Option 3 : 1

Given:

x + 1/x = 1

Formula Used:

a3 + b3 = (a + b)(a2 - ab + b2)

Calculation:

x + 1/x = 1

⇒ x2 + 1 = x

⇒ x2 - x + 1 = 0 ---(1)

Multiply (x + 1) into both side of eq (1)

⇒ (x + 1)( x2 - x + 1) = 0

⇒ x3 + 1 = 0

⇒ x3 = - 1

x95 + 1/x95 + x44 + 1/x44 + x1025 + 1/x1025

⇒ (x93)x2 + 1/(x93)x2 + (x^{42})x^{2} + 1/(x^{42})x^{2} + (x^{1023})x^{2} + 1/(x^{1023})x^{2}

⇒ [{(x3)31}x2] + 1/[{(x3)31}x2] + [{(x^{3})14}x2] + 1/[{(x3)14}x2] + [{(x3)341}x2] + 1/[{(x3)341}x2]

⇒ [{(-1)31}x2] + 1/[{(-1)31}x2] + [{(-1)14}x2] + 1/[{(-1)14}x2] + [{(-1)341}x2] + 1/[{(-1)341}x2]

⇒ – x2 - 1/x2 + x^{2} + 1/x^{2} - x2 - 1/x2

⇒ – x2 - 1/x2

Now multiply and divide by x

⇒ (-x3/x) - (x/x3)

⇒ {-(-1)/x} - {x/(-1)}

⇒ (1/x) + x

⇒ 1

**∴ The value of x95 + 1/x95 + x44 + 1/x44 + x1025 + 1/x1025 is 1.**