Question : If $x^2-11x+1=0$, what is the value of $x^8-14159x^4+11$?
Option 1: 9
Option 2: 10
Option 3: 12
Option 4: 11
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Correct Answer: 10
Solution :
Given:
$x^2-11x+1=0$
Divide both sides by $x$, we get
$x-11+\frac{1}{x}=0$
⇒ $x+\frac{1}{x}=11$
Squaring both sides we get
⇒ $x^2+\frac{1}{x^2}+2×x×\frac{1}{x}=11^2$
⇒ $x^2+\frac{1}{x^2}=121-2$
⇒ $x^2+\frac{1}{x^2}=119$
Again squaring both sides we get,
⇒ $x^4+\frac{1}{x^4}+2×x^2×\frac{1}{x^2}=119^2$
⇒ $x^4+\frac{1}{x^4}=14161-2$
⇒ $x^4+\frac{1}{x^4}-14159=0$
Multiplying with $x^4$ we get
⇒ $x^8-14159x^4+1=0$
$\therefore x^8-14159x^4+11=10$
Hence, the correct answer is 10.
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