In each question below, some statements are given followed by two conclusions numbered I and II. You have to take the given statements to be true even if they seem to be at variance with commonly known facts. Read all the conclusions and then decide which of the given conclusion logically follows from the given statements, disregarding commonly known facts. Giveanswer-

Statements:
All Rose are Pink.
Only a few Black is Pink.
No White is Rose.

Conclusions:
I. Some Rose is Black.
II. Some Pink can never be White.

Correct Answer: If only conclusion II follows

Let us analyze the statements and verify the conclusions step-by-step using Venn diagrams and logical reasoning:

1. Analyze the Statements:
- "All Rose are Pink": The set of Rose is entirely contained within the set of Pink ($R\subseteq P$).
- "Only a few Black is Pink": This statement carries a dual meaning:
- Some Black is Pink ($B\cap P\ne \varnothing$). - Some Black is NOT Pink ($B⊈P$).
- "No White is Rose": The set of White and the set of Rose have no elements in common ($W\cap R=\varnothing$).

2. Evaluate Conclusion I: "Some Rose is Black"
- From the statements, we know that all Rose are Pink and there is some overlap between Black and Pink. However, there is no direct relationship or mandatory overlap specified between Black and Rose.
- It is possible to draw a Venn diagram where the set of Black and the set of Rose do not intersect. Since this conclusion does not hold true in all possible scenarios, Conclusion I does not logically follow.

3. Evaluate Conclusion II: "Some Pink can never be White"
- The phrase "can never be" means "are definitely not". Thus, the conclusion is equivalent to stating: "Some Pink is not White".
- We are given that "All Rose are Pink", which means the entire set of Rose is inside Pink.
- We are also given that "No White is Rose" (or "No Rose is White"). This means none of the elements in the Rose set can belong to the White set.
- Since all Rose are Pink, those Pink elements which are Rose can never be White.
- Therefore, there is definitely a portion of Pink (specifically, the portion that is Rose) that can never be White.
- Hence, Conclusion II logically follows.

Conclusion:
Only conclusion II follows.