The system of equations
$kx+(k+1)y+(k-1)z=0$
$(k+1)x+ky+(k+2)z=0$
$(k-1)x+(k+2)y+kz=0$ has a non-trivial solution for

The correct option is C Exactly one real value of k
For non-trivial solution

$\left|\begin{array}{ccc}k& k+1& k-1\\ k+1& k& k+2\\ k-1& k+2& k\end{array}\right|=0$
$R_2\to R_2-R_1,R_3\to R_3-R_1$
$\left|\begin{array}{ccc}k& k+1& k-1\\ 1& -1& 3\\ -1& 1& 1\end{array}\right|=0k=-\frac{1}{2}$