A mathematical theorem is:

Options

A. a statement that has been proven by logical arguments based on axioms.
B. a statement which is always true and doesn't need proof.
C. a statement whose truth or falsity is not known.
D. is a statement without sufficient evidence for proof.

Answer

To solve the question regarding the definition of a mathematical theorem, we will analyze each option provided and determine which one correctly describes a mathematical theorem.

### Step-by-Step Solution:

1. **Understanding the Definition of a Theorem**:
A mathematical theorem is a statement that has been proven based on previously established statements, such as other theorems, and axioms.

2. **Analyzing Option A**:
- **Statement**: "A statement that has been proven by logical arguments based on axioms."
- This option aligns with the definition of a theorem because it emphasizes that the statement is proven using logical reasoning and foundational principles (axioms).
- **Conclusion**: This option is likely correct.

3. **Analyzing Option B**:
- **Statement**: "A statement which is always true and doesn't need proof."
- This option describes a mathematical truth or axiom, not a theorem. Theorems require proof.
- **Conclusion**: This option is incorrect.

4. **Analyzing Option C**:
- **Statement**: "A statement whose truth and falsity is not known."
- This option refers to conjectures or hypotheses, not theorems. Theorems have established truth values.
- **Conclusion**: This option is incorrect.

5. **Analyzing Option D**:
- **Statement**: "A statement without sufficient evidence for proof."
- This option suggests a lack of proof, which contradicts the very nature of a theorem, as theorems are proven statements.
- **Conclusion**: This option is incorrect.

6. **Final Conclusion**:
- After evaluating all options, we find that Option A is the only one that accurately defines a mathematical theorem. Therefore, the correct answer is Option A.

### Final Answer:
**Option A: A statement that has been proven by logical arguments based on axioms.**