If x² - 5x + 1 = 0, the the value of ( x⁴+1/(x²) ) ÷ ( x²+1 ) is :

If x² - 5x + 1 = 0, the the value of ( x⁴+1/(x²) ) ÷ (…

If x² - 5x + 1 = 0, the the value of ( $x^4+\dfrac{1}{x^2}$ ) ÷ ( $x^2+1$ ) is :

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Correct answer: D

Explanation

The correct answer is option 4 i.e. 22

x² - 5x + 1 = 0

⇒ x(x - 5 +1/x) = 0

⇒ x + (1/x) = 5

Cube on both side :

⇒ x³ + (1/x³) = 125 - 15

⇒ x³ + (1/x³) = 110

Now,

⇒ ( $x^4+\dfrac{1}{x^2}$ ) ÷ ( $x^2+1$ )

⇒ x(x³ + 1/x³) ÷ x ( x + 1/x)

⇒ 110 ÷ 5 = 22

So, the value of ( $x^4+\dfrac{1}{x^2}$ ) ÷ ( $x^2+1$ ) = 22