Question

Let $0\le x<4$, $-2\le y<3$ and $-1\le z<5$. If [a] denotes the greatest integer$\le a$, then maximum possible value of

$\Delta =\left|\begin{array}{ccc}[x+2]& \left[y\right]& \left[z\right]\\ \left[x\right]& [y+1]& \left[z\right]\\ \left[x\right]& \left[y\right]& [z+1]\end{array}\right|$
is

• A. $13$
• B. $15$
• C. $17$
• D. $19$

Answer 1

The correct option is C $17$

$\Delta =\left|\begin{array}{ccc}[x+2]& \left[y\right]& \left[z\right]\\ \left[x\right]& [y+1]& \left[z\right]\\ \left[x\right]& \left[y\right]& [z+1]\end{array}\right|=\left|\begin{array}{ccc}\left[x\right]+2& \left[y\right]& \left[z\right]\\ \left[x\right]& \left[y\right]+1& \left[z\right]\\ \left[x\right]& \left[y\right]& \left[z\right]+1\end{array}\right|$

Applying $R_2\to R_2-R_1,R_3\to R_3-R_1$

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

$\Delta =(\left[x\right]+2)\left(1\right)-\left[y\right](-2)+\left[z\right]\left(2\right)=\left[x\right]+2+2\left[y\right]+2\left[z\right]$

This is maximum when $x,y,z$ are maximum

$\Delta =3+2+2\left(2\right)+2\left(4\right)=17$