Questions FD1 AS - A-Level Maths

1. Use Dijkstra's algorithm to find the shortest time needed to travel from A to H .
2. State the quickest route. For a network with $n$ vertices, Dijkstra's algorithm has order $n ^ { 2 }$
If it takes 1.5 seconds to run the algorithm when $n = 250$, calculate approximately how long it will take, in seconds, to run the algorithm when $n = 9500$. You should make your method and working clear.
Explain why your answer to part (b) is only an approximation.

Edexcel FD1 AS 2018 June Q2

2. A simply connected graph is a connected graph in which any two vertices are directly connected by at most one arc and no vertex is directly connected to itself.

Given that a simply connected graph has exactly four vertices,
1. write down the minimum number of arcs it can have,
2. write down the maximum number of arcs it can have.
1. Draw a simply connected graph that has exactly four vertices and exactly five arcs.
2. State, with justification, whether your graph is Eulerian, semi-Eulerian or neither.
By considering the orders of the vertices, explain why there is only one simply connected graph with exactly four vertices and exactly five arcs.

Edexcel FD1 AS 2018 June Q3

Activity	Time taken (days)	Immediately preceding activities
A	5	-
B	8	-
C	4	-
D	14	A
E	10	A
F	3	B, C, E
G	7	C
H	5	D, F, G
I	7	H
J	9	H

The table above shows the activities required for the completion of a building project. For each activity, the table shows the time it takes, in days, and the immediately preceding activities. Each activity requires one worker. The project is to be completed in the shortest possible time. \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{3e853c6d-e90e-4a09-b990-1c2c146b54e1-4_486_1161_1194_551} \captionsetup{labelformat=empty} \caption{Figure 2}

\end{figure} Figure 2 shows a partially completed activity network used to model the project. The activities are represented by the arcs and the number in brackets on each arc is the time taken, in days, to complete the corresponding activity.

Add the missing activities and necessary dummies to Diagram 1 in the answer book.
Complete Diagram 1 in the answer book to show the early event times and the late event times.
State the critical activities. At the beginning of the project it is decided that activity G is no longer required.
Explain what effect, if any, this will have on
1. the shortest completion time of the project if activity G is no longer required,
2. the timing of the remaining activities.

Edexcel FD1 AS 2018 June Q4

4. The manager of a factory is planning the production schedule for the next three weeks for a range of cabinets. The following constraints apply to the production schedule.

The total number of cabinets produced in week 3 cannot be fewer than the total number produced in weeks 1 and 2
At most twice as many cabinets must be produced in week 3 as in week 2
The number of cabinets produced in weeks 2 and 3 must, in total, be at most 125

The production cost for each cabinet produced in weeks 1,2 and 3 is $\pounds 250 , \pounds 275$ and $\pounds 200$ respectively.
The factory manager decides to formulate a linear programming problem to find a production schedule that minimises the total cost of production. The objective is to minimise $250 x + 275 y + 200 z$

Explain what the variables $x , y$ and $z$ represent.
Write down the constraints of the linear programming problem in terms of $x , y$ and $z$. Due to demand, exactly 150 cabinets must be produced during these three weeks. This reduces the constraints to $$\begin{gathered} x + y \leqslant 75
x + 3 y \geqslant 150
x \geqslant 25
y \geqslant 0 \end{gathered}$$ which are shown in Diagram 1 in the answer book.
Given that the manager does not want any cabinets left unfinished at the end of a week,
1. use a graphical approach to solve the linear programming problem and hence determine the production schedule which minimises the cost of production. You should make your method and working clear.
2. Find the minimum total cost of the production schedule.

Edexcel FD1 AS 2021 June Q2

2. \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{d3f5dcb4-3e23-4d78-965a-a1acaac13819-03_885_1493_226_287} \captionsetup{labelformat=empty} \caption{Figure 1}

\end{figure} A project is modelled by the activity network shown in Figure 1. The activities are represented by the arcs. The number in brackets on each arc gives the time, in hours, to complete the corresponding activity. The exact duration, $x$, of activity N is unknown, but it is given that $5 < x < 10$ Each activity requires one worker. The project is to be completed in the shortest possible time.

Complete the precedence table in the answer book.
Complete Diagram 1 in the answer book to show the early event times and the late event times.
List the critical activities. It is given that activity J can be delayed by up to 4 hours without affecting the shortest possible completion time of the project.
Determine the value of $x$. You must make the numbers used in your calculation clear.
Draw a cascade chart for this project on Grid 1 in the answer book.

Edexcel FD1 AS 2021 June Q3

3. Donald plans to bake and sell cakes. The three types of cake that he can bake are brownies, flapjacks and muffins. Donald decides to bake 48 brownies and muffins in total.
Donald decides to bake at least 5 brownies for every 3 flapjacks.
At most $40 \%$ of the cakes will be muffins.
Donald has enough ingredients to bake 60 brownies or 45 flapjacks or 35 muffins.
Donald plans to sell each brownie for $\pounds 1.50$, each flapjack for $\pounds 1$ and each muffin for $\pounds 1.25$ He wants to maximise the total income from selling the cakes. Let $x$ represent the number of brownies, let $y$ represent the number of flapjacks and let $z$ represent the number of muffins that Donald will bake. Formulate this as a linear programming problem in $x$ and $y$ only, stating the objective function and listing the constraints as simplified inequalities with integer coefficients. You should not attempt to solve the problem.

Edexcel FD1 AS 2021 June Q4

4. \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{d3f5dcb4-3e23-4d78-965a-a1acaac13819-05_712_1433_223_315} \captionsetup{labelformat=empty} \caption{Figure 2}

\end{figure} Dijkstra's algorithm has been applied to the network in Figure 2.
A working value has only been replaced at a node if the new working value is smaller.

State the length of the shortest path from A to G .
Complete the table in the answer book giving the weight of each arc listed. (Note that arc CE and arc EF are not in the table.)
State the shortest path from A to G. It is now given that
- when Prim's algorithm, starting from A, is applied to the network, the order in which the arcs are added to the tree is $\mathrm { AB } , \mathrm { BC } , \mathrm { CD } , \mathrm { CE } , \mathrm { EF }$ and FG
- the weight of the corresponding minimum spanning tree is 80
- the shortest path from A to F via E has weight 67
- Determine the weight of arc CE and the weight of arc EF , making your reasoning clear.

Edexcel FD1 AS 2022 June Q1

55534345928373452334247

The list of eleven numbers shown above is to be sorted into ascending order.

Carry out a quick sort to produce the sorted list. You should show the result of each pass and identify your pivots clearly.
(4) \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{c8134d3b-71cb-4b92-ac54-81a4ff8f3011-03_814_1545_614_260} \captionsetup{labelformat=empty} \caption{Figure 1}
\end{figure}
Use Kruskal's algorithm to find the minimum spanning tree for the network in Figure 1. You should list the arcs in the order in which you consider them. For each arc, state whether or not you are adding it to your minimum spanning tree.
1. Draw the minimum spanning tree on Diagram 1 in the answer book.
2. State the total weight of the tree.

Edexcel FD1 AS 2022 June Q2

Activity	Immediately preceding activities
A	-
B	-
C	-
D	-
E	A
F	A, B, C
G	C
H	C
I	E
J	E, F, G
K	D, H

Draw the activity network described in the precedence table above, using activity on arc. Your activity network must contain the minimum number of dummies only.
Explain why it is necessary to draw a dummy from the end of activity A . Every activity shown in the precedence table has the same duration.
State which activity cannot be critical, justifying your answer.

Edexcel FD1 AS 2022 June Q3

3. \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{c8134d3b-71cb-4b92-ac54-81a4ff8f3011-05_702_1479_201_293} \captionsetup{labelformat=empty} \caption{Figure 2}

\end{figure} [The total weight of the network is 120]

Explain what is meant by the term "path".
State, with a reason, whether the network in Figure 2 is Eulerian, semi-Eulerian or neither. Figure 2 represents a network of cycle tracks between eight villages, $\mathrm { A } , \mathrm { B } , \mathrm { C } , \mathrm { D } , \mathrm { E } , \mathrm { F } , \mathrm { G }$ and H . The number on each arc represents the length, in km , of the corresponding track. Samira lives in village A, and wishes to visit her friend, Daisy, who lives in village H.
Use Dijkstra's algorithm to find the shortest path that Samira can take. An extra cycle track of length 9 km is to be added to the network. It will either go directly between C and D or directly between E and G . Daisy plans to cycle along every track in the new network, starting and finishing at H .
Given that the addition of either track CD or track EG will not affect the final values obtained in (c),
use a suitable algorithm to find out which of the two possible extra tracks will give Daisy the shortest route, making your method and working clear. You must
- state which tracks Daisy will repeat in her route
- state the total length of her route

Edexcel FD1 AS 2022 June Q4

4. \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{c8134d3b-71cb-4b92-ac54-81a4ff8f3011-06_1504_1733_210_173} \captionsetup{labelformat=empty} \caption{Figure 3}

\end{figure} Figure 3 shows the constraints of a maximisation linear programming problem in $x$ and $y$, where $x \geqslant 0$ and $y \geqslant 0$. The unshaded area, including its boundaries, forms the feasible region, $R$. An objective line has been drawn and labelled on the graph.

List the constraints as simplified inequalities with integer coefficients. The optimal value of the objective function is 216
1. Calculate the exact coordinates of the optimal vertex.
2. Hence derive the objective function. Given that $x$ represents the number of small flower pots and $y$ represents the number of large flower pots supplied to a customer,
deduce the optimal solution to the problem. TOTAL FOR DECISION MATHEMATICS 1 IS 40 MARKS END

Edexcel FD1 AS 2023 June Q1

1. $$\begin{array} { l l l l l l l l l l l } 67 & 59 & 46 & 71 & 40 & 48 & 53 & 63 & 45 & 54 & 56 \end{array}$$ The list of eleven numbers shown above is to be sorted into descending order.
Carry out a quick sort to produce the sorted list. You should show the result of each pass and identify the pivots clearly.

Edexcel FD1 AS 2023 June Q2

2. \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{9edb5209-4244-4916-b3ee-d77e395e8cab-03_750_1490_262_285} \captionsetup{labelformat=empty} \caption{Figure 1}

\end{figure} A project is modelled by the activity network shown in Figure 1. The activities are represented by the arcs. The number in brackets on each arc gives the time required, in hours, to complete the corresponding activity. The numbers in circles are the event numbers. Each activity requires one worker, and the project is to be completed in the shortest possible time.

Explain the significance of the dummy activity from event 3 to event 4
Complete Diagram 1 in the answer book to show the early event times and the late event times.
State the critical activities.
Calculate a lower bound for the number of workers needed to complete the project in the minimum time. You must show your working.
Draw a Gantt chart for this project on Grid 1 in the answer book.

Edexcel FD1 AS 2023 June Q3

3. \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{9edb5209-4244-4916-b3ee-d77e395e8cab-04_977_1472_259_294} \captionsetup{labelformat=empty} \caption{Figure 2}

\end{figure} Figure 2 represents a network of train tracks. The number on each edge represents the length, in kilometres, of the corresponding track.
Dyfan wishes to travel from A to J via C. Dyfan wishes to minimise the distance they travel. Given that Dijkstra's algorithm is to be applied only once to find Dyfan's route,

explain why the algorithm should begin at C.
Use Dijkstra's algorithm to find the shortest route from A to J via C. State this route and its length.
Use Prim's algorithm, starting at C , to find a minimum spanning tree for the network. You must clearly state the order in which you select the edges of your tree.
State the total length, in km , of the minimum spanning tree.

Edexcel FD1 AS 2023 June Q4

4. \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{9edb5209-4244-4916-b3ee-d77e395e8cab-05_997_1379_260_456} \captionsetup{labelformat=empty} \caption{Figure 3}

\end{figure} Figure 3 shows the constraints of a linear programming problem in $x$ and $y$. The unshaded area, including its boundaries, forms the feasible region, $R$. An objective line has been drawn and labelled on the graph.

State the inequalities that define the feasible region. The maximum value of the objective function is $\frac { 160 } { 3 }$ The minimum value of the objective function is $\frac { 883 } { 41 }$
Determine the objective function, showing your working clearly.

Edexcel FD1 AS 2023 June Q5

5. \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{9edb5209-4244-4916-b3ee-d77e395e8cab-06_873_739_178_664} \captionsetup{labelformat=empty} \caption{Figure 4}

\end{figure} [The weight of the network is $20 x + 3$ ] Figure 4 shows a graph G that contains 8 arcs and 6 vertices.

State the minimum number of arcs that would need to be added to make G into an Eulerian graph.
Explain whether or not the route $\mathrm { A } - \mathrm { C } - \mathrm { F } - \mathrm { E } - \mathrm { C } - \mathrm { D } - \mathrm { B }$ is an example of a path on G. Figure 4 represents a network of 8 roads in a city. The expression on each arc gives the time, in minutes, to travel along the corresponding road. You are given that $x > 1.6$
A route is required that
- starts and finishes at the same vertex
- traverses each road at least once
- minimises the total time taken
The route inspection algorithm is applied to the network in Figure 4 and the time taken for the route is found to be at most 189 minutes. Given that the inspection route contains two roads that need to be traversed twice,
determine the range of possible values of $x$, making your reasoning clear.

Edexcel FD1 AS 2024 June Q1

1. $$\begin{array} { l l l l l l l l l l l } 4 & 6.5 & 7 & 1.3 & 2 & 5 & 1.5 & 6 & 4.5 & 6 & 1 \end{array}$$ The list of eleven numbers shown above is to be sorted into descending order.

Carry out a quick sort to produce the sorted list. You should show the result of each pass and identify the pivots clearly.
Use the first-fit decreasing bin packing algorithm to pack the numbers into bins of size 10
Determine whether your answer to part (b) uses the minimum number of bins. You must justify your answer. A different list of eleven numbers is to be sorted into descending order using a bubble sort. The list after the second pass is
1.6
1.7
1.5
3.8
3.3
4.5
4.8
5.6
5.4
6.7
9.1
Explain how you know that at least one of the first two passes of the bubble sort was not carried out correctly.

Edexcel FD1 AS 2024 June Q2

2. A company manages an awards evening. The table below lists the activities required to set up the room for the evening, and their immediately preceding activities. Each activity requires exactly one person.

Activity	Immediately preceding activities
A	-
B	A
C	A
D	C
E	C
F	B, D, E
G	E
H	B
J	H, F, G

Figure 1 shows a partially completed activity network used to model the project. Each activity is represented by an arc. \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{ca57c64b-0b33-4179-be7f-684bd6ea2162-04_440_813_1689_726} \captionsetup{labelformat=empty} \caption{Figure 1}

\end{figure}

Add the remaining five activities to Diagram 1 in the answer book to complete the activity network, using exactly two dummies. In addition to setting up the room, the company must prepare the meals for the guests. Figure 2 shows the activity network for preparing the main courses. The numbers in brackets represent the time, in minutes, to complete each task. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{ca57c64b-0b33-4179-be7f-684bd6ea2162-05_793_1515_451_373} \captionsetup{labelformat=empty} \caption{Figure 2}
\end{figure}
Complete Diagram 2 in the answer book to show the early event times and the late event times for the activity network shown in Figure 2.
State the critical activities.
Given that the main courses need to be ready to be served (with all activities completed) at 8 pm , state the latest time that activity $R$ can start.

Edexcel FD1 AS 2024 June Q3

3. \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{ca57c64b-0b33-4179-be7f-684bd6ea2162-06_764_1547_314_355} \captionsetup{labelformat=empty} \caption{Figure 3}

\end{figure} [The total weight of the network is $139 + x + y$ ]

Explain what is meant by the term "tree". Figure 3 represents a network of walkways in a warehouse.
The arcs represent the walkways and the nodes represent junctions between them.
The number on each arc represents the length, in metres, of the corresponding walkway.
The values $x$ and $y$ are unknown, however it is known that $x$ and $y$ are integers and that $$9 < x < y < 14$$
1. Use Dijkstra's algorithm to find the shortest route from A to M.
2. State an expression for the length of the shortest route from A to M . The warehouse manager wants to check that all of the walkways are in good condition.
  Their inspection route starts at B and finishes at C .
  The inspection route must traverse each walkway at least once and be as short as possible.
State the arcs that are traversed twice.
State the number of times that H appears in the inspection route. The warehouse manager finds that the total length of the inspection route is 172 metres.
Determine the value of $x$ and the value of $y$

Edexcel FD1 AS 2024 June Q4

4. \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{ca57c64b-0b33-4179-be7f-684bd6ea2162-07_1105_1249_312_512} \captionsetup{labelformat=empty} \caption{Figure 4}

\end{figure} Figure 4 shows three of the six constraints for a linear programming problem in $x$ and $y$ The unshaded region and its boundaries satisfy these three constraints.

State these three constraints as simplified inequalities with integer coefficients. The variables $x$ and $y$ represent the number of orange fish and the number of blue fish, respectively, that are to be kept in an aquarium. The number of fish in the aquarium is subject to these three further constraints
- there must be at least one blue fish
- the orange fish must not outnumber the blue fish by more than ten
- there must be no more than five blue fish for every orange fish
- Write each of these three constraints as a simplified inequality with integer coefficients.
- Represent these three constraints by adding lines and shading to Diagram 1 in the answer book, labelling the feasible region, $R$
The total value (in pounds) of the fish in the aquarium is given by the objective function $$\text { Maximise } P = 3 x + 5 y$$
1. Use the objective line method to determine the optimal point of the feasible region, giving its coordinates as exact fractions.
2. Hence find the maximum total value of the fish in the aquarium, stating the optimal number of orange fish and the optimal number of blue fish. \begin{table}[h]
  \captionsetup{labelformat=empty} \caption{Please check the examination details below before entering your candidate information}
  Candidate surname Other names
  Centre Number Candidate Number
  □ □
  \end{table} \section*{Pearson Edexcel Level 3 GCE} \section*{Friday 17 May 2024} Afternoon \section*{Further Mathematics} Advanced Subsidiary
  Further Mathematics options
  27: Decision Mathematics 1
  (Part of options D, F, H and K) \section*{D1 Answer Book} Do not return the question paper with the answer book.
  1.
  $\begin{array} { l l l l l l l l l l l } 4 & 6.5 & 7 & 1.3 & 2 & 5 & 1.5 & 6 & 4.5 & 6 & 1 \end{array}$ 2.
  \includegraphics[max width=\textwidth, alt={}]{ca57c64b-0b33-4179-be7f-684bd6ea2162-12_435_815_392_463}
  \section*{Diagram 1} Use this diagram only if you need to redraw your activity network.
  \includegraphics[max width=\textwidth, alt={}, center]{ca57c64b-0b33-4179-be7f-684bd6ea2162-12_442_820_2043_458} Copy of Diagram 1
  VJYV SIHI NI JIIYM ION OC V346 SIHI NI JLIYM ION OC V34V SIHI NI IIIIM ION OC
  Key: \begin{figure}[h]
  \includegraphics[alt={},max width=\textwidth]{ca57c64b-0b33-4179-be7f-684bd6ea2162-13_1217_1783_451_236} \captionsetup{labelformat=empty} \caption{Diagram 2}
  \end{figure} 3.
  \includegraphics[max width=\textwidth, alt={}, center]{ca57c64b-0b33-4179-be7f-684bd6ea2162-14_2463_1240_339_465}
  Shortest route from A to M:
  Length of shortest route from A to M:
  
  \includegraphics[max width=\textwidth, alt={}]{ca57c64b-0b33-4179-be7f-684bd6ea2162-16_3038_2264_0_0}
  
  \includegraphics[max width=\textwidth, alt={}]{ca57c64b-0b33-4179-be7f-684bd6ea2162-17_1103_1247_397_512}
  \section*{Diagram 1} \section*{There is a copy of Diagram 1 on page 11 if you need to redraw your graph.}
  VJYV SIHI NI JIIIM ION OC V341 S1H1 NI JLIYM ION OA V34V SIHI NI IIIVM ION OC
  Use this diagram only if you need to redraw your graph.
  \includegraphics[max width=\textwidth, alt={}, center]{ca57c64b-0b33-4179-be7f-684bd6ea2162-19_1108_1252_1606_509} Copy of Diagram 1

Edexcel FD1 AS Specimen Q1

1. \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{1e2c1dc4-3724-4bba-961c-1c2ae7e649c4-2_698_1173_447_443} \captionsetup{labelformat=empty} \caption{Figure 1}

\end{figure} [The total weight of the network is 189]
Figure 1 represents a network of pipes in a building. The number on each arc is the length, in metres, of the corresponding pipe.

Use Dijkstra's algorithm to find the shortest path from A to F . State the path and its length. On a particular day, Gabriel needs to check each pipe. A route of minimum length, which traverses each pipe at least once and which starts and finishes at A, needs to be found.
Use an appropriate algorithm to find the pipes that will need to be traversed twice. You must make your method and working clear.
State the minimum length of Gabriel's route. A new pipe, BG, is added to the network. A route of minimum length that traverses each pipe, including BG, needs to be found. The route must start and finish at A. Gabriel works out that the addition of the new pipe increases the length of the route by twice the length of BG .
Calculate the length of BG. You must show your working.

Edexcel FD1 AS Specimen Q2

2. \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{1e2c1dc4-3724-4bba-961c-1c2ae7e649c4-3_1463_1194_239_440} \captionsetup{labelformat=empty} \caption{Figure 2}

\end{figure} A teacher buys pens and pencils. The number of pens, $x$, and the number of pencils, $y$, that he buys can be represented by a linear programming problem as shown in Figure 2, which models the following constraints: $$\begin{aligned} 8 x + 3 y & \leqslant 480
8 x + 7 y & \geqslant 560
y & \geqslant 4 x
x , y & \geqslant 0 \end{aligned}$$ The total cost, in pence, of buying the pens and pencils is given by $$C = 12 x + 15 y$$ Determine the number of pens and the number of pencils which should be bought in order to minimise the total cost. You should make your method and working clear.

Edexcel FD1 AS Specimen Q3

Activity	Time taken (days)	Immediately preceding activities
A	5	-
B	7	-
C	3	-
D	4	A, B
E	4	D
F	2	B
G	4	B
H	5	C, G
I	10	C, G

The table above shows the activities required for the completion of a building project. For each activity, the table shows the time taken in days to complete the activity and the immediately preceding activities. Each activity requires one worker. The project is to be completed in the shortest possible time.

Draw the activity network described in the table, using activity on arc. Your activity network must contain the minimum number of dummies only.
1. Show that the project can be completed in 21 days, showing your working.
2. Identify the critical activities.

Edexcel FD1 AS Specimen Q4

4. (a) Explain why it is not possible to draw a graph with exactly 5 nodes with orders $1,3,4,4$ and 5 A connected graph has exactly 5 nodes and contains 18 arcs. The orders of the 5 nodes are $2 ^ { 2 x } - 1,2 ^ { x } , x + 1,2 ^ { x + 1 } - 3$ and $11 - x$.
(b) (i) Calculate X .
(ii) State whether the graph is Eulerian, semi-Eulerian or neither. You must justify your answer.
(c) Draw a graph which satisfies all of the following conditions:

The graph has exactly 5 nodes.
The nodes have orders 2, 2, 4, 4 and 4
The graph is not Eulerian.

Edexcel FD1 AS Specimen Q5

Jonathan makes two types of information pack for an event, Standard and Value.

Each Standard pack contains 25 posters and 500 flyers.
Each Value pack contains 15 posters and 800 flyers.
He must use at least 150000 flyers.
Between $35 \%$ and $65 \%$ of the packs must be Standard packs.
Posters cost 20p each and flyers cost 4p each.
Jonathan wishes to minimise his costs.
Let x and y represent the number of Standard packs and Value packs produced respectively.
Formulate this as a linear programming problem, stating the objective and listing the constraints as simplified inequalities with integer coefficients. You should not attempt to solve the problem. \section*{(Total for Question 5 is 5 marks)} TOTAL IS 40 MARKS

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