**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)`

#### Solution

Z = 2x + 3y

The given inequalities are 1 ≤ x + 2y ≤ 10

i.e. x + 2y ≥ 1 and x + 2y ≤ 10

consider lines L_{1 }and L_{2} where L_{1} : x + 2y = 1, L_{2} : x + 2y = 10.

For line L_{1} plot A`(0, 1/2)`, B(1, 0)

For line L_{2} 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 L_{1}.

The coordinates of origin O(0, 0) satisfies the inequalities x + 2y ≤ 10.

Required region lies on the origin side of L_{2}.

Lines L_{1} and L_{2 }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)`**