Question : If $x+\frac{1}{x}=-2$, then what is the value of $x^{17}+x^{-17}+x^{12}+x^{-12} ?(x<0)$
Option 1: –2
Option 2: –1
Option 3: 1
Option 4: 0
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Correct Answer: 0
Solution :
Given $x+\frac{1}{x}=-2$
$⇒x^2+1=-2x$
$⇒x^2+2x+1=0$
$⇒(x+1)^2=0$
$⇒x=-1$
Now, putting the value of $x=-1$ in $x^{17}+x^{-17}+x^{12}+x^{-12}$ we get:
$=(-1)^{17}+(-1)^{-17}+(-1)^{12}+(-1)^{-12}$
$=-1-1+1+1=0$
Hence, the correct answer is 0.
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