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Question

The corner points of the feasible region for optimization problem determined by a system of linear constraints are (6,5),(9,7),(15,0),(0,10). Let z=px+qy and the maximum of z occurs at both the points (6,5) and (9,7), then which of the following is true:

A
3p+2q=0 and p>0,q>0
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B
3p+2q=0 and p>0,q<0
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C
3p+2q=0 and p<0,q<0
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D
3p+2q=0 and p<0,q>0
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Solution

The correct option is C 3p+2q=0 and p<0,q<0
Let z0 be the maximum value of z in the feasible region. Since maximum occurs at both (6,5) and (9,7), the value z0​ is attained at both (6,5) and (9,7).
z0=p(6)+q(5)(i)
and z0=p(9)+q(7)(ii)
From (i) and (ii), we get:
6p+5q=9p+7q
3p+2q=0(iii)
and the values at other corner points (15,0) and (0,10) should be less than the z0
6p+5q>15pp<0(A) [from(iii)]
and 6p+5q>10qq<0(B) [from(iii)]

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