Which of the following statements (s) is/are TRUE ?
I. `sqrt(11)+sqrt(7) lt sqrt(10)+sqrt(8)`
II. `sqrt(17)+sqrt(11) gt sqrt(15)+sqrt(13)`

Options

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

Answer

To determine the truth of the statements given, we will analyze each statement step by step.

### Statement I: \( \sqrt{11} + \sqrt{7} < \sqrt{10} + \sqrt{8} \)

1. **Square both sides**:
\[ (\sqrt{11} + \sqrt{7})^2 < (\sqrt{10} + \sqrt{8})^2 \]
Expanding both sides:
\[ 11 + 7 + 2\sqrt{11 \cdot 7} < 10 + 8 + 2\sqrt{10 \cdot 8} \]
This simplifies to:
\[ 18 + 2\sqrt{77} < 18 + 2\sqrt{80} \]

2. **Subtract 18 from both sides**:
\[ 2\sqrt{77} < 2\sqrt{80} \]

3. **Divide both sides by 2**:
\[ \sqrt{77} < \sqrt{80} \]

4. **Square both sides again**:
\[ 77 < 80 \]
This is true.

Thus, Statement I is **TRUE**.

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### Statement II: \( \sqrt{17} + \sqrt{11} > \sqrt{15} + \sqrt{13} \)

1. **Square both sides**:
\[ (\sqrt{17} + \sqrt{11})^2 > (\sqrt{15} + \sqrt{13})^2 \]
Expanding both sides:
\[ 17 + 11 + 2\sqrt{17 \cdot 11} > 15 + 13 + 2\sqrt{15 \cdot 13} \]
This simplifies to:
\[ 28 + 2\sqrt{187} > 28 + 2\sqrt{195} \]

2. **Subtract 28 from both sides**:
\[ 2\sqrt{187} > 2\sqrt{195} \]

3. **Divide both sides by 2**:
\[ \sqrt{187} > \sqrt{195} \]

4. **Square both sides again**:
\[ 187 > 195 \]
This is false.

Thus, Statement II is **FALSE**.

### Conclusion:
- Statement I is TRUE.
- Statement II is FALSE.

The answer to the question is that only the first statement is true.

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