which of the following statement(s) is/are CORRECT?
I. `(sqrt(11) + sqrt(2)) gt (sqrt(8) + sqrt(5))`
II. `(sqrt(10) + sqrt(3)) gt (sqrt(7) + sqrt(6))`

Options

A. Only I
B. Only II
C. Neither I nor II
D. Both I and II

Answer

To determine which of the statements is correct, we will analyze each statement step by step.

### Step 1: Analyze Statement I
We need to compare \( \sqrt{11} + \sqrt{2} \) and \( \sqrt{8} + \sqrt{5} \).

1. **Square both sides**:
\[ (\sqrt{11} + \sqrt{2})^2 \quad \text{and} \quad (\sqrt{8} + \sqrt{5})^2 \]

2. **Calculate the squares**:
- For \( \sqrt{11} + \sqrt{2} \):
\[ (\sqrt{11} + \sqrt{2})^2 = 11 + 2 + 2\sqrt{11 \cdot 2} = 13 + 2\sqrt{22} \]
- For \( \sqrt{8} + \sqrt{5} \):
\[ (\sqrt{8} + \sqrt{5})^2 = 8 + 5 + 2\sqrt{8 \cdot 5} = 13 + 2\sqrt{40} \]

3. **Compare the two results**:
We need to compare \( 2\sqrt{22} \) and \( 2\sqrt{40} \). Since both terms have a factor of 2, we can simplify our comparison to:
\[ \sqrt{22} \quad \text{and} \quad \sqrt{40} \]
Since \( 40 > 22 \), it follows that \( \sqrt{40} > \sqrt{22} \).

4. **Conclusion for Statement I**:
Since \( 2\sqrt{40} > 2\sqrt{22} \), we conclude:
\[ \sqrt{11} + \sqrt{2} < \sqrt{8} + \sqrt{5} \]
Therefore, Statement I is **incorrect**.

### Step 2: Analyze Statement II
We need to compare \( \sqrt{10} + \sqrt{3} \) and \( \sqrt{7} + \sqrt{6} \).

1. **Square both sides**:
\[ (\sqrt{10} + \sqrt{3})^2 \quad \text{and} \quad (\sqrt{7} + \sqrt{6})^2 \]

2. **Calculate the squares**:
- For \( \sqrt{10} + \sqrt{3} \):
\[ (\sqrt{10} + \sqrt{3})^2 = 10 + 3 + 2\sqrt{10 \cdot 3} = 13 + 2\sqrt{30} \]
- For \( \sqrt{7} + \sqrt{6} \):
\[ (\sqrt{7} + \sqrt{6})^2 = 7 + 6 + 2\sqrt{7 \cdot 6} = 13 + 2\sqrt{42} \]

3. **Compare the two results**:
We need to compare \( 2\sqrt{30} \) and \( 2\sqrt{42} \). Again, we can simplify our comparison to:
\[ \sqrt{30} \quad \text{and} \quad \sqrt{42} \]
Since \( 42 > 30 \), it follows that \( \sqrt{42} > \sqrt{30} \).

4. **Conclusion for Statement II**:
Since \( 2\sqrt{42} > 2\sqrt{30} \), we conclude:
\[ \sqrt{10} + \sqrt{3} < \sqrt{7} + \sqrt{6} \]
Therefore, Statement II is also **incorrect**.

### Final Conclusion
Both statements I and II are incorrect. The correct answer is that neither statement is correct.