Question

If \( A = 4x + \frac{1}{x} \), then the value of \( A + \frac{1}{A} \) is:

• A. \( \frac{1}{4x^3 + x} \)
• B. \( 4x^2 + 1 \)
• C. \( x \)
• D. \( \frac{x}{4x^2 + 1} \)

Hint:

When given expressions involving variables and fractions, simplify each term first and then combine them to get the final result.

Answer 1

The Correct Option is D

Given \( A = 4x + \frac{1}{x} \), we find \( A + \frac{1}{A} \). First, compute \( \frac{1}{A} \): \[\nA = 4x + \frac{1}{x} \quad \Rightarrow \quad \frac{1}{A} = \frac{1}{4x + \frac{1}{x}} = \frac{x}{4x^2 + 1}\n\] Then, add \( A \) and \( \frac{1}{A} \): \[\nA + \frac{1}{A} = \left( 4x + \frac{1}{x} \right) + \frac{x}{4x^2 + 1} = \frac{x}{4x^2 + 1}\n\] Therefore, the result is \( \frac{x}{4x^2 + 1} \).