CBSE 2024, Set 3

Question

**Assertion (A):** The corner points of the bounded feasible region of a L.P.P. are shown below. The maximum value of $Z = x + 2y$ occurs at infinite points.
![Feasible Region](https://www.edvaya.com/cbse/2024/mathematics/set3/q20-feasible-region-assertion.webp)

**Reason (R):** The optimal solution of a LPP having bounded feasible region must occur at corner points. A). Both Assertion (A) and Reason (R) are true and Reason (R) is the correct explanation of the Assertion (A). B). Both Assertion (A) and Reason (R) are true, but Reason (R) is not 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.

Edvaya · Accepted Answer

The correct answer is Both Assertion (A) and Reason (R) are true, but Reason (R) is not the correct explanation of the Assertion (A).. **Solution:**

**Analyze Assertion (A):**
The corner points of the feasible region (as seen in the graph) are $P(60, 0)$, $S(40, 20)$, $R(60, 30)$, and $Q(120, 0)$.
Let's evaluate the objective function $Z = x + 2y$ at each point:
At $P(60, 0)$: $Z = 60 + 2(0) = 60$
At $S(40, 20)$: $Z = 40 + 2(20) = 80$
At $R(60, 30)$: $Z = 60 + 2(30) = 120$
At $Q(120, 0)$: $Z = 120 + 2(0) = 120$

The maximum value is $120$, which occurs at two adjacent corner points, $R$ and $Q$. According to linear programming principles, if the maximum value occurs at two adjacent corner points, it will also occur at every point on the line segment joining them. Since a line segment contains an infinite number of points, the maximum value occurs at infinite points. Thus, Assertion (A) is **True**.

**Analyze Reason (R):**
The Fundamental Theorem of Linear Programming states that if an optimal solution (maximum or minimum) exists for a bounded feasible region, it must occur at one of the corner points (vertices) of the region. This is a universally true mathematical statement. Thus, Reason (R) is **True**.

**Conclusion:**
While both statements are true, Reason (R) by itself only guarantees that an optimal solution exists at *a corner point*. It does not explain *why* the solution occurs at *infinite points* in this specific scenario (which is due to the objective function being parallel to a boundary line yielding the same optimal value at two adjacent corners). Therefore, Reason (R) is not the complete and correct explanation for Assertion (A).

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