Suppose R is relation whose graph is symmetric to both X-axis and Y-axis and that the point (1,2) is on the graph of R. Which one of the following is not necessarily on the graph of R?

Options

A. (-1,2)
B. (1,-2)
C. (-1,-2)
D. (2,1)

Answer

To solve the problem, we need to analyze the properties of the relation R, which is symmetric to both the X-axis and Y-axis, given that the point (1, 2) is on the graph of R.

### Step-by-Step Solution:

1. **Understanding Symmetry**:
- A graph is symmetric about the X-axis if for every point (a, b) on the graph, the point (a, -b) is also on the graph.
- A graph is symmetric about the Y-axis if for every point (a, b) on the graph, the point (-a, b) is also on the graph.

2. **Starting Point**:
- We are given that the point (1, 2) is on the graph of R.

3. **Finding Points from Symmetry**:
- From the symmetry about the X-axis:
- The point (1, 2) implies that (1, -2) is also on the graph.
- From the symmetry about the Y-axis:
- The point (1, 2) implies that (-1, 2) is also on the graph.
- From the symmetry about both axes:
- The point (-1, 2) implies that (-1, -2) is also on the graph.

4. **Listing the Points**:
- From the above analysis, we have the following points on the graph:
- (1, 2)
- (1, -2)
- (-1, 2)
- (-1, -2)

5. **Identifying the Options**:
- Now, we need to check which of the given options is not necessarily on the graph of R. Let's denote the options as A, B, C, and D.
- We need to evaluate each option based on the points we derived.

6. **Conclusion**:
- After checking the options, we find that the point (2, 1) does not correspond to any of the points derived from the symmetry of the graph. Therefore, it is not necessarily on the graph of R.

### Final Answer:
The point that is not necessarily on the graph of R is **(2, 1)**.