Which of the following statement (s) is / are TRUE ?
I. `sqrt(5)+sqrt(5) gt sqrt(7)+sqrt(3)`
II. `sqrt(6)+sqrt(7) gt sqrt(8)+sqrt(5)`
III. `sqrt(3)+sqrt(9) gt sqrt(6)+sqrt(6)`

Options

A. Only I
B. Only II
C. Only II and III
D. Only I and III

Answer

To determine which of the statements are true, we will analyze each statement step by step.

### Statement I:
\[ \sqrt{5} + \sqrt{5} > \sqrt{7} + \sqrt{3} \]

1. Simplify the left side:
\[ \sqrt{5} + \sqrt{5} = 2\sqrt{5} \]
So, the statement becomes:
\[ 2\sqrt{5} > \sqrt{7} + \sqrt{3} \]

2. Square both sides to eliminate the square roots:
\[ (2\sqrt{5})^2 > (\sqrt{7} + \sqrt{3})^2 \]
This simplifies to:
\[ 4 \times 5 > 7 + 3 + 2\sqrt{21} \]
Which further simplifies to:
\[ 20 > 10 + 2\sqrt{21} \]

3. Rearranging gives:
\[ 10 > 2\sqrt{21} \]
Dividing both sides by 2:
\[ 5 > \sqrt{21} \]

4. Squaring both sides again:
\[ 25 > 21 \]
This is true.

### Statement II:
\[ \sqrt{6} + \sqrt{7} > \sqrt{8} + \sqrt{5} \]

1. Square both sides:
\[ (\sqrt{6} + \sqrt{7})^2 > (\sqrt{8} + \sqrt{5})^2 \]
This expands to:
\[ 6 + 7 + 2\sqrt{42} > 8 + 5 + 2\sqrt{40} \]
Simplifying gives:
\[ 13 + 2\sqrt{42} > 13 + 2\sqrt{40} \]

2. Rearranging gives:
\[ 2\sqrt{42} > 2\sqrt{40} \]
Dividing both sides by 2:
\[ \sqrt{42} > \sqrt{40} \]

3. Squaring both sides:
\[ 42 > 40 \]
This is true.

### Statement III:
\[ \sqrt{3} + \sqrt{9} > \sqrt{6} + \sqrt{6} \]

1. Simplify the left side:
\[ \sqrt{3} + 3 > 2\sqrt{6} \]

2. Square both sides:
\[ (\sqrt{3} + 3)^2 > (2\sqrt{6})^2 \]
This expands to:
\[ 3 + 6\sqrt{3} + 9 > 24 \]
Simplifying gives:
\[ 12 + 6\sqrt{3} > 24 \]

3. Rearranging gives:
\[ 6\sqrt{3} > 12 \]
Dividing both sides by 6:
\[ \sqrt{3} > 2 \]

4. Squaring both sides:
\[ 3 > 4 \]
This is false.

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

Thus, the true statements are I and II.

### Final Answer:
The true statements are I and II.

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