The corner points of the feasible region for an L.P.P. are (0, 10), (5, 5), (5, 15) and (0, 30). If the objective function is \(Z = \alpha x + \beta y,\ \alpha, \beta \gt 0\), the condition on \(\alpha\) and \(\beta\) so that maximum of Z occurs at corner points (5, 5) and (0, 20) is:
Answer & explanation
Correct answer: option 3
The correct answer is Option (3) → \(\alpha = 3\beta\)
When the maximum of Z occurs at two points, Z must have the same value at both points:
\(Z(5, 5) = 5\alpha + 5\beta\)
\(Z(0, 20) = 20\beta\)
Equating the two values:
\(5\alpha + 5\beta = 20\beta\)
\(5\alpha = 15\beta\)
\(\alpha = 3\beta\)
None of the other conditions makes the two values equal. For example, \(\alpha = 5\beta\) gives \(Z(5, 5) = 30\beta\), not \(20\beta\).
Note: It seems that there is a mistake in the original NTA paper, which asks for the maximum at (0, 20), but (0, 20) is not one of the given corner points; also, with \(\alpha, \beta \gt 0\), the value of Z at (5, 15) is always larger than at (5, 5), so the maximum cannot be at (5, 5). The question can still be solved by equating the values of Z at the two points named in the question.