Three Statements followed by three Conclusions are given below. You have to take the Statements to be true even if they seem to be at variance from the commonly known facts. Read all the Conclusions and then decide which of the given Conclusions logically follows/follow from the Statements, disregarding the commonly known facts:
Statement- 1: Some doctors are teachers.
Statement- 2: All teachers are engineers.
Statement- 3: All engineers are scientists.
Conclusion-I: Some scientists are doctors.
Conclusion-II: All engineers are doctors.
Conclusion-III: Some engineers are doctors.
Which one of the following is correct?

The correct option is: Both Conclusion-I and Conclusion-III.

To determine which conclusions logically follow from the given statements, we can analyze them step-by-step using set relations and Venn diagrams.

Step 1: Analyze the Statements
Let us represent the categories as sets:
- Let D represent the set of Doctors.
- Let T represent the set of Teachers.
- Let E represent the set of Engineers.
- Let S represent the set of Scientists.

From the statements, we establish the following relations:
1. Statement-1: Some doctors are teachers.
This means there is a non-empty intersection between the set of Doctors and Teachers.

$D\cap T\ne \varnothing$

2. Statement-2: All teachers are engineers.
This means the set of Teachers is a subset of the set of Engineers.

$T\subseteq E$

3. Statement-3: All engineers are scientists.
This means the set of Engineers is a subset of the set of Scientists.

$E\subseteq S$

Step 2: Evaluate the Conclusions
- Conclusion-I: Some scientists are doctors.
Since some doctors are teachers, and all teachers are engineers, those doctors who are teachers must also be engineers. Furthermore, since all engineers are scientists, those doctors must also be scientists. Therefore, there is definitely an overlap between the set of Scientists and the set of Doctors.

$S\cap D\ne \varnothing$

Hence, Conclusion-I logically follows.

- Conclusion-II: All engineers are doctors.
While we know that those engineers who are teachers are also doctors, we cannot conclude that all engineers are doctors. The set of Engineers can be larger and contain individuals who are not doctors. Thus, Conclusion-II does not logically follow.

- Conclusion-III: Some engineers are doctors.
As established earlier, the doctors who are teachers are also engineers (because all teachers are engineers). Therefore, there is a guaranteed overlap between the set of Engineers and the set of Doctors.

$E\cap D\ne \varnothing$

Hence, Conclusion-III logically follows.

Conclusion:
Since only Conclusion-I and Conclusion-III logically follow from the given statements, the correct option is "Both Conclusion-I and Conclusion-III".