If ( a + 1/a )^2 = 3 , then the value of a18 + a12 + a6 + 1 is?

Answer: Option C

$$\begin{array}{rl}& {\left(a+\frac{1}{a}\right)}^2=3\\ & \Rightarrow a+\frac{1}{a}=\sqrt{3}\\ & \Rightarrow a^3+\frac{1}{a^3}=0\\ & \Rightarrow a^6=-1\\ & \Rightarrow \left(a^6+1\right)=0\\ & \text{So,}{a}^{18}+a^{12}+a^6+\text{ 1}\\ & \Rightarrow a^{12}\left(a^6+1\right)+a^6+\text{ 1}\\ & \Rightarrow a^{12}\left(0\right)+0\\ & \Rightarrow 0\end{array}$$