Question : If $x+\frac{1}{x}=0$, then the value of $x^{5}+\frac{1}{x^{5}}$ is:
Option 1: 2
Option 2: –1
Option 3: 1
Option 4: 0
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Correct Answer: 0
Solution :
Given: $x+\frac{1}{x}=0$............................................ $(i)$
Now,
$(x^2+\frac{1}{x^2})(x+\frac{1}{x})$ = 0
⇒ $x^3+\frac{1}{x^3} + x+\frac{1}{x}$= 0
⇒ $x^3+\frac{1}{x^3}=0$
Also,
$(x^4+\frac{1}{x^4})(x+\frac{1}{x})$ = 0
⇒ $x^5+\frac{1}{x^5} + x^3+\frac{1}{x^3}$= 0
⇒ $ x^5+\frac{1}{x^5} + 0 = 0$
⇒ $x^5+\frac{1}{x^5} = 0$
Hence, the correct answer is 0.
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