Consider the LPP
Maximize: Z= 5x+3y subject to constraints
$2x+5y ≤15; 5x +2y ≤ 10; x, y ≥0.$ Which of the following is true ?
A. The feasible region lies in the $1^{st}$ quadrant.
B. The maximum value of Z is 9.
C. The feasible region has 4 corner points.
D. It has three optimal solutions.
E. The maximum value of Z lies at $\left(\frac{20}{19}, \frac{45}{19}\right) $ and $(2, 0)$
Choose the correct answer from the options given below :
Answer & explanation
Correct answer: option 3
The correct answer is Option (3) → A, C only
Statement A (True): The non-negativity constraints $x \geq 0, y \geq 0$ mean the feasible region is in the $1^{st}$ quadrant.
Statement B (False): The maximum value of Z is 9.
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$Z$ at $(2, 0) = 10$, and $Z$ at $(\frac{20}{21}, \frac{55}{21}) \approx 12.62$.
- Since both values are greater than 9, Statement B is false.
Statement C (True): The region is bounded by the axes and the two lines, having 4 corner points: $(0,0)$, $(0,3)$, $(2,0)$, and the intersection point $(\frac{20}{21}, \frac{55}{21})$.
Statement D (False): It has three optimal solutions.
-
In this case, the maximum value of $12.62$ occurs only at the unique intersection point $(\frac{20}{21}, \frac{55}{21})$.
-
There is no other corner point or line segment that produces this same maximum value, so there is only one optimal solution.
Statement E (False): The maximum value of Z lies at $\left(\frac{20}{19}, \frac{45}{19}\right) $ and $(2, 0)$. The actual intersection point derived from the constraints $2x + 5y = 15$ and $5x + 2y = 10$ is $(\frac{20}{21}, \frac{55}{21})$, which contradicts the coordinates provided in the statement. Furthermore, the value of $Z$ at this intersection point ($\approx 12.62$) is not equal to the value of $Z$ at $(2, 0)$, which is $10$.