Question 5 - A-Level Maths

Edexcel D1 — Question 5

Exam Board	Edexcel
Module	D1 (Decision Mathematics 1)
Paper	Download PDF ↗
Mark scheme	Download PDF ↗
Topic	Critical Path Analysis
Type	Schedule with limited workers - create schedule/chart
Difficulty	Standard +0.3 This is a standard Critical Path Analysis question covering routine D1 techniques: forward/backward pass for early/late times, identifying critical path, and basic resource scheduling. While multi-part with 13 marks total, each component follows textbook procedures with no novel problem-solving required. Slightly easier than average due to the algorithmic nature of CPA methods.
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

5. This question should be answered on the sheet provided in the answer booklet. \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{12f9ae59-b2ff-4a03-9ac9-c61dbaf8c9f5-006_542_1389_483_352} \captionsetup{labelformat=empty} \caption{Fig. 2}

\end{figure} Figure 2 shows the activity network used to model a small building project. The activities are represented by the edges and the number in brackets on each edge represents the time, in hours, taken to complete that activity.

Calculate the early time and the late time for each event. Write your answers in the boxes on the answer sheet.
(6 marks)
Hence determine the critical activities and the length of the critical path.
(2 marks)
Each activity requires one worker. The project is to be completed in the minimum time.
Schedule the activities for the minimum number of workers using the time line on the answer sheet. Ensure that you make clear the order in which each worker undertakes his activities.
(5 marks)

Show mark scheme Show mark scheme source

Question 5:

Part (a) - Kruskal's Algorithm:

Answer	Marks	Guidance
Answer	Marks	Guidance
BC, AB, (not AC), DE, CD, DF, (not BF/CE), EJ, FH, (not HJ), (not BD), GH	M1 A1 A1	M1: first four arcs BC, AB, DE, CD in order + at least one rejection; A1: all eight arcs in correct order, no additional arcs; A2: all selections and rejections correct in order and at correct time

Part (b) - Prim's Algorithm:

Answer	Marks	Guidance
Answer	Marks	Guidance
GH, FH, DF, DE; CD, BC; AB, EJ	M1; A1; A1	M1: first four arcs correct in order; A1: CSO — all arcs correctly stated and chosen in correct order

Part (c):

Answer	Marks	Guidance
Answer	Marks	Guidance
98 km	B1	Condone lack of units

Part (d)(i):

Answer	Marks	Guidance
Answer	Marks	Guidance
\(\dfrac{m}{2}\)	B1

Part (d)(ii):

Answer	Marks	Guidance
Answer	Marks	Guidance
\(n-1\)	B1

Part (d)(iii):

Answer	Marks	Guidance
Answer	Marks	Guidance
\(m \geq 2(n-1)\)	B1	oe; must include correct bracketing; do not accept strict inequality

# Question 5:

## Part (a) - Kruskal's Algorithm:

| Answer | Marks | Guidance |
|--------|-------|----------|
| BC, AB, (not AC), DE, CD, DF, (not BF/CE), EJ, FH, (not HJ), (not BD), GH | M1 A1 A1 | M1: first four arcs BC, AB, DE, CD in order + at least one rejection; A1: all eight arcs in correct order, no additional arcs; A2: all selections and rejections correct in order and at correct time |

## Part (b) - Prim's Algorithm:

| Answer | Marks | Guidance |
|--------|-------|----------|
| GH, FH, DF, DE; CD, BC; AB, EJ | M1; A1; A1 | M1: first four arcs correct in order; A1: CSO — all arcs correctly stated and chosen in correct order |

## Part (c):

| Answer | Marks | Guidance |
|--------|-------|----------|
| 98 km | B1 | Condone lack of units |

## Part (d)(i):

| Answer | Marks | Guidance |
|--------|-------|----------|
| $\dfrac{m}{2}$ | B1 | |

## Part (d)(ii):

| Answer | Marks | Guidance |
|--------|-------|----------|
| $n-1$ | B1 | |

## Part (d)(iii):

| Answer | Marks | Guidance |
|--------|-------|----------|
| $m \geq 2(n-1)$ | B1 | oe; must include correct bracketing; do not accept strict inequality |

Show LaTeX source

5. This question should be answered on the sheet provided in the answer booklet.

\begin{figure}[h]
\begin{center}
  \includegraphics[alt={},max width=\textwidth]{12f9ae59-b2ff-4a03-9ac9-c61dbaf8c9f5-006_542_1389_483_352}
\captionsetup{labelformat=empty}
\caption{Fig. 2}
\end{center}
\end{figure}

Figure 2 shows the activity network used to model a small building project. The activities are represented by the edges and the number in brackets on each edge represents the time, in hours, taken to complete that activity.
\begin{enumerate}[label=(\alph*)]
\item Calculate the early time and the late time for each event. Write your answers in the boxes on the answer sheet.\\
(6 marks)
\item Hence determine the critical activities and the length of the critical path.\\
(2 marks)\\
Each activity requires one worker. The project is to be completed in the minimum time.
\item Schedule the activities for the minimum number of workers using the time line on the answer sheet. Ensure that you make clear the order in which each worker undertakes his activities.\\
(5 marks)
\end{enumerate}

\hfill \mbox{\textit{Edexcel D1  Q5}}

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