Question : Which of the following statement(s) is/are true? I. $\sqrt{12}<\sqrt[3]{16}<\sqrt[4]{24}$ II. $\sqrt[3]{25}>\sqrt[4]{32}>\sqrt[6]{48}$ III. $\sqrt[4]{9}>\sqrt[3]{15}>\sqrt[6]{24}$
Option 1: Only II
Option 2: Only I and III
Option 3: Only I
Option 4: All are true
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Correct Answer: Only II
Solution : By evaluating each statement: I. $\sqrt{12}<\sqrt[3]{16}<\sqrt[4]{24}$ ⇒ $12^\frac{1}{2}<16^\frac{1}{3}<24^\frac{1}{4}$ LCM of 2, 3, and 4 = 12 ⇒ $(12^\frac{1}{2})^{12}<(16^\frac{1}{3})^{12}<(24^\frac{1}{4})^{12}$ ⇒ $12^6<16^4<24^3$ ⇒ $(1728)^2<65536<13824$ So, the first statement is false. II. $\sqrt[3]{25}>\sqrt[4]{32}>\sqrt[6]{48}$ Similarly, ⇒ $25^4>32^3>48^2$ ⇒ $390625>32768>2304$ So, the second statement is true. III. $\sqrt[4]{9}>\sqrt[3]{15}>\sqrt[6]{24}$ Similarly, ⇒ $9^3>15^4>24^2$ ⇒ $729>50625>576$ So, the third statement is false. Hence, the correct answer is 'Only II'.
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