GATE Linear Programming: Duality – Previous Year Questions

10 GATE previous year questions on Duality (Linear Programming, Mathematics) with answers and explanations, from every paper.

GATE 2021 MA Q33 (1 mark, Numerical answer) – For max 2x₁+3x₂ subject to 2x₁+x₂≤6,−x₁+x₂≤1,x₁+x₂≤3,x₁,x₂≥0, find the optimal value of its dual.
GATE 2021 MA Q42 (2 marks, Multiple choice) – Maximize c₁x₁+c₂x₂ under three inequalities a i₁x₁+a i₂x₂≤b i (i=1,2,3),x≥0. A feasible (p,q) has c₁p+c₂q=6, and all feasible objectives lie in…
GATE 2021 MA Q65 (2 marks, Numerical answer) – The point x̄=(11/3,2/3,0)ᵀ is optimal for max 4x₁+x₂−3x₃ subject to 2x₁+4x₂+ax₃≤10,x₁−x₂+bx₃≤3,2x₁+3x₂+5x₃≤11,x≥0, where a,b∈ℝ. If (p,q,r)ᵀ is dual…
GATE 2022 MA Q48 (2 marks, Multiple choice) – For max 5x₁+12x₂ subject to x₁+2x₂+x₃≤10,2x₁−x₂+3x₃=8 and x≥0, suppose y=(α,−1)ᵀ is dual feasible. Which must hold?
GATE 2022 MA Q55 (2 marks, Multiple select) – For the primal min cᵀx subject to Ax=b,x≥0, let x⁰,y⁰ be primal and dual feasible. Which hold?
GATE 2022 MA Q63 (2 marks, Numerical answer) – Maximize 6x₁+5x₂ subject to 3x₁+2x₂≤12,−x₁+x₂≤1,x₁,x₂≥0. Find the optimum.
GATE 2023 MA Q65 (2 marks, Numerical answer) – Maximize 3x₁+2x₂+5x₃ subject to x₁+2x₂+x₃≤44,x₁+2x₃≤48,x₁+4x₂≤52 and x₁,x₂,x₃≥0. Find the optimal value.
GATE 2024 MA Q58 (2 marks, Numerical answer) – Maximize 10x+6y−8z with 5x−2y+6z≤20, 10x+4y−6z≤30, x,y,z≥0. Find maximum to two decimals.
GATE 2025 MA Q49 (2 marks, Multiple select) – Maximize Z=3x₁+5x₂ with x₁+x₃=4, 2x₂+x₄=12, 3x₁+2x₂+x₅=18, x≥0. Optimal basis is (x₃,x₂,x₁), with inverse B-1=pmatrix&&-\\0&&0\\0&-&pmatrix. Let…
GATE 2026 MA Q63 (2 marks, Numerical answer) – (4,0,2,β) is optimal for minimizing x₁+3x₂+2x₃−αx₄, with 4x₁+x₂+x₃=18, −3x₁+2x₃+x₄=2 and all variables nonnegative. For real α,β, find maximum 22(α+β)…