Correct Answer - Option 3 : 488
Calculation:
Given: x = 4 + √15,
⇒ 1/x = 1/(4 + √15) = 4 - √15
\( \Rightarrow {\rm{\;}}x\; + \;\frac{1}{x}\; = \;4{\rm{\;}} + {\rm{\;}}\sqrt {15} \; + \;4 - {\rm{\;}}\sqrt {15} \; = \;8\)
\( \Rightarrow {\rm{\;}}{\left( {x\; + \;\frac{1}{x}} \right)^3}\; = \;{x^3}\; + \;\frac{1}{{{x^3}}}\; + \;3\left( {x\; + \;\frac{1}{x}} \right)\)
\( \Rightarrow {\rm{\;}}{\left( 8 \right)^3}\; = \;{x^3}\; + \;\frac{1}{{{x^3}}}\; + \;3\left( 8 \right)\)
\( \Rightarrow {\rm{\;}}{x^3}\; + \;\frac{1}{{{x^3}}}\; = \;512 - 24\; = \;488\)
∴ Required answer is 488.
