KCET · Maths · Linear Programming

The corner points of the feasible region determined by the system of linear constraints are \((0, 10), (5, 5), (15, 15), (0, 20)\). Let \(z = px + qy\) where \(p, q > 0\). The relation between \(p\) and \(q\), so that the maximum \(z\) occurs at both points \((15, 15)\) and \((0, 20)\) is

A \(p = q\)
B \(p = 2q\)
C \(q = 2p\)
D \(q = 3p\)

Verified Solution

Answer & Solution

Correct Answer

(D) \(q = 3p\)

Step-by-step Solution

Detailed explanation

The value of the objective function \(z = px + qy\) at the given corner points is calculated as follows:

At \((15, 15)\), \(z = 15p + 15q\)

At \((0, 20)\), \(z = 0p + 20q = 20q\)

Since the maximum value of \(z\) occurs at both \((15, 15)\) and \((0, 20)\), the values of \(z\) at these points must be equal.

\(15p + 15q = 20q\)

\(15p = 5q\)

\(q = 3p\)

Answer: \(q = 3p\)

Apply the relevant identity from the chapter and substitute the values established in the previous step, keeping the original constraint in view.

Use the simplified expression to isolate the unknown, then plug the result back into the question setup to confirm the working is internally consistent.

Cross-check against a boundary or limiting case. Both the dimensional balance and the qualitative behaviour at the extreme should agree with the candidate answer.

Combine the steps above to identify the correct option. The remaining choices each fail at least one of the consistency, dimensional, or boundary checks.

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