Question:
Directions: Professor Mukhopadhay works only on Mondays, Tuesdays, Wednesdays, Fridays, and Saturdays. She performs four different activities: lecturing, conducting quizzes, evaluating quizzes, and working on consultancy projects. Each working day she performs exactly one activity in the morning and exactly one activity in the afternoon. Her weekly schedule must satisfy all of the following restrictions:
1. She conducts quizzes on exactly three mornings.
2. If she conducts a quiz on Monday, she does not conduct a quiz on Tuesday.
3. She lectures in the afternoon on exactly two consecutive calendar days.
4. She evaluates quizzes on exactly one morning and three afternoons.
5. She works on a consultancy project on exactly one morning.
6. On Saturday, she neither lectures nor conducts a quiz.
Which of the following statements must be true?
Directions: Professor Mukhopadhay works only on Mondays, Tuesdays, Wednesdays, Fridays, and Saturdays. She performs four different activities: lecturing, conducting quizzes, evaluating quizzes, and working on consultancy projects. Each working day she performs exactly one activity in the morning and exactly one activity in the afternoon. Her weekly schedule must satisfy all of the following restrictions:
1. She conducts quizzes on exactly three mornings.
2. If she conducts a quiz on Monday, she does not conduct a quiz on Tuesday.
3. She lectures in the afternoon on exactly two consecutive calendar days.
4. She evaluates quizzes on exactly one morning and three afternoons.
5. She works on a consultancy project on exactly one morning.
6. On Saturday, she neither lectures nor conducts a quiz.
Which of the following statements must be true?
1. She conducts quizzes on exactly three mornings.
2. If she conducts a quiz on Monday, she does not conduct a quiz on Tuesday.
3. She lectures in the afternoon on exactly two consecutive calendar days.
4. She evaluates quizzes on exactly one morning and three afternoons.
5. She works on a consultancy project on exactly one morning.
6. On Saturday, she neither lectures nor conducts a quiz.
Which of the following statements must be true?
Show Hint
Notice the whole puzzle really has only two open choices, which three mornings get the quiz and which two afternoons get the lecture, then check each statement against every combination.
Updated On: Jul 10, 2026
- There is one day on which she evaluates quizzes both in the morning and in the afternoon
- She works on the consultancy project on one of the days on which she lectures
- She works on the consultancy project on one of the days on which she evaluates quizzes
- She lectures on one of the days on which she conducts a quiz
Show Solution
Verified By Collegedunia
The Correct Option is D
Solution and Explanation
Step 1: Understanding the Question.
We need to find the one statement, among five, that is guaranteed to be true in every schedule the professor could possibly follow, not just true in one particular week.
Step 2: Key Formula or Approach.
From the restrictions, the three quiz mornings can only be Monday, Wednesday, Friday (if Monday is a quiz day) or Tuesday, Wednesday, Friday (if it is not). The two lecture afternoons, being on consecutive calendar days among Monday, Tuesday, Wednesday, Friday, Saturday, can only be Monday-Tuesday or Tuesday-Wednesday, since Wednesday-Friday is not consecutive because Thursday is skipped, and Friday-Saturday is ruled out since Saturday never lectures. A statement is must be true only if it holds for all four combinations of these two choices.
Step 3: Detailed Explanation.
Check option (EE) first, since it is the simplest to test: does the professor always lecture on a day she also conducts a quiz? Pair the quiz set Monday, Wednesday, Friday with the lecture pair Monday, Tuesday: they share Monday. Pair the quiz set Tuesday, Wednesday, Friday with Monday, Tuesday: they share Tuesday. Pair Monday, Wednesday, Friday with Tuesday, Wednesday: they share Wednesday. Pair Tuesday, Wednesday, Friday with Tuesday, Wednesday: they share both Tuesday and Wednesday. In all four combinations there is at least one overlapping day, so option (EE) always holds.
Now rule out the rest using two concrete, fully valid schedules. Schedule 1: Monday (quiz, lecture), Tuesday (evaluate, lecture), Wednesday (quiz, evaluate), Friday (quiz, evaluate), Saturday (consultancy, evaluate). This satisfies every restriction, so it is a valid week. In Schedule 1, evaluation happens on Tuesday morning and on Wednesday, Friday, Saturday afternoons, with no single day getting evaluation both morning and afternoon, so option (AA) fails here. Consultancy sits on Saturday morning while lectures sit on Monday and Tuesday, different days, so option (BB) fails here too.
Schedule 2: Monday (quiz, lecture), Tuesday (consultancy, lecture), Wednesday (quiz, evaluate), Friday (quiz, evaluate), Saturday (evaluate, evaluate). This is also a fully valid week. In Schedule 2, consultancy sits on Tuesday, which is not an evaluation day at all, since Tuesday is consultancy in the morning and lecture in the afternoon, so option (CC) fails here. Lectures sit on Monday and Tuesday, and evaluation sits on Wednesday, Friday, Saturday afternoons plus Saturday morning, with no overlap with Monday or Tuesday, so option (DD) fails here too.
Step 4: Final Answer.
Options (AA), (BB), (CC) and (DD) can each be broken by at least one valid schedule, while option (EE) survives every possible combination of quiz mornings and lecture afternoons. The statement that must be true is that she lectures on one of the days on which she conducts a quiz. \[ \boxed{\text{Option (EE)}} \]
We need to find the one statement, among five, that is guaranteed to be true in every schedule the professor could possibly follow, not just true in one particular week.
Step 2: Key Formula or Approach.
From the restrictions, the three quiz mornings can only be Monday, Wednesday, Friday (if Monday is a quiz day) or Tuesday, Wednesday, Friday (if it is not). The two lecture afternoons, being on consecutive calendar days among Monday, Tuesday, Wednesday, Friday, Saturday, can only be Monday-Tuesday or Tuesday-Wednesday, since Wednesday-Friday is not consecutive because Thursday is skipped, and Friday-Saturday is ruled out since Saturday never lectures. A statement is must be true only if it holds for all four combinations of these two choices.
Step 3: Detailed Explanation.
Check option (EE) first, since it is the simplest to test: does the professor always lecture on a day she also conducts a quiz? Pair the quiz set Monday, Wednesday, Friday with the lecture pair Monday, Tuesday: they share Monday. Pair the quiz set Tuesday, Wednesday, Friday with Monday, Tuesday: they share Tuesday. Pair Monday, Wednesday, Friday with Tuesday, Wednesday: they share Wednesday. Pair Tuesday, Wednesday, Friday with Tuesday, Wednesday: they share both Tuesday and Wednesday. In all four combinations there is at least one overlapping day, so option (EE) always holds.
Now rule out the rest using two concrete, fully valid schedules. Schedule 1: Monday (quiz, lecture), Tuesday (evaluate, lecture), Wednesday (quiz, evaluate), Friday (quiz, evaluate), Saturday (consultancy, evaluate). This satisfies every restriction, so it is a valid week. In Schedule 1, evaluation happens on Tuesday morning and on Wednesday, Friday, Saturday afternoons, with no single day getting evaluation both morning and afternoon, so option (AA) fails here. Consultancy sits on Saturday morning while lectures sit on Monday and Tuesday, different days, so option (BB) fails here too.
Schedule 2: Monday (quiz, lecture), Tuesday (consultancy, lecture), Wednesday (quiz, evaluate), Friday (quiz, evaluate), Saturday (evaluate, evaluate). This is also a fully valid week. In Schedule 2, consultancy sits on Tuesday, which is not an evaluation day at all, since Tuesday is consultancy in the morning and lecture in the afternoon, so option (CC) fails here. Lectures sit on Monday and Tuesday, and evaluation sits on Wednesday, Friday, Saturday afternoons plus Saturday morning, with no overlap with Monday or Tuesday, so option (DD) fails here too.
Step 4: Final Answer.
Options (AA), (BB), (CC) and (DD) can each be broken by at least one valid schedule, while option (EE) survives every possible combination of quiz mornings and lecture afternoons. The statement that must be true is that she lectures on one of the days on which she conducts a quiz. \[ \boxed{\text{Option (EE)}} \]
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