# If the system of equations $x+2y-3z=1$, $(p+2)z=3$, $(2p+1)y+z=2$ is inconsistent, then the value of $p$ is

$(a)\;-2\qquad(b)\;1\qquad(c)\;0\qquad(d)\;2$

The augmented matrix is C=[A B]
$\Rightarrow \begin{vmatrix}1&2&-3&1\\0&0&p+2&3\\0&2p+1&1&2\end{vmatrix}$
$R_2\leftrightarrow R_3$
$\Rightarrow \begin{vmatrix}1&2&-3&1\\0&2p+1&1&2\\0&0&p+2&3\end{vmatrix}$
The system of equations is inconsistent if $r(A)=2$ and $r(C)=3$
Now,$r(A)=2$ if $p+2=0$
(i.e) $p=-2$
Hence (a) is the correct answer.