Question 2 - A-Level Maths

Edexcel FD1 AS 2023 June — Question 2 10 marks

Exam Board	Edexcel
Module	FD1 AS (Further Decision 1 AS)
Year	2023
Session	June
Marks	10
Paper	Download PDF ↗
Mark scheme	Download PDF ↗
Topic	Critical Path Analysis
Type	Explain dummy activities
Difficulty	Moderate -0.5 This is a standard Critical Path Analysis question covering routine techniques (dummy activities, early/late times, critical path, lower bound, Gantt chart). Part (a) tests basic understanding of dummy activities, while other parts involve mechanical application of well-practiced algorithms. Slightly easier than average due to being a straightforward multi-part question with no novel problem-solving required.
Spec	7.05a Critical path analysis: activity on arc networks 7.05b Forward and backward pass: earliest/latest times, critical activities 7.05c Total float: calculation and interpretation

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.

Show mark scheme Show mark scheme source

Question 2:

Part 2(a):

Answer	Marks	Guidance
Answer	Mark	Guidance
The dummy from event 3 to event 4 is required as activity F depends only on activity C, but activities G, H and J depend on activities C, B and E	B1	Must mention activities F and C (twice or clearly implied twice), at least one of B/E, and at least one of G/H/J. e.g. 'F relies on C, but G relies on C and E'

Total: (1 mark)

Part 2(b):

Answer	Marks	Guidance
Answer	Mark	Guidance
All top boxes and bottom boxes completed with values generally increasing left to right (top) and decreasing right to left (bottom)	M1	Condone missing 0s at source node or 29 at sink node. Condone one rogue value in top boxes and one in bottom boxes
Top boxes correct (including zero at source node)	A1	cao
Bottom boxes correct (including zero at source node and 29 at sink node)	A1	cao

Total: (3 marks)

Part 2(c):

Answer	Marks	Guidance
Answer	Mark	Guidance
Critical activities are A, E, J and K	B1	cao — correct four critical activities and no others

Total: (1 mark)

Part 2(d):

Answer	Marks	Guidance
Answer	Mark	Guidance
\(\dfrac{6+10+7+7+8+11+5+9+2+6+9+5}{29} = \dfrac{85}{29} = 2.931\ldots\) so a lower bound of 3 workers	B1ft	Correct deduction from correct calculation. Follow through on their 29 only. Must see \(\frac{85}{29}\) or awrt 2.9 followed by 3. Answer of 3 with no working scores no marks

Total: (1 mark)

Part 2(e):

Answer	Marks	Guidance
Answer	Mark	Guidance
At least nine different activities labelled including at least five floats shown	M1	A scheduling diagram with no floats shown scores M0
Critical activities (A, E, J, K) dealt with correctly appearing just once, and three non-critical activities with correct duration and total float	A1
Any six non-critical activities correct	A1	Independent of previous A mark
Completely correct Gantt chart — all twelve activities appearing exactly once	A1	cso

Useful float data:

Answer	Marks	Guidance
Activity	Duration + Float	Activity
A	0 to 6, Critical	F
B	0 to 10, F: 10 to 14	G
C	0 to 7, F: 7 to 9	H
D	6 to 13, F: 13 to 22	I
E	6 to 14, Critical	J

Total: (4 marks)

## Question 2:

### Part 2(a):
| Answer | Mark | Guidance |
|--------|------|----------|
| The dummy from event 3 to event 4 is required as activity F depends only on activity C, but activities G, H and J depend on activities C, B and E | B1 | Must mention activities F and C (twice or clearly implied twice), at least one of B/E, and at least one of G/H/J. e.g. 'F relies on C, but G relies on C and E' |

**Total: (1 mark)**

### Part 2(b):
| Answer | Mark | Guidance |
|--------|------|----------|
| All top boxes and bottom boxes completed with values generally increasing left to right (top) and decreasing right to left (bottom) | M1 | Condone missing 0s at source node or 29 at sink node. Condone one rogue value in top boxes and one in bottom boxes |
| Top boxes correct (including zero at source node) | A1 | cao |
| Bottom boxes correct (including zero at source node and 29 at sink node) | A1 | cao |

**Total: (3 marks)**

### Part 2(c):
| Answer | Mark | Guidance |
|--------|------|----------|
| Critical activities are A, E, J and K | B1 | cao — correct four critical activities and no others |

**Total: (1 mark)**

### Part 2(d):
| Answer | Mark | Guidance |
|--------|------|----------|
| $\dfrac{6+10+7+7+8+11+5+9+2+6+9+5}{29} = \dfrac{85}{29} = 2.931\ldots$ so a lower bound of 3 workers | B1ft | Correct deduction from correct calculation. Follow through on their 29 only. Must see $\frac{85}{29}$ or awrt 2.9 followed by 3. Answer of 3 with no working scores no marks |

**Total: (1 mark)**

### Part 2(e):
| Answer | Mark | Guidance |
|--------|------|----------|
| At least nine different activities labelled including at least five floats shown | M1 | A scheduling diagram with no floats shown scores M0 |
| Critical activities (A, E, J, K) dealt with correctly appearing just once, and three non-critical activities with correct duration and total float | A1 | |
| Any six non-critical activities correct | A1 | Independent of previous A mark |
| Completely correct Gantt chart — all twelve activities appearing exactly once | A1 | cso |

**Useful float data:**
| Activity | Duration + Float | Activity | Duration + Float | Activity | Duration + Float |
|----------|-----------------|----------|-----------------|----------|-----------------|
| A | 0 to 6, Critical | F | 7 to 18, F: 18 to 20 | K | 20 to 29, Critical |
| B | 0 to 10, F: 10 to 14 | G | 14 to 19, F: 19 to 22 | L | 23 to 28, F: 28 to 29 |
| C | 0 to 7, F: 7 to 9 | H | 14 to 23, F: 23 to 24 | | |
| D | 6 to 13, F: 13 to 22 | I | 19 to 21, F: 21 to 24 | | |
| E | 6 to 14, Critical | J | 14 to 20, Critical | | |

**Total: (4 marks)**

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Show LaTeX source

2.

\begin{figure}[h]
\begin{center}
  \includegraphics[alt={},max width=\textwidth]{9edb5209-4244-4916-b3ee-d77e395e8cab-03_750_1490_262_285}
\captionsetup{labelformat=empty}
\caption{Figure 1}
\end{center}
\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.
\begin{enumerate}[label=(\alph*)]
\item Explain the significance of the dummy activity from event 3 to event 4
\item Complete Diagram 1 in the answer book to show the early event times and the late event times.
\item State the critical activities.
\item Calculate a lower bound for the number of workers needed to complete the project in the minimum time. You must show your working.
\item Draw a Gantt chart for this project on Grid 1 in the answer book.
\end{enumerate}

\hfill \mbox{\textit{Edexcel FD1 AS 2023 Q2 [10]}}

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