Question

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

• A. $\left[x\right]$
• B. $\left[y\right]$
• C. $\left[z\right]$
• D. none of these

Answer 1

Answer: (C)

$\left[z\right]$

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