In must-be-false questions, we need to identify which statement directly contradicts the established facts. The stimulus establishes clear constraints: 100 total residents, with exactly 60 musicians and therefore exactly 40 artists (since everyone is either one or the other, but not both). Choice A claims there are more artists than musicians, which would require more than 60 artists. However, we know definitively that there are only 40 artists and 60 musicians. This directly contradicts the mathematical certainty established by the facts. Choice D might seem tempting because it states the obvious (60 > 40), but the question asks what must be FALSE, not what must be true. The key insight for must-be-false questions is to look for statements that create mathematical impossibilities given the constraints, not merely unlikely scenarios.

With 50 total members and 35 students, exactly 15 must be teachers. Choice A claims teachers outnumber students, requiring more than 35 teachers. Since we know there are exactly 15 teachers, this statement is mathematically impossible (15 cannot be greater than 35). Choice E accurately reflects the established relationship (students > teachers), but this confirms rather than contradicts our facts. The fundamental principle in must-be-false questions is identifying statements that demand impossible numerical relationships. We need answers that require quantities exceeding the established constraints, not those that merely state obvious or redundant information about the given scenario.

This question establishes precise numerical constraints: 75 total employees with 45 in sales, leaving exactly 30 in marketing. Choice A states that marketing has more employees than sales, which would require marketing to have more than 45 employees. Since we know marketing has exactly 30 employees, this statement creates a mathematical impossibility. The established facts make it impossible for 30 to be greater than 45. Choice D might appear redundant since it states what we already know (sales > marketing), but remember that must-be-false questions seek statements that contradict established facts, not those that confirm them. The critical distinction in must-be-false reasoning is identifying statements that violate the numerical certainties rather than those that merely restate given information.

The facts establish 300 total animals with 180 mammals, leaving exactly 120 reptiles. Choice A claims there are more reptiles than mammals, requiring more than 180 reptiles. However, we know definitively there are only 120 reptiles compared to 180 mammals. This creates a direct mathematical contradiction (120 cannot exceed 180). Choice D might seem redundant because it states what we can deduce (mammals > reptiles), but it remains consistent with our established facts. Must-be-false questions demand we identify statements that violate numerical certainties, not those that confirm obvious relationships. The answer must create an arithmetically impossible scenario given the constraints, requiring numbers that exceed what the established totals allow.

With 150 total participants and 90 juniors, we know there are exactly 60 seniors. Choice E states that seniors outnumber juniors, which would require more than 90 seniors. Since we have established that there are only 60 seniors, this creates a mathematical impossibility (60 cannot be greater than 90). Choice A correctly identifies that juniors outnumber seniors (90 > 60), but this confirms the established relationship rather than contradicting it. The crucial skill in must-be-false questions is distinguishing between statements that violate the numerical constraints versus those that accurately describe the given relationships. We need statements that create impossible mathematical scenarios, not merely obvious or redundant ones.

The stimulus establishes 200 total meals with 120 vegetarian, leaving exactly 80 non-vegetarian meals. Choice A states that non-vegetarian meals outnumber vegetarian meals, which would require more than 120 non-vegetarian meals. Since we know there are only 80 non-vegetarian meals compared to 120 vegetarian ones, this creates a mathematical impossibility (80 cannot exceed 120). Choice D correctly identifies that vegetarian meals outnumber non-vegetarian meals, but this confirms the established relationship. Must-be-false questions require identifying statements that create numerical contradictions with the given facts. The answer must demand quantities that violate the established constraints, making the statement arithmetically impossible rather than merely incorrect or redundant.

The established facts show 250 total participants with 150 chess players, leaving exactly 100 checkers players. Choice A claims checkers players outnumber chess players, requiring more than 150 checkers players. Since we know there are only 100 checkers players compared to 150 chess players, this statement is mathematically impossible (100 cannot exceed 150). Choice E correctly states that chess players outnumber checkers players (150 > 100), confirming rather than contradicting our established facts. Must-be-false questions require us to identify statements that create numerical impossibilities with the given constraints. The correct answer must demand quantities that exceed what the established totals permit, making the statement arithmetically contradictory rather than merely obvious.

The stimulus creates definitive constraints: 120 students total, 70 on soccer, therefore exactly 50 on basketball. Choice A claims basketball has more players than soccer, requiring basketball to have more than 70 players. However, we know with mathematical certainty that basketball has exactly 50 players while soccer has 70. This creates a direct contradiction with the established facts (50 cannot be greater than 70). Choice D states the obvious relationship (70 > 50), but this confirms rather than contradicts our facts. In must-be-false questions, we must distinguish between statements that violate mathematical certainties versus those that seem redundant but remain consistent with the given information. The answer must create an impossible scenario, not merely an obvious one.

By inclusion–exclusion, 25 attended only A (38−13) and 22 attended only B (35−13), so exactly 30 attending only A is impossible. The other statements are either true on these numbers or consistent with them.

B failed math, so B cannot be among those who passed both tests. The other statements are consistent with exactly two students passing both and everyone passing at least one test.