Is D the brother of F? Study the following two statements and decide whether the data in them are sufficient to answer the question: I. B has two sons, and F is one of them. II. The mother of D is married to B.

Introduction / Context:
This data sufficiency problem tests understanding of family relationships and the precise meaning of the word brother. The question asks whether D is a brother of F, and the learner must decide whether the given statements provide enough information to confirm or reject that relationship. The focus is on analysis of sufficiency rather than on constructing a single family tree.

Given Data / Assumptions:

• Brother means a male child of the same parents, or at minimum a male child of the same father, within the context of typical reasoning questions.
• Statement I says that B has two sons and that F is one of those sons.
• Statement II says that the mother of D is married to B, which makes D a child of B but does not specify the sex of D.
• There may be other children of B, including daughters or additional sons, because no upper bound on the number of children is provided.

Concept / Approach:
In data sufficiency questions, each statement is examined alone and then in combination. The key is to ask whether the statements force a unique conclusion in all possible scenarios that satisfy them. If it is possible to construct two different valid family structures, one where D is a brother of F and another where D is not, then the data are not sufficient to answer the question.

Step-by-Step Solution:
Step 1: Consider statement I alone. We know that B has exactly two sons and that F is one of those sons. The other son could be D, or it could be another boy entirely. D is not mentioned at all, so we cannot say anything about D relationship to F from statement I alone. Step 2: Consider statement II alone. The mother of D is married to B, so D is a child of B. However, the statement does not tell us whether D is male or female, and it does not mention F. Therefore, statement II alone does not determine whether D is a brother of F. Step 3: Combine statements I and II. From I, B has two sons including F. From II, D is a child of B but the sex is not specified. There is no claim that B has only two children, so D could be one of the two sons, or D could be a daughter while the second son is someone else. Step 4: Build two consistent examples. In the first example, let D be the second son of B. Then B sons are F and D, and D is clearly a brother of F. In the second example, let B have sons F and some boy X, and a daughter D. Then D is not a brother of F. Both examples satisfy statements I and II, yet they give opposite answers to the main question. Step 5: Because at least two different consistent family structures exist, one with a Yes answer and one with a No answer, the combined data are not sufficient to decide.

Verification / Alternative check:
A useful check is to ask whether the sex of D is fixed by the statements. Here, even when both statements are used, the sex of D remains unknown. Whenever the question depends critically on knowing whether a person is male or female, and the statements leave that point open, the data cannot be sufficient. This confirms that the correct classification is that even both statements together do not allow a definite answer.

Why Other Options Are Wrong:
Option A, which claims statement I alone is sufficient, is incorrect because statement I never mentions D. Option B is wrong because statement II alone does not mention F and does not fix the sex of D. Option C, which suggests either statement alone is sufficient, fails for the same reasons. Option D, which says both statements together are necessary and sufficient, is also wrong because the combined data still permit more than one valid arrangement of the children of B. Only option E correctly states that even together the statements are not sufficient.

Common Pitfalls:
Many learners mistakenly assume that if a man has two sons, then those must be his only two children. That assumption is not justified by the wording. Others overlook the importance of gender and behave as if every child of a man is a brother, ignoring daughters. Data sufficiency questions require careful reading of exactly what is and is not stated, and they reward constructing counterexamples to test sufficiency.

Final Answer:
The data even in both statements together are not sufficient to answer the question, so the correct option is E.