If a = 2 + 3 , then the value of ( a^2 + 1a^2 ) = \,?

Answer: Option B

$$\begin{array}{rl}& a=2+\sqrt{3}\\ & \Rightarrow a^2={\left(2+\sqrt{3}\right)}^2\\ & \Rightarrow a^2=4+3+4\sqrt{3}\\ & \Rightarrow a^2=7+4\sqrt{3}\\ & \frac{1}{a^2}=\frac{1}{7+4\sqrt{3}}\\ & \Rightarrow \frac{1}{a^2}=\frac{7-4\sqrt{3}}{\left(7+4\sqrt{3}\right)\left(7-4\sqrt{3}\right)}\\ & \Rightarrow \frac{1}{a^2}=\frac{7-4\sqrt{3}}{1}\\ & \Rightarrow \frac{1}{a^2}=7-4\sqrt{3}\\ & \therefore a^2+\frac{1}{a^2}\\ & =7+4\sqrt{3}+7-4\sqrt{3}\\ & =14\end{array}$$