A is the brother of B. B is the brother of C. D is the father of A. Based on these three facts, which of the following statements cannot be definitely true?

Introduction / Context:
This question checks careful reading of family statements and the difference between what is definitely true and what is only possibly true. Logical reasoning tests often ask whether a statement must be true in every valid case or whether there is some scenario where it could fail. Here the unknown gender of C plays a central role.

Given Data / Assumptions:

• A is the brother of B, so A is male and A and B are siblings.
• B is the brother of C, so B is male and B and C are siblings.
• D is the father of A, so D is male and parent of A.
• In such questions, siblings are usually assumed to share the same parents unless otherwise mentioned.
• The gender of C is not given anywhere.

Concept / Approach:
The method is to first list everything known for sure, then test each option against that information. A and B are sons of D. Since B is the brother of C, C is another child of D, but we do not know if C is male or female. The word "brother" describes the gender of the subject of the relation (the one who is called brother), not the other person. This detail decides which statements are definite and which are not.

Step-by-Step Solution:
Step 1: From "A is the brother of B" we know A is male and sibling of B.Step 2: From "B is the brother of C" we know B is male and sibling of C. C could be male or female.Step 3: From "D is the father of A" and standard exam assumptions, D is also the father of B and C, so all three share the same parents.Step 4: Check option a: "B is the brother of A." B is male and sibling of A, so this is definitely true.Step 5: Check option b: "B is the son of D." B is male child of D, so this is also definitely true.Step 6: Check option c: "A is the brother of C." A is male and sibling of C, so A is indeed the brother of C in every valid case. This is definitely true.Step 7: Check option d: "C is the brother of A." For C to be a brother, C must be male. But the gender of C is unknown. C could be a sister instead. So this statement is not definitely true in all possible cases.

Verification / Alternative check:
Construct two examples. In the first, let C be male. Then C is brother of A and all four options become true. In the second, let C be female. Then C is sister of A, not brother, and option d fails while the others remain true. Because it is possible to build a valid family where option d is false, it cannot be classified as definitely true.

Why Other Options Are Wrong:
Options a, b and c all follow directly from the given statements and the standard understanding of siblings and parentage. There is no variation in gender or family structure that can make them false without contradicting the given information. Only option d depends on an unknown gender for C and therefore fails the "definitely true" test.

Common Pitfalls:
Many candidates treat sibling and brother as automatically symmetric, assuming that if A is brother of B then B is brother of A, and extend that idea without checking gender. However, the exam carefully controls which person is described as brother or sister. Always pay attention to which person is on which side of the relation and whether the gender for that person is actually specified.

Final Answer:
The statement that cannot be definitely true is "C is the brother of A."