Correct option is A

Given:

​$$x + \frac{1}{x} = 1$$​​

Formula Used:  

$$a^3 + b^3 = (a+b)(a^2 +b^2 - ab)$$ 

Solution:

​$$​x + \frac{1}{x} = 1$$   

​$$x^2 - x + 1 = 0$$​​

​$$x^3 + 1^3 = (x + 1)(x^2 - x + 1) = 0$$​​

​$$x^3 = -1$$

Then the value $$x^{12}+x^9 +x^6 +x^3 +1$$​​

Thus,  $$x^9 (x^3 + 1) + x^3 (x^3 + 1) + 1 = 1$$​​