Comprehension
A library is equipped with a system of pneumatic tubes for sending documents from one to another of exactly six departments-G, H, L, M, S, and T. A tube line is a pair of tubes that connects one department with exactly one other department, with documents moving in one direction in one tube and in the opposite direction in the other tube. The library's system consists of the following seven tube lines and no others.
Line 1 connects H and L.
Line 2 connects H and S.
Line 3 connects L and T.
Line 4 connects S and T.
Line 5 connects M and T.
Line 6 connects L and M.
Line 7 connects G and H.
Use of the system is subject to the following restrictions:
Documents to be sent between departments that are not connected by a tube line can be transferred from one line to another at departments served by two or more lines, until the document reaches its destination.
A document cannot use any tube line more than once on its way to its destination, nor can the document return to its department of origin on its way to its destination.
Question: 1
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
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
Show Hint
When solving network or pathway problems, drawing a simple diagram of the nodes (departments) and connections (lines) is extremely helpful. This allows you to visually trace the paths and quickly spot impossible transfers. If you find multiple incorrect options in an EXCEPT question, double-check your diagram and the rules, but be aware that test questions can sometimes be flawed.
Updated On: Oct 1, 2025
- line 4, line 5
- line 2, line 3, line 5
- line 2, line 1, line 6
- line 4, line 1, line 6
- line 2, line 1, line 3, line 5
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The Correct Option is D
Solution and Explanation
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.
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.
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Question: 2
Which of the following is a complete and accurate list of the lines any one of which could be the second line used by a document sent from T to G?
Which of the following is a complete and accurate list of the lines any one of which could be the second line used by a document sent from T to G?
Show Hint
For pathway questions, work methodically from the start point. List all possible first steps, and then for each first step, list all possible subsequent steps, tracing the path toward the destination.
Updated On: Oct 1, 2025
- Lines 1, 2, and 3
- Lines 1, 2, and 4
- Lines 1, 2, and 6
- Lines 2, 3, and 4
- Lines 2, 3, and 6
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The Correct Option is C
Solution and Explanation
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.
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.
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Question: 3
If line 3 cannot be used, a document to be sent from T to H that uses as few tube lines as possible must use line
If line 3 cannot be used, a document to be sent from T to H that uses as few tube lines as possible must use line
Show Hint
When looking for the shortest path, first identify all possible paths and count the number of steps (lines) in each. Then, select the one with the minimum number of steps.
Updated On: Oct 1, 2025
- 1
- 2
- 5
(D) 6
(E) 7
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The Correct Option is B
Solution and Explanation
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.
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.
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Question: 4
A pathway from M to H that includes as many tube lines as possible must include lines
A pathway from M to H that includes as many tube lines as possible must include lines
Show Hint
To find the longest simple path in a small network, systematically trace all possible routes from start to finish without reusing any lines. List them out and count the steps to find the maximum.
Updated On: Oct 1, 2025
- 1 and 2
- 1 and 3
- 3 and 4
- 4 and 5
- 5 and 6
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The Correct Option is C
Solution and Explanation
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.
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.
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