6 Answer parts (i), (ii) and (iii) of this question on the insert provided. The activity network for a project is shown below. The durations are in minutes. The events are numbered 1, 2, 3, etc. for reference.
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1. Complete the table in the insert to show the immediate predecessors for each activity.
2. Explain why the dummy activity is needed between event 2 and event 3, and why the dummy activity is needed between event 4 and event 5 .
3. Carry out a forward pass to find the early event times and a backward pass to find the late event times. Record your early event times and late event times in the table in the insert. Write down the minimum project completion time and the critical activities. Suppose that the duration of activity \(K\) changes to \(x\) minutes.
4. Find, in terms of \(x\), expressions for the early event time and the late event time for event 9 .
5. Find the maximum duration of activity \(K\) that will not affect the minimum project completion time found in part (iii). \section*{ADVANCED GCE
   MATHEMATICS} Decision Mathematics 2
   INSERT for Questions 5 and 6
6. Dummy activity is needed between event 2 and event 3 because \(\_\_\_\_\)
   Dummy activity is needed between event 4 and event 5 because \(\_\_\_\_\)

7. Event	1	2	3	4	5	6	7	8	9	10
   Early event time
   Late event time

   Minimum project completion time = \(\_\_\_\_\) minutes Critical activities: \(\_\_\_\_\) \section*{Answer part (iv) and part (v) in your answer booklet.} OCR
   RECOGNISING ACHIEVEMENT