Statement-I The number of vectors of unit length and perpendicular to both the vectors `hat(i)+hat(j) and hat(j)+hat(k)` is zero.
Statement-II a and b are two non-zero and non-parallel vectors it is true that `axxb` is perpendicular to the plane containing a and b

Options

A. Both Statement-I and Statement-II are correct and Statement-II is the correct explanation of Statement-I
B. Both Statement-I and Statement-II are correct but Statement-II is not the correct explanation of Statement-I
C. Statement-I is correct but Statement-II is incorrect
D. Statement-II is correct but Statement-I is incorrect

Answer

To solve the problem, we need to analyze both statements provided in the question.

### Statement I:
The number of vectors of unit length and perpendicular to both the vectors \( \hat{i} + \hat{j} \) and \( \hat{j} + \hat{k} \) is zero.

1. **Identify the vectors**: Let \( \mathbf{a} = \hat{i} + \hat{j} \) and \( \mathbf{b} = \hat{j} + \hat{k} \).

2. **Find the cross product**: The cross product \( \mathbf{a} \times \mathbf{b} \) gives a vector that is perpendicular to both \( \mathbf{a} \) and \( \mathbf{b} \).
\[ \mathbf{a} \times \mathbf{b} = (\hat{i} + \hat{j}) \times (\hat{j} + \hat{k}) \]

3. **Calculate the cross product**:
- Using the distributive property of the cross product:
\[ \mathbf{a} \times \mathbf{b} = \hat{i} \times \hat{j} + \hat{i} \times \hat{k} + \hat{j} \times \hat{j} + \hat{j} \times \hat{k} \]
- We know that \( \hat{j} \times \hat{j} = \mathbf{0} \) (cross product of any vector with itself is zero).
- Thus, we compute:
\[ \hat{i} \times \hat{j} = \hat{k}, \quad \hat{i} \times \hat{k} = -\hat{j}, \quad \hat{j} \times \hat{k} = \hat{i} \]
- Therefore,
\[ \mathbf{a} \times \mathbf{b} = \hat{k} - \hat{j} + \hat{i} = \hat{i} - \hat{j} + \hat{k} \]

4. **Magnitude of the cross product**: The magnitude of \( \mathbf{a} \times \mathbf{b} \) is:
\[ |\mathbf{a} \times \mathbf{b}| = \sqrt{(1)^2 + (-1)^2 + (1)^2} = \sqrt{3} \]

5. **Unit vectors**: The unit vector in the direction of \( \mathbf{a} \times \mathbf{b} \) is:
\[ \hat{n} = \frac{\mathbf{a} \times \mathbf{b}}{|\mathbf{a} \times \mathbf{b}|} = \frac{\hat{i} - \hat{j} + \hat{k}}{\sqrt{3}} \]
- There are two unit vectors in opposite directions (one in the direction of \( \hat{n} \) and one in the opposite direction).

6. **Conclusion for Statement I**: Since there are two unit vectors perpendicular to both \( \hat{i} + \hat{j} \) and \( \hat{j} + \hat{k} \), Statement I is **false**.

### Statement II:
It is true that \( \mathbf{a} \times \mathbf{b} \) is perpendicular to the plane containing \( \mathbf{a} \) and \( \mathbf{b} \).

1. **Understanding the property**: The cross product of two vectors \( \mathbf{a} \) and \( \mathbf{b} \) is always perpendicular to the plane formed by those two vectors.

2. **Conclusion for Statement II**: Since this is a well-known property of vector cross products, Statement II is **true**.

### Final Conclusion:
- Statement I is false.
- Statement II is true.

Thus, the correct answer is that Statement II is correct while Statement I is incorrect.