Answer: C

We need to determine whether $x$ is even.

Statement I: $x^2y^2$ is even.

This means at least one of $x^2$ or $y^2$ is even. Hence, at least one of $x$ or $y$ is even.

But this does not definitely tell us whether $x$ is even. For example, $x$ could be odd and $y$ could be even.

So, Statement I alone is not sufficient.

Statement II: $1+x^2+y^2$ is odd.

For $1+x^2+y^2$ to be odd, $x^2+y^2$ must be even.

This means $x^2$ and $y^2$ have the same parity. Therefore, $x$ and $y$ have the same parity: either both are even or both are odd.

So, Statement II alone is not sufficient.

Using both statements together:

From Statement I, at least one of $x$ or $y$ is even.

From Statement II, $x$ and $y$ have the same parity.

Therefore, both $x$ and $y$ must be even.

Hence, $x$ is definitely even.

So, the question can be answered using both statements together, but not using either statement alone.