Question

Which of the following can be the schedule for the six experiments?

• A. Aug: P, R | Sep: T, X | Oct: Y, Z
• B. Aug: R, T | Sep: X, Y | Oct: P, Z
• C. Aug: R, X | Sep: T, Y | Oct: P, Z
• D. Aug: X, Y | Sep: P, Z | Oct: R, T
• E. Aug: Y, Z | Sep: R, T | Oct: P, X
• A. P
• B. R
• C. X
• D. Y
• J. Z
• A. P is performed in August.
• B. R is performed in September.
• C. X is performed in August.
• D. Y is performed in September.
• O. Z is performed in October.
• A. P and X
• B. P and Y
• C. R and Z
• D. T and Y
• T. X and Y
• A. P and R
• B. P and Y
• C. R and Y
• D. R and Z
• Y. X and Y
• A. R is performed in the same month as T.
• B. R is performed in the same month as X.
• C. T is performed in August.
• D. X is performed in August.
• ^. Y is performed in October.
• A. P is performed in August.
• B. R is performed in September.
• C. T is performed in September.
• D. X is performed in August.
• c. Z is performed in October.

Answer 1

The Correct Option is C

Step 1: Understanding the Concept:
This is an "acceptable schedule" question. We need to check each option against the rules until we find one that is valid.
Step 2: Detailed Explanation:
- (A) Aug: P, R | Sep: T, X | Oct: Y, Z: This violates Rule 3, as T and X are in the same month (September).
- (B) Aug: R, T | Sep: X, Y | Oct: P, Z: This violates Rule 2, as T is in August, but it must be in September or October.
- (C) Aug: R, X | Sep: T, Y | Oct: P, Z: - Rule 1 (R in Aug/Sep): Met, R is in August. - Rule 2 (T in Sep/Oct): Met, T is in September. - Rule 3 (T \(\neq\) X): Met, T is in September, X is in August. - Rule 4 (X \textless Z): Met, X is in August, Z is in October. Since all rules are satisfied, this is a valid schedule.
- (D) Aug: X, Y | Sep: P, Z | Oct: R, T: This violates Rule 1, as R is in October, but it must be in August or September.
- (E) Aug: Y, Z | Sep: R, T | Oct: P, X: This violates Rule 4, as X is in October, which is later than Z in August.
Step 3: Final Answer:
The only schedule that satisfies all the rules is (C).

Answer 2

Step 1: Understanding the Concept:
This question asks which experiment can NEVER be in August. We need to check the rules to see which experiment is forbidden from being in the first month.
Step 2: Detailed Explanation:
The most relevant rule is Rule 4: "X must be performed in an earlier month than the month in which Z is performed."
- This means X must come before Z in the sequence of months. - If Z were in August (the earliest month), there would be no earlier month available for X to be performed in. - Therefore, Z cannot be in August.
To confirm the others are possible in August, we can refer to the valid schedule from question 1: - Aug: R, X | Sep: T, Y | Oct: P, Z - This shows R and X can be in August. We can also easily construct a schedule with P and Y in August. For example: - Aug: P, Y | Sep: R, X | Oct: T, Z. This is valid. So, P, R, X, and Y can all be in August.
Step 3: Final Answer:
Z is the only experiment that cannot be performed in August due to the rule that X must precede it.

Answer 3

Step 1: Understanding the Concept:
This is a conditional question. We add the new information (T is in September) to our rules and deduce what must necessarily follow.
Step 2: Detailed Explanation:
1. New Condition: T is in September.
2. Rule 3 (T \(\neq\) X): Since T is in September, X cannot be in September.
3. Rule 4 (X \textless Z): X must be performed in an earlier month than Z.
4. Combining these, X cannot be in September. If X were in October, there would be no later month for Z. Therefore, X must be in August.
5. Let's check the other options to see if they are necessarily true. - If X is in August and T is in September, then Z must be in October to satisfy X \textless Z. So (E) is also true. Let's re-read the question. - Ah, a subtlety. X must be in an earlier month. So if X is in August, Z can be in September or October. If X is in September, Z must be in October. - Let's re-do the deduction. 1. T is in September. 2. By Rule 3, X is not in September. 3. X cannot be in October, because then there would be no later month for Z (Rule 4). 4. Therefore, X must be in August. This makes option (C) a necessary truth.
6. Now let's re-evaluate (E) based on this. If X is in August, Z can be in September or October. If Z is in September, the September experiments would be T and Z. This is a possible scenario (e.g., Aug: R, X | Sep: T, Z | Oct: P, Y). Since Z does not HAVE to be in October, (E) is not a "must be true".
7. Let's check the other options. - (A) P can be in October. (See the valid scenario just constructed). - (B) R can be in August. (See the valid scenario). - (D) Y can be in August or October. (See the valid scenario).
Step 3: Final Answer:
Given that T is in September, X cannot be in September. To satisfy the condition that X is before Z, X must be in August.

Answer 4

Step 1: Understanding the Concept:
This is a conditional question. We add the condition that R and Z are in the same month and then see what possible pair can occupy the October slots.
Step 2: Detailed Explanation:
1. New Condition: R and Z are in the same month.
2. Rule 1 (R in Aug/Sep): R must be in August or September. Therefore, the RZ pair must be in August or September.
3. Rule 4 (X \textless Z): X must be in an earlier month than Z. - If the RZ pair is in September, X must be in August. - If the RZ pair is in August, this violates Rule 4, as there is no earlier month for X. - Therefore, the RZ pair must be in September, and X must be in August.
4. Summary of fixed experiments: - August: X and one other experiment. - September: R and Z. - October: Two remaining experiments.
5. Rule 2 (T in Sep/Oct): T must be in September or October. September is full with R and Z, so T must be in October.
6. Conclusion about October: One of the experiments in October must be T. The other experiment will be one of the remaining "floater" experiments (P or Y). 7. Checking the options: - (A) P and X: X must be in August. - (B) P and Y: T must be in October. - (C) R and Z: This pair is in September. - (D) T and Y: This is a possible pairing for October. - (E) X and Y: X must be in August.
Step 3: Final Answer:
Based on the deductions, the pair of experiments in October must include T. Option (D), T and Y, is a possible valid pair. A full valid schedule could be: Aug: X, P | Sep: R, Z | Oct: T, Y.

Answer 5

Step 1: Understanding the Concept:
This is a conditional question. We add the new information "T is performed in the month before Z" and then determine which pair of experiments could possibly be scheduled together. This is a "could be true" question.
Initial Rules Recap: 1. R \(\in\) \{Aug, Sep\} 2. T \(\in\) \{Sep, Oct\} 3. T \(\neq\) X (in the same month) 4. X \textless Z (X is in an earlier month than Z)
Step 2: Applying the New Condition and Deducing the Schedule:
1. New Condition: T is in the month before Z (T \textless Z).
2. Deduction 1: From Rule 2, T is in Sep or Oct. If T were in Oct, there would be no later month for Z. Therefore, T must be in September, and Z must be in October.
3. Deduction 2: From Rule 4 (X \textless Z) and our finding that Z is in Oct, X can be in Aug or Sep. However, Rule 3 states T and X cannot be in the same month. Since T is in Sep, X cannot be in Sep. Therefore, X must be in August.
4. Summary of fixed experiments: The schedule must have X in August, T in September, and Z in October. This gives us the following frame: - August: {X, \_} - September: {T, \_} - October: {Z, \_}
5. Placing remaining experiments: The remaining experiments are P, R, and Y. They must fill the three empty slots. Rule 1 states R must be in Aug or Sep. - Scenario A: R is in August. The August pair is {X, R}. The remaining P and Y fill the slots in Sep and Oct. A possible valid schedule is Aug:{X,R}, Sep:{T,P}, Oct:{Z,Y}. - Scenario B: R is in September. The September pair is {T, R}. The remaining P and Y fill the slots in Aug and Oct. A possible valid schedule is Aug:{X,P}, Sep:{T,R}, Oct:{Z,Y}.
Step 3: Evaluating the Options:
We are looking for a pair that can be performed together. Let's try to build a valid schedule for one of the options. - (A) P and R: In both scenarios above, P and R are in different months. - (B) P and Y: In both scenarios above, P and Y are in different months. - (C) R and Y: In both scenarios above, R and Y are in different months. - (D) R and Z: Z is in October, R must be in August or September. They can't be together. - (E) X and Y: Let's see if this is possible. X must be in August. Let's try to place Y in August as well. - August: {X, Y} - We already deduced T is in Sep and Z is in Oct. - September: {T, \_} - October: {Z, \_} - The remaining experiments are P and R. From Rule 1, R must be in Aug or Sep. August is full, so R must be in September. The September pair becomes {T, R}. - The last experiment, P, must go in the last available slot in October. The October pair becomes {Z, P}. - The full schedule is: Aug:{X,Y}, Sep:{T,R}, Oct:{Z,P}. Let's check all rules. R is in Sep (ok). T is in Sep (ok). T is not with X (ok). X is before Z (ok). - This is a valid schedule. Therefore, X and Y can be performed together.

Answer 6

Step 1: Understanding the Concept:
This is a conditional question asking what must be true if P and Y are scheduled in the same month. We must consider all possible months for the P-Y pair and see what conclusion holds in every case.
Step 2: Detailed Explanation by Cases:
Let's analyze the three possible months for the {P, Y} pair. The other four experiments are R, T, X, Z.
- Case 1: {P, Y are in August.} - August: {P, Y} - Rule 1 (R in Aug/Sep): August is full, so R must be in September. - Rule 2 (T in Sep/Oct): T must be in September or October. - The remaining experiments to place are R, T, X, Z in Sep and Oct. We know R is in Sep. - If T is in September, then Sep would be {R, T}. The remaining two experiments, X and Z, would go in October. This violates Rule 4 (X \textless Z), as they would be in the same month. So T cannot be in September. - T must be in October. - The schedule so far: Aug:{P,Y}, Sep:{R,\_}, Oct:{T,\_}. The remaining experiments are X and Z. - To satisfy Rule 4 (X \textless Z), X must go in September and Z must go in October. - The only possible schedule is: Aug:{P,Y}, Sep:{R,X}, Oct:{T,Z}. In this case, R and X are in the same month.
- Case 2: {P, Y are in September.} - September: {P, Y} - Rule 1 (R in Aug/Sep): September is full, so R must be in August. - Rule 2 (T in Sep/Oct): September is full, so T must be in October. - The remaining experiments are X and Z. The open slots are one in August and one in October. - To satisfy Rule 4 (X \textless Z), X must be in August and Z must be in October. - The only possible schedule is: Aug:{R,X}, Sep:{P,Y}, Oct:{T,Z}. In this case, R and X are in the same month.
- Case 3: {P, Y are in October.} - October: {P, Y} - Rule 2 (T in Sep/Oct): October is full, so T must be in September. - Rule 3 (T \(\neq\) X): Since T is in Sep, X cannot be in Sep. X must be in August. - The remaining experiments are R and Z. R must be in Aug or Sep (Rule 1). - If R were in September, Sep would be {T, R}. August would have X and Z. But this violates Rule 4 (X \textless Z). So R cannot be in September. - R must be in August. - The schedule so far: Aug:{X,R}, Sep:{T,\_}, Oct:{P,Y}. The only remaining experiment is Z, which must go in September. - The only possible schedule is: Aug:{R,X}, Sep:{T,Z}, Oct:{P,Y}. In this case, R and X are in the same month.
Step 3: Final Answer:
In all three possible cases, R and X must be performed in the same month. Therefore, statement (B) must be true.

Answer 7

Step 1: Understanding the Concept:
This is another "must be true" question with a new conditional statement. We need to find a conclusion that is a necessary consequence of the new rule and the original rules.
Step 2: Detailed Explanation:
1. New Condition: X is performed in the month before Y is performed (X \textless Y).
2. Original Rules Reminder: We also know from Rule 4 that X must be performed in a month before Z (X \textless Z).
3. Analysis of X's Position: Let's test the possible months for X. - Can X be in October? No, because it must come before both Y and Z, and there are no months after October. - Can X be in September? Let's assume X is in September. - If X is in September, then according to the new condition (X \textless Y) and the old rule (X \textless Z), both Y and Z must be performed in October. - This means the two experiments in October are {Y, Z}. - Now consider Rule 2: T must be in September or October. October is now full with {Y, Z}. So, T must be in September. - This would mean that the two experiments in September are {X, T}. - However, this violates Rule 3, which states that T and X cannot be performed in the same month. - This creates a contradiction. Therefore, the initial assumption that X can be in September must be false.
4. Conclusion: Since X cannot be in October and X cannot be in September, the only remaining possibility is that X must be performed in August.
Step 3: Final Answer:
The combination of the new condition and the original rules forces X to be in August. Therefore, the statement "X is performed in August" must be true.