In this question on statements and conclusions, you are given information about girls who are clever and hardworking. Accept the statements as true even if they seem unrealistic, then read all the conclusions carefully and decide which of them logically follow. Statements: (I) Some girls are clever. (II) All clever are hardworking. Conclusions: (I) Some girls are not hardworking. (II) Some girls are hardworking. (III) Some hardworking are not girls.

Introduction / Context:
This question explores logical relationships involving girls, clever individuals, and hardworking individuals. The statements link cleverness with hard work and assert that some girls are clever. You are asked to determine which of the three possible conclusions necessarily follow from this information. The key is to use Venn style reasoning and avoid assuming more about the groups than what is stated.

Given Data / Assumptions:

• Statement (I): Some girls are clever.
• Statement (II): All clever are hardworking.
• Conclusion (I): Some girls are not hardworking.
• Conclusion (II): Some girls are hardworking.
• Conclusion (III): Some hardworking are not girls.
• We accept the statements as true and do not introduce extra relationships.

Concept / Approach:
The two statements can be chained: some girls belong to the clever group, and every clever person belongs to the hardworking group. Therefore, those clever girls must also belong to the hardworking group. This directly supports one conclusion. However, the information does not specify anything about girls who are not clever or about hardworking individuals who are not girls. To test each conclusion, we construct possible diagrams and see whether a conclusion fails in at least one of them.

Step-by-Step Solution:
Step 1: Draw a large set representing hardworking individuals.Step 2: Place the clever set completely inside the hardworking set, because all clever are hardworking.Step 3: Represent the girls set so that at least some girls lie inside the clever region, capturing “Some girls are clever.” There may be other girls outside the clever region about whom nothing is said.Step 4: Evaluate conclusion (II): “Some girls are hardworking.” The girls who are clever are automatically hardworking, since all clever are hardworking. Therefore, there is at least one girl who is hardworking. Conclusion (II) must follow.Step 5: Evaluate conclusion (I): “Some girls are not hardworking.” This would require at least one girl outside the hardworking set. However, it is logically possible that all girls happen to be hardworking, with some of them clever and some not clever. The statements do not prevent that. In such a scenario, conclusion (I) would be false while statements remain true. Therefore, conclusion (I) does not necessarily follow.Step 6: Evaluate conclusion (III): “Some hardworking are not girls.” This requires at least one hardworking individual who is not a girl. But the statements allow the possibility that the only hardworking individuals are girls. For example, all clever people could be girls and all hardworking people could be exactly those clever girls. In that case, conclusion (III) fails, yet both statements are satisfied. So conclusion (III) also does not follow.

Verification / Alternative check:
Symbolically, from statement (I) we know there exists a person g such that g is a girl and g is clever. From statement (II), for any person x, if x is clever then x is hardworking. Putting these together, g is a girl and g is hardworking. This guarantees that some girls are hardworking, which supports conclusion (II). However, nothing in the statements guarantees the existence of a girl who is not hardworking or the existence of a hardworking individual who is not a girl. Those are possibilities, not necessities.

Why Other Options Are Wrong:
Option A says that conclusions (I) and (II) follow, but conclusion (I) can be false in a valid arrangement where every girl is hardworking. Option C says that conclusions (I) and (III) follow, yet we have seen that both are merely possible, not forced. Option D claims that all three conclusions follow, which is clearly incorrect because we find diagrams where two of them fail. Only option B, which accepts conclusion (II) alone, is consistent with exact logical reasoning.

Common Pitfalls:
Students often confuse “some” and “all,” and may think that if some girls are clever, then some girls must also be not clever, or if some people are hardworking, then some must not be girls. These are not valid inferences unless explicitly stated. Another common mistake is to import real world notions that most hardworking people are not girls or vice versa, which should be avoided in formal logic questions.

Final Answer:
Thus, the only conclusion that logically follows is that some girls are hardworking, corresponding to conclusion (II) and option B.