| Exam Board | AQA |
|---|---|
| Module | Further Paper 3 Discrete (Further Paper 3 Discrete) |
| Year | 2019 |
| Session | June |
| Marks | 10 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Critical Path Analysis |
| Type | Calculate early and late times |
| Difficulty | Moderate -0.3 This is a standard Critical Path Analysis question requiring routine application of forward and backward passes to find early/late times, identification of critical paths, and basic resource histogram construction. While it involves multiple parts and careful bookkeeping with 12 activities, the techniques are algorithmic and well-practiced in Further Maths Decision modules, making it slightly easier than average overall. |
| Spec | 7.05a Critical path analysis: activity on arc networks7.05b Forward and backward pass: earliest/latest times, critical activities |
| Activity | Duration (days) | Immediate Predecessors |
| \(A\) | 7 | - |
| B | 6 | - |
| C | 15 | - |
| D | 9 | \(A , B\) |
| \(E\) | 8 | D |
| \(F\) | 6 | C, D |
| G | 7 | C |
| H | 14 | \(E\) |
| \(I\) | 17 | \(F , G\) |
| \(J\) | 9 | H, I |
| K | 8 | \(I\) |
| L | 12 | J, K |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| Constructs activity network with at least 10 labelled activities drawn and at least 4 connections | M1 | — |
| Activity network fully correct with all activities and connections | A1 | Condone omission of arrows |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| Finds correctly earliest start times for activities \(E\), \(F\) and \(G\) | M1 | — |
| Finds correctly earliest start time for each activity on the network | A1 | — |
| Finds correctly latest finish time for each activity: \(A\): \([0\ | 7\ | 7]\), \(B\): \([0\ |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| Critical paths: \(A\)–\(D\)–\(F\)–\(I\)–\(J\)–\(L\) and \(C\)–\(G\)–\(I\)–\(J\)–\(L\) | B1 | Must identify both critical paths and no others |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| Draws resource histogram with consistent vertical scale, at least 10 labelled regions, correct critical path on bottom row | M1 | Condone floats but not overhangs |
| Draws correct resource histogram | A1 | — |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| \(A\) and \(B\) cannot be done simultaneously as this would require 3 team members; therefore \(B\) must follow \(A\), increasing earliest start time for \(D\) to 13 | M1 | PI by figure or by increase of 6 days |
| Minimum completion time for the project increases to 66 days | R1 | Activities \(E\), \(F\) and \(G\) will only take two team members to complete |
## Question 8:
### Part (a)(i):
| Answer | Mark | Guidance |
|--------|------|----------|
| Constructs activity network with at least 10 labelled activities drawn and at least 4 connections | M1 | — |
| Activity network fully correct with all activities and connections | A1 | Condone omission of arrows |
### Part (a)(ii):
| Answer | Mark | Guidance |
|--------|------|----------|
| Finds correctly earliest start times for activities $E$, $F$ and $G$ | M1 | — |
| Finds correctly earliest start time for each activity on the network | A1 | — |
| Finds correctly latest finish time for each activity: $A$: $[0\|7\|7]$, $B$: $[0\|6\|7]$, $C$: $[0\|15\|15]$, $D$: $[7\|9\|16]$, $E$: $[16\|8\|25]$, $F$: $[16\|6\|22]$, $G$: $[15\|7\|22]$, $H$: $[24\|14\|39]$, $I$: $[22\|17\|39]$, $J$: $[39\|9\|48]$, $K$: $[39\|8\|48]$, $L$: $[48\|12\|60]$ | B1 | — |
### Part (b):
| Answer | Mark | Guidance |
|--------|------|----------|
| Critical paths: $A$–$D$–$F$–$I$–$J$–$L$ and $C$–$G$–$I$–$J$–$L$ | B1 | Must identify both critical paths and no others |
### Part (c)(i):
| Answer | Mark | Guidance |
|--------|------|----------|
| Draws resource histogram with consistent vertical scale, at least 10 labelled regions, correct critical path on bottom row | M1 | Condone floats but not overhangs |
| Draws correct resource histogram | A1 | — |
### Part (c)(ii):
| Answer | Mark | Guidance |
|--------|------|----------|
| $A$ and $B$ cannot be done simultaneously as this would require 3 team members; therefore $B$ must follow $A$, increasing earliest start time for $D$ to 13 | M1 | PI by figure or by increase of 6 days |
| Minimum completion time for the project increases to 66 days | R1 | Activities $E$, $F$ and $G$ will only take two team members to complete |8 A motor racing team is undertaking a project to build next season's racing car. The project is broken down into 12 separate activities $A , B , \ldots , L$, as shown in the precedence table below. Each activity requires one member of the racing team.
\begin{center}
\begin{tabular}{|l|l|l|}
\hline
Activity & Duration (days) & Immediate Predecessors \\
\hline
$A$ & 7 & - \\
\hline
B & 6 & - \\
\hline
C & 15 & - \\
\hline
D & 9 & $A , B$ \\
\hline
$E$ & 8 & D \\
\hline
$F$ & 6 & C, D \\
\hline
G & 7 & C \\
\hline
H & 14 & $E$ \\
\hline
$I$ & 17 & $F , G$ \\
\hline
$J$ & 9 & H, I \\
\hline
K & 8 & $I$ \\
\hline
L & 12 & J, K \\
\hline
\end{tabular}
\end{center}
8
\begin{enumerate}[label=(\alph*)]
\item (i) Complete the activity network for the project on Figure 3.
8 (a) (ii) Find the earliest start time and the latest finish time for each activity and show these values on Figure 3.
8
\item Write down the critical path(s).\\
\section*{Figure 3}
Figure 3\\
\includegraphics[max width=\textwidth, alt={}, center]{22f11ce2-8d07-4f51-9326-b578d1e454f9-15_469_1360_356_338}
8
\item (i) Using Figure 4, draw a resource histogram for the project to show how the project can be completed in the shortest possible time. Assume that each activity is to start as early as possible.
\begin{figure}[h]
\begin{center}
\captionsetup{labelformat=empty}
\caption{Figure 4}
\includegraphics[alt={},max width=\textwidth]{22f11ce2-8d07-4f51-9326-b578d1e454f9-16_698_1534_541_251}
\end{center}
\end{figure}
8 (c) (ii) The racing team's boss assigns two members of the racing team to work on the project.
Explain the effect this has on the minimum completion time for the project.\\
You may use Figure 5 in your answer.
\begin{figure}[h]
\begin{center}
\captionsetup{labelformat=empty}
\caption{Figure 5}
\includegraphics[alt={},max width=\textwidth]{22f11ce2-8d07-4f51-9326-b578d1e454f9-16_704_1539_1695_248}
\end{center}
\end{figure}
\end{enumerate}
\hfill \mbox{\textit{AQA Further Paper 3 Discrete 2019 Q8 [10]}}