Question

Which of the following statement(s) is/are correct?

1. \(\frac{3}{7} < \frac{5}{9} < \frac{7}{{11}}\)

2. \(\sqrt 6 > \sqrt[3]{{12}}\)

1. Only 2
2. Only 1
3. Both 1 and 2
4. Neither 1 nor 2

Answer 1

Correct Answer - Option 3 : Both 1 and 2

Calculation:

Statement 1: \(\frac{3}{7} < \frac{5}{9} < \frac{7}{{11}}\)

LCM  of (7, 9 , 11) = 693

3/7 = (3 × 99)/693 = 297/693

5/9 = (5 × 77)/693 = 385/693

7/11 = (7 × 63)/693 = 441/693

That is, (3/7) < (5/9) < 7/11

Statement 1 is correct.

Statement 2: \(√ 6 > √[3]{{12}}\)

The order of the surds √6  and ∛12  is 6 and 12 respectively.

LCM of 6 and 12 = 12

Let us change the surds to their 12 th root so that both the LHS and RHS can be compared.

6^1/2 → (6⁶)^1/12

(12)^1/3 → (12⁴)^1/12

Now, as powers are the same, let us compare 6⁶ and 12⁴.

We know that,  66 >124

Then, \(\sqrt 6 > \sqrt[3]{{12}}\)

Statement 2 is correct.

∴ Both statements are correct.