| Exam Board | OCR |
|---|---|
| Module | Further Discrete (Further Discrete) |
| Year | 2019 |
| Session | June |
| Marks | 7 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Critical Path Analysis |
| Type | Complete precedence table from network |
| Difficulty | Standard +0.3 This is a multi-part question covering standard critical path analysis techniques (precedence tables, latest start times) plus basic combinatorics. Parts (a)-(c) are routine applications of CPA algorithms requiring careful reading of diagrams but no novel insight. Part (d) involves straightforward combinations summing over cases. While multi-step, each component is a textbook exercise making this slightly easier than average for Further Maths Decision content. |
| Spec | 5.01a Permutations and combinations: evaluate probabilities5.01b Selection/arrangement: probability problems7.01e Permutations: ordered subsets of r from n elements7.01g Arrangements in a line: with repetition and restriction7.05a Critical path analysis: activity on arc networks7.05b Forward and backward pass: earliest/latest times, critical activities |
| Activity | Workers |
| A | 2 |
| B | X |
| C | |
| D | |
| E | |
| F |
| Answer | Marks | Guidance |
|---|---|---|
| 2 | (a) | Activity Immediate predecessors |
| Answer | Marks |
|---|---|
| F D | B1 |
| B1 | 1.1 |
| 1.1 | Rows C and F correct |
| Answer | Marks | Guidance |
|---|---|---|
| (b) | Activity | Latest start time (days) |
| B | 1 | |
| C | 5 | |
| F | 8 | |
| (c) | Construction | B1 |
| Vertices | A, B, C, D, E | V = 5 |
| Edges | AB, AC, AE, BC, BD, BE | E = 6 |
| Regions | ABC, ABE, ACBE | R = 3 |
| Activity | Immediate predecessors | |
| A | - | |
| B | - | |
| C | A | |
| D | A, B | |
| E | C, D | |
| (d) | (i) | 8C = 28 ways to choose 2 workers for A |
| 2 | B1 | 1.1 |
| (ii) | Number of workers for B can be 1, 2 or 3 |
| Answer | Marks |
|---|---|
| 1 2 3 | M1 |
| A1 | 3.1a |
| 1.1 | 1, 2, or 3 workers for B |
| 41 | Final answer 63 = SC1 |
Question 2:
2 | (a) | Activity Immediate predecessors
A -
B -
C A
D A, B
E C, D
F D | B1
B1 | 1.1
1.1 | Rows C and F correct
Rows D and E correct
With nothing in rows A and B
(b) | Activity | Latest start time (days) | B1 | 1.1 | All correct
B | 1
C | 5
F | 8
(c) | Construction | B1 | 1.2
Vertices | A, B, C, D, E | V = 5
Edges | AB, AC, AE, BC, BD, BE | E = 6
Regions | ABC, ABE, ACBE | R = 3
Activity | Immediate predecessors
A | -
B | -
C | A
D | A, B
E | C, D
(d) | (i) | 8C = 28 ways to choose 2 workers for A
2 | B1 | 1.1 | 28
(ii) | Number of workers for B can be 1, 2 or 3
6C + 6C + 6C = 6 + 15 + 20 = 41
1 2 3 | M1
A1 | 3.1a
1.1 | 1, 2, or 3 workers for B
41 | Final answer 63 = SC1
[7]2 A project is represented by the activity network and cascade chart below. The table showing the number of workers needed for each activity is incomplete. Each activity needs at least 1 worker.\\
\includegraphics[max width=\textwidth, alt={}, center]{7717b4ca-45ab-4111-9f59-5a3abb04b388-2_202_565_1605_201}\\
\includegraphics[max width=\textwidth, alt={}, center]{7717b4ca-45ab-4111-9f59-5a3abb04b388-2_328_560_1548_820}
\begin{center}
\begin{tabular}{ | c | c | }
\hline
Activity & Workers \\
\hline
A & 2 \\
\hline
B & X \\
\hline
C & \\
\hline
D & \\
\hline
E & \\
\hline
F & \\
\hline
\end{tabular}
\end{center}
\begin{enumerate}[label=(\alph*)]
\item Complete the table in the Printed Answer Booklet to show the immediate predecessors for each activity.
\item Calculate the latest start time for each non-critical activity.
The minimum number of workers needed is 5 .
\item What type of problem (existence, construction, enumeration or optimisation) is the allocation of a number of workers to the activities?
There are 8 workers available who can do activities A and B .
\item \begin{enumerate}[label=(\roman*)]
\item Find the number of ways that the workers for activity A can be chosen.
\item When the workers have been chosen for activity A , find the total number of ways of choosing the workers for activity B for all the different possible values of x , where $\mathrm { x } \geqslant 1$.
\end{enumerate}\end{enumerate}
\hfill \mbox{\textit{OCR Further Discrete 2019 Q2 [7]}}