Read the statements and conclusions carefully: Statements: Some actors are singers. All the singers are dancers. Conclusions: (i) Some actors are dancers. (ii) No singer is an actor. Disregarding commonly known facts and using only the statements, which conclusion logically follows?

Introduction / Context:
Syllogism questions test your ability to draw valid conclusions from given statements, treating them as logically perfect, even if they conflict with real world facts. Here you are given two statements about actors, singers and dancers, and two proposed conclusions. Your task is to decide which conclusion must be true if the statements are assumed to be true.

Given Data / Assumptions:
- Statement 1: Some actors are singers. - Statement 2: All the singers are dancers. - Conclusion (i): Some actors are dancers. - Conclusion (ii): No singer is an actor. - We must ignore outside knowledge and focus only on the logical structure of the statements.

Concept / Approach:
The phrase some actors are singers means there is at least one person who is both an actor and a singer. The phrase all the singers are dancers means that every singer automatically belongs to the group of dancers. When we combine these, any person who is both an actor and a singer must also be a dancer. This shows that some actors are dancers. At the same time, conclusion (ii) directly contradicts the first statement and so cannot follow. A simple Venn diagram with three overlapping sets can help visualise this reasoning.

Step-by-Step Solution:
Step 1: From the first statement, identify that there is at least one member in the intersection of actors and singers. Step 2: From the second statement, note that the set of singers is completely contained within the set of dancers. Step 3: Combine the information. Any person who is in the singers set must also be in the dancers set. Step 4: The person or people who are both actors and singers are therefore also dancers. Step 5: This means that at least one actor is a dancer. So conclusion (i) "Some actors are dancers" must be true. Step 6: Now examine conclusion (ii) "No singer is an actor". This would mean there is no overlap between singers and actors. Step 7: However, the first statement clearly says that some actors are singers, which means there is at least some overlap. Step 8: Therefore conclusion (ii) directly contradicts the given statement and cannot follow. Step 9: As a result, only conclusion (i) follows logically from the statements.

Verification / Alternative check:
Draw three circles labelled actors, singers and dancers. Mark a region where actors and singers overlap. Then shade all of the singers circle and place it fully inside the dancers circle, because all singers are dancers. You can clearly see that the overlap region between actors and singers lies inside the dancers circle, confirming that some actors are dancers. There is no way to draw the diagram so that no singer is an actor without breaking the first statement, so conclusion (ii) is impossible under the given conditions.

Why Other Options Are Wrong:
Neither (i) nor (ii) follows: Incorrect because conclusion (i) definitely follows from combining the statements. Either (i) or (ii) follows: Incorrect because only (i) is compatible with the premises; they cannot be alternatives. Only conclusion (ii) follows: Incorrect because (ii) contradicts the first statement and therefore cannot be true.

Common Pitfalls:
A common mistake is ignoring the word some and treating it as many or confusing it with all. Another frequent error is not drawing or imagining the sets carefully, which can lead to accepting contradictory conclusions. Students also sometimes forget that in these questions, we are not allowed to add new assumptions beyond the given statements. Keeping a clear mental or drawn Venn diagram can prevent these mistakes.

Final Answer:
The only logically valid conclusion is that some actors are dancers, so only conclusion (i) follows.