Question

Any of the following is an acceptable pathway for a document to be sent from S to M, listing all lines used in order from the line first used to the line last used, EXCEPT

• A. line 4, line 5
• B. line 2, line 3, line 5
• C. line 2, line 1, line 6
• D. line 4, line 1, line 6
• E. line 2, line 1, line 3, line 5
• A. Lines 1, 2, and 3
• B. Lines 1, 2, and 4
• C. Lines 1, 2, and 6
• D. Lines 2, 3, and 4
• J. Lines 2, 3, and 6
• A. 1
• B. 2
• C. 5
  (D) 6
  (E) 7
• A. 1 and 2
• B. 1 and 3
• C. 3 and 4
• D. 4 and 5
• R. 5 and 6

Answer 1

The Correct Option is D

Step 1: Understanding the Concept:
This is an EXCEPT question. We need to trace each proposed path from S to M and find the one that is invalid. A path is invalid if it doesn't follow the connections defined by the lines.
Step 2: Detailed Explanation:
Let's trace each path from the starting department S to the destination M: - (A) line 4, line 5: - Start at S. Take line 4, which connects S and T. Arrive at T. - From T, take line 5, which connects T and M. Arrive at M. - Path: S \(\xrightarrow{4}\) T \(\xrightarrow{5}\) M. This is a valid path.
- (B) line 2, line 3, line 5: - This path is impossible. Start at S. Take line 2, arriving at H. The next line listed is line 3, which connects L and T. The department H is not connected to line 3. Therefore, this path is invalid.
- (C) line 2, line 1, line 6: - Start at S. Take line 2, arriving at H. - From H, take line 1, arriving at L. - From L, take line 6, arriving at M. - Path: S \(\xrightarrow{2}\) H \(\xrightarrow{1}\) L \(\xrightarrow{6}\) M. This is a valid path.
- (D) line 4, line 1, line 6: - Start at S. Take line 4, arriving at T. - The next line listed is line 1, which connects H and L. The department T is not connected to line 1. Therefore, this path is invalid.
- (E) line 2, line 1, line 3, line 5: - Start at S. Take line 2, arriving at H. - From H, take line 1, arriving at L. - From L, take line 3, arriving at T. - From T, take line 5, arriving at M. - Path: S \(\xrightarrow{2}\) H \(\xrightarrow{1}\) L \(\xrightarrow{3}\) T \(\xrightarrow{5}\) M. This is a valid path.
Note on Question Flaw: Both paths (B) and (D) are invalid as described. This indicates a flaw in the question's design. However, in a forced-choice scenario, we must select one. Both are equally impossible. We will select (D) as the answer to be explained.
Step 3: Final Answer:
The path described in (D) is not possible. A document starting at S and taking line 4 arrives at T. From T, it is impossible to take line 1, as line 1 only connects H and L. Therefore, this is not an acceptable pathway.

Answer 2

Step 1: Understanding the Concept:
We need to find all possible paths from T to G and then collect all the unique lines that appear as the second step in any of these paths.
Step 2: Detailed Explanation:
1. Identify the start (T) and end (G) points. The final connection to G must be via Line 7 from H. So, the goal for any path from T is to reach H.
2. List the possible first moves from T. T is connected to L (line 3), S (line 4), and M (line 5).
3. Trace the paths for each possible first move and identify the second line. - \rightarrowPath starts with Line 3 (T \(\rightarrow\) L):\rightarrow
- After the first step (Line 3), we are at department L. From L, we need to get to H. Line 1 connects L directly to H. - So, the second line in this path is Line 1. (Full path: T \(\xrightarrow{3}\) L \(\xrightarrow{1}\) H \(\xrightarrow{7}\) G). - \rightarrowPath starts with Line 4 (T \(\rightarrow\) S):\rightarrow
- After the first step (Line 4), we are at department S. From S, we need to get to H. Line 2 connects S directly to H. - So, the second line in this path is Line 2. (Full path: T \(\xrightarrow{4}\) S \(\xrightarrow{2}\) H \(\xrightarrow{7}\) G). - \rightarrowPath starts with Line 5 (T \(\rightarrow\) M):\rightarrow
- After the first step (Line 5), we are at department M. From M, we need to get to H. Line 6 connects M to L. - So, the second line in this path is Line 6. (Full path: T \(\xrightarrow{5}\) M \(\xrightarrow{6}\) L \(\xrightarrow{1}\) H \(\xrightarrow{7}\) G).
4. Consolidate the findings. The possible second lines used in a path from T to G are Line 1, Line 2, and Line 6.
Step 3: Final Answer:
The complete and accurate list of possible second lines is Lines 1, 2, and 6.

Answer 3

Step 1: Understanding the Concept:
We have a new condition (Line 3 is unavailable) and an objective (find the shortest path from T to H). We need to identify one of the lines that is essential for this shortest path.
Step 2: Detailed Explanation:
1. Goal: Find the shortest path from T to H.
2. New Condition: Line 3 (L-T) is out of service.
3. Analyze possible paths from T to H. - Without Line 3, T's connections are to S (Line 4) and M (Line 5). - \rightarrowPath 1 (via S):\rightarrow Go from T to S using Line 4.
From S, go to H using Line 2. The path is T \(\xrightarrow{4}\) S \(\xrightarrow{2}\) H. This path uses 2 lines.
- \rightarrowPath 2 (via M):\rightarrow Go from T to M using Line 5. From M, go to L using Line 6. From L, go to H using Line 1. The path is T \(\xrightarrow{5}\) M \(\xrightarrow{6}\) L \(\xrightarrow{1}\) H. This path uses 3 lines.
4. Identify the shortest path. The shortest path is Path 1, which uses only 2 lines (Line 4 and Line 2).
5. Answer the question. The question asks which line the shortest path "must use". The shortest path uses Line 4 and Line 2. Looking at the options, Line 2 is listed. The shortest path does not use Line 1, Line 5, Line 6, or Line 7 (which connects to G, not from T).
Step 3: Final Answer:
The shortest path from T to H without using line 3 is via lines 4 and 2. Therefore, the path must use line 2.

Answer 4

Step 1: Understanding the Concept:
The goal is to find the longest possible simple path (no repeated lines) from M to H. Then we need to identify which pair of lines from the options is part of this longest path.
Step 2: Detailed Explanation:
1. Find all possible simple paths from M to H. - Path 1: M \(\xrightarrow{6}\) L \(\xrightarrow{1}\) H. (Length: 2 lines) - Path 2: M \(\xrightarrow{5}\) T \(\xrightarrow{3}\) L \(\xrightarrow{1}\) H. (Length: 3 lines) - Path 3: M \(\xrightarrow{5}\) T \(\xrightarrow{4}\) S \(\xrightarrow{2}\) H. (Length: 3 lines) - Path 4: M \(\xrightarrow{6}\) L \(\xrightarrow{3}\) T \(\xrightarrow{4}\) S \(\xrightarrow{2}\) H. (Length: 4 lines) - Can we make a longer path? Let's trace the longest path (Path 4) again: M to L, L to T, T to S, S to H. We have visited M, L, T, S, H. The only unvisited department is G, but we cannot reach it without going through H, which is our destination. So, this 4-line path is the longest possible simple path.
2. Identify the longest path. The longest path is M \(\rightarrow\) L \(\rightarrow\) T \(\rightarrow\) S \(\rightarrow\) H, which uses lines {6, 3, 4, 2}.
3. Check the options against the longest path. We are looking for a pair of lines that are both in the set {6, 3, 4, 2}. - (A) 1 and 2: The path does not use line 1. - (B) 1 and 3: The path does not use line 1. - (C) 3 and 4: The path {6, 3, 4, 2} includes both line 3 and line 4. - (D) 4 and 5: The path does not use line 5. - (E) 5 and 6: The path does not use line 5.
Step 3: Final Answer:
The longest pathway from M to H is M-L-T-S-H. This path includes lines 3 and 4.