Correct Option (3)
The problem requires evaluating the logical validity of conclusions based on the given statements. We will analyze each conclusion:
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Conclusion-I: Some scientists are doctors.
From Statement-1 (Some doctors are teachers) and Statement-2 (All teachers are engineers), it can be deduced that some doctors are engineers. This is because if some doctors belong to the set of teachers, and the entire set of teachers is contained within the set of engineers, then those doctors must also be engineers.
Now, combining "some doctors are engineers" with Statement-3 (All engineers are scientists), it logically follows that some doctors are scientists. If some individuals are engineers, and all engineers are scientists, then those individuals must also be scientists.
Therefore, Conclusion-I is logically valid.
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Conclusion-III: Some engineers are doctors.
From Statement-1 (Some doctors are teachers) and Statement-2 (All teachers are engineers), it is directly inferred that some doctors are engineers. Since some doctors fall into the category of teachers, and all teachers are engineers, it implies that these specific doctors are also engineers.
This statement is equivalent to "Some engineers are doctors."
Therefore, Conclusion-III is logically valid.
Since both Conclusion-I and Conclusion-III are logically valid, option (3) is the correct answer.
Incorrect Options:
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Conclusion-II: All engineers are doctors.
While it is established that some engineers are doctors (as derived for Conclusion-III), the statements do not provide sufficient information to conclude that *all* engineers are doctors. The relationship is "some," not "all." There could be engineers who are not teachers and, consequently, not doctors based on the given premises.
Therefore, Conclusion-II is not logically valid.
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Options (1), (2), and (4) are incorrect because they either fail to include all valid conclusions (like option 1) or include an invalid conclusion (like options 2 and 4).