Question : If $x^{2} - 11 x + 1 = 0$ , what is the value of $x^{8} - 14159 x^{4} + 11$ ?
Option 1: 9
Option 2: 10
Option 3: 12
Option 4: 11

Question

Question : If $x^{2} + \frac{1}{x^{2}} = 1$ , then, what is the value of $x^{48} + x^{42} + x^{36} + x^{30} + x^{24} + x^{18} + x^{12} + x^{6} + 1$ ?

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Answer 1

Correct Answer: 10

Solution : Given:
$x^{2} - 11 x + 1 = 0$
Divide both sides by $x$ , we get
$x - 11 + \frac{1}{x} = 0$
⇒ $x + \frac{1}{x} = 11$
Squaring both sides we get
⇒ $x^{2} + \frac{1}{x^{2}} + 2 \times x \times \frac{1}{x} = 11^{2}$
⇒ $x^{2} + \frac{1}{x^{2}} = 121 - 2$
⇒ $x^{2} + \frac{1}{x^{2}} = 119$
Again squaring both sides we get,
⇒ $x^{4} + \frac{1}{x^{4}} + 2 \times x^{2} \times \frac{1}{x^{2}} = 119^{2}$
⇒ $x^{4} + \frac{1}{x^{4}} = 14161 - 2$
⇒ $x^{4} + \frac{1}{x^{4}} - 14159 = 0$
Multiplying with $x^{4}$ we get
⇒ $x^{8} - 14159 x^{4} + 1 = 0$
$∴ x^{8} - 14159 x^{4} + 11 = 10$
Hence, the correct answer is 10.

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Question : If x2−11x+1=0, what is the value of x8−14159x4+11?Option 1: 9Option 2: 10Option 3: 12Option 4: 11

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Question : If $x^{2} - 11 x + 1 = 0$ , what is the value of $x^{8} - 14159 x^{4} + 11$ ?
Option 1: 9
Option 2: 10
Option 3: 12
Option 4: 11