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For an objective function 𝑍 = π‘Žπ‘₯ + 𝑏𝑦, where π‘Ž, 𝑏 > 0; the corner points of the feasible region determined by a set of constraints (linear inequalities) are (0, 20), (10, 10), (30, 30) and (0, 40). The condition on a and b such that the maximum Z occurs at both the points (30, 30) and (0, 40) is:

(a) 𝑏 βˆ’ 3π‘Ž = 0Β  (b) π‘Ž = 3𝑏

(c) π‘Ž + 2𝑏 = 0Β  (d) 2π‘Ž βˆ’ 𝑏 = 0

Β 

This question is inspired from Ex 12.2, 11 (MCQ) - Chapter 12 Class 12 - Linear Programming

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Question 41 For an objective function 𝑍 = π‘Žπ‘₯ + 𝑏𝑦, where π‘Ž, 𝑏 > 0; the corner points of the feasible region determined by a set of constraints (linear inequalities) are (0, 20), (10, 10), (30, 30) and (0, 40). The condition on a and b such that the maximum Z occurs at both the points (30, 30) and (0, 40) is: (a) 𝑏 βˆ’ 3π‘Ž = 0 (b) π‘Ž = 3𝑏 (c) π‘Ž + 2𝑏 = 0 (d) 2π‘Ž βˆ’ 𝑏 = 0 Since maximum Z occurs at both the points (30, 30) and (0, 40) Value of Z at both these points will be same Therefore 30a + 30b = 40b 30a = 40b βˆ’ 30b 30a = 10b Dividing by 10 both sides 3a = b 0 = b βˆ’ 3a b βˆ’ 3a = 0 So, the correct answer is (A)

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