# Question 1:

## Part (a)

| Answer/Working | Marks | Guidance |
|---|---|---|
| All top boxes complete, values generally increasing in direction of arrows (left to right), condone one rogue | M1 | a1M1: All top boxes complete |
| CAO (top boxes) — correct values: 0,4,7,9,12,13,18,24 etc. | A1 | a1A1: CAO top boxes |
| All bottom boxes complete, values generally decreasing in opposite direction of arrows (right to left), condone one rogue. Condone missing 0 and/or 24 at end event for M mark only | M1 | a2M1 |
| CAO (bottom boxes) | A1 **(4)** | a2A1 |

## Part (b)

| Answer/Working | Marks | Guidance |
|---|---|---|
| Activity D has a total float of $12 - 4 - 5 = 3$ (hours) | B1ft **(1)** | Correct calculation for their event times for activity D (all three figures must be seen) |

## Part (c)

| Answer/Working | Marks | Guidance |
|---|---|---|
| Lower bound $= \dfrac{4+3+7+\ldots+5+12}{24} = \dfrac{67}{24} = 2.791\ldots = 3$ workers | M1 A1 **(2)** | |

## Part (d)

| Answer/Working | Marks | Guidance |
|---|---|---|
| Gantt chart — correct rows for C, F, G, J (critical activities) | M1 | |
| Correct placement of A, B | A1 | |
| Correct placement of D, E | A1 | |
| Correct placement of H, I, K, L | A1 **(4)** | |

## Part (e)

| Answer/Working | Marks | Guidance |
|---|---|---|
| Minimum is 4 workers e.g. activities F, H, I and L together with $12 < \text{time} < 13$ | dM1 A1 **(2)** | |

**Total: 13 marks**

# Question 1 (Critical Path Analysis):

## Part (c) - Lower Bound:
| Answer | Mark | Guidance |
|--------|------|----------|
| Attempt to find lower bound: value in interval $[55-79]$ / their finish time, or sum of activities (12 values) / their finish time, or awrt 2.8 | c1M1 | Condone one missing value |
| Answer of 3, requires correct calculation or awrt 2.8 seen | c1A1 | CSO - answer of 3 with no working scores no marks |

## Part (d) - Scheduling Diagram:
| Answer | Mark | Guidance |
|--------|------|----------|
| At least eight different activities labelled including at least five floats | d1M1 | A scheduling diagram with no floats scores M0 |
| Critical activities C, F, G, J correct (appearing just once) and three non-critical activities with correct duration and total float | d1A1 | |
| Any five non-critical activities correct | d2A1 | Not dependent on previous A mark |
| Completely correct Gantt chart with exactly twelve activities appearing just once | d3A1 | CSO |

## Part (e) - Workers:
| Answer | Mark | Guidance |
|--------|------|----------|
| Statement with correct number of workers (4) and correct activities (F, H, I, L) with numerical time $12 \leq t \leq 13$ | e1depM1 | Mark numerical value only, dependent on M mark in (d) |
| Completely correct statement with both time and activities; time within $12 < t < 13$ (e.g. 12.5) and activities F, H, I and L | e1A1 | Condone 'days' instead of 'hours'; strict inequalities mean time of exactly 12 is incorrect |

**Float Table:**
| Activity | Duration + Float | Activity | Duration + Float | Activity | Duration + Float |
|----------|-----------------|----------|-----------------|----------|-----------------|
| A | 0 to 4, F: 4 to 7 | F | 7 to 13, Critical | K | 13 to 18, F: 18 to 24 |
| B | 0 to 3, F: 3 to 13 | G | 13 to 18, Critical | L | 9 to 21, F: 21 to 24 |
| C | 0 to 7, Critical | H | 7 to 13, F: 13 to 18 | | |
| D | 4 to 9, F: 9 to 12 | I | 7 to 13, F: 13 to 18 | | |
| E | 4 to 6, F: 6 to 18 | J | 18 to 24, Critical | | |

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