The system of linear equations

$x+y + z= 2 $

$2x+ 3y + 2z= 5 $

$2x+ 3y + (a^2 -1) z= a+1$

is inconsistent $|a|= \sqrt{3}$

The correct answer is option (2) : is inconsistent $|a|= \sqrt{3}$

We find that

$D=\begin{vmatrix}1 & 1 & 1\\2 & 3 & 2\\2 & 3 & a^2-1\end{vmatrix}=a^2-3$

$D_1= \begin{vmatrix}2 & 1 & 1\\5 & 3 & 2\\a+1 & 3 & a^2-1\end{vmatrix}=a^2-a+1$

$D_2= \begin{vmatrix}1 & 2 & 1\\2 & 5 & 2\\2 & a+1 & a^2-1\end{vmatrix}=a^2-3, D_3=\begin{vmatrix}1 & 1 & 2\\2 & 3 & 5\\2 & 3 & a+1\end{vmatrix}=a-4$

When $|a|=\sqrt{3}, $ i.e. $a^2 = 3,$ we obtain $D=0$ and $D_1≠0, D_3 ≠ 0.$

Hence, the given system is inconsistent when $|a|= \sqrt{3}.$