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Linear Programming — Maths STD 12 Science — Question

Gujarat BoardEnglish MediumSTD 12 ScienceMathsLinear Programming1 Mark
MCQ
Let $X_1$ and $X_2$ are optimal solutions of a $\ce{LPP},$ then:
  • A
    $\text{X}=\lambda\ \text{X}_1+(1-\lambda)\text{X}_2,\lambda\in R$ is also an optimal solution
  • ✓
    $\text{X}=\lambda\ \text{X}_1+(1-\lambda)\text{X}_2,0\leq\lambda\leq1$ given an optimal solution
  • C
    $\text{X}=\lambda\ \text{X}_1+(1+\lambda)\text{X}_2,0\leq\lambda\leq1$ given an optimal solution
  • D
    $\text{X}=\lambda\ \text{X}_1+(1+\lambda)\text{X}_2,\lambda\in R$ given an optimal solution
✓

Answer

Correct option: B.
$\text{X}=\lambda\ \text{X}_1+(1-\lambda)\text{X}_2,0\leq\lambda\leq1$ given an optimal solution
A set $A$ is convex if, for any two points $X_1, X_2 \in\text{A}$ and $\lambda\in0,1$ imply that $\lambda\times1+1-\lambda\times2\in\text{A}$.
Since, here $X_1$ and $X_2$ are optimal solution
Therefore, their convex combination will also be an optimal solution
Thus, $\text{X}=\lambda\ \text{X}_1+(1-\lambda)\text{X}_2,0\leq\lambda\leq1$ gives an optimal solution.

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