Question 4 - A-Level Maths

AQA Further AS Paper 2 Discrete 2023 June — Question 4 8 marks

Exam Board	AQA
Module	Further AS Paper 2 Discrete (Further AS Paper 2 Discrete)
Year	2023
Session	June
Marks	8
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 forward and backward passes to find early/late times and identify critical activities. Part (c) adds mild conceptual understanding about when critical paths can be summed, but the calculations are routine algorithmic procedures typical of Decision Maths. Slightly easier than average A-level due to being a standard textbook exercise 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

4 A community project consists of 10 activities \(A , B , \ldots , J\), as shown in the activity network below. \includegraphics[max width=\textwidth, alt={}, center]{372edcfa-c3cd-4c83-89e9-2bb5fd9825f1-06_899_1083_367_466} The duration of each activity is shown in days. 4

1. Complete the activity network in the diagram above, showing the earliest start time and latest finish time for each activity. 4
  1. (ii) State the minimum completion time for the community project.
    4
2. Write down the critical activities of the network.
  4
3. Glyn claims that a project's activity network can be used to determine its minimum completion time by adding together the durations of all the project's critical activities. 4
4. 1. Show that Glyn's claim is false for this community project's activity network.
    4
5. (ii) Describe a situation in which Glyn's claim would be true.

Show mark scheme Show mark scheme source

Question 4(a)(i):

Answer	Marks	Guidance
Answer	Marks	Guidance
Finds correctly the earliest start time for activity \(H\) or activity \(J\)	M1	Method mark for finding EST for H or J
Finds correctly the earliest start time for each activity on the network (see diagram)	A1	All ESTs correct
Finds correctly the latest finish time for each activity on the network	B1	Condone inclusion of 'END' activity with zero duration

Network values:

- \(A\): \([0 \mid 12 \mid 12]\)

- \(B\): \([0 \mid 6 \mid 8]\)

- \(C\): \([12 \mid 8 \mid 20]\)

- \(D\): \([12 \mid 4 \mid 16]\)

- \(E\): \([6 \mid 8 \mid 16]\)

- \(F\): \([20 \mid 10 \mid 30]\)

- \(G\): \([16 \mid 12 \mid 28]\)

- \(H\): \([30 \mid 6 \mid 36]\)

- \(I\): \([30 \mid 4 \mid 36]\)

- \(J\): \([28 \mid 8 \mid 36]\)

Question 4(a)(ii):

Answer	Marks	Guidance
Answer	Marks	Guidance
\(36\) days	B1F	FT their values in activity network; condone missing units

Question 4(b):

Answer	Marks	Guidance
Answer	Marks	Guidance
\(A, C, D, F, G, H\) and \(J\)	B1	States all correct critical activities and no others

Question 4(c)(i):

Answer	Marks	Guidance
Answer	Marks	Guidance
\(12 + 8 + 4 + 10 + 12 + 6 + 8 = 60\), \(36 \neq 60\)	E1	Finds that 60 days is the sum of the durations for each correct critical activity, and compares this with 36 days (the minimum completion time)

Question 4(c)(ii):

Answer	Marks	Guidance
Answer	Marks	Guidance
Glyn's statement is true for an activity network with all critical activities on a single critical path	E1	Explains that Glyn's statement is true for an activity network with all critical activities on a single critical path

# Question 4(a)(i):

| Answer | Marks | Guidance |
|--------|-------|----------|
| Finds correctly the earliest start time for activity $H$ or activity $J$ | M1 | Method mark for finding EST for H or J |
| Finds correctly the earliest start time for each activity on the network (see diagram) | A1 | All ESTs correct |
| Finds correctly the latest finish time for each activity on the network | B1 | Condone inclusion of 'END' activity with zero duration |

Network values:
- $A$: $[0 \mid 12 \mid 12]$
- $B$: $[0 \mid 6 \mid 8]$
- $C$: $[12 \mid 8 \mid 20]$
- $D$: $[12 \mid 4 \mid 16]$
- $E$: $[6 \mid 8 \mid 16]$
- $F$: $[20 \mid 10 \mid 30]$
- $G$: $[16 \mid 12 \mid 28]$
- $H$: $[30 \mid 6 \mid 36]$
- $I$: $[30 \mid 4 \mid 36]$
- $J$: $[28 \mid 8 \mid 36]$

---

# Question 4(a)(ii):

| Answer | Marks | Guidance |
|--------|-------|----------|
| $36$ days | B1F | FT their values in activity network; condone missing units |

---

# Question 4(b):

| Answer | Marks | Guidance |
|--------|-------|----------|
| $A, C, D, F, G, H$ and $J$ | B1 | States all correct critical activities and no others |

---

# Question 4(c)(i):

| Answer | Marks | Guidance |
|--------|-------|----------|
| $12 + 8 + 4 + 10 + 12 + 6 + 8 = 60$, $36 \neq 60$ | E1 | Finds that 60 days is the sum of the durations for each correct critical activity, and compares this with 36 days (the minimum completion time) |

---

# Question 4(c)(ii):

| Answer | Marks | Guidance |
|--------|-------|----------|
| Glyn's statement is true for an activity network with all critical activities on a single critical path | E1 | Explains that Glyn's statement is true for an activity network with all critical activities on a single critical path |

---

Show LaTeX source

4 A community project consists of 10 activities $A , B , \ldots , J$, as shown in the activity network below.\\
\includegraphics[max width=\textwidth, alt={}, center]{372edcfa-c3cd-4c83-89e9-2bb5fd9825f1-06_899_1083_367_466}

The duration of each activity is shown in days.

4
\begin{enumerate}[label=(\alph*)]
\item (i) Complete the activity network in the diagram above, showing the earliest start time and latest finish time for each activity.

4 (a) (ii) State the minimum completion time for the community project.\\

4
\item Write down the critical activities of the network.\\

4
\item Glyn claims that a project's activity network can be used to determine its minimum completion time by adding together the durations of all the project's critical activities.

4 (c) (i) Show that Glyn's claim is false for this community project's activity network.\\

4 (c) (ii) Describe a situation in which Glyn's claim would be true.
\end{enumerate}

\hfill \mbox{\textit{AQA Further AS Paper 2 Discrete 2023 Q4 [8]}}

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