i is larger than 1 then go to step 2.
Step 8 Stop.
Apply this algorithm to sort the following list. $$\begin{array} { l l l l l } 109 & 32 & 3 & 523 & 58 \end{array} .$$ Count the number of comparisons and the number of swaps which you make in applying the algorithm.
Put the five values into the order which maximises the number of swaps made in applying the algorithm, and give that number.
Bubble sort has quadratic complexity. Using bubble sort it takes a computer 1.5 seconds to sort a list of 1000 values. Approximately how long would it take to sort a list of 100000 values? (Give your answer in hours and minutes.) 3 A bag contains five pieces of paper labelled A, B, C, D and E. One piece is drawn at random from the bag. If the piece is labelled with a vowel (A or E) then the process stops. Otherwise the piece of paper is replaced, the bag is shaken, and the process is repeated. You are to simulate this process to estimate the mean number of draws needed to get a vowel.
Show how to use single digit random numbers to simulate the process efficiently. You need to describe exactly how your simulation will work.
Use the random numbers in your answer book to run your simulation 5 times, recording your results.
From your results compute an estimate of the mean number of draws needed to get a vowel.

State how you could produce a more accurate estimate. Section B (48 marks)
4 Cassi is managing the building of a house. The table shows the major activities that are involved, their durations and their precedences.

Activity		Duration (days)	Immediate predecessors
A	Build concrete frame	10	-
B	Lay bricks	7	A
C	Lay roof tiles	10	A
D	First fit electrics	5	B
E	First fit plumbing	4	B
F	Plastering	6	C, D, E
G	Second fit electrics	3	F
H	Second fit plumbing	2	F
I	Tiling	10	G, H
J	Fit sanitary ware	2	H
K	Fit windows and doors	5	I

Draw an activity-on-arc network to represent this information.
Find the early time and the late time for each event. Give the project duration and list the critical activities.
Calculate total and independent floats for each non-critical activity. Cassi's clients wish to take delivery in 42 days. Some durations can be reduced, at extra cost, to achieve this.
- The tiler will finish activity I in 9 days for an extra $\pounds 250$, or in 8 days for an extra $\pounds 500$.
- The bricklayer will cut his total of 7 days on activity B by up to 3 days at an extra cost of $\pounds 350$ per day.
- The electrician could be paid $\pounds 300$ more to cut a day off activity D, or $\pounds 600$ more to cut two days.
- What is the cheapest way in which Cassi can get the house built in 42 days?
5 Leone is designing her new garden. She wants to have at least $1000 \mathrm {~m} ^ { 2 }$, split between lawn and flower beds. Initial costs are $\pounds 0.80$ per $\mathrm { m } ^ { 2 }$ for lawn and $\pounds 0.40$ per $\mathrm { m } ^ { 2 }$ for flowerbeds. Leone's budget is $\pounds 500$.
Leone prefers flower beds to lawn, and she wants the area for flower beds to be at least twice the area for lawn. However, she wants to have at least $200 \mathrm {~m} ^ { 2 }$ of lawn. Maintenance costs each year are $\pounds 0.15$ per $\mathrm { m } ^ { 2 }$ for lawn and $\pounds 0.25$ per $\mathrm { m } ^ { 2 }$ for flower beds. Leone wants to minimize the maintenance costs of her garden.
Formulate Leone's problem as a linear programming problem.
Produce a graph to illustrate the inequalities.
Solve Leone's problem.
If Leone had more than $\pounds 500$ available initially, how much extra could she spend to minimize maintenance costs?

OCR MEI D1 2007 January Q7

7
(8)
(O) The symbol representing zero has three regions, the outside, that between the two boundaries and the inside. To classify the symbols a graph is produced for each one. The graph has a vertex for each region, with arcs connecting regions which share a boundary. Thus the graph for
\includegraphics[max width=\textwidth, alt={}, center]{a56f79f4-3b71-4ae3-a88b-5a88b0197433-2_110_81_900_767}
is
\includegraphics[max width=\textwidth, alt={}, center]{a56f79f4-3b71-4ae3-a88b-5a88b0197433-2_49_350_959_959}

Produce the graph for the symbol
Give two symbols each having the graph
\includegraphics[max width=\textwidth, alt={}, center]{a56f79f4-3b71-4ae3-a88b-5a88b0197433-2_52_199_1187_1030}
Produce the graph for the symbol .
Produce a single graph for the composite symbol
\includegraphics[max width=\textwidth, alt={}, center]{a56f79f4-3b71-4ae3-a88b-5a88b0197433-2_112_158_1398_1165}
Give the name of a connected graph with $n$ nodes and $n - 1 \operatorname { arcs }$. 2 The following algorithm is a version of bubble sort.
Step 1 Store the values to be sorted in locations $\mathrm { L } ( 1 ) , \mathrm { L } ( 2 ) , \ldots , \mathrm { L } ( \mathrm { n } )$ and set i to be the number, n , of values to be sorted. Step $2 \quad$ Set $\mathrm { j } = 1$.
Step 3 Compare the values in locations $\mathrm { L } ( \mathrm { j } )$ and $\mathrm { L } ( \mathrm { j } + 1 )$ and swap them if that $\mathrm { in } \mathrm { L } ( \mathrm { j } )$ is larger than that in $\mathrm { L } ( \mathrm { j } + 1 )$. Step 4 Add 1 to j.
Step 5 If j is less than i then go to step 3.
Step 5 Write out the current list, $\mathrm { L } ( 1 ) , \mathrm { L } ( 2 ) , \ldots , \mathrm { L } ( \mathrm { n } )$.
Step 6 Subtract 1 from i .
Step 7 If i is larger than 1 then go to step 2.
Step 8 Stop.
Apply this algorithm to sort the following list. $$\begin{array} { l l l l l } 109 & 32 & 3 & 523 & 58 \end{array} .$$ Count the number of comparisons and the number of swaps which you make in applying the algorithm.
Put the five values into the order which maximises the number of swaps made in applying the algorithm, and give that number.
Bubble sort has quadratic complexity. Using bubble sort it takes a computer 1.5 seconds to sort a list of 1000 values. Approximately how long would it take to sort a list of 100000 values? (Give your answer in hours and minutes.) 3 A bag contains five pieces of paper labelled A, B, C, D and E. One piece is drawn at random from the bag. If the piece is labelled with a vowel (A or E) then the process stops. Otherwise the piece of paper is replaced, the bag is shaken, and the process is repeated. You are to simulate this process to estimate the mean number of draws needed to get a vowel.
Show how to use single digit random numbers to simulate the process efficiently. You need to describe exactly how your simulation will work.
Use the random numbers in your answer book to run your simulation 5 times, recording your results.
From your results compute an estimate of the mean number of draws needed to get a vowel.

State how you could produce a more accurate estimate. Section B (48 marks)
4 Cassi is managing the building of a house. The table shows the major activities that are involved, their durations and their precedences.

Activity		Duration (days)	Immediate predecessors
A	Build concrete frame	10	-
B	Lay bricks	7	A
C	Lay roof tiles	10	A
D	First fit electrics	5	B
E	First fit plumbing	4	B
F	Plastering	6	C, D, E
G	Second fit electrics	3	F
H	Second fit plumbing	2	F
I	Tiling	10	G, H
J	Fit sanitary ware	2	H
K	Fit windows and doors	5	I

Draw an activity-on-arc network to represent this information.
Find the early time and the late time for each event. Give the project duration and list the critical activities.
Calculate total and independent floats for each non-critical activity. Cassi's clients wish to take delivery in 42 days. Some durations can be reduced, at extra cost, to achieve this.
- The tiler will finish activity I in 9 days for an extra $\pounds 250$, or in 8 days for an extra $\pounds 500$.
- The bricklayer will cut his total of 7 days on activity B by up to 3 days at an extra cost of $\pounds 350$ per day.
- The electrician could be paid $\pounds 300$ more to cut a day off activity D, or $\pounds 600$ more to cut two days.
- What is the cheapest way in which Cassi can get the house built in 42 days?
5 Leone is designing her new garden. She wants to have at least $1000 \mathrm {~m} ^ { 2 }$, split between lawn and flower beds. Initial costs are $\pounds 0.80$ per $\mathrm { m } ^ { 2 }$ for lawn and $\pounds 0.40$ per $\mathrm { m } ^ { 2 }$ for flowerbeds. Leone's budget is $\pounds 500$.
Leone prefers flower beds to lawn, and she wants the area for flower beds to be at least twice the area for lawn. However, she wants to have at least $200 \mathrm {~m} ^ { 2 }$ of lawn. Maintenance costs each year are $\pounds 0.15$ per $\mathrm { m } ^ { 2 }$ for lawn and $\pounds 0.25$ per $\mathrm { m } ^ { 2 }$ for flower beds. Leone wants to minimize the maintenance costs of her garden.
Formulate Leone's problem as a linear programming problem.
Produce a graph to illustrate the inequalities.
Solve Leone's problem.
If Leone had more than $\pounds 500$ available initially, how much extra could she spend to minimize maintenance costs? 6 In a factory a network of pipes connects 6 vats, A, B, C, D, E and F. Two separate connectors need to be chosen from the network The table shows the lengths of pipes (metres) connecting the 6 vats.
A B C D E F
A - 7 - - 12 -
B 7 - 5 3 6 6
C - 5 - 8 4 7
D - 3 8 - 1 5
E 12 6 4 1 - 7
F - 6 7 5 7 -
Use Kruskal's algorithm to find a minimum connector. Show the order in which you select pipes, draw your connector and give its total length.
Produce a new table excluding the pipes which you selected in part (i). Use the tabular form of Prim's algorithm to find a second minimum connector from this reduced set of pipes. Show your working, draw your connector and give its total length.
The factory manager prefers the following pair of connectors:
$\{ \mathrm { AB } , \mathrm { BC } , \mathrm { BD } , \mathrm { BE } , \mathrm { BF } \}$ and $\{ \mathrm { AE } , \mathrm { BF } , \mathrm { CE } , \mathrm { DE } , \mathrm { DF } \}$.
Give two possible reasons for this preference.

OCR MEI D1 2013 June Q1

1 The adjacency graph for a map has a vertex for each country. Two vertices are connected by an arc if the corresponding countries share a border.

Draw the adjacency graph for the following map of four countries. The graph is planar and you should draw it with no arcs crossing.
\includegraphics[max width=\textwidth, alt={}, center]{e528b905-7419-44b6-b700-4c04ad96c816-2_531_1486_561_292}
Number the regions of your planar graph, including the outside region. Regarding the graph as a map, draw its adjacency graph.
Repeat parts (i) and (ii) for the following map.
\includegraphics[max width=\textwidth, alt={}, center]{e528b905-7419-44b6-b700-4c04ad96c816-2_533_1484_1361_294}

OCR MEI D1 2013 June Q2

2 The instructions labelled 1 to 7 describe the steps of a sorting algorithm applied to a list of six numbers.
1 Let $i$ equal 1.
2 Repeat lines 3 to 7, stopping when $i$ becomes 6 .

OCR MEI D1 2013 June Q3

3 Let $j$ equal 1.

OCR MEI D1 2013 June Q4

4 Repeat lines 5 and 6 , until $j$ becomes $7 - i$.

OCR MEI D1 2013 June Q5

5 If the $j$ th number in the list is bigger than the $( j + 1 )$ th, then swap them.

OCR MEI D1 2013 June Q6

6 Let the new value of $j$ be $j + 1$.

OCR MEI D1 2013 June Q7

7 Let the new value of $i$ be $i + 1$.

Apply the sorting algorithm to the list of numbers shown below. Record in the table provided the state of the list after each pass. Record the number of comparisons and the number of swaps that you make in each pass. (The result of the first pass has already been recorded.) List: $\begin{array} { l l l l l l } 9 & 11 & 7 & 3 & 13 & 5 \end{array}$
Suppose now that the list is split into two sublists, $\{ 9,11,7 \}$ and $\{ 3,13,5 \}$. The sorting algorithm is adapted to apply to three numbers, and is applied to each sublist separately. This gives $\{ 7,9,11 \}$ and $\{ 3,5,13 \}$. How many comparisons and swaps does this need?
How many further swaps would the original algorithm need to sort the revised list $\{ 7,9,11,3,5,13 \}$ into increasing order? 3 The network below represents a number of villages together with connecting roads. The numbers on the arcs represent distances (in miles).
\includegraphics[max width=\textwidth, alt={}, center]{e528b905-7419-44b6-b700-4c04ad96c816-3_684_785_1612_625}
Use Dijkstra's algorithm to find the shortest routes from A to each of the other villages. Give these shortest routes and the corresponding distances. Traffic in the area travels at 30 mph . An accident delays all traffic passing through C by 20 minutes.
Describe how the network can be adapted and used to find the fastest journey time from A to F .

	\(\boldsymbol { A }\)	\(\boldsymbol { B }\)	\(\boldsymbol { C }\)	\(\boldsymbol { D }\)	\(\boldsymbol { E }\)	\(\boldsymbol { F }\)
\(\boldsymbol { A }\)	-	7
\(\boldsymbol { B }\)	7	-	5
\(\boldsymbol { C }\)		5	-	4
\(\boldsymbol { D }\)			4	-	6
\(\boldsymbol { E }\)				6	-	10
\(\boldsymbol { F }\)					10	-

Questions D1 (899 questions)

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