Question

Assertion (A): The shaded portion of the graph represents the feasible region for the given Linear Programming Problem (LPP).
Reason (R): The region representing \( Z = 50x + 70y \) such that \( Z < 380 \) does not have any point common with the feasible region.

• A. Both Assertion (A) and Reason (R) are true and the Reason (R) is the correct explanation of the Assertion (A)
• B. Both Assertion (A) and Reason (R) are true and the Reason (R) is the correct explanation of the Assertion (A)
• C. Assertion (A) is true, but Reason (R) is false.
• D. Assertion (A) is false, but Reason (R) is true.

Hint:

When solving Linear Programming Problems, always check the intersection of the feasible region with the objective function's value to identify the point at which the minimum or maximum occurs.

Answer 1

The Correct Option is A

- The feasible region is the area that satisfies all the given constraints for the Linear Programming Problem (LPP).
This region is bounded by the lines representing the constraints. The graph shows this feasible region as the shaded portion. Therefore, Assertion (A) is correct.
- The objective function is \( Z = 50x + 70y \). The given condition \( Z = 50x + 70y \) has a minimum value of 380 at the point \( B(2, 4) \). 
- If \( Z < 380 \), this means the values of \( x \) and \( y \) are outside the feasible region, as the region representing \( Z = 50x + 70y \) less than 380 does not intersect the feasible region. Thus, Reason (R) correctly explains why the region \( Z < 380 \) does not intersect the feasible region.
Hence, both Assertion (A) and Reason (R) are correct, and Reason (R) is the correct explanation for Assertion (A).