Question : Which of the following statement(s) is/are TRUE?
I. $\frac{2}{3\sqrt5}<\frac{3}{2\sqrt5}<\frac{5}{4\sqrt3}$
II. $\frac{3}{2\sqrt5}<\frac{2}{3\sqrt3}<\frac{7}{4\sqrt5}$
Option 1: Only I
Option 2: Only II
Option 3: Both I and II
Option 4: Neither I nor II
Correct Answer: Only I
Solution :
Let's simplify the fractions in each statement to see if they are true:
I. $\frac{2}{3\sqrt5}<\frac{3}{2\sqrt5}<\frac{5}{4\sqrt3}$
Rationalising each fraction, we get:
$\frac{2\sqrt5}{15}<\frac{3\sqrt5}{10}<\frac{5\sqrt3}{12}$
Converting these fractions to decimal form, we get:
0.29 < 0.47 < 0.72, where $\sqrt5$ = 2.236 and $\sqrt3$ = 1.732
So, Statement I is true.
II. $\frac{3}{2\sqrt5}<\frac{2}{3\sqrt3}<\frac{7}{4\sqrt5}$
Rationalising each fraction, we get:
$\frac{3\sqrt5}{10}<\frac{2\sqrt3}{9}<\frac{7\sqrt5}{20}$
Converting these fractions to decimal form, we get:
0.67 < 0.38 < 0.78, where $\sqrt5$ = 2.236 and $\sqrt3$ = 1.732
So, Statement II is false.
Hence, the correct answer is 'Only I'.
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