For α ≠ 0, the nonnegative system x 1-2x 2+ x 3=8, x 1-x 2+x 4=, x 1,x 2,x 3,x 40, has NO basic feasible solution if

GATE 2020 MA – Question 52

Linear Programming · Models and Solution Methods · 2 marks · Multiple choice

For α ≠ 0, the nonnegative system $x_1-2x_2+\alpha x_3=8$, $x_1-x_2+x_4=\beta$, $x_1,x_2,x_3,x_4\geq0$, has NO basic feasible solution if

$\alpha<0,\beta>8$
$\alpha>0,0<\beta<8$
$\alpha>0,\beta<0$
$\alpha<0,\beta<8$

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Correct answer: (D) $\alpha<0,\beta<8$

Explanation

If α < 0, the first equation gives $x_1-2x_2=8-\alpha x_3\geq8$. Consequently $\beta=x_1-x_2+x_4\geq8+x_2+x_4\geq8$. Thus β < 8 makes the system infeasible and in particular rules out a basic feasible solution. This is D. For α > 0, increasing x₂ and adjusting nonnegative x₃ permits the other listed β ranges.

GATE 2020 MA – Question 52

Explanation

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