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Questions › IIT JAM › IIT JAM 2021 › Q56
2 marksMathematics › Constrained optimisation and linear programming › Linear programming: formulation and graphical solutionHardabout 3 min

IIT JAM 2021 Question 56: Constrained optimisation and linear programming

For some β>0\beta > 0β>0, let x1x_1x1​ and x3x_3x3​ be the optimal basic feasible solution of: maximize x1+2x2+3x3x_1 + 2x_2 + 3x_3x1​+2x2​+3x3​ subject to 2x1−x2+x3=92x_1 - x_2 + x_3 = 92x1​−x2​+x3​=9, x1+2x2−βx3=1x_1 + 2x_2 - \beta x_3 = 1x1​+2x2​−βx3​=1, x1,x2,x3≥0x_1, x_2, x_3 \ge 0x1​,x2​,x3​≥0. If the optimal value is 7, then β\betaβ equals _______ (in integer).

Answer and worked solution

Answer: 3

Since x1x_1x1​ and x3x_3x3​ are basic, x2=0x_2 = 0x2​=0. Then the first constraint is 2x1+x3=92x_1 + x_3 = 92x1​+x3​=9 and the objective is x1+3x3=7x_1 + 3x_3 = 7x1​+3x3​=7.

Solving these gives x3=1x_3 = 1x3​=1 and x1=4x_1 = 4x1​=4. The second constraint x1−βx3=1x_1 - \beta x_3 = 1x1​−βx3​=1 becomes 4−β=14 - \beta = 14−β=1, so β=3\beta = 3β=3. The answer is 333.

Plain-text version

For some β > 0, let x₁ and x₃ be the optimal basic feasible solution of: maximize x₁ + 2x₂ + 3x₃ subject to 2x₁ - x₂ + x₃ = 9, x₁ + 2x₂ - β x₃ = 1, x₁, x₂, x₃ ≥ 0. If the optimal value is 7, then β equals _ (in integer).

Also written as: For some beta > 0, let x_1 and x_3 be the optimal basic feasible solution of: maximize x_1 + 2x_2 + 3x_3 subject to 2x_1 - x_2 + x_3 = 9, x_1 + 2x_2 - beta x_3 = 1, x_1, x_2, x_3 >= 0. If the optimal value is 7, then beta equals _ (in integer).

Constrained optimisation and linear programming has come up 44 times in 7 exams since 2007.

What this question tests

This question is on Linear programming: formulation and graphical solution, part of Constrained optimisation and linear programming in Mathematics.

Setting up a linear program and finding its optimum at a corner of the feasible region.

TestStop has 15 past-paper questions on this subtopic; the nearest in year are below.

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