| Exam Board | AQA |
|---|---|
| Module | D2 (Decision Mathematics 2) |
| Year | 2006 |
| Session | June |
| Marks | 14 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Critical Path Analysis |
| Type | Draw cascade/Gantt chart |
| Difficulty | Moderate -0.8 This is a standard textbook critical path analysis question requiring systematic application of well-defined algorithms (forward pass, backward pass, float calculation) with no novel problem-solving. The multi-part structure and drawing requirements add length but not conceptual difficulty, making it easier than average A-level maths. |
| Spec | 7.05a Critical path analysis: activity on arc networks7.05b Forward and backward pass: earliest/latest times, critical activities7.05c Total float: calculation and interpretation7.05d Latest start and earliest finish: independent and interfering float7.05e Cascade charts: scheduling and effect of delays |
| Activity | Immediate Predecessors | Duration (days) |
| A | - | 2 |
| B | A | 5 |
| C | A | 8 |
| D | B | 8 |
| E | B | 10 |
| F | B | 4 |
| G | \(C , F\) | 7 |
| \(H\) | D, E | 4 |
| I | \(G , H\) | 3 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| Activity network with correct structure showing nodes with earliest start time, latest finish time, duration | M1 | SCA |
| Almost correct (up to 2 slips) | A1 | |
| Fully correct diagram | A1 | Total: 3 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| Forward pass for earliest start times | M1 | |
| All correct | A1 | Total: 2 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| Backward pass for latest finish times | M1 | |
| All correct | A1 | Total: 2 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| Critical path \(A\ B\ E\ H\ I\) | B1 | Total: 1 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| Non-critical activities \(C, D, F, G\) with floats \(4, 2, 3, 3\) respectively | M1 | At least one float time correct |
| All correct | A1 | Total: 2 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| Their critical path on chart | \(\text{B1}\sqrt{}\) | |
| \(C\) from 6 to 14 (with space 2–6) | M1 | One other activity (condone no slack or earliest start) |
| \(D\) from 9 to 17 (with slack 7–9) | A1 | 2 other non-critical activities |
| \(F\) and \(G\) from 10 to 21 with appropriate slack | A1 | All correct; Total: 4 |
# Question 1:
## Part (a)
| Answer | Marks | Guidance |
|--------|-------|----------|
| Activity network with correct structure showing nodes with earliest start time, latest finish time, duration | M1 | SCA |
| Almost correct (up to 2 slips) | A1 | |
| Fully correct diagram | A1 | Total: 3 |
## Part (b)
| Answer | Marks | Guidance |
|--------|-------|----------|
| Forward pass for earliest start times | M1 | |
| All correct | A1 | Total: 2 |
## Part (c)
| Answer | Marks | Guidance |
|--------|-------|----------|
| Backward pass for latest finish times | M1 | |
| All correct | A1 | Total: 2 |
## Part (d)
| Answer | Marks | Guidance |
|--------|-------|----------|
| Critical path $A\ B\ E\ H\ I$ | B1 | Total: 1 |
## Part (e)
| Answer | Marks | Guidance |
|--------|-------|----------|
| Non-critical activities $C, D, F, G$ with floats $4, 2, 3, 3$ respectively | M1 | At least one float time correct |
| All correct | A1 | Total: 2 |
## Part (f)
| Answer | Marks | Guidance |
|--------|-------|----------|
| Their critical path on chart | $\text{B1}\sqrt{}$ | |
| $C$ from 6 to 14 (with space 2–6) | M1 | One other activity (condone no slack or earliest start) |
| $D$ from 9 to 17 (with slack 7–9) | A1 | 2 other non-critical activities |
| $F$ and $G$ from 10 to 21 with appropriate slack | A1 | All correct; Total: 4 |
---1 [Figures 1 and 2, printed on the insert, are provided for use in this question.] A construction project is to be undertaken. The table shows the activities involved.
\begin{center}
\begin{tabular}{|l|l|l|}
\hline
Activity & Immediate Predecessors & Duration (days) \\
\hline
A & - & 2 \\
\hline
B & A & 5 \\
\hline
C & A & 8 \\
\hline
D & B & 8 \\
\hline
E & B & 10 \\
\hline
F & B & 4 \\
\hline
G & $C , F$ & 7 \\
\hline
$H$ & D, E & 4 \\
\hline
I & $G , H$ & 3 \\
\hline
\end{tabular}
\end{center}
\begin{enumerate}[label=(\alph*)]
\item Complete the activity network for the project on Figure 1.
\item Find the earliest start time for each activity.
\item Find the latest finish time for each activity.
\item Find the critical path.
\item State the float time for each non-critical activity.
\item On Figure 2, draw a cascade diagram (Gantt chart) for the project, assuming each activity starts as late as possible.
\end{enumerate}
\hfill \mbox{\textit{AQA D2 2006 Q1 [14]}}