Which of the following statement(s) is/are TRUE I. \(\frac{2}{{3\sqrt 5 }} < \frac{3}{{2\sqrt 5 }} < \frac{5}{{4\sqrt 3 }}\)  II. \(\frac{3}{{2\sqrt 5 }} < \frac{2}{{3\sqrt 3 }} < \frac{7}{{4\sqrt 5 }}\) ?

Option 1 : Only I

Statement I:

We need to make the denominators equal,

(2 × 4 × √3) / (3√5 × 4 × √3) < (3 × 3 × 2 × √3) / (2√5 × 3 × 2 × √3) < (5 × 3 × √5) / (4√3 × 3 × √5)

= (8√3) / (12√15) < (18√3) / (12√15) < (15√5) / (12√15)

Now denominators are equal and by approximation we can say 8√3 < 18√3 < 15√5 is true (√192 < √972 < √1125 after transforming the square root)

So statement I is true.

Statement II:

Similarly we need to make denominator equal here too.

So (3 × 6 × √3) / (3√5 × 4 × √3) < (4 × 2 × √5) / (2√5 × 3 × 2 × √3) < (7 × 3 × √5) / (4√3 × 3 × √5)

= (18√3) / (12√15) < (8√5) / (12√15) < (21√3) / (12√15)

Now denominators are equal and by approximation we can say 18√3 < 8√5 < 21√3, is not true (√972 < √320 < √1323 after transforming the square root)

So statement II is not true.

∴ Only Option 1 is correct.