`A` and `B` are isotopes. `B` and `C` are isobars. All three are radioactive. Which one of the following is true.

Options

A. A, B and C must belong to the same element
B. A, B and C may belong to the same element
C. it is possible that A will change to B through a radioactive decay process
D. it is possible that B will change to C through a radioactive decay process

Answer

To solve the question regarding the isotopes and isobars, we need to analyze the definitions and relationships between the elements A, B, and C based on the information given.

### Step-by-Step Solution:

1. **Understanding Isotopes**:
- Isotopes are atoms of the same element that have the same number of protons (atomic number) but different numbers of neutrons (and thus different atomic masses).
- Since A and B are isotopes, they must belong to the same element.

2. **Understanding Isobars**:
- Isobars are atoms of different elements that have the same atomic mass (total number of protons and neutrons).
- Since B and C are isobars, they have the same atomic mass but are different elements.

3. **Analyzing Relationships**:
- From the above definitions, we conclude:
- A and B are the same element (isotopes).
- B and C are different elements (isobars) but have the same atomic mass.

4. **Evaluating the Options**:
- **Option 1**: A, B, and C must belong to the same element.
- This is incorrect because while A and B are the same element, C is a different element.
- **Option 2**: A, B, and C may belong to the same element.
- This is also incorrect for the same reason as above.
- **Option 3**: It is possible that A will change to B through a radioactive decay process.
- This is incorrect because A and B are already isotopes of the same element and do not change into each other through decay.
- **Option 4**: It is possible that B will change to C through a radioactive decay process.
- This is correct because isobars can undergo radioactive decay to form more stable nuclei.

5. **Conclusion**:
- The only true statement among the options provided is option 4, which states that it is possible for B to change into C through a radioactive decay process.

### Final Answer:
**Option 4 is correct: It is possible that B will change to C through a radioactive decay process.**