Question

lf $[.]$ denotes the greatest integer less than or equal to the real number under consideration, and $-1\le x<0,0\le y<1,1\le z<2$, then the value of the determinant $\left|\begin{array}{ccc}\left[x\right]+1& \left[y\right]& \left[z\right]\\ \left[x\right]& \left[y\right]+1& z\\ \left[x\right]& \left[y\right]& \left[z\right]+1\end{array}\right|$ is

• A. $\left[x\right]$
• B. $\left[y\right]$
• C. $\left[z\right]$
• D. $\left[x\right]+\left[y\right]+\left[z\right]$

Answer 1

The correct option is C $\left[z\right]$

$-1x\ge 0$ $\Rightarrow \left[x\right]=-1$
$0y\ge 1$ $\Rightarrow \left[y\right]=0$
$1z\ge 2$ $\Rightarrow \left[z\right]=1$
Now, $\left|\begin{array}{ccc}\left[x\right]+1& \left[y\right]& \left[z\right]\\ \left[x\right]& \left[y\right]+1& z\\ \left[x\right]& \left[y\right]& \left[z\right]+1\end{array}\right|=\left|\begin{array}{ccc}0& 0& 1\\ -1& 1& 1\\ -1& 0& 2\end{array}\right|=1=\left[z\right]$