The corner points of the bounded feasible region determined by a set of constraints in an LPP are P(0, 5) Q(3, 5), R(4, 0) and S(4, 1). If the objective function is \(z = ax + 2by\), where, \(a, b \gt 0\), then the condition on a and b such that the maximum value of z occurs at Q and S is
Answer & explanation
Correct answer: option 4
The correct answer is Option (4) → \(a-8b=0\)
Values of z at the corner points:
\(z(P) = 10b\)
\(z(Q) = 3a + 10b\)
\(z(R) = 4a\)
\(z(S) = 4a + 2b\)
For the maximum to occur at both Q and S, the values at Q and S must be equal:
\(3a + 10b = 4a + 2b\)
\(a = 8b\), that is, \(a - 8b = 0\)
Check that this common value is the largest: with \(a = 8b\),
\(z(Q) = z(S) = 34b,\quad z(R) = 32b,\quad z(P) = 10b\)
So the maximum, 34b, occurs at Q and S.
Note: It seems that there is a mistake in the original NTA question, which gives the corner R as (5, 0). With that corner, \(a = 8b\) gives \(z(R) = 40b\), which is larger than \(z(Q) = z(S) = 34b\), so the maximum could never occur at Q and S. Also, S(4, 1) lies on the segment PR, so it could not be a corner. For the benefit of students and to ensure that the question is correct and has a single valid answer, the corner R has been changed to (4, 0), so that Option (4) gives the maximum at Q and S.