Question : If for non-zero $x,x^{2}-4x-1=0,$ then the value of $x^{2}+\frac{1}{x^{2}}$ is:
Option 1: 12
Option 2: 4
Option 3: 18
Option 4: 10
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Correct Answer: 18
Solution :
$x^{2}-4x-1=0$
Dividing with $x$,
$x-4- \frac{1}{x}=0$
⇒ $x- \frac{1}{x}=4$
Squaring on both sides,
⇒ $(x- \frac{1}{x})^2=4^2$
⇒ $x^{2}+\frac{1}{x^{2}}-2×x×\frac{1}{x} = 16$
⇒ $x^{2}+\frac{1}{x^{2}}= 16+2 = 18$
Hence, the correct answer is 18.
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