1.

If x2 - 5x + 1 = 0, then the value of \(\left(x^4 + \frac 1 {x^2}\right) \div \left(x^2 + 1\right)\) is:1. 252. 213. 224. 24

Answer» Correct Answer - Option 3 : 22

Given:

x2 - 5x + 1 = 0

Formula used:

(a + b)3 = a3 + b3 + 3ab(a + b)

Calculations:

x2 - 5x + 1 = 0

⇒ x(x - 5 + 1/x) = 0

⇒ (x - 5 + 1/x) = 0

⇒ (x + 1/x) = 5

Taking cube both sides,

⇒ (x + 1/x)3 = (5)3

⇒ x3 + 1/x3 + 3(x)(1/x)(x + 1/x) = 125

⇒ x3 + 1/x3 + 3(5) = 125

⇒ x3 + 1/x3 + 15 = 125

⇒ x3 + 1/x3 = 125 - 15

⇒ x3 + 1/x3 = 110

(x4 + 1/x2)/(x2 + 1)

Dividing the numerator and denominator by x,

(1/x)(x4 + 1/x2)/(x2 + 1)(1/x)

⇒ (x3 + 1/x3)/(x + 1/x)

⇒ 110/5

⇒ 22

∴ The value of \(\left(x^4 + \frac 1 {x^2}\right) \div \left(x^2 + 1\right)\) is 22.


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