Question 5 - A-Level Maths

AQA Further AS Paper 2 Discrete 2018 June — Question 5 9 marks

Exam Board	AQA
Module	Further AS Paper 2 Discrete (Further AS Paper 2 Discrete)
Year	2018
Session	June
Marks	9
Paper	Download PDF ↗
Mark scheme	Download PDF ↗
Topic	Critical Path Analysis
Type	Complete precedence table from network
Difficulty	Moderate -0.5 This is a standard Critical Path Analysis question requiring students to complete a precedence table from a given network diagram, perform forward and backward passes, and identify the critical path. While it involves multiple steps, these are routine algorithmic procedures taught in Decision Maths with no novel problem-solving required. The context is straightforward and the calculations are mechanical, making it slightly easier than an average A-level question.
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 A group of friends want to prepare a meal. They start preparing the meal at 6:30 pm Activities to prepare the meal are shown in Figure 1 below. \begin{table}[h]

\captionsetup{labelformat=empty} \caption{Figure 1}

Label	Activity	Duration (mins)	Immediate predecessors
A	Weigh rice	1	-
\(B\)	Cook rice	18	\(A\)
C	Drain rice	1	B
D	Chop vegetables	10	-
\(E\)	Fry vegetables	12
\(F\)	Combine fried vegetables and drained rice	1
G	Prepare sauce ingredients	4	-
\(H\)	Boil sauce	12
\(I\)	Serve meal on plates	2

\end{table} 5

1. Use Figure 2 shown below to complete Figure 1 above. 5
  1. (ii) Complete Figure 2 showing the earliest start time and latest finishing time for each activity. \begin{figure}[h]
    \captionsetup{labelformat=empty} \caption{Figure 2} \includegraphics[alt={},max width=\textwidth]{5a826f8b-4751-4589-ad0a-109fc5c821f2-06_700_1650_1781_194}
    \end{figure} 5
2. 1. State the activity which must be started first so that the meal is served in the shortest possible time. Fully justify your answer.
    5
3. (ii) Determine the earliest possible time at which the preparation of the meal can be completed.
  Question 5 continues on the next page 5
4. The group of friends want to cook spring rolls so that they are served at the same time as the rest of the meal. This requires the additional activities shown in Figure 3. \begin{table}[h]
  \captionsetup{labelformat=empty} \caption{Figure 3}
  Label Activity Duration Immediate predecessors
  J Switch on and heat oven -
  K Put spring rolls in oven and cook
  \(L\) Transfer spring rolls to serving dish
  \end{table} It takes 15 seconds to switch on the oven. The oven must be allowed to heat up for 10 minutes before the spring rolls are put in the oven. It takes 15 seconds to put the spring rolls in the oven.
  The spring rolls must cook in the hot oven for 8 minutes.
  It takes 30 seconds to transfer the spring rolls to a serving dish.
  5
5. 1. Complete Figure 3 above. 5
6. (ii) Determine the latest time at which the oven can be switched on in order for the spring rolls to be served at the same time as the rest of the meal.
  [0pt] [2 marks] \includegraphics[max width=\textwidth, alt={}, center]{5a826f8b-4751-4589-ad0a-109fc5c821f2-09_2488_1716_219_153}

Show mark scheme Show mark scheme source

Question 5(a)(i):

Answer	Marks	Guidance
Answer/Working	Mark	Guidance
\(D\); \(C, E\); \(G\); \(F, H\)	B1	Correctly completes precedence table

Question 5(a)(ii):

Answer	Marks	Guidance
Answer/Working	Mark	Guidance
Earliest start times: \(A: 0\), \(B: 1\), \(C: 19\), \(D: 0\), \(E: 10\), \(F: 22\), \(G: 0\), \(H: 4\), \(I: 23\)	B1	Correctly states earliest start times for all activities
Latest finish times: \(A: 3\), \(B: 21\), \(C: 22\), \(D: 10\), \(E: 22\), \(F: 23\), \(G: 11\), \(H: 23\), \(I: 25\)	B1	Correctly states latest finish times for all activities

Question 5(b)(i):

Answer	Marks	Guidance
Answer/Working	Mark	Guidance
\(D\), Chop vegetables	B1F	Only FT if answer is \(A\), \(D\) or \(G\)
\(D\) is the first activity on the critical path	E1F	Identifies activity as critical OR states float is zero

Question 5(b)(ii):

Answer	Marks	Guidance
Answer/Working	Mark	Guidance
\(6{:}55\) pm	B1F	\(6{:}30\) pm \(+\) their minimum completion time; must see indication of pm

Question 5(c)(i):

Answer	Marks	Guidance
Answer/Working	Mark	Guidance
Durations: \(10.25\), \(8.25\), \(0.5\); Predecessors: \(-\), \(J\), \(K\)	B1	Condone correct durations in seconds

Question 5(c)(ii):

Answer	Marks	Guidance
Answer/Working	Mark	Guidance
\(25 - 19 = 6\) minutes	M1	Uses minimum completion time and 19 in consistent units
\(6{:}36\) pm	A1	Condone omission of pm

## Question 5(a)(i):

| Answer/Working | Mark | Guidance |
|---|---|---|
| $D$; $C, E$; $G$; $F, H$ | B1 | Correctly completes precedence table |

## Question 5(a)(ii):

| Answer/Working | Mark | Guidance |
|---|---|---|
| Earliest start times: $A: 0$, $B: 1$, $C: 19$, $D: 0$, $E: 10$, $F: 22$, $G: 0$, $H: 4$, $I: 23$ | B1 | Correctly states earliest start times for all activities |
| Latest finish times: $A: 3$, $B: 21$, $C: 22$, $D: 10$, $E: 22$, $F: 23$, $G: 11$, $H: 23$, $I: 25$ | B1 | Correctly states latest finish times for all activities |

## Question 5(b)(i):

| Answer/Working | Mark | Guidance |
|---|---|---|
| $D$, Chop vegetables | B1F | Only FT if answer is $A$, $D$ or $G$ |
| $D$ is the first activity on the critical path | E1F | Identifies activity as critical OR states float is zero |

## Question 5(b)(ii):

| Answer/Working | Mark | Guidance |
|---|---|---|
| $6{:}55$ pm | B1F | $6{:}30$ pm $+$ their minimum completion time; must see indication of pm |

## Question 5(c)(i):

| Answer/Working | Mark | Guidance |
|---|---|---|
| Durations: $10.25$, $8.25$, $0.5$; Predecessors: $-$, $J$, $K$ | B1 | Condone correct durations in seconds |

## Question 5(c)(ii):

| Answer/Working | Mark | Guidance |
|---|---|---|
| $25 - 19 = 6$ minutes | M1 | Uses minimum completion time and 19 in consistent units |
| $6{:}36$ pm | A1 | Condone omission of pm |

---

Show LaTeX source

5 A group of friends want to prepare a meal. They start preparing the meal at 6:30 pm Activities to prepare the meal are shown in Figure 1 below.

\begin{table}[h]
\begin{center}
\captionsetup{labelformat=empty}
\caption{Figure 1}
\begin{tabular}{|l|l|l|l|}
\hline
Label & Activity & Duration (mins) & Immediate predecessors \\
\hline
A & Weigh rice & 1 & - \\
\hline
$B$ & Cook rice & 18 & $A$ \\
\hline
C & Drain rice & 1 & B \\
\hline
D & Chop vegetables & 10 & - \\
\hline
$E$ & Fry vegetables & 12 &  \\
\hline
$F$ & Combine fried vegetables and drained rice & 1 &  \\
\hline
G & Prepare sauce ingredients & 4 & - \\
\hline
$H$ & Boil sauce & 12 &  \\
\hline
$I$ & Serve meal on plates & 2 &  \\
\hline
\end{tabular}
\end{center}
\end{table}

5
\begin{enumerate}[label=(\alph*)]
\item (i) Use Figure 2 shown below to complete Figure 1 above.

5 (a) (ii) Complete Figure 2 showing the earliest start time and latest finishing time for each activity.

\begin{figure}[h]
\begin{center}
\captionsetup{labelformat=empty}
\caption{Figure 2}
  \includegraphics[alt={},max width=\textwidth]{5a826f8b-4751-4589-ad0a-109fc5c821f2-06_700_1650_1781_194}
\end{center}
\end{figure}

5
\item (i) State the activity which must be started first so that the meal is served in the shortest possible time.

Fully justify your answer.\\

5 (b) (ii) Determine the earliest possible time at which the preparation of the meal can be completed.\\

Question 5 continues on the next page

5
\item The group of friends want to cook spring rolls so that they are served at the same time as the rest of the meal. This requires the additional activities shown in Figure 3.

\begin{table}[h]
\begin{center}
\captionsetup{labelformat=empty}
\caption{Figure 3}
\begin{tabular}{|l|l|l|l|}
\hline
Label & Activity & Duration & Immediate predecessors \\
\hline
J & Switch on and heat oven &  & - \\
\hline
K & Put spring rolls in oven and cook &  &  \\
\hline
$L$ & Transfer spring rolls to serving dish &  &  \\
\hline
\end{tabular}
\end{center}
\end{table}

It takes 15 seconds to switch on the oven.

The oven must be allowed to heat up for 10 minutes before the spring rolls are put in the oven.

It takes 15 seconds to put the spring rolls in the oven.\\
The spring rolls must cook in the hot oven for 8 minutes.\\
It takes 30 seconds to transfer the spring rolls to a serving dish.\\
5 (c) (i) Complete Figure 3 above.

5 (c) (ii) Determine the latest time at which the oven can be switched on in order for the spring rolls to be served at the same time as the rest of the meal.\\[0pt]
[2 marks]\\

\includegraphics[max width=\textwidth, alt={}, center]{5a826f8b-4751-4589-ad0a-109fc5c821f2-09_2488_1716_219_153}
\end{enumerate}

\hfill \mbox{\textit{AQA Further AS Paper 2 Discrete 2018 Q5 [9]}}

This paper (7 questions)

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Q1 1 Q2 1 Q3 4 Q4 6 Q5 9 Q6 5 Q7 14

Label	Activity	Duration	Immediate predecessors
J	Switch on and heat oven		-
K	Put spring rolls in oven and cook
\(L\)	Transfer spring rolls to serving dish