PUC Science 2nd PUC Class 12 - Karnataka Board PUC Question Bank Solutions

Minimize Z = 2x + 4y
Subject to 

\[x + y \geq 8\]
\[x + 4y \geq 12\]
\[x \geq 3, y \geq 2\]

Minimize Z = 5x + 3y
Subject to 

\[2x + y \geq 10\]
\[x + 3y \geq 15\]
\[ x \leq 10\]
\[ y \leq 8\]
\[ x, y \geq 0\]

Minimize Z = 30x + 20y
Subject to 

\[x + y \leq 8\]
\[ x + 4y \geq 12\]
\[5x + 8y = 20\]
\[ x, y \geq 0\]

Maximize Z = 4x + 3y
Subject to 

\[3x + 4y \leq 24\]
\[8x + 6y \leq 48\]
\[ x \leq 5\]
\[ y \leq 6\]
\[ x, y \geq 0\]

Minimize Z = x − 5y + 20
Subject to

\[x - y \geq 0\]
\[ - x + 2y \geq 2\]
\[ x \geq 3\]
\[ y \leq 4\]
\[ x, y \geq 0\]

Maximize Z = 3x + 5y
Subject to

\[x + 2y \leq 20\]
\[x + y \leq 15\]
\[ y \leq 5\]
\[ x, y \geq 0\]

Minimize Z = 3x₁ + 5x₂
Subject to

\[x_1 + 3 x_2 \geq 3\]
\[ x_1 + x_2 \geq 2\]
\[ x_1 , x_2 \geq 0\]

Maximize Z = 2x + 3y
Subject to

\[x + y \geq 1\]
\[10x + y \geq 5\]
\[x + 10y \geq 1\]
\[ x, y \geq 0\]

Maximize Z = −x₁ + 2x₂
Subject to

\[- x_1 + 3 x_2 \leq 10\]
\[ x_1 + x_2 \leq 6\]
\[ x_1 - x_2 \leq 2\]
\[ x_1 , x_2 \geq 0\]

Maximize Z = x + y
Subject to

\[- 2x + y \leq 1\]
\[ x \leq 2\]
\[ x + y \leq 3\]
\[ x, y \geq 0\]

Maximize Z = 3x₁ + 4x₂, if possible,
Subject to the constraints 

\[x_1 - x_2 \leq - 1\]

\[ - x_1 + x_2 \leq 0\]

\[ x_1 , x_2 \geq 0\]

Maximize Z = 3x + 3y, if possible,
Subject to the constraints

\[x - y \leq 1\]
\[x + y \geq 3\]
\[ x, y \geq 0\]

Show the solution zone of the following inequalities on a graph paper:

Find the maximum and minimum value of 2x + y subject to the constraints:
x + 3y ≥ 6, x − 3y ≤ 3, 3x + 4y ≤ 24, − 3x + 2y ≤ 6, 5x + y ≥ 5, x, y ≥ 0.

Find the minimum value of 3x + 5y subject to the constraints
− 2x + y ≤ 4, x + y ≥ 3, x − 2y ≤ 2, x, y ≥ 0.

Solved the following linear programming problem graphically:
Maximize Z = 60x + 15y
Subject to constraints

\[x + y \leq 50\]
\[3x + y \leq 90\]
\[ x, y \geq 0\]

Find graphically, the maximum value of Z = 2x + 5y, subject to constraints given below:

2x + 4y ≤ 8

3x + y ≤ 6

x + y ≤ 4 

x ≥ 0, y ≥ 0   

Solve the following LPP graphically:
Maximize Z = 20 x + 10 y 
Subject to the following constraints 

\[x +\]2\[y \leq\]28 
3x+ \[y \leq\]24 
\[x \geq\] 2x.
\[y \geq\]  0

 Solve the following linear programming problem graphically:
Minimize  z = 6 x + 3 y
Subject to the constraints:

4 x + \[y \geq\] 80
x + 5 \[y \geq\] 115 

3 x + 2 \[y \leq\] 150
\[x \geq\] 0  , \[y \geq\] 0

A diet of two foods F₁ and F₂ contains nutrients thiamine, phosphorous and iron. The amount of each nutrient in each of the food (in milligrams per 25 gms) is given in the following table:

The minimum requirement of the nutrients in the diet are 1.00 mg of thiamine, 7.50 mg of phosphorous and 10.00 mg of iron. The cost of F₁ is 20 paise per 25 gms while the cost of F₂ is 15 paise per 25 gms. Find the minimum cost of diet.