Question 2 - A-Level Maths

Edexcel FD1 AS 2020 June — Question 2 14 marks

Exam Board	Edexcel
Module	FD1 AS (Further Decision 1 AS)
Year	2020
Session	June
Marks	14
Paper	Download PDF ↗
Mark scheme	Download PDF ↗
Topic	Critical Path Analysis
Type	Complete precedence table from network
Difficulty	Moderate -0.3 This is a standard Critical Path Analysis question covering routine techniques (precedence table, early/late times, critical path, float calculation, lower bound, cascade chart). While multi-part with 7 sub-questions, each part follows textbook procedures with no novel problem-solving required. The network diagram interpretation and calculations are straightforward for Further Maths students who have studied this topic.
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 7.05d Latest start and earliest finish: independent and interfering float

2. \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{a2a6e659-aab5-4eec-9af4-ca6ab895f1c8-03_693_1379_233_342} \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. 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.
1. State the minimum project completion time.
2. List the critical activities.
Calculate the maximum number of hours by which activity H could be delayed without affecting the shortest possible completion time of the project. You must make the numbers used in your calculation clear.
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 cascade chart for this project on Grid 1 in the answer book.
Using the answer to (f), explain why it is not possible to complete the project in the shortest possible time using the number of workers found in (e).

Show mark scheme Show mark scheme source

Question 2:

Part 2(a):

Answer	Marks	Guidance
Answer	Mark	Guidance
5 non-empty rows correct (any 5 of rows D to K)	B1	1.1b
All 11 rows correct: I preceded by D; J preceded by D, E, F, G, H; K preceded by H	B1	1.1b

Part 2(b):

Answer	Marks	Guidance
Answer	Mark	Guidance
All boxes completed, numbers generally increasing L to R and decreasing R to L (condone one "rogue")	M1	2.1
All top boxes correct: early times 0, 0, 5, 5, 6, 5, 6, 13, 13, 12, 13 giving final value 22	A1	1.1b
All bottom boxes correct	A1	1.1b

Part 2(c)(i):

Answer	Marks	Guidance
Answer	Mark	Guidance
Minimum project completion time is 22 hours	B1ft	1.1b - CAO following through from completed top boxes in (b)

Part 2(c)(ii):

Answer	Marks	Guidance
Answer	Mark	Guidance
Critical activities are A, E and J	B1	1.1b - CAO (A, E and J only)

Part 2(d):

Answer	Marks	Guidance
Answer	Mark	Guidance
H could be delayed by \(13 - 5 - 7 = 1\) hour	B1ft	3.4 - Correct calculation for activity H from (b); must see all 3 numbers (so just \(13-12=1\) is B0)

Part 2(e):

Answer	Marks	Guidance
Answer	Mark	Guidance
\(\dfrac{5+3+4+\ldots+8+9+6}{22}\)	M1	1.1b - (55 to 73 inclusive) / their duration; answers to (b) and (c)(i) must be consistent
\(= 2.909\ldots\) so a lower bound of 3 workers	A1	2.2a - Correct deduction; answer of 3 with no working scores no marks

Part 2(f):

Answer	Marks	Guidance
Answer	Mark	Guidance
At least 9 activities including at least 6 floats shown on Gantt chart	M1	2.1
All correct critical activities present (A, E, J) and 5 non-critical activities correct	A1	1.1b
All non-critical activities correct	A1	1.1b

Part 2(g):

Answer	Marks	Guidance
Answer	Mark	Guidance
e.g. between times 5 and 13, activities E, D, F, G and H must all be happening. Total time for these five activities is 29 hours and \(29/8 > 3\) so not possible with lower bound of 3 workers. Or: at time 8.5, activities E, D, F and H must be happening so not possible with only 3 workers	B1	3.4 - Must refer to both times and activities (as indication they have used (f))

## Question 2:

**Part 2(a):**

| Answer | Mark | Guidance |
|--------|------|----------|
| 5 non-empty rows correct (any 5 of rows D to K) | B1 | 1.1b |
| All 11 rows correct: I preceded by D; J preceded by D, E, F, G, H; K preceded by H | B1 | 1.1b |

**Part 2(b):**

| Answer | Mark | Guidance |
|--------|------|----------|
| All boxes completed, numbers generally increasing L to R and decreasing R to L (condone one "rogue") | M1 | 2.1 |
| All top boxes correct: early times 0, 0, 5, 5, 6, 5, 6, 13, 13, 12, 13 giving final value 22 | A1 | 1.1b |
| All bottom boxes correct | A1 | 1.1b |

**Part 2(c)(i):**

| Answer | Mark | Guidance |
|--------|------|----------|
| Minimum project completion time is 22 hours | B1ft | 1.1b - CAO following through from completed top boxes in (b) |

**Part 2(c)(ii):**

| Answer | Mark | Guidance |
|--------|------|----------|
| Critical activities are A, E and J | B1 | 1.1b - CAO (A, E and J only) |

**Part 2(d):**

| Answer | Mark | Guidance |
|--------|------|----------|
| H could be delayed by $13 - 5 - 7 = 1$ hour | B1ft | 3.4 - Correct calculation for activity H from (b); must see all 3 numbers (so just $13-12=1$ is B0) |

**Part 2(e):**

| Answer | Mark | Guidance |
|--------|------|----------|
| $\dfrac{5+3+4+\ldots+8+9+6}{22}$ | M1 | 1.1b - (55 to 73 inclusive) / their duration; answers to (b) and (c)(i) must be consistent |
| $= 2.909\ldots$ so a lower bound of 3 workers | A1 | 2.2a - Correct deduction; answer of 3 with no working scores no marks |

**Part 2(f):**

| Answer | Mark | Guidance |
|--------|------|----------|
| At least 9 activities including at least 6 floats shown on Gantt chart | M1 | 2.1 |
| All correct critical activities present (A, E, J) and 5 non-critical activities correct | A1 | 1.1b |
| All non-critical activities correct | A1 | 1.1b |

**Part 2(g):**

| Answer | Mark | Guidance |
|--------|------|----------|
| e.g. between times 5 and 13, activities E, D, F, G and H must all be happening. Total time for these five activities is 29 hours and $29/8 > 3$ so not possible with lower bound of 3 workers. Or: at time 8.5, activities E, D, F and H must be happening so not possible with only 3 workers | B1 | 3.4 - Must refer to both times and activities (as indication they have used (f)) |

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Show LaTeX source

2.

\begin{figure}[h]
\begin{center}
  \includegraphics[alt={},max width=\textwidth]{a2a6e659-aab5-4eec-9af4-ca6ab895f1c8-03_693_1379_233_342}
\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, in hours, to complete the corresponding activity. Each activity requires one worker. The project is to be completed in the shortest possible time.
\begin{enumerate}[label=(\alph*)]
\item Complete the precedence table in the answer book.
\item Complete Diagram 1 in the answer book to show the early event times and the late event times.
\item \begin{enumerate}[label=(\roman*)]
\item State the minimum project completion time.
\item List the critical activities.
\end{enumerate}\item Calculate the maximum number of hours by which activity H could be delayed without affecting the shortest possible completion time of the project. You must make the numbers used in your calculation clear.
\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 cascade chart for this project on Grid 1 in the answer book.
\item Using the answer to (f), explain why it is not possible to complete the project in the shortest possible time using the number of workers found in (e).
\end{enumerate}

\hfill \mbox{\textit{Edexcel FD1 AS 2020 Q2 [14]}}

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