Question : If $x+\frac{1}{x}=2 \sqrt{5}$ where $x>1$, then the value of $x^3-\frac{1}{x^3}$ is:
Option 1: 82
Option 2: 76
Option 3: 86
Option 4: 78
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Correct Answer: 76
Solution :
Given: $x+\frac{1}{x}=2 \sqrt{5}$
⇒ $(x+\frac{1}{x})^2=(2 \sqrt{5})^2$
⇒ $x^2+\frac{1}{x^2}+2=20$
⇒ $x^2+\frac{1}{x^2}=18$ -------------(i)
⇒ $x^2+\frac{1}{x^2}-2=18-2=16$
⇒ $(x-\frac{1}{x})^2=(4)^2$
⇒ $x-\frac{1}{x}=4$ -------------(ii)
So, $x^3-\frac{1}{x^3}$ = $(x-\frac{1}{x})(x^2+\frac{1}{x^2}+x×\frac{1}{x})$
= 4 × (18 + 1)
= 76
Hence, the correct answer is 76.
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