Question

Choose the correct alternative :

Solution of LPP to minimize z = 2x + 3y st. x ≥ 0, y ≥ 0, 1≤ x + 2y ≤ 10 is

Options

• x = 0, y = `(1)/(2)`

• x = `(1)/(2)`, y = 0

• x = 1, y = – 2

• x = y = `(1)/(2)`

Answer 1

Z = 2x + 3y
The given inequalities are 1 ≤ x + 2y ≤ 10
i.e. x + 2y ≥ 1 and x + 2y ≤ 10
consider lines L₁ and L₂ where L₁ : x + 2y = 1, L₂ : x + 2y = 10.
For line L₁ plot A`(0, 1/2)`, B(1, 0)
For line L₂ plot P (0, 5), Q (10, 0).
The coordinates of origin O (0, 0) do not satisfy x + 2y ≥ 1.
Required region lies on non – origin side of L₁.
The coordinates of origin O(0, 0) satisfies the inequalities x + 2y ≤ 10.
Required region lies on the origin side of L₂.
Lines L₁ and L₂ are parallel.
ABQPA is the required feasible region
At `"A"(0, 1/2), "Z" =  0+ 3(1/2)` = 1.5
At B (1, 0), Z = 2 (1) + 0 = 2
At P (0, 5), Z = 0 + 3(5) = 15
At Q (10, 0), Z = 2 (10) + 0 = 20
The maximum value of Z is 1.5 and it occurs at `"A"(0, 1/2)` i.e. x = 0, y = `(1)/(2)`