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Question

Let $$0\leq x< 4$$, $$-2\leq y< 3$$ and $$-1\leq z< 5$$. If [a] denotes the greatest integer$$\leq a$$, then maximum possible value of
$$\Delta =\begin{vmatrix}
\left [ x+2 \right ] & \left [ y \right ] & \left [ z \right ]\\
\left [ x \right ] & \left [ y+1 \right ] & \left [ z \right ]\\
\left [ x \right ] & \left [ y \right ] & \left [ z+1 \right ]
\end{vmatrix}$$
is


A
13
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B
15
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C
17
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D
19
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Solution

The correct option is C $$17$$
$$\Delta =\begin{vmatrix} \left[ x+2 \right]  & \left[ y \right]  & \left[ z \right]  \\ \left[ x \right]  & \left[ y+1 \right]  & \left[ z \right]  \\ \left[ x \right]  & \left[ y \right]  & \left[ z+1 \right]  \end{vmatrix}=\begin{vmatrix} \left[ x \right] +2 & \left[ y \right]  & \left[ z \right]  \\ \left[ x \right]  & \left[ y \right] +1 & \left[ z \right]  \\ \left[ x \right]  & \left[ y \right]  & \left[ z \right] +1 \end{vmatrix}$$
Applying $${ R }_{ 2 }\rightarrow { R }_{ 2 }-{ R }_{ 1 },{ R }_{ 3 }\rightarrow { R }_{ 3 }-{ R }_{ 1 }$$
$$\Delta =\begin{vmatrix} \left[ x \right] +2 & \left[ y \right]  & \left[ z \right]  \\ -2 & 1 & 0 \\ -2 & 0 & 1 \end{vmatrix}\\$$

$$ \Delta =\left( \left[ x \right] +2 \right) \left( 1 \right) -\left[ y \right] \left( -2 \right) +\left[ z \right] \left( 2 \right) =\left[ x \right] +2+2\left[ y \right] +2\left[ z \right] $$
This is maximum when $$x,y,z$$ are maximum
$$\Delta =3+2+2\left( 2 \right) +2\left( 4 \right) =17$$

Mathematics

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