If [ ] denotes the greatest integer less than or equal to the real number under consideration, and $-1 ≤ x <0,0≤y <1, 1 ≤z<2$, then the value of the determinant $\begin{vmatrix}[x]+1&[y]&{[z]}\\{[x]}&[y]+1&{[z]}\\{[x]}&[y]&[z]+1\end{vmatrix}$, is

$[z]$

We have, $[x]=-1, [y]=0$ and $[z]=1$

$∴\begin{vmatrix}[x]+1&[y]&{[z]}\\{[x]}&[y]+1&{[z]}\\{[x]}&[y]&[z]+1\end{vmatrix}=\begin{vmatrix}0&0&1\\-1&1&1\\-1&0&2\end{vmatrix}=1=[z]$