For the linear programming problem(LPP),

Maximize $Z = 4x + y$

$x + y \leq 5$

$3x + y \leq 9$

$x,y \geq 0$.

Which of the following are NOT true?

(A) The given LPP has unbounded feasible region.

(B) The corner points of the feasible region are (0,0), (0, 5), (3, 2) and (3, 0).

(C) The optimal value of the objective function is 12.

(D) The given LPP has a unique optimal solution.

Choose the correct answer from the options given below:

We need to maximize $Z = 4x + y$, subject to:

$x + y \leq 5$

$3x + y \leq 9$

$x, y \geq 0$

We'll find the feasible region, its corner points, and check which statements are NOT true.

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Since all constraints are of the type $\leq$ with $x, y \geq 0$, the feasible region lies in the first quadrant and is enclosed from all sides. The region does not stretch to infinity in any direction, so it is bounded.

Statement (A) says the feasible region is unbounded — this is NOT true. ✅

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Now, let's find the corner points of the feasible region.

At the origin: $(0, 0)$

$x + y = 5$ meets the $y$-axis ($x = 0$): $y = 5$ → $(0, 5)$

$3x + y = 9$ meets the $x$-axis ($y = 0$): $x = 3$ → $(3, 0)$

Intersection of $x + y = 5$ and $3x + y = 9$:

Subtracting the first from the second:

$3x + y - (x + y) = 9 - 5$

$2x = 4$

$x = 2$

Substituting back: $2 + y = 5$ → $y = 3$

So the intersection point is $(2, 3)$, NOT $(3, 2)$.

The correct corner points are: $(0, 0),\ (0, 5),\ (2, 3),\ (3, 0)$

Statement (B) lists $(3, 2)$ as a corner point — this is NOT true. ✅

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Evaluating $Z = 4x + y$ at each corner point:

At $(0, 0)$: $Z = 4(0) + 0 = 0$

At $(0, 5)$: $Z = 4(0) + 5 = 5$

At $(2, 3)$: $Z = 4(2) + 3 = 11$

At $(3, 0)$: $Z = 4(3) + 0 = 12$

Maximum $Z = 12$ at the point $(3, 0)$.

Statement (C) says the optimal value is $12$ — this is TRUE. ✗

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Since the maximum value $Z = 12$ occurs at only one corner point $(3, 0)$, the optimal solution is unique.

(If the maximum occurred at two or more corner points, every point on the line segment joining them would also be optimal, making the solution non-unique. That is not the case here.)

Statement (D) says the LPP has a unique optimal solution — this is TRUE. ✗

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The statements that are NOT true are (A) and (B).

$\boxed{\text{(A) and (B)}}$