1
GATE ME 2009
+2
-0.6
Consider the following Linear Programming problem $$(LLP)$$

Maximize: $$Z = 3{x_1} + 2{x_2}$$
$$\,\,$$ Subject $$\,\,$$ to
\eqalign{ & \,\,\,\,\,\,\,{x_1} \le 4 \cr & \,\,\,\,\,\,\,{x_2} \le 6 \cr & 3{x_1} + 2{x_2} \le 18 \cr & {x_1} \ge 0,\,\,{x_2} \ge 0 \cr}

A
The $$LPP$$ has a unique optimal solution.
B
The $$LPP$$ is infeasible
C
The $$LPP$$ is unbounded
D
The $$LPP$$ has multiple optimal solutions.
2
GATE ME 2008
+2
-0.6
Consider the Linear programme $$(LP)$$
Max $$4x$$ + $$6y$$
Subject to
\eqalign{ & \,\,\,\,\,\,\,\,\,\,\,3x + 2y \le 6 \cr & \,\,\,\,\,\,\,\,\,\,\,2x + 3y \le 6 \cr & \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,x,y \ge 0 \cr}

The dual for the $$LP$$ is

A
\eqalign{ & {Z_{\min }} = 6u + 6v \cr & \,\,subjuect\,\,to\,\, \cr & 3u + 2v \ge 4 \cr & 2u + 3v \ge 6 \cr & u,v \ge 0 \cr}
B
\eqalign{ & {Z_{\max }} = 6u + 6v \cr & \,\,subjuect\,\,to\,\, \cr & 3u + 2v \le 4 \cr & 2u + 3v \le 6 \cr & u,v \ge 0 \cr}
C
\eqalign{ & {Z_{\max }} = 4u + 6v \cr & \,\,subjuect\,\,to\,\, \cr & 3u + 2v \ge 6 \cr & 2u + 3v \ge 6 \cr & u,v \ge 0 \cr}
D
\eqalign{ & {Z_{\max }} = 4u + 6v \cr & \,\,subjuect\,\,to\,\, \cr & 3u + 2v \le 6 \cr & 2u + 3v \le 6 \cr & u,v \ge 0 \cr}
3
GATE ME 2008
+2
-0.6
Consider the Linear programme $$(LP)$$
Max $$4x$$ + $$6y$$
Subject to
\eqalign{ & \,\,\,\,\,\,\,\,\,\,\,3x + 2y \le 6 \cr & \,\,\,\,\,\,\,\,\,\,\,2x + 3y \le 6 \cr & \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,x,y \ge 0 \cr}

After introducing slack variables $$s$$ and $$t$$, the initial basic feasible solution is represented by the table below (basic variables are $$s=6$$ $$t=6,$$ and the objective function value is $$0$$).

After some simplex iterations, the following table is obtained

From this, one can conclude that

A
The $$LP$$ has a unique optimal solution
B
The $$LP$$ has an optimal solution that is not unique
C
The $$LP$$ is infeasible
D
The $$LP$$ is unbounded
4
GATE ME 2005
+2
-0.6
Consider a linear programming problem with two variables and two constraints. The objective function is: Maximize $${x_1} + {x_2}.$$ The corner points of the feasible region are $$(0,0), (0,2), (2,0)$$ and $$(4/3, 4/3).$$

If an additional constraint $${x_1} + {x_2} \le 5$$ is added, the optimal solution is

A
$$\left( {{5 \over 3},{5 \over 3}} \right)$$
B
$$\left( {{4 \over 3},{4 \over 3}} \right)$$
C
$$\left( {{5 \over 2},{5 \over 2}} \right)$$
D
$$(5,0)$$
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