Which of the following statements (s) is/are TRUE ?
I. `(2)/(3sqrt(5)) lt (3)/(2sqrt(5)) lt (5)/(4sqrt(3))`
II. `(3)/(2sqrt(5)) lt (2)/(3sqrt(3)) lt (7)/(4sqrt(5))`

Options

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

Answer

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

### Statement I:
\[ \frac{2}{3\sqrt{5}} < \frac{3}{2\sqrt{5}} < \frac{5}{4\sqrt{3}} \]

1. **Compare \(\frac{2}{3\sqrt{5}}\) and \(\frac{3}{2\sqrt{5}}\)**:
- To compare these fractions, we can cross-multiply:
\[ 2 \cdot 2\sqrt{5} < 3 \cdot 3\sqrt{5} \]
- This simplifies to:
\[ 4\sqrt{5} < 9\sqrt{5} \]
- Since \(\sqrt{5} > 0\), we can divide both sides by \(\sqrt{5}\):
\[ 4 < 9 \quad \text{(True)} \]

2. **Compare \(\frac{3}{2\sqrt{5}}\) and \(\frac{5}{4\sqrt{3}}\)**:
- Again, we cross-multiply:
\[ 3 \cdot 4\sqrt{3} < 5 \cdot 2\sqrt{5} \]
- This simplifies to:
\[ 12\sqrt{3} < 10\sqrt{5} \]
- Squaring both sides to eliminate the square roots:
\[ 144 \cdot 3 < 100 \cdot 5 \]
- This simplifies to:
\[ 432 < 500 \quad \text{(True)} \]

Since both comparisons in Statement I are true, we conclude that Statement I is true.

### Statement II:
\[ \frac{3}{2\sqrt{5}} < \frac{2}{3\sqrt{3}} < \frac{7}{4\sqrt{5}} \]

1. **Compare \(\frac{3}{2\sqrt{5}}\) and \(\frac{2}{3\sqrt{3}}\)**:
- Cross-multiply:
\[ 3 \cdot 3\sqrt{3} < 2 \cdot 2\sqrt{5} \]
- This simplifies to:
\[ 9\sqrt{3} < 4\sqrt{5} \]
- Squaring both sides:
\[ 81 \cdot 3 < 16 \cdot 5 \]
- This simplifies to:
\[ 243 < 80 \quad \text{(False)} \]

Since the first comparison in Statement II is false, we do not need to check the second comparison. Therefore, Statement II is false.

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

Thus, the answer is that only Statement I is true.

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