For the linear programming problem (LPP):

Maximize $Z = x + 1.5y$, subject to constraints,

$x + 2y \leq 40$, $2x + y \leq 40$, $x + y \leq 25$, $x \geq 0$, $y \geq 0$.

Which of the following is NOT correct?

The corner points are where the constraint lines intersect.

Corner Point 1: Intersection of $x = 0$ and $y = 0$

Point: $(0, 0)$

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Corner Point 2: Where $y = 0$ and $2x + y = 40$

$2x + 0 = 40$

$x = 20$

Point: $(20, 0)$

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Corner Point 3: Where $x = 0$ and $x + 2y = 40$

$0 + 2y = 40$

$y = 20$

Point: $(0, 20)$

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Corner Point 4: Intersection of $x + 2y = 40$ and $x + y = 25$

$x + 2y = 40$

$x + y = 25$

Subtracting the second from the first:

$y = 15$

Substituting back:

$x = 25 - 15$

$x = 10$

Point: $(10, 15)$

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Corner Point 5: Intersection of $2x + y = 40$ and $x + y = 25$

$2x + y = 40$

$x + y = 25$

Subtracting the second from the first:

$x = 15$

Substituting back:

$y = 25 - 15$

$y = 10$

Point: $(15, 10)$

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Evaluating $Z = x + 1.5y$ at each corner point:

At $(0, 0)$: $Z = 0 + 1.5(0) = 0$

At $(20, 0)$: $Z = 20 + 1.5(0) = 20$

At $(15, 10)$: $Z = 15 + 1.5(10) = 15 + 15 = 30$

At $(10, 15)$: $Z = 10 + 1.5(15) = 10 + 22.5 = 32.5$

At $(0, 20)$: $Z = 0 + 1.5(20) = 30$

Maximum value $= 32.5$ at point $(10, 15)$

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Checking each option:

Option 1: "The feasible region is bounded"

This is correct. The five finite corner points form a closed polygon.

Option 2: "The corner points are $(0,0)$, $(20,0)$, $(15,10)$, $(10,15)$ and $(0,20)$"

This is correct. These match the corner points found.

Option 3: "The optimal value is attained at point $(15, 10)$"

This is NOT correct. The optimal value $(32.5)$ is at $(10, 15)$, not $(15, 10)$.

Option 4: "The LPP has a unique optimal solution"

This is correct. Only one point $(10, 15)$ gives the maximum value.

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Option 3 is NOT correct.

The optimal solution is at $(10, 15)$ with $Z = 32.5$, not at $(15, 10)$.