Question

If ‘x’ is positive , is  $^4 \sqrt{x}   >  ^3 \sqrt{x} $
Statement 1: 0.5 < x < 2 
Statement 2:  $12x^2 - 7x +1 = 0$

• A. statement (1) alone is sufficient to answer the question
• B. statement (2) alone is sufficient to answer the question
• C. both the statements together are needed to answer the question
• D. statement (1) alone or statement (2) alone is sufficient to answer the question
• E. neither statement (1) nor statement (2) suffices to answer the question

Answer 1

The Correct Option is B

To determine if \(^4 \sqrt{x} > ^3 \sqrt{x}\) given the statements, we need to analyze each statement individually.

1. Analyzing Statement 1: \(0.5 < x < 2\)
   • The inequality compares the fourth root and cube root of a positive number \(x\). 
   • To explore when \(^4 \sqrt{x} > ^3 \sqrt{x}\), consider the expressions:
     \(x^{\frac{1}{4}} > x^{\frac{1}{3}}\)
   • Simplifying the inequality, we get \(x^{\frac{3}{12}} > x^{\frac{4}{12}}\), which implies \(x^3 > x^4\).
   • Subtracting \(x^3\) from both sides gives: \(0 > x^4 - x^3\) or \(0 > x^3(x - 1)\).
   • This indicates \(x^3 \gt 0\) and \((x-1) < 0\), which simplifies to \(x < 1\).
   • Since \(0.5 < x < 2\) includes \(x > 1\), statement 1 is not conclusive. We cannot determine the outcome for all values in the range.
2. Analyzing Statement 2: \(12x^2 - 7x + 1 = 0\)
   • This is a quadratic equation. To find possible values for \(x\), use the quadratic formula: \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\), where \(a = 12, b = -7, c = 1\).
   • The discriminant is \(b^2 - 4ac = 49 - 48 = 1\) (positive, implying two distinct real roots).
   • Applying the quadratic formula, we find:
     \(x = \frac{7 \pm 1}{24}\), giving solutions \(x = \frac{8}{24} = \frac{1}{3}\) and \(x = \frac{6}{24} = \frac{1}{4}\).
   • Both solutions, \(x = \frac{1}{3}\) and \(x = \frac{1}{4}\), satisfy \(x < 1\).
   • Thus, for both values, \(^4 \sqrt{x} > ^3 \sqrt{x}\) holds true since \(x < 1\).

Conclusion: Statement 2 alone is sufficient to determine that \(^4 \sqrt{x} > ^3 \sqrt{x}\), while Statement 1 is inconclusive. Therefore, the correct answer is: statement (2) alone is sufficient to answer the question.