The maximum value of the objective function $Z = 8x + 2y$ of an LPP subject to constraints $2x + y \leq 3, 2x + 3y \leq 6, x \geq 0, y \geq 0$ is:

The objective function is $Z = 8x + 2y$ subject to constraints $2x + y \leq 3$, $2x + 3y \leq 6$, $x \geq 0$, $y \geq 0$.

Since $x \geq 0$ and $y \geq 0$, the feasible region lies in the first quadrant. The maximum of $Z$ occurs at a corner point of the feasible region.

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Corner point $O$ (origin): $(0,\ 0)$

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Corner point $A$ — line $2x + y = 3$ meets the $x$-axis ($y = 0$):

$2x = 3$

$x = \tfrac{3}{2}$

$A = \left(\tfrac{3}{2},\ 0\right)$

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Corner point $C$ — line $2x + 3y = 6$ meets the $y$-axis ($x = 0$):

$3y = 6$

$y = 2$

$C = (0,\ 2)$

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Corner point $B$ — intersection of $2x + y = 3$ and $2x + 3y = 6$:

$2x + y = 3 \quad \cdots (i)$

$2x + 3y = 6 \quad \cdots (ii)$

$(ii) - (i)$:

$2y = 3$

$y = \tfrac{3}{2}$

From $(i)$:

$2x + \tfrac{3}{2} = 3$

$2x = \tfrac{3}{2}$

$x = \tfrac{3}{4}$

$B = \left(\tfrac{3}{4},\ \tfrac{3}{2}\right)$

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Evaluating $Z = 8x + 2y$ at each corner point:

At $O(0,\ 0)$: $Z = 8(0) + 2(0) = 0$

At $A\left(\tfrac{3}{2},\ 0\right)$: $Z = 8\left(\tfrac{3}{2}\right) + 2(0) = 12$

At $B\left(\tfrac{3}{4},\ \tfrac{3}{2}\right)$: $Z = 8\left(\tfrac{3}{4}\right) + 2\left(\tfrac{3}{2}\right) = 6 + 3 = 9$

At $C(0,\ 2)$: $Z = 8(0) + 2(2) = 4$

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The maximum value of $Z$ is $\boxed{12}$ at the point $\left(\tfrac{3}{2},\ 0\right)$.