Question : If $x^2-4 x-1=0$, then find the value of $x^2+\frac{1}{x^2}-5$.
Option 1: 17
Option 2: 12
Option 3: 15
Option 4: 13
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Correct Answer: 13
Solution :
$x^2-4 x-1=0$
$x-\frac{1}{x} = 4$
Squaring on both sides,
$ x^{2} + \frac{1}{x^{2}} - 2×x×\frac{1}{x} = 16$
$ x^{2} + \frac{1}{x^{2}} = 16+2 = 18$ -----------(i)
Now, $x^2+\frac{1}{x^2}-5$
= 18 – 5
= 13
Hence, the correct answer is 13.
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