If x² - 4x + 1 = 0, then what is the value of x⁹ + x⁷ - 194x⁵ - 194x³?

A. 4

B. -4

C. 1

D. -1

Answer: Option B

Solution(By Examveda Team)

$$\eqalign{ & {x^2} - 4x + 1 = 0 \cr & x + \frac{1}{x} = 4 \cr & {x^2} + \frac{1}{{{x^2}}} = 14 \cr & {x^4} + \frac{1}{{{x^4}}} = 194 \cr & {x^9} + {x^7} - 194{x^5} - 194{x^3} \cr & {x^9} + {x^7} - \left( {{x^4} + \frac{1}{{{x^4}}}} \right){x^5} - \left( {{x^4} + \frac{1}{{{x^4}}}} \right){x^3} \cr & = {x^9} + {x^7} - {x^9} - x - {x^7} - \frac{1}{x} \cr & = - \left( {x + \frac{1}{x}} \right) \cr & = - 4 \cr} $$