Question : If $x^2-5 x+1=0$, then the value of $\frac{x^6+x^4+x^2+1}{5 x^3}=?$
Option 1: 30
Option 2: 25
Option 3: 23
Option 4: 28
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Correct Answer: 23
Solution :
Given, $x^2-5 x+1=0$
$⇒x^2+1=5 x$----(i)
Multiplying whole equation by $x^4$, we get:
$⇒x^6+x^4=5 x^5$
Also dividing equation (i) by $x$, we get,
$x+\frac{1}{x}=5$
Now, $\frac{x^6+x^4+x^2+1}{5 x^3}$
= $\frac{5 x^5+5 x}{5 x^3}$
= $x^2+\frac{1}{x^2}$
= $(x+\frac{1}{x})^2-2$
= $5^2-2$
= $23$
Hence, the correct answer is 23.
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