10) Linear Programming Problems

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Linear Programming

1. For the following shaded region, the linear in equations are:

a) x+ y ≤ 5, x ≥2, y ≥1 b)

x+ y ≥ 5, x ≥2, y ≥1

c) x+ y ≥ 5, x ≥2, y ≤1 d)

x+ y ≤ 5, x ≥2, y ≤1

a)

2 x + y ≤2, y−x ≥−1, x+2 y ≥ 8, x ≥0, y ≥ 0

b) 2 x + y ≥ 2, y− x ≥−1, x+2 y ≤ 8, x ≥0, y ≥ 0 c)

2 x + y ≥ 2, y− x ≥−1, x+2 y ≤ 8, x ≥0, y ≥ 0

d) 2 x + y ≤2, y−x ≥−1, x+2 y ≥ 8, x ≥0, y ≥ 0 3. For the following shaded region, the linear in equations are:

a)

x ≥ 2, y ≥ 3, x ≤6, y ≤5

b) x ≥ 2, y ≥ 3, x ≥6, y ≤5 c)

x ≥ 2, y ≥ 3, x ≤6, y ≤5

d) x ≥ 2, y ≥ 3, x ≤6, y ≥5

4. For the following shaded region, the linear constraints are:

a)

2 x +3 y ≥3,−5 x +4 y ≤0, 3 x+4 y ≤18, x ≥ 0, y ≥ 0

b)

2 x +3 y ≥3,−5 x +4 y ≤0, 3 x+4 y ≥18, x ≥ 0, y ≥ 0

c)

2 x +3 y ≥3,−5 x +4 y ≥0, 3 x+4 y ≤18, x ≥ 0, y ≥ 0

d)

2 x +3 y ≤3,−5 x +4 y ≤ 0, 3 x+4 y ≤18, x ≥ 0, y ≥ 0

a)

y−x ≥3, y−x ≥6, 0 ≤ x ≤ 4, y ≥ 0

b) y−x ≥3, y−x ≤6, 0 ≤ x ≤ 4, y ≥ 0

c)

y−x ≤3, y−x ≥6, 0 ≤ x ≤ 4, y ≥ 0

d) y−x ≥3, x− y ≤6, 0 ≤ x ≤ 4, y ≥ 0

a) x+ y ≤ 0, 2 x + y ≤ 4, x ≥0, y ≤2 b)

x+ y ≥ 0, 2 x + y ≥ 4, x ≥0, y ≤2

c) x+ y ≤ 0, 2 x + y ≥ 4, x ≥0, y ≥ 0

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d)

x+ y ≥ 0, 2 x + y ≤ 4, x ≥0, y ≤2

a) x+ y ≤ 4, 3 x+ y ≥ 3, x +4 y ≥ 4, x ≥ 0, y ≥ 0 b)

x+ y ≤ 4, 3 x+ y ≥ 3, x +4 y ≤ 4, x ≥ 0, y ≥ 0

c) x+ y ≤ 4, 3 x+ y ≤ 3, x +4 y ≥ 4, x ≥ 0, y ≥ 0 d)

x+ y ≥ 4, 3 x+ y ≥ 3, x +4 y ≥ 4, x ≥ 0, y ≥0

a)

3 x+2 y ≥20, 3 x+ y ≥ 15, x ≥0, y ≥ 0

b) 3 x+2 y ≥20, 3 x+ y ≤ 15, x ≥0, y ≥ 0 c)

3 x+2 y ≤20, 3 x+ y ≤ 15, x ≥0, y ≥0

d) 3 x+2 y ≤20, 3 x+ y ≥ 15, x ≥0, y ≥ 0 9. For the following shaded region, the linear constraints are:

a) 2 x + y ≥12, x + y ≤ 7, x+2 y ≤10, x , y ≥0 b)

2 x + y ≤12, x + y ≥ 7, x+2 y ≤10, x , y ≥0

c)

2 x + y ≤12, x + y ≤ 7, x +2 y ≤10, x , y ≥0

d) 2 x + y ≤12, x + y ≤ 7, x +2 y ≥10, x , y ≥0 10. For the following shaded region, the linear constraints are:

a)

2 x +5 y ≥80, x + y ≤ 20, x ≥ 0, y ≥ 0

b) 2 x +5 y ≤80, x+ y ≤ 20, x ≥ 20, y ≥ 0 c)

2 x +5 y ≤80, x+ y ≥ 20, x ≥ 0, y ≥ 0

d) 2 x +5 y ≥80, x + y ≥ 20, x ≥ 0, y ≥ 0

a) x+ y ≥ 5,2 x + y ≥ 4,0 ≤ x ≤ 3,0 ≤ y ≤3 b)

x+ y ≤ 5,2 x + y ≤ 4, 0 ≤ x ≤ 3,0 ≤ y ≤ 3

c) x+ y ≥ 5,2 x + y ≤ 4, 0 ≤ x ≤ 3,0 ≤ y ≤3 d)

x+ y ≤ 5,2 x + y ≥ 4, 0 ≤ x ≤ 3,0 ≤ y ≤3

a)

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b) x+2 y ≥16, x + y ≤ 12, 2 x + y ≥14, x ≥ 0, y ≥ 0 c) x+2 y ≥16, x + y ≥ 12, 2 x + y ≥14, x ≥ 0, y ≥ 0 d) x+2 y ≥16, x + y ≥ 12, 2 x + y ≤14, x ≥ 0, y ≥ 0 13. For the following shaded region, the linear constraints are:

a)

x+ y ≤ 5, x +2 y ≥ 4, 4 x+ y ≥ 12, x ≥0, y ≥ 0

b) x+ y ≤ 5, x +2 y ≥ 4, 4 x+ y ≤ 12, x ≥0, y ≥ 0 c)

x+ y ≤ 5, x +2 y ≤ 4, 4 x+ y ≤ 12, x ≥0, y ≥0

d) x+ y ≥ 5, x +2 y ≥ 4, 4 x+ y ≤ 12, x ≥0, y ≥ 0 14. For the following shaded region, the linear constraints are:

a)

3 x+4 y ≤12, 4 x +3 y ≤12, x ≥ 0, y ≥ 0

b) 3 x+4 y ≥12, 4 x +3 y ≤12, x ≥ 0, y ≥ 0 c)

3 x+4 y ≤12, 4 x +3 y ≤12, x ≥ 0, y ≥ 0

d) 3 x+4 y ≥12, 4 x +3 y ≥12, x ≥ 0, y ≥ 0 15. For the following shaded region, the linear constraints are:

a)

x+ y ≥ 5, 4 x + y ≥ 4, x +5 y ≥ 5, x ≤ 4, y ≤ 4

b) x+ y ≤ 5, 4 x + y ≤ 4, x +5 y ≥ 5, x ≤ 4, y ≤ 4 c)

x+ y ≤ 5, 4 x + y ≥ 4, x +5 y ≤ 5, x ≤ 4, y ≤ 4

d) x+ y ≤ 5, 4 x + y ≥ 4, x +5 y ≥ 5, x ≤ 4, y ≤ 4 16. For the following shaded region, the linear constraints are:

a) x+2 y ≤ 8,2 x+ y ≤ 2, x − y ≤ 1, x ≥ 0, y ≥ 0 b)

x+2 y ≤ 8,2 x+ y ≥ 2, x − y ≥ 1, x ≥ 0, y ≥ 0

c) x+2 y ≤ 8,2 x+ y ≥ 2, x − y ≤ 1, x ≥ 0, y ≥ 0 d)

x+2 y ≥ 8,2 x+ y ≥ 2, x − y ≤ 1, x ≥ 0, y ≥ 0

a)

2 x +3 y ≥3, 3 x+4 y ≤18,−7 x+4 y ≥14, x , y ≥ 0

b)

2 x +3 y ≥3, 3 x+4 y ≥18,−7 x+4 y ≤14, x , y ≥ 0

c)

2 x +3 y ≥3, 3 x+4 y ≥18,−7 x+4 y ≤14, x , y ≥ 0

d)

2 x +3 y ≥3, 3 x+4 y ≤18,−7 x+4 y ≤14, x , y ≥ 0

a)

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b) 2 x +3 y ≥6, 4 x+6 y ≤ 24,−3 x+2 y ≥ 3, x ≥0, y ≥0 c) 2 x +3 y ≥6, 4 x+6 y ≥ 24,−3 x+2 y ≤ 3, x ≥0, y ≥0 d) 2 x +3 y ≤6, 4 x+6 y ≤ 24,−3 x+2 y ≤3, x ≥ 0, y ≥0 19. Find the linear inequalities for which the solution set is the shaded region given in figure below:-a) x+ y ≤ 4, x+5 y ≥ 4, 6 x+2 y ≤ 8,0≤ x ≤ 3,0 ≤ y ≤ 3 b) x+ y ≤ 4, x+5 y ≤ 4, 6 x+2 y ≥ 8,0≤ x ≤ 3,0 ≤ y ≤ 3 c) x+ y ≥ 4, x+5 y ≥ 4, 6 x+2 y ≥ 8,0≤ x ≤ 3,0 ≤ y ≤ 3 d) x+ y ≤ 4, x+5 y ≥ 4, 6 x+2 y ≥ 8,0≤ x ≤ 3,0 ≤ y ≤ 3 20. The solution set of the linear inequalities

2 x + y ≥ 8 andx+2 y ≥10 is

a)

b)

c)

d) None of these

a) 2 x −3 y ≤ 6,2 x− y ≥1, x ≥ 1, y ≤ 0 b)

2 x −3 y ≤ 6,2 x− y ≥1, x ≤ 1, y ≥ 0

c) 2 x −3 y ≤ 6,2 x− y ≤1, x ≥ 1, y ≤ 0 d)

2 x −3 y ≥ 6,2 x− y ≥1, x ≥ 1, y ≤ 0

22. The minimum value of

z=10 x+8 ysubjectto 4 x+ y ≥ 4, x+3 y ≥ 6, x + y ≥ 3, x ≥ 0, y ≥0 is

a) 60 b) 27

c) 74/3 d) 32

23. For the following shaded region, the linear constraints except

x ≥ 0, y ≥ 0

are :

a)

2 x + y ≥ 2, x− y ≥ 1, x +2 y ≤ 8

b) 2 x + y ≥ 2, x− y ≤1, x +2 y ≤ 8 c)

2 x + y ≤2, x− y ≤1, x +2 y ≤ 8

d) 2 x + y ≥ 2, x− y ≥ 1, x +2 y ≥ 8

24. For the following feasible region, the linear constraints except x ≥ 0, y ≥ 0 are :

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a)

x ≥ 20, y ≤ 10, x +2 y ≥ 30

b) x ≤ 20, y ≥ 10, x +2 y ≥ 30 c)

x ≤ 20, y ≤ 10, x +2 y ≤ 30

d) x ≥ 20, y ≥ 10, x +2 y ≤ 30

a)

x

1

+

x

2

≥1,

x

₁

8 +

x

₂

3 ≤1, x

1

≥ 0, x

2

≥ 0

b)

x

1

+

x

2

≤1,

x

₁

8 +

x

₂

3 ≥1, x

1

≥ 0, x

2

≥ 0

c)

x

1

+

x

2

≤1,

x

₁

8 +

x

₂

3 ≤1, x

1

≥ 0, x

2

≥ 0

d)

x

1

+

x

2

≥1,

x

₁

8 +

x

₂

3 ≥1, x

1

≥ 0, x

2

≥ 0

a)

4 x +6 y ≥ 24,5 x+3 y ≥ 15,2 y ≤5, x ≥ 0

b) 4 x +6 y ≤ 24,5 x+3 y ≤ 15,2 y ≤5, x ≥ 0 c)

4 x +6 y ≤ 24,5 x+3 y ≥ 15,2 y ≤5, x ≥ 0

d) 4 x +6 y ≥ 24,5 x+3 y ≤ 15,2 y ≤5, x ≥ 0 27. For the following shaded region, the linear constraints are:

a) x+ y ≥ 60,5 x + y ≥100, x ≥ 0, y ≥ 0 b)

x+ y ≤ 60,5 x + y ≤100, x ≥ 0, y ≥ 0

c) x+ y ≥ 60,5 x + y ≤100, x ≥ 0, y ≥ 0 d)

x+ y ≥ 60,5 x + y ≥100, x ≥ 0, y ≥ 0

28. For the following shaded region in following figure, the linear constraints( except

x ≥ 0, y ≥ 0

_{)are :}

a) 2 x + y ≤2, x− y ≤1, x + y ≤ 8 b)

2 x + y ≥ 2, x− y ≤1, x +2 y ≤ 8

c) 2 x + y ≤2, x− y ≥1, x +2 y ≤ 8 d)

2 x + y ≤2, x− y ≤1, x +2 y ≥ 8

29. For the following shaded region in following figure, the linear constraints are :

a)

x+2 y ≤ 4, 2 x + y ≤ 4,1 ≤ x ≤2, y ≥0

b) x+2 y ≤ 4, 2 x + y ≥ 4,1 ≤ x ≤2, y ≥0 c)

x+2 y ≥ 4, 2 x + y ≤ 4,1 ≤ x ≤2, y ≥0

d) x+2 y ≥ 4, 2 x + y ≥ 4,1 ≤ x ≤2, y ≥ 0 30. For the following shaded region in following figure, the linear constraints are :

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a)

2 x + y ≤ 6, x + y ≤ 4,1 ≤ x ≥ 0, y ≥ 0

b) 2 x + y ≥ 6, x + y ≤ 4,1 ≤ x ≥ 0, y ≥ 0 c)

2 x + y ≥ 6, x + y ≥ 4,1 ≤ x ≥ 0, y ≥ 0

d) 2 x + y ≤ 6, x + y ≥ 4,1 ≤ x ≥ 0, y ≥ 0 31. For the following shaded region in following figure, the linear constraints are :

a)

x+ y ≤ 4, 3 x+8 y ≤ 24,10 x +7 y ≤ 35, x ≥ 0, y ≥ 0

b)

x+ y ≤ 4, 3 x+8 y ≤ 24,10 x +7 y ≥ 35, x ≥ 0, y ≥ 0

c)

x+ y ≤ 4, 3 x+8 y ≥ 24,10 x +7 y ≤ 35, x ≥ 0, y ≥ 0

d)

x+ y ≥ 4, 3 x+8 y ≤ 24,10 x +7 y ≤ 35, x ≥ 0, y ≥ 0

a)

5 x+3 y ≤150, 3 x+4 y ≤120, x ≥ 0, y ≥ 0

b) 5 x+3 y ≥150, 3 x+4 y ≤120, x ≥ 0, y ≥ 0 c)

5 x+3 y ≤150, 3 x+4 y ≥120, x ≥ 0, y ≥ 0

d) 5 x+3 y ≥150, 3 x+4 y ≥120, x ≥ 0, y ≥ 0 33. For the following shaded region in following figure, the linear constraints are :

a)

x

50 +

y

25 ≥ 1, x + y ≤35,

x

20 +

y

40 ≥1, x ≥ 0, y ≥ 0

b)

x

50 +

y

25 ≥ 1, x + y ≥35,

x

20 +

y

40 ≥1, x ≥ 0, y ≥ 0

c)

x

50 +

y

25 ≤ 1, x + y ≥35,

x

20 +

y

40 ≥1, x ≥ 0, y ≥ 0

d)

x

50 +

y

25 ≥ 1, x + y ≥35,

x

20 +

y

40 ≤1, x ≥ 0, y ≥ 0

a) 7 x+9 y ≤ 63, x + y ≥1.5, x ≤ 6, y ≤ 5 b)

7 x+9 y ≤ 63, x + y ≥1.5, x ≤ 6, y ≤ 5

c) 7 x+9 y ≥ 63, x + y ≥1.5, x ≤ 6, y ≤ 5 d)

7 x+9 y ≥ 63, x + y ≤1.5, x ≤ 6, y ≤ 5

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b)

8 x+5 y ≤ 200,2 x+5 y ≥ 100, x ≥ 0, y ≥0

c) 8 x+5 y ≥ 200,2 x+5 y ≤ 100, x ≥ 0, y ≥0 d)

8 x+5 y ≥ 200,2 x+5 y ≥ 100, x ≥0, y ≥0

36. For the following shaded region in following figure, the linear constraintsare :

a)

3 x+5 y ≥15, x ≥ 1, y ≥ 1

b) 3 x+5 y ≥15, x ≤ 1, y ≥ 1 c)

3 x+5 y ≤15, x ≥ 1, y ≥ 1

d) 3 x+5 y ≤15, x ≤ 1, y ≤ 1

a)

2 x +3 y ≥6, x +2 y ≥ 8, 0≤ x ≤ 4, y ≥ 0

b) 2 x +3 y ≤6, x+2 y ≤ 8,0 ≤ x ≤ 4, y ≥0 c)

2 x +3 y ≥6, x +2 y ≤ 8, x ≤ 4, x ≥ 0, y ≥ 0

d) 2 x +3 y ≤6, x+2 y ≥ 8, 0≤ x ≤ 4, y ≥0 38. For the following shaded region in following figure, the linear constraints are :

a)

x+2 y ≥ 2,3 x+5 y ≥ 15, x ≥0, y ≥ 0

b) x+2 y ≥ 2,3 x+5 y ≤ 15, x ≥0, y ≥0

c)

x+2 y ≤2, 3 x+5 y ≥ 15, x ≥0, y ≥0

d) x+2 y ≤2, 3 x+5 y ≥ 15, x ≥0, y ≥0

39. The common region represented by the in equalities x+ y ≤ 3, y ≥ 2,−2 x+ y ≤ 1, x ≥ 0, y ≥ 0is a) Triangle b) Quadrilateral c) Pentagon d) None of these

40. The common region represented by the in equalities 0 ≤ x ≤6,0 ≤ y ≤ 4 is

a) Triangle b) rectangle c) square d) Pentagon

41. The common region represented by the in equalities x ≤ 5,2 y ≤9, 9 x +10 y ≥ 45, is

a) Triangle b) Quadrilateral c) Pentagon d) None of these

42. The graph of the in equalities

3 x−4 y ≤ 12, x ≤1, x ≥ 0, y ≤ 0 _{lies fully in} a) First quadrant

b) second quadrant c) Third quadrant d) Fourth quadrant

a)

x− y ≤2, 2 x + y ≥ 4, x+2 y ≤10, x ≥ 0, y ≥ 0

b) x− y ≤2, 2 x + y ≥ 4, x+2 y ≥10, x ≥ 0, y ≥ 0 c)

x− y ≤2, 2 x + y ≤ 4, x+2 y ≤10, x ≥ 0, y ≥ 0

d) x− y ≥2, 2 x + y ≥ 4, x+2 y ≤10, x ≥ 0, y ≥ 0 44. For the following shaded region in following figure, the linear constraints are :

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a) 7 x+2 y ≤14, x+ y ≤ 5, x +5 y ≥5, y ≤3, x ≥ 0, y ≥ 0 b) 7 x+2 y ≤14, x+ y ≥ 5, x +5 y ≥5, y ≤3, x ≥ 0, y ≥ 0 c) 7 x+2 y ≥14, x+ y ≤ 5, x +5 y ≥5, y ≤3, x ≥ 0, y ≥ 0 d) 7 x+2 y ≥14, x+ y ≤ 5, x +5 y ≤5, y ≤3, x ≥ 0, y ≥ 0 45. For the following shaded region in following figure, the linear constraints are :

a)

7 x+2 y ≤16, 2 x+7 y ≥14, x + y ≤ 9, x ≥ 0, y ≥ 0

b)

7 x+2 y ≥14, 2 x+7 y ≥ 14, x + y ≤ 9, x ≥ 0, y ≥ 0

c)

7 x+2 y ≥14, 2 x+7 y ≤14, x + y ≤ 9, x ≥ 0, y ≥ 0

d)

7 x+2 y ≥14, 2 x+7 y ≥ 14, x + y ≥ 9, x ≥ 0, y ≥ 0

a)

2 x + y ≤14, 6 x +8 y ≤ 48, x +2 y ≤ 10, x ≥0, y ≥0

b)

2 x + y ≤14, 6 x +8 y ≥ 48, x +2 y ≤ 10, x ≥0, y ≥0

c)

2 x + y ≤14, 6 x +8 y ≤ 48, x +2 y ≥ 10, x ≥0, y ≥0

d)

2 x + y ≥14, 6 x +8 y ≤ 48, x +2 y ≤ 10, x ≥0, y ≥0

a) x+ y ≤ 7.5,2 x+ y ≥ 4,2 x +9 y ≥ 18,0 ≤ x ≤5, 0 ≤ y ≤ 3 b) x+ y ≤ 7.5,2 x+ y ≥ 4,2 x +9 y ≤ 18,0 ≤ x ≤5, 0 ≤ y ≤ 3 c) x+ y ≤ 7.5,2 x+ y ≤ 4,2 x +9 y ≥ 18,0 ≤ x ≤5, 0 ≤ y ≤ 3 d) x+ y ≥ 7.5,2 x + y ≥ 4,2 x +9 y ≥ 18,0 ≤ x ≤5, 0 ≤ y ≤ 3 48. For the following shaded region in following figure, the linear constraints are :

a) x+ y ≥ 0, 3 x +2 y ≥ 6, x ≥ 0, y ≤ 2 b)

x+ y ≤ 0, 3 x +2 y ≤ 6, x ≥ 0, y ≤ 2

c) x+ y ≤ 0, 3 x +2 y ≥ 6, x ≥ 0, y ≤ 2 d)

x+ y ≥ 0, 3 x +2 y ≤ 6, x ≥ 0, y ≤ 2

a)

6 x+ y ≥ 6, 3 x +7 y ≥21,4 x + y ≤ 8, x ≥0, y ≥0

b)

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c)

6 x+ y ≥ 6, 3 x +7 y ≤21,4 x + y ≤ 8, x ≥0, y ≥0 d)

6 x+ y ≥ 6, 3 x +7 y ≥21,4 x + y ≥ 8, x ≥0, y ≥ 0 50. For the following shaded region in following figure, the linear constraints are :

a)

2 x +3 y ≤6, 4 x+3 y ≤24,2 x− y ≤2, x ≥ 0, y ≥ 0

b)

2 x +3 y ≥6, 4 x+3 y ≤ 24,2 x− y ≤ 2, x ≥ 0, y ≥0

c)

2 x +3 y ≥6, 4 x+3 y ≥ 24,2 x− y ≤ 2, x ≥0, y ≥0

d)

2 x +3 y ≤6, 4 x+3 y ≥ 24,2 x− y ≤ 2, x ≥ 0, y ≥0

a)

2 x + y ≥3, x + y ≥ 5,−x + y ≤1, x ≥ 0, y ≥ 0

b) 2 x + y ≥3, x + y ≤ 5,−x + y ≤1, x ≥ 0, y ≥ 0 c)

2 x + y ≤3, x + y ≤ 5,−x + y ≤1, x ≥ 0, y ≥ 0

d) 2 x + y ≥3, x + y ≤ 5,−x + y ≤1, x ≥ 0, y ≥ 0 52. The solution set of the inequalities

2 x +3 y ≤6 _is

a) Wholly xy plane

b) Open half plane which does not contain the origin.

c) Half plane those contain the origin. d) Open half planes which contain the origin.

53. Maximize value z=15x+10y subject to the constraints 3 x+2 y ≤12, 2 x +3 y ≤15, x ≥ 0, y ≥ 0 is

a) 61.5 b) 50

c) 60 d) 62.5

54. The minimum value of p=10x+15y subject to the constraints

2 x +3 y ≥12, 2 x + y ≥ 6, x ≥0, y ≥0 _is

a) 90 b) 70

c) 60 d) None of these

55. The minimum value of z=4x+5y subject to the constraints

x ≤ 30, y ≤ 40, andx ≥ 0, y ≥ 0 is

a) 320 b) 200

c) 120 d) 0

56. The maximize value of p=40x+50y subject to the constraints

3 x+ y ≤ 9, x +2 y ≤ 8 andx ≥0, y ≥ 0is _is

a) 0 b) 120

c) 230 d) 200

57. The maximize value of z=48x+40y subject to the constraints

2 x + y ≤ 90, x +2 y ≤ 80, x+ y ≤ 50 andx ≥ 0, y ≥ 0 is is

a) 0 b) 2320

c) 2160 d) 1600

58. The minimum value of c=3x+y subject to the constraints

2 x +3 y ≤6, x+ y ≥ 1, andx ≥ 0, y ≥ 0 is _is

a) 0 b) 3

c) 2 d) 1

59. The maximize value of p=6x+3y subject to the constraints

4 x + y ≤12, 2 x+2 y ≥10, andx ≥ 0, y ≥0 is

a) 18 b) 16

c) 23 d) 36

60. The minimum value of c=6x+7y subject to the constraints

5 x+8 y ≤ 40, 3 x + y ≤ 6,andx ≥ 0, y ≥2 is

a) 12 b) 14

c) 16 d) 20

61. The maximize value of z=5x+7y subject to following in equations given by

3 x+2 y ≤12, x+ y ≤ 5, andx ≥ 0, y ≥ 0 is _is

a) 25 b) 30

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62. The minimize value of

12 x

1

+42 x

2 subject to the constraints

x

₁

+2 x

₂

≥ 3, x

₁

+4 x

₂

≥ 4, 3 x

₁

+

x

₂

≥ 3∧x

₁

≥ 0, x

₂

≥ 0 is

a) 40 b) 45

c) 50 d) 55

63. The maximize value of

z=5 x

1

+

3 x

2 subject to

x

₁

≤ 4, x

₂

≤ 8,∧x

₁

+

x

₂

≤8, x

₁

≥ 0, x

₂

≥ 0is

a) 20 b) 32

c) 40 d) 50

64. The maximize value of z=5x+10y subject to following in equations given by

x+2 y ≤10, 3 x+ y ≤ 12, andx ≥ 0, y ≥ 0 is

a) 40 b) 45

c) 55 d) 50

z=22 x

1

+

18 x

2 subject to

x

₁

+

x

₂

≤20, 3 x

₁

+2 x

₂

≤ 48, x

₁

≥ 0, x

₂

≥ 0 is

a) 390 b) 392

c) 391 d) 395

z=8 x

1

+

3 x

2 subject to constraints

3 x

₁

+

x

₂

≤ 45, x

₁

+

x

₂

≤25, 2 x

₁

+

x

₂

≤ 40 is

a) 100 b) 115

c) 125 d) 150

z=40 x

1

+60 x

2 subject to constraints

4 x

1

+

x

2

≥ 10,3 x

1

+2 x

2

≥12, x

2

≥ 5,is

a) 300 b) 350

c) 400 d) 425

z=6 x

1

−2 x

2 _such that

2 x

₁

−

x

₂

≤2, x

₁

≤ 3, x

₁

≥ 0, x

₂

≥ 0 thevaluesof x

₁

, x

₂

are

a) (3,4) b) (2,3)

c) (1,2) d) None of these

69. The maximize value of z=4x+8y subject to constraints x− y ≥0, 3 x+ y ≥ 3, andx ≤ 4 is is

a) 46 b) 48

c) 47 d) 50

70. The minimum value of z=4x+8y under the following constraints

20 x+10 y ≥200, x + y ≥ 35, x+2 y ≥50∧2 x + y ≥ 20is

is

a) 250 b) 220

c) 200 d) 210

71. The minimum value of z=28x+28y subject to constraints

x+ y ≥ 2,7 x +8 y ≤56, x ≤ 6 andy ≤5 is

_is

a) 215 b) 217

c) 220 d) 218

72. The maximum value of z=2x+3y under the following constraints

x+ y ≤ 40, x+2 y ≤ 60, x ≥ 0, y ≥0 is

_is

a) 110 b) 100

c) 105 d) 115

73. The minimum value of z=3x+4y subject to the following constraints

x

1

−x

2

≥ 1,−x

1

+

3 x

2

≤ 15, x

1

≥ 0, x

2

≥ 0 is

_is

a) 3 b) 3.5

c) 4 d) 5

74. The constraints in equation are given by x+2 y ≤ 8,3 x +2 y ≤ 12, x ≥0, y ≥0 _{then the} maximum value of z=3x+4y is

a) 15 b) 18

c) 16 d) 17

75. The maximum value of z=3x+4y subject to the set of linear in equation given by

x+ y ≤ 4.5, 2 x + y ≤7, x +2 y ≤ 8, x ≥ 0, y ≥ 0is _is

a) 15 b) 16

c) 17 d) 17.5

76. The maximum value of P=20x+30y

subject to the constraints given by following in equations

3 x+2 y ≤210, 2 x +4 y ≤300, y ≤ 65, x ≥ 0, y ≥ 0 is

a) 2400 b) 2450

c) 2500 d) 2600

z=40 x

1

+60 x

2 subject to the following constraints

4 x

₁

+

x

₂

≥ 10,3 x

₁

+2 x

₂

≥12, x

₁

≥ 0, x

₂

≥5 is

_is

a) 300 b) 350

c) 400 d) 425

a)

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b)

x

₁

+2 x

₂

≥ 2,3 x

₁

+

x

₂

≥ 3, 4 x

₁

+3 x

₂

≥6, x

₁

, x

₂

≥ 0

c)

x

₁

+2 x

₂

≥ 2,3 x

₁

+

x

₂

≤ 3, 4 x

₁

+3 x

₂

≥ 6, x

₁

, x

₂

≥ 0

d)

x

₁

+2 x

₂

≥ 2,3 x

₁

+

x

₂

≥ 3, 4 x

₁

+3 x

₂

≤6, x

₁

, x

₂

≥ 0

79. The maximum value of z=4x+5y subject to the constraints

x+ y ≤ 20, x +2 y ≤ 35, x−3 y ≤ 12is

a) 90 b) 93 c) 95 d) 100

80. The objective function z=4x+3y can be maximized subject to the constraints

3 x+4 y ≤24, 8 x+6 y ≤ 48, 0 ≤ x ≤5∧0 ≤ y ≤ 6 is

a) At only one point. b) Two points only

c) At an infinite number of points d) None of these

81. The maximum value of z=5x+10y subject to the following set of in equations

2 x +3 y ≤6, x+4 y ≤ 4, x ≥ 0, y ≥ 0 is

a) 15 b) 16 c) 16.5 d) 17

82. A firm manufactures headache pills of two sizes A & B. size A contains 2 units of aspirin , 5 units of bicarbonate and one unit of codeine, while size B contains 1 nit of aspirin, 8 units of bicarbonate and 6 unit of codeine. it is found that it requires 12 units of aspirin, 74 units of bicarbonate and 24 units of codeine for the relief of headache. It is require determining the last number of pills a patient must take to get relief. a)

2 x + y ≥12, 5 x+8 y ≥ 74, x +6 y ≥24, x , y ≥ 0

b)

2 x + y ≤12, 5 x+8 y ≥ 74, x +6 y ≥24, x , y ≥ 0

c)

2 x + y ≥12, 5 x+8 y ≤ 74, x +6 y ≥24, x , y ≥ 0

d)

2 x + y ≥12, 5 x+8 y ≥ 74, x +6 y ≤24, x , y ≥ 0

83. Two kind of food A and B are being considered to form a weekly diet. The minimum weekly requirement of fats,

carbohydrates, and proteins are 16, 23 and 16 units respectively. One kg of food A has 5, 15 and 7 units of these ingredients respectively and one kg of food B has 12, 5 and 8 units of ingredients respectively. The price of food A is 4 per kg and food B is Rs. 3 per kg. Formulate this LLP to minimize the cost and set of

solution.

a) Minimize z=4x+3y subject to

5 x+12 y ≥16, 15 x +5 y ≥23, 7 x+8 y ≥ 16, x ≥ 0, y ≥ 0 b) Minimize z=4x+3y subject to

5 x+12 y ≤16, 15 x +5 y ≥23, 7 x+8 y ≥ 16, x ≥ 0, y ≥ 0 c) Minimize z=4x+3y subject to

5 x+12 y ≥16, 15 x +5 y ≤23, 7 x+8 y ≥ 16, x ≥ 0, y ≥ 0 d) Minimize z=4x+3y subject to

5 x+12 y ≥16, 15 x +5 y ≥23, 7 x+8 y ≤ 16, x ≥ 0, y ≥ 0 84. The maximum value of z=5x+4y subject to constraints 2 x + y ≤2, x− y ≥ 3, x ≥ 0 is

a) 6 b) 8

c) -12 d) 12

85. The maximum value of z=3x+2y subject to the following set of in equations

x+ y ≥ 1, x− y ≥−1, y −5 x ≤ 0, x + y ≤6, 0 ≤ x ≤3, y ≥0 is

a) 10 b) 12

c) 15 d) 9

86. The minimum value of z=5(x)+7(y) subject to the following linear in equations representing the common feasible region

5 x+8 y ≤ 40, 3 x + y ≤ 6, x ≥ 0, y ≥2 is

a) 16 b) 14

c) 15 d) 14.5

87. The maximum value of z=48x+40y subject to the following constraints

2 x + y ≤ 90, x +2 y ≤ 80, x+ y ≤ 50, x ≥0, y ≥0 is

a) 2160 b) 2320

c) 2300 d) 2175

88. The minimum value of z=20x+30y subject to the following constraints

2 x +5 y ≤80, x+ y ≤ 20, x ≥ 0, y ≥ 0 is

a) 300 b) 400

(13)

89. For the following shaded region in

following figure, the linear constraints( except x ≥ 0, y ≥ 0 _{)are :}

a)

x+ y ≥ 2, x − y ≥ 2

b) x+ y ≥ 2, x − y ≤ 2 c)

x+ y ≤ 2, x − y ≤ 2

d) x+ y ≤ 2, x − y ≥ 2

a)

x+ y ≥ 1,2 x− y ≥ 0, x ≥0, y ≥0

b) x+ y ≤ 1,2 x− y ≤ 0, x ≥0, y ≥0 c)

x+ y−1 ≥0, 2 x− y ≤ 0, x ≥ 0, y ≥ 0

d) x+ y−1 ≤0, 2 x− y ≥ 0, x ≥ 0, y ≥ 0

91. The maximum value of z=4x+7y in the feasible region constraints by in equations

2 x + y ≤14, x +2 y ≥ 16, x ≥0, y ≥0 is

a) 56 b) 58

c) 28 d) 42

a) x+2 y ≥ 4, 2 x− y ≤ 6, y ≥ 0 b)

x+2 y ≤ 4, 2 x− y ≤ 6, y ≥ 0

c) x+2 y ≥ 4, 2 x− y ≥ 6, y ≥ 0 d)

x+2 y ≥ 4, 2 x− y ≤ 6, x ≥0, y ≥0

a)

3 x+4 y ≤12, x−2 y ≤ 2, x ≥ 0, y ≤ 0

b) 3 x+4 y ≤12, x−2 y ≥ 2, x ≥ 0, y ≤ 0 c)

3 x+4 y ≤12, x−2 y ≥ 2, x ≥ 0, y ≥ 0

d) 3 x+4 y ≥12, x−2 y ≥ 2, x ≥ 0, y ≤ 0 94. For the following shaded region in following figure, the linear constraints are :

a) 4 x +2 y ≤ 40,2 x+5 y ≤ 90, x ≥ 0, y ≥ 0 b)

4 x +2 y ≥ 40,2 x+5 y ≤ 90, x ≥ 0, y ≥ 0

c) 4 x +2 y ≤ 40,2 x+5 y ≥ 90, x ≥ 0, y ≥ 0 d)

4 x +2 y ≥ 40,2 x+5 y ≥ 90, x ≥ 0, y ≥ 0

(14)

a)

3 x+4 y ≤12, 2 x +5 y ≤10, x ≥ 0, y ≥ 0

b) 3 x+4 y ≥12, 2 x +5 y ≥10, x ≥ 0, y ≥ 0 c)

3 x+4 y ≤12, 2 x +5 y ≥10, x ≥ 0, y ≥ 0

d) 3 x+4 y ≥12, 2 x +5 y ≤10, x ≥ 0, y ≥ 0 96. For the following shaded region in following figure, the linear constraints are :

a)

x+2 y ≥10, x + y ≤ 6, 0 ≤ x ≥ 4, y ≥ 0

b) x+2 y ≤10, x + y ≥ 6, 0 ≤ x ≥ 4, y ≥ 0 c)

x+2 y ≤10, x + y ≤ 6, 0 ≤ x ≤ 4, y ≥ 0

d) x+2 y ≥10, x + y ≥ 6, 0 ≤ x ≤ 4, y ≥ 0 97. For the following shaded region in following figure, the linear constraints are :

a)

2 x + y ≥ 2, x− y ≤1, x +2 y ≤ 8, x ≥ 0, y ≥ 0

b) 2 x + y ≤2, x− y ≤1, x +2 y ≤ 8, x ≥ 0, y ≥ 0 c)

2 x + y ≥ 2, x− y ≥ 1, x +2 y ≤ 8, x ≥ 0, y ≥ 0

d) 2 x + y ≥ 2, x− y ≤1, x +2 y ≥ 8, x ≥ 0, y ≥ 0 98. A manufacturer produces two items A and B. both are processes on two machines I and II.A need 1 and 2 hours on the machine and B need 3 hours on machine I and 1 hour on machine II. If machine I can run for maximum 12 hrs per day and II for 8 hours per day. Construct problem in form of in equalities and find feasible solution graphically.

a)

2 x +3 y ≤12, 2 x + y ≥ 8, andx ≥ 0, y ≥ 0

b) 2 x +3 y ≥12, 2 x + y ≤ 8, andx ≥ 0, y ≥ 0 c)

2 x +3 y ≥12, 2 x + y ≥ 8, andx ≥ 0, y ≥ 0

d) 2 x +3 y ≤12, 2 x + y ≤ 8,andx ≥ 0, y ≥ 0 99. For the following shaded region in

following figure, the linear constraints(except x ≥ 0, y ≥ 0 _{)are :}

a)

2 x +3 y ≤12, 2 x + y ≥ 8

b) 2 x +3 y ≥12, 2 x + y ≤ 8 c)

2 x +3 y ≥12, 2 x + y ≥ 8

d) 2 x +3 y ≤12, 2 x + y ≤ 8

100. The maximum value of z=6x+4y subject to

2 x +3 y ≤30, 3 x+2 y ≤24, x+4 y ≥ 3, x ≥0, y ≥0 is

a) 50 b) 48

(15)

0 ≤ x ≤3, 0 ≤ y ≤ 3, x + y ≤5, 2 x+ y ≥ 4 is

a) 12 b) 13

c) 14 d) 15

102. The minimum value of z=4x+5y subject to following in equations

5 x+ y ≥ 1,0, x + y ≥6, x+4 y ≥ 12, x ≥ 0, y ≥ 0 is

a) 25 b) 25.5

c) 26 d) 27

103. The maximum value of z=40x+50y subject to 3 x+ y ≤ 9, x +2 y ≤ 8, x ≥ 0, y ≥ 0 is

a) 220 b) 230

c) 235 d) 240

x+4 y ≤ 24,3 x+ y ≤ 21, x + y ≤ 9, x ≥ 0, y ≥ 0is

a) 33 b) 32

105. The maximum value of z=5x+4y subject to x+2 y ≥ 6,5 x+3 y ≤ 15, x ≤6, y ≤ 4 is

a) 46 b) 48

106. A company manufactures two products electric iron and ceiling fan. Each of these undergoes assembly and finishing processes. The time in hours required for each unit of the products for different processes and time available for each process in day given as Product Assembly Process Finishing Process Electric Iron 4 2 Ceiling fan 2 6 Hours available 16 18 per day

The profit on each unit electric iron and ceiling fan sold are Rs.40 and Rs.50 respectively. Then the maximum value of profit is

a) 250 b) 240

c) 230 d) 220

107. A carpenter has 90, 80, and 50 running feet of teak, plywood and rosewood

respectively. The product A requires 2,1 and 1 running feet and product B requires 1,2 and 1

running feet of teak, plywood and rosewood respectively. If A would sell for Rs.48 per unit and B would sell for Rs.40 per unit. The maximum income would be

a) 2000 b) 2100

c) 2160 d) None of These

108. The minimum value of z=2x+3y subject to linear constraints

5 x+3 y ≥15, x +2 y ≥ 6, x ≥ 0, y ≥ 0is

a) 69/7 b) 65/7

c) 64/7 d) None of these

109. The standard weight of a trick has to be at least 5 Kg and has to contain two basic ingredients

B

1

∧B

2

. B

1 _{cost Rs. 5/Kg and}

B

₂ _{cost Rs. 8/Kg. strength consideration}

dictate that the trick should not contain more than 4 Kg of

B

1 _{and should contain a}

minimum of 2 Kg of

B

2 _{.The minimum cost} of trick is

a) 30 b) 32

c) 31 d) 31.5

110. A firm produces two products A and B. The profit on each unit of product A is Rs.40 and that of the B is Rs. 50.Both of the

products are processed on three machines M1, M2, and M3.The time required in hours by each product and total time available in hours per week on each machine is as follows. Product machines M1 M2 M3 A 3 4 2 B 4 5 8 x=no. of product A Total available 36 48 70 y=no. of product B Hrs

Formulate the problem as L.P.P. to maximize profit z=40x+50y subject to linear constraints.

a) 3 x+4 y ≤36, 4 x +5 y ≤ 48,2 x +8 y ≤70, x ≥ 0, y ≥ 0 b) 3 x+4 y ≥36, 4 x+5 y ≤ 48,2 x +8 y ≤70, x ≥ 0, y ≥ 0 c) 3 x+4 y ≤36, 4 x +5 y ≥ 48,2 x +8 y ≤70, x ≥ 0, y ≥ 0

(16)

d)

3 x+4 y ≤36, 4 x +5 y ≤ 48,2 x +8 y ≥70, x ≥ 0, y ≥ 0 111. An agriculturist wishes to mix fertilizer that will provide minimum of 15 units of potash, 20 units of nitrates and 24 units of phosphates. One unit of A provides 3 units of potash, 1unit of nitrate and 3 unit of

phosphates and cost Rs. 120 per unit. One unit of B provides 1 unit of potash, 5 unit of nitrates and 2 units of phosphate. It costs Rs.60 per unit. Formulate the problem as L.P.P. to minimize the cost

a) 3 x+ y ≥ 15, x +5 y ≥ 20,3 x+2 y ≥ 24, x ≥ 0, y ≥0 b) 3 x+ y ≥ 15, x +5 y ≥ 20,3 x+2 y ≤24, x ≥ 0, y ≥ 0 c) 3 x+ y ≥ 15, x +5 y ≤ 20,3 x+2 y ≥24, x ≥ 0, y ≥ 0 d) 3 x+ y ≤ 15, x +5 y ≥ 20,3 x+2 y ≥24, x ≥ 0, y ≥ 0 112. A firm makes two types of furniture’s namely tables and chairs. The profit is Rs.20 per chair and Rs.30 per table. Both are

processed on three machines M1, M2, M3.The time required in hours by each product and total available time in hours per week on each machines are as follows.

Machines Chairs Tables Available time in hours

M1 3 3 36 M2 5 2 50 M3 2 6 60

The maximum value of profit is

a) 320 b) 325

113. The maximum value of z=10x+15y subject to linear constraints

2 x +3 y ≤36, 5 x+2 y ≤50,2 x+6 y ≤ 60, x ≥ 0, y ≥ 0is

a) 160 b) 170

114. The maximum value of z=40x+100y subject to following linear in equation

12 x +6 y ≤3000, 4 x +10 y ≤2000,2 x+3 y ≤ 900, x ≥ 0, y ≥ 0is

a) 18000 b) 20000

c) 21000 d) 20,500

115. A company produces cars of two types A and B. To stay in business it must produce at least 5o models of A per month. However it does not have the facilities to produce more than 200 of that model per month. Also it does not have the facilities to produce more than 150 of model B per month, while the total demand does not exceed 300 per month. If the profit is Rs. 400 on each model A and Rs. 300 on each model B. The maximum profit is

a) 1,05,000 b) 1,10,000

c) 1,15,000 d) 1,20,000

116. Vijay want to invest Rs.15, 000 in saving certificate (SC) and fixed deposit (FD).He want to invest at least Rs.3000 in SC and at least Rs. 5000 in fixed deposit. The rates of interest on SC is 8% and that on FD is 10%.formulate this as LPP problem, taking Rs. X and y interest in SC and FD respectively.

a)

x+ y ≤ 15000, x ≥3000, y ≤ 5000

b) x+ y ≥ 15000, x ≥3000, y ≥ 5000

c)

x+ y ≤ 15000, x ≥3000, y ≥ 5000

d) x+ y ≤ 15000, x ≤3000, y ≥ 5000 117. If Ajay drives a car at speed of 60 km/hr, he has to spend Rs.5 per km on petrol. If he drives at a greater speed of 90 km/hr. The petrol cost increases to Rs. 8 per km. he has Rs. 600 to spend on petrol and wishes to travel the maximum distance within an hour. Formulate the above problem as LPP

.assuming

X= distance travelled at speed of 60 km/hr y= distance travelled at speed of 90 km/hr a) maximum value of z=x+y subject constraints

x /60+ y /90 ≤1, 5 x+8 y ≤ 600, x ≥0, y ≥0 b) maximum value of z=x+y subject constraints

(17)

c) maximum value of z=x+y subject constraints

x /60+ y /90 ≥1, 5 x+8 y ≤ 600, x ≥0, y ≥ 0 d) maximum value of z=x+y subject constraints

x /60+ y /90 ≥1, 5 x+8 y ≥ 600, x ≥0, y ≥ 0 118. In order to ensure optimal health of rabbits, a lab technician nee to feed the rabbits a daily diet containing a minimum of 24 gm of fats, 36 gm of carbohydrates and 4 gm of protein. Food X contains 8 gm of fats, 12 gm of carbohydrates and 2 gm of protein per gram and cost Rs. 20 per gram. Food y contains 12 gm of fats 12 gm of

carbohydrates and 1 gm of protein per gram at cost of Rs. 30 per gram. Formulate the problem as LPP in order to ensure optimal health of rabbits at minimum cost assuming. Minimum cost

z=20 x

1

+30 x

2 _{subject to the} following constraints.

Let

x

1 _{= quantity (in grams) of food x} Let

x

2 _{= quantity (in grams) of food y}

a)

8 x

₁

+12 x

₂

≥ 24,12 x

₁

+

12 x

₂

≥36, 2 x

₁

+

x

₂

≤ 4, x

₁

≥ 0, x

₂

≥0

b)

8 x

₁

+12 x

₂

≥ 24,12 x

₁

+

12 x

₂

≥36, 2 x

₁

+

x

₂

≥ 4, x

₁

≥ 0, x

₂

≥0

c)

8 x

₁

+12 x

₂

≥ 24,12 x

₁

+

12 x

₂

≤36, 2 x

₁

+

x

₂

≤ 4, x

₁

≥ 0, x

₂

≥0

d)

8 x

₁

+12 x

₂

≤ 24,12 x

₁

+

12 x

₂

≥36, 2 x

₁

+

x

₂

≤ 4, x

₁

≥ 0, x

₂

≥0

119. Vijay wants to buy some filling cabinets for his office, the cost of cabinet x is Rs. 1000 per unit, requires 6 sq. feet of floor space and hold eight cubic feet of files. Cabinet y casts Rs.2000 per unit, requires eight square feet of floor space and hold 12 cubic feet of files. He has Rs. 8000 for this purchase through he does not have to spend that much. The office has room for no more than 72 square feet of cabinets. Formulate this problem as LLP in order to maximize storage volume.

Let x= no of cabinets x

Maximum storage volume z=8x+12y subject to following linear to constraints.

a) 1000 x +2000 y ≤8000, 6 x +8 y ≤ 72, x ≥ 0, y ≥ 0 b) 1000 x +2000 y ≤8000, 6 x +8 y ≥ 72, x ≥ 0, y ≥ 0 c) 1000 x +2000 y ≥8000, 6 x +8 y ≥ 72, x ≥ 0, y ≥ 0 d) 1000 x +2000 y ≥8000, 6 x +8 y ≤ 72, x ≥ 0, y ≥ 0 120. A manager of a hotel plans an

extension not more than 50 room. At least 5 must be executive single rooms. The number of executive double rooms should be at least 3 times the number of executive single room. He charges Rs.3000 for an executive double room and Rs. 1800 for an executive single room per day. Formulate the above problem as LPP to obtain maximum income.

Let x= number of executive double rooms Let y= number of executive single room Maximum z=300x+1800y subject to constraints.

a)

x+ y ≤ 50, y ≤ 3 x , 0 ≤ x ≤5, y ≥ 0

b) x+ y ≤ 50, y ≥ 3 x , 0 ≤ x ≤5, y ≥ 0 c)

x+ y ≥ 50, y ≥ 3 x , 0 ≤ x ≤5, y ≥0

d) x+ y ≥ 50, y ≤ 3 x , 0 ≤ x ≤5, y ≥ 0 121. Two products x and y when

manufactured must pass through machine operations I, II, III, are 46,100 and 300 Hrs respectively. Product A gives a profit of Rs. 6 per unit and product B gives a profit of Rs. 4 per unit. Formulate this problem as LPP problem to maximize the profit assuming Let x= number of product A

Let y= number of product B

Maximum z=6x+4y subject to constraints. a)

2 x +

1

2 y ≤ 46, 4 x+2 y ≤ 100,3 x+ y ≤ 300, x ≥ 0, y ≥ 0

b)

2 x +

1

2 y ≤ 46, 4 x+2 y ≤ 100,3 x+ y ≥ 300, x ≥ 0, y ≥ 0

(18)

c)

2 x +

1

2 y ≤ 46, 4 x+2 y ≥ 100,3 x+ y ≤ 300, x ≥ 0, y ≥ 0

d)

2 x +

1

2 y ≥ 46, 4 x +2 y ≤ 100,3 x+ y ≤ 300, x ≥ 0, y ≥ 0

122. A company produces two type of goods A and B, that requires gold and silver. Each unit of type A requires 3 gm of silver and 1 gm of gold while that B requires 2 gm of silver and 8 gm of gold. If each unit of type A bring out a profit of Rs.35 and that of type B Rs. 50, formulate this problem as LPP to get the maximum profit given

Let x= number of good of type A y= number of good of type B

a) maximizez=35 x +50 ysubjectto 3 x +2 y ≥ 12, x +2 y ≤8, x ≥ 0, y ≥ 0 b) maximizez=35 x +50 ysubjectto 3 x +2 y ≤ 12, x +2 y ≤8, x ≥ 0, y ≥ 0 c) maximizez=35 x +50 ysubjectto 3 x +2 y ≤ 12, x +2 y ≥8, x ≥ 0, y ≥ 0 d) maximizez=35 x +50 ysubjectto 3 x +2 y ≥ 12, x +2 y ≥8, x ≥ 0, y ≥ 0 123. A company produces bicycle and

tricycles. Each of which must be produces through two types of machines A and B. machine A has a maximum of 120 hrs

available and machine B has maximum of 180 Hrs available. Manufacturing tricycle requires 6 Hrs on machine A and 3 Hrs on machine B. manufacturing bicycle requires 4 hours of machine A and 10 hours on machine B. If profits are Rs. 50 for tricycle and 70 for bicycle. solve graphically and show that the maximum profit is

a) 1500 b) 1550

c) 1600 d) 1650

124. At form horses are fed on various products having certain nutrient constituent necessary. The contents of the product (A and B) per unit in nutritional constituents are given below in the table.

Nutrients Nutrient content in product Minimum quantity required

A B N1 36 6 108 N2 3 12 36 N3 20 10 If the product A and B cost Rs. 20 and Rs.40 per unit respectively. Find how many each of these two products should be bought to minimize cost assuming

Let

x

1 _{= number of product A} Let

x

2 _{= number of product B}

a) 155 b) 150

c) 160 d) 162

125. The maximum value of z=48x+40y subject to linear in equations

2 x + y ≤ 90, x +2 y ≤ 80, x+ y =50, x ≥0, y ≥ 0is

a) 2300 b) 2320

c) 2350 d) 2360

2 x +3 y ≥12, 2 x + y ≥ 6, x ≥0, y ≥0 is

a) 50 b) 55

c) 60 d) 65

3 x+2 y ≤12, 2 x +3 y ≤15 is

a) 61 b) 60

c) 61.5 d) 60.5

128. For the following shaded region the linear constraints (except x ≥ 0, y ≥ 0 ) are

a)

3 x+3 y ≤50, 3 x+5 y ≥ 80

b) 3 x+3 y ≤50, 3 x+5 y ≤ 80 c)

3 x+3 y ≥50, 3 x+5 y ≤ 80

d) 3 x+3 y ≥50, 3 x+5 y ≥ 80

129. The vertex of common graph of in equalities 2 x + y ≥ 2andx− y ≤ 3 is

a) (5/3,4/3) b) (5/3,-4/3) c) (-5/3,4/3) d) (-5/3,-4/3)

(19)

a) x+2 y ≤11,3 x +4 y ≤ 30,2 x +5 y ≤ 30, x ≥ 0, y ≥ 0 b)

x+2 y ≥11,3 x +4 y ≤ 30, 2 x +5 y ≤ 30, x ≥ 0, y ≥ 0

c)

x+2 y ≥11,3 x +4 y ≥ 30, 2 x +5 y ≤ 30, x ≥ 0, y ≥ 0

d)

x+2 y ≥11,3 x +4 y ≤ 30, 2 x +5 y ≥ 30, x ≥ 0, y ≥ 0

131. The true statement for the graph of in equations

3 x+2 y ≤6, 6 x+4 y ≥ 20, x ≥0, y ≥0

a) Both graph are disjoint b) Both do not contain origin c) Both contain point

d) None of these

132. The minimum value of z=12x+8y subject to constraints

4 x + y ≥ 4, x+3 y ≥6, x + y ≥ 3, x ≥ 0, y ≥ 0 is

a) 76/6 b) 76/5

c) 76/3 d) 74/3

133. The minimum value of z=x+y subject to constraints

x+ y ≤ 8, 6 x+4 y ≥12,5 x+8 y ≥ 20, x ≥ 0, y ≥ 0 is

a) 16/7 b) 3

c) 19/7 d) 18/7

134. The maximum value of z=x+y subject to following in equations

x− y ≥0, 3 x+ y ≥ 3, x ≥ 0, y ≥ 0is

a) 8 b) 8.5

c) 7 d) 7.5

135. The maximum value of z=5x+5y subject to constraints

10 x+7 y ≤ 35,3 x+8 y ≤ 24, x + y ≤ 4, x ≥ 0, y ≥ 0 is

a) 15 b) 20

c) 18 d) 16

136. The maximum value of z=3x+5y subject to constraints

x+ y ≤ 4, 2 x +3 y ≤12,0 ≤ x ≤ 2,0 ≤ y ≤3 is

a) 16 b) 18

c) 17 d) 15

137. The minimum value of z=12x+8y subject to linear in equations

4 x + y ≥ 4, x+ y ≥ 3, x +3 y ≥ 6, x ≥ 0, y ≥ 0 is

a) 25 b) 78/3

c) 76/3 d) 74/3

138. The minimum value of z=2x+3y subject to linear in equations

2 x +7 y ≥22, x + y ≤ 6,5 x+ y ≥ 10 is

a) 13 b) 14

c) 15 d) 16

139. Solve graphically z=15x+12y subject to linear in equations 2 x + y ≥ 2, x− y ≤ 3, x ≥ 0 then vertex of in equalities is

a) (5/3,4/3) b) (-5/3,4/3) c) (-5/3,-4/3) d) (5/3,-4/3) 140. The maximum value of z=40x+50y subject to linear constraints is

3 x+ y ≤ 9, x +2 y ≤ 8, x ≥ 0, y ≥ 0 is

a) 230 b) 220

c) 215 d) 225

141. The maximum value of z=60x+15y subject to linear constraints is

x+ y ≤ 50,3 x + y ≤90, x ≥ 0, y ≥ 0 is

a) 1850 b) 1800

c) 1700 d) 1750

142. Two types of food packets A and B are available. Each contains vitamins A1 and B1.A person needs 4 decigrams of A1 and 12 decigrams of B1 every day. Food packets A contains 2 decigram of vitamin A1 and 4 decigram of vitamin B1.Food packets A and B cost Rs.15 and Rs 10 for A and B respectively. solve LLP graphically to minimize the cost.

a) 30 b) 40

c) 35 d) 32

143. If a man rides his motor cycle at 25km/hr, he has to spend Rs.2 per km on petrol. If he rides it a faster speed of 40 km/hr, the petrol cost increases to Rs. 5 per km. He has Rs. 100 to spend on petrol and wishes to find what is the maximum distance he travel within 1 hour.

a) 20 b) 30

144. A machine can produces by using 2 units of chemicals and 1 units of a compound and can produce product B by using unit of chemicals and 2 units of compound. Only 800 units of chemical and 1000 units of compound are available. The profit per unit of A and B

(20)

LLP and solve graphically to maximize profit.max. profit is

a) 12000 b) 13000

c) 14000 d) 15000

145. A diet for the seek person must contain at least 4000 units of vitamins, 50 units of minerals and 1400 units of calories. Two foods A and B are available at a cost Rs. 4 and Rs.3 per unit respectively. If one unit of A contains 200 units of vitamins, 1 units minerals and 40 units of calories. One unit of food B contains 100 units of vitamins, 2 units of minerals and 40 units of calories.

Minimum value of z=4x+3y is

a) 120 b) 110

c) 1005 d) 115

146. A has Rs.1500 for purchase of rice and wheat. A bag of rice and wheat cost Rs.180 and Rs.120 respectively. He has storage capacity of 10 bags only. He earns a profit of Rs. 11 and Rs. 9 per bag of rice and wheat. Maximum value of profit is

a) 150 b) 100

c) 120 d) 110

147. A firm manufactures two products A and B on which profit earned per unit Rs.3 and Rs. 4 respectively. each product is processed on two machines M1 and M2.the product A requires one minute of processing time on M1 and 2 minute on M2.B requires one minute of M1 and one minute of M2.M1 is available for not more than 600 minutes during any day. The value of maximum profit is

a) 1600 b) 1620

c) 1650 d) None

148. If one gram of wheat provides 0.1 gm of proteins and 0.25 gm of carbohydrates and the corresponding values of rice are 0.05 gm and 0.5 gm of carbohydrates. Wheat costs Rs.2 per Kg and rice Rs. 8 per Kg. The minimum requirements of proteins and carbohydrates for an average child are 50 g and 200 gm respectively. Find the quantities in which wheat and rice should be mixed in the daily diet to provide minimum daily

requirements of proteins and carbohydrates. The value of minimum cost is

a) 2.4 b) 2.5

c) 2.6 d) 2.3

149. A carpenter has 20 and 15 sq. meters of plywood and sun mica respectively. He

produces A and B. Product A requires 2 and 1 sq. meters and product B requires 1 and 3 sq. meters of plywood and sun mica respectively, if the profit on one piece of product A is Rs.30 and that on the piece of product B is

Rs.20,The no. of pieces of product A and B to be made to maximize his profit are

a) (9,2) b) (2,9)

c) (2,8) d) (8,2)

150. The extreme values of the function

f ( x , y )=2 x + y

_{over the convex polygon x}

subject to

x ≤ 2, y ≥−2, x +2 y ≤ 4, is

a) 4,-8 b) 4,8

c) 8,4 d) -8,4

151. Which of the following is the common region for

x+ y ≥ 2, x +3 y ≥ 9, x ≤2, x ≥ 0, y ≥ 0

a)

b)

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d)

152. A business firm produces two types of product I and II. The profit for product I

Rs.100 per ton and that for product II is Rs. 70 per ton. The plant of three production

department A, B and C. The equipment in each department can be used for 8 h0urs a day. Production I requires 2 hours in

department C per ton. Product II requires 1 hour in department B and 1 hour in

department C per ton. Formulate the problem as LPP to maximize the profit is

a) 690 b) 680

153. A company manufacturer two type of toy A and B, each type of toy requires 2 minutes for cutting and 1 minute for assembling. Each toy of type B requires 3 minute for cutting and 4 minutes of

assembling. There are 3 hours available for cutting and 2 hours 40 minutes available for assembling. On selling a toy of type A , he get a profit Rs. 10 and that on selling a toy of type B , he gets profit of Rs.20The value of

maximum profit is

a) 17 b)

17

21

c)

17

31 _{d) 18}

154. A company produces two types of vehicles car and jeeps. Each of these are produced on three machine I,II and III.A car needs 4 hours of machine I, 3 hours of

machine II and 2 hours of machine III and jeep needs 3 hours, 2 hours and 3 hours of

machine I, II and III. machine I, II, and III are to be used at least 16 hrs, 12 hrs and 10 hrs a day respectively to minimize the cost .the production cost of a car and jeep are Rs. 4 lacs and Rs.3 Lacs respectively. construct the

a) 15 b) 16

c) 17 d) 14

155. Solve the following LPP graphically, Maximize

z=3 x+4 ysubjectto 0 ≤ x ≤ 4, y ≥

7

2 , x+ y ≤ 5,2 x + y ≥ 4, x ≥ 0, y ≥0

.the maximum value of Z is

a) 18.5 b) 18

c) 17.5 d) 17

156. The maximum value of

z=5 x+3 ysubjectto 2 x+3 y ≤ 30,3 x+2 y ≤ 24, x +4 y ≥ 3, 0≤ x ≤ , 0 ≤ y ≤ 9 is

a) 40 b) 41

c) 39 d) 34.5

z=x + y

subject to linear constraints

3 x+4 y ≤18, x−6 y ≤3, 2 x +3 y ≥3, x , y ≥ 0

_is

a) 155/22 b) 125/22

c) 130/22 d) 128/22

z=6 x +9 ysubjecttolinearconstraintsx+ y ≤ 20,, 2 x +5 y ≤ 80, x , y ≥0

is

a) 155 b) 160

c) 150 d) 165

159. The extreme value of linear function z(x,y)=2x+5y over the convex polygon given by

x+3 y ≤ 3, x− y ≤ 3, y ≥−1, x ≥ 1 are

a) 3,6 b) -6,3

c) -3,6 d) 6,-3

z=2 x+3 ysubjecttolinearequationsx+ y ≥5, 5 x+3 y ≥ 15,2 x+ y ≥ 6, x , y ≥ 0

is

a) 10 b) 12

c) 11 d) 10.5

z=2 x+ ysubjecttolinearequations 2 x + y ≥ 4, x + y ≤3, 2 x−3 y ≥ 6, x ≥ 0

is

a) 3 b) 4

c) 4.5 d) 3.5

z=3 x+3 ysubjecttolinearequations 7 x+4 y ≥28, x+ y ≤ 6, x ≥ 0, y ≥ 0

is

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z=2 x+3 ysubjecttolinearequationsx + y ≥1, y ≥2 x , x +2 y ≤ 6, is

a) 8/3 b) 7/3

c) 5/3 d) None of these

z=3 x+6 ysubjecttolinearequations 8 x+9 y ≤ 72, x + y ≥6,4 x+7 y ≥ 28, x ≥0, y ≥0 is

a) 20 b) 21

z=5 x+10 ysubjecttolinearequationsx −2 y ≤ 4, x+3 y ≥ 3,2 x + y ≥ 2, x ≥ 0, y ≥ 0 is

a) 10 b) 10.5

c) 11 d) 11.5

z=2 x+3 ysubjecttolinearequationsx −2 y ≤−1, 2 x − y ≥ 1, x +4 y ≤ 4, x ≥0, y ≥0 is

a) 2 b) 3

c) 5 d) 29/3

z=3 x+2 ysubjecttolinearconstraints 3 x +4 y ≥ 12, x−2 y ≤3, x ≥ 0, y ≥ 0 is

a) 16 b) 17

168. Solve graphically linear in equations and shade common feasible region.

2 x +3 y ≥3, 3 x+4 y ≤18,−7 x+4 y ≤14, x−6 y ≤3, x ≥ 0, y ≥ 0 The maximum value of z=x+y is

a) 129/11 b) 129/22

c) 131/11 d) 133/22

169. The shaded common feasible region is given by linear constraints other than

x ≥ 0, y ≥ 0 _are

a)

3 x−2 y ≥−3, 4 x+6 y ≤ 24,2 x +3 y ≤ 6

b)

3 x−2 y ≤−3, 4 x+6 y ≥ 24,2 x +3 y ≥ 6

c) 3 x−2 y ≥−3, 4 x+6 y ≤ 24,2 x +3 y ≥ 6 d) 3 x−2 y ≤−3, 4 x+6 y ≤ 24,2 x +3 y ≥ 6 170. Solve graphically following linear in equations and minimize z=2x+3y subject to

2 x −3 y ≤ 6,2 x + y ≥ 4, x + y ≤ 3

a) 7 b) 5

c) 4 d) 6

171. A person wants to invest up to an amount of Rs. 30,000 in fixed income

securities. His friend recommended investing in two bonds, bond A yielding 7% per annum and bond B yielding 10 % per annum. he decides to invest at the most Rs.1200 in bond B and at least Rs.6000 in bond A. he also want that the amount invested in bond A should be more than the amount invested in bond B. maximum value of profit is

a) 2360 b) 2460

c) 2820 d) None

172. Solve graphically following linear in equations and minimize z=5x+8y subject to

x+ y ≥ 5,3 x +3 y ≥ 12, x ≤ 4, y ≥ 2, x , y ≥ 0 _The minimum value of z is

a) 30 b) 30.5

c) 31 d) 32

173. Solve graphically following linear in equations and maximize z=4x+y subject to

3 x+2 y ≤30, x+ y ≥ 10,2 x + y ≥ 10, x , y ≥ 0 _and shade the common feasible region. The maximum value of z is

a) 40 b) 45

c) 30 d) None of these.

12 x +6 y ≤3000, 4 x +10 y ≤2000, 2 x +3 y ≤ 900, x , y ≥ 0 and shade the common feasible region. The minimum value of z is

a) 16000 b) 17000

x+ y ≤ 8, x+2 y ≥ 4,6 x +4 y ≥ 12, x , y ≥ 0 _and shade the common feasible region. The minimum value of z is

a) 60 b) 50

176. Maximize z=2x+5y subject to constraints

3 x+ y ≤ 21, x +4 y ≤ 24, x + y ≥ 9, x , y ≥ 0 _is

a) 31 b) 33

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177. For the following shaded region the linear constraints except x , y ≥0 are

a)

x+ y ≤ 4, 3 x+8 y ≤ 24,5 x +3 y ≥ 18

b) x+ y ≤ 4, 3 x+8 y ≥ 24,5 x +3 y ≤ 18 c)

x+ y ≤ 4, 3 x+8 y ≤ 24,5 x +3 y ≤ 18

d) x+ y ≥ 4, 3 x+8 y ≤ 24,5 x +3 y ≤ 18 178. The maximum value of

z=3 x+4 ysubjecttolinearequationsx + y ≤ 40, x +2 y ≤60, x+2 y ≤70, x ≥ 0, y ≥ 0 is

a) 120 b) 130

c) 135 d) 140

z=3 x+4 ysubjecttolinearequationsx + y ≤ 300, x +2 y ≤ 400, x+2 y ≤500, x ≥ 0, y ≥ 0 is

a) 1010 b) 1000

c) 1020 d) 1015

z=3 x+4 ysubjecttolinearequationsx +2 y ≤8, 2 x + y ≤7, x + y ≤ 4.5, x ≥ 0, y ≥1 is a) 16 b) 18 c) 17 d) 20 181. Minimize

z=21 x

1

+15 x

2

subjectto x

1

+2 x

2

≤6, 4 x

1

+3 x

2

≤ 12, x

1

, x

2

≥ 0

Express above LPP in standard form. All basic

feasible solutions are

a) 2 b) 3

c) 4 d) 6

182. A company manufacture two products namely toaster and mixture. Each of them are first assembled and then finished. The time required for each unit of product for assembly and finishing processes in a day are given below Product assembly finishing Toaster 4 3 Electric mixture 2 6 Hrs available 16 18 per day

The profit on each toaster and mixture are Rs.50 and Rs.100.The basic feasible solution are a)

4 x +2 y +s

₁

≤ 16, 2 x +6 y +s

₂

≤18, x

₁

x

₂

, s

₁

, s

₂

≥ 0,

b)

4 x +2 y +s

₁

≤ 16, 2 x +6 y +s

₂

≥18, x

₁

x

₂

, s

₁

, s

₂

≥ 0,

c)

4 x +2 y +s

₁

≥ 16, 2 x +6 y +s

₂

≤18, x

₁

x

₂

, s

₁

, s

₂

≥ 0,

d)

4 x +2 y +s

₁

≥ 16, 2 x +6 y +s

₂

≥18, x

₁

x

₂

, s

₁

, s

₂

≥ 0,

183. To formulate the problem for solution by the simplex method, we must add artificial variables to

a) Only equality constraints b) Only greater than constraints. c) Both (a) and (b)

d) None of these.

184. If for a given solution, a slack variable is equal to zero then

a) The solution is optimal. b) The solution is infeasible.

c) The entire amount of resources with the constraints in which the slack variable

appears has assumed. d) All the above.

185. For maximization LP model, the simplex model is terminated when all values

a)

C

j

−

z

j

≥ 0

_b)

C

j

−

z

j

≤ 0

c)

C

j

−

z

j

=0

_d)

z

j

≤0

z=4 x+5 ysubjecttolinearequations 2 x +3 y ≤ 12,2 x + y ≤ 8, x+ y ≥ 4, x ≥ 0, y ≥ 1 is

a) 20 b) 22

c) 21 d) 24

187. A company manufactures two types of cricket bats B1 and B2.Both the products pass through machines I and II. The time required for processing each product on the machine I and II and availability of these machines is given below.

Product machine I II 4 5

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Available 1800 1500 Material is enough to produce 300 types of bats B1 and 400 types of B2.each unit of bat gives a profit of Rs.50 and Rs.100.Formulate the above A LPP, The constraints are

a) 4 x +3 y ≤1800, 5 x+7 y ≥ 1500,0 ≤ x ≤ 300,0≤ y ≤ 400 b) 4 x +3 y ≤1800, 5 x+7 y ≤1500, 0 ≤ x ≤ 300,0≤ y ≤ 400 c) 4 x +3 y ≥1800, 5 x+7 y ≤ 1500,0 ≤ x ≤ 300,0≤ y ≤ 400 d) 4 x +3 y ≥1800, 5 x+7 y ≥ 1500,0 ≤ x ≤ 300,0≤ y ≤ 400 188. The maximum value of the linear

objective function z in given feasible region is z=4 x+5 ysubjecttox+ y ≤ 8, x ≥ 2 y , x+ y ≤ 8, x ≥ 0, y ≥ 0

a) 17 b) 17.5

c)

17

3 1

d) 18

a)

x−2 y ≥ 0, 4 x +3 y ≥24, 0 ≤ x ≤5, y ≥0

b) x−2 y ≥ 0, 4 x +3 y ≤24, 0 ≤ x ≤5, y ≥0 c)

x−2 y ≤ 0, 4 x +3 y ≤ 24, 0 ≤ x ≤5, y ≥0

d) x−2 y ≤ 0, 4 x +3 y ≥ 24, 0 ≤ x ≤5, y ≥0 190. The maximum value of

z=3 x+6 ysubjecttolinearequationsy≥

x

2 , y ≤ 2 x , x + y ≤ 8, x ≥0, y ≥ 0

is

a) 50 b) 60

c) 40 d) 55

191. Laura is to buy some orange and

lemons. Lemans are Rs. 50 and orange are for Rs. 25.she must not buy more than 2 kg of lemons and she must buy at least 4 kg of orange. She is told to at least 6 kg of fruits together. Minimum value of cost z is

a) 140 b) 150

c) 160 d) 170

192. A agriculturalist wishes to mix two brands of fertilizer type A and B. Type A contains 1 unit of potash , 3 units of phosphates and 5 units of nitrate. Type B

contains 1 units of potash, 8 units of

phosphate and 3 units of nitrate. Formulate the problem to maximize profit, given that type A gives Rs.50 and type B gives Rs.70 profit per unit. The maximum value of profit z is

a) 250 b) 249

c) 248 d) 252

193. A firm produces two types of gadgets A and B. They are produced at foundry and then sent to finishing to the machine shop. The no of man hours of labor required in each shop for production of A and B, the available man hours are as follows.

Product foundry machining A 10 5 B 6 4 Time available 60 35

Formulate the above problem as an LLP. a)

10 x+6 y ≤ 60,5 x +4 y ≥ 35, x ≥0, y ≥ 0

b) 10 x+6 y ≤ 60,5 x +4 y ≥ 35, x ≥0, y ≥ 0 c)

10 x+6 y ≥ 60,5 x +4 y ≤ 35, x ≥0, y ≥ 0

d) 10 x+6 y ≥ 60,5 x +4 y ≥ 35, x ≥ 0, y ≥ 0 194. A shopkeeper buys 2 types of food for his shop .tea at Rs. 4o a packet and coffee at Rs. 60 a tin. He has Rs. 1500 available and decides that at least 30 tins should be bought and one third of the tins should be of coffee. He makes a profit of Rs.10 on tins of tea and Rs 20 on a tin of coffee. Formulate the

problem as LPP for maximum profit.

a)

2 x +3 y ≥75, x+ y ≥ 30, 2 y ≥ x , x ≥ 0, y ≥ 0

b) 2 x +3 y ≤75, x+ y ≤ 30,2 y ≥ x , x ≥ 0, y ≥ 0 c)

2 x +3 y ≤75, x+ y ≥ 30,2 y ≤ x , x ≥ 0, y ≥ 0

d) 2 x +3 y ≤75, x+ y ≥ 30,2 y ≥ x , x ≥ 0, y ≥ 0 195. A firm produces two types of health pills A and B.A contains 2 units of proteins 5 units of vitamin D and one unit of vitamin B1.B contains 1,8 and 6 units of protein, vitamin D and vitamin B1.it is found that it requires 12 unit of proteins ,26 units of vitamin D and 24 units of vitamin D. formulate this problem as LLP .

a)

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b)

2 x + y ≤12, 5 x+8 y ≥ 26, x +6 y ≥24, x , y ≥ 0 c)

2 x + y ≥12, 5 x+8 y ≥ 26, x +6 y ≤24, x , y ≥ 0 d) None of these

196. A farm is engaged in feeding pigs, who are fed on various farm products .in view of need to ensure certain nutrient constituent it is necessary to buy 2 product A and B. The content of products A and B are given below. Nutrient nutrient content in minimum amount

A B

M1 36 6 108 M2 3 12 36 M3 20 10 100 Formulate this problem as LLP

a) 36 x+6 y ≥ 108,3 x +12 y ≥ 36,20 x +10 y ≥ 100, x , y ≥ 0 b) 36 x+6 y ≥ 108,3 x +12 y ≥ 36,20 x +10 y ≤ 100, x , y ≥ 0 c) 36 x+6 y ≥ 108,3 x +12 y ≤ 36,20 x +10 y ≥ 100, x , y ≥ 0 d) 36 x+6 y ≤ 108,3 x +12 y ≥ 36,20 x +10 y ≥ 100, x , y ≥ 0 197. A former want to make sure that his herd gets the minimum daily requirement of three basic nutrients A, B and C. daily

requirements are 15 units of A , 20 units of B and one unit of C. one gram of product P has 2 units of A , one unit of B and one unit of C. The cost of P is Rs.12 per gram and the cost of Q is Rs.18 per gram. Formulate this problem as LPP to determine units of p and Q should the farmer buy so that the cost is minimum. Then the system of in equations are

a)

2 x + y ≤15, x + y ≥ 20, x+3 y ≥ 0, x , y ≥ 0

b) 2 x + y ≥15, x + y ≤ 20, x+3 y ≥ 0, x , y ≥ 0 c)

2 x + y ≥15, x + y ≥ 20, x+3 y ≥ 0, x , y ≥ 0

d) 2 x + y ≥15, x + y ≥ 20, x+3 y ≤ 0, x , y ≥ 0 198. A dealer wishes to purchase a no. of fans and sewing machines. He has Rs. 4000 to invest and has space of 25 items. A fan costs

Rs.350 and a sewing machine Rs.250.He sells a fan at a profit of Rs.20 and a machine at the profit of Rs.15.formulate this problem as LPP to maximize profit solution.

a)

350 x+250 y ≤ 4000, x + y ≤25, x , y ≥ 0

b) 350 x+250 y ≤ 4000, x + y ≥25, x , y ≥ 0 c)

350 x+250 y ≥ 4000, x + y ≤25, x , y ≥ 0

d) 350 x+250 y ≥ 4000, x + y ≥25, x , y ≥ 0 199. A foundry produces two kinds of steel casting P and Q. The product is to be fettled, machined and then finishing process. The time in hrs required is as follows.

P Q Fettling 2 ½ Machining 4 2 Finishing 3 1

Total hrs available for fettling. Machining and finishing are 45, 150 and 250 respectively. Profit P costs Rs.5 and B gives a profit of Rs.4 per unit. Formulate this problem as LPP to maximize profit. a)

2 x

1

+

1

2 x

2

≤ 45, 4 x

1

+2 x

2

≤150, 3 x

1

+

x

2

≥ 250 x

1

, x

2

≥ 0

b)

2 x

1

+

1

2 x

2

≤ 45, 4 x

1

+2 x

2

≤150, 3 x

1

+

x

2

≤ 250 x

1

, x

2

≥ 0

c)

2 x

1

+

1 ₂

x

2

≤ 45, 4 x

1

+2 x

2

≥150, 3 x

1

+

x

2

≤ 250 x

1

, x

2

≥ 0

d)

2 x

₁

+

1

2 x

2

≥ 45, 4 x

1

+2 x

2

≤150, 3 x

1

+

x

2

≤ 250 x

1

, x

2

≥ 0

200. A company produces two types of food stuffs F1 and F2 which contains vitamins V1, V2 and V3.F1 contains 1 mg of V1 , and 2 mg of V2, and 1 mg of V3,whereas F2 contains 1mg of V1 , 1 mg of V2 and 4 mg of

V3.minimum daily requirements of these vitamins are 6 mg of V1, 7 mg of V2 and 8 mg of V3.The cost of 1unit of F1 is Rs.2 and that of F2 is Rs.3.write the linear problem to find the least expensive diet that would supply to body .

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a)

x+ y ≥ 6, 2 x + y ≥7, x+4 y ≤ 8, x , y ≥ 0

b) x+ y ≥ 6, 2 x + y ≤7, x+4 y ≥ 8, x , y ≥ 0 c)

x+ y ≥ 6, 2 x + y ≥7, x+4 y ≥ 8, x , y ≥ 0

d) x+ y ≤ 6, 2 x + y ≥ 7, x+4 y ≥ 8, x , y ≥ 0 201. The maximize value of

z=5 x+10 y

subject to the constraints

2 x +3 y ≤18, x+4 y ≤16, x ≤ 6, x ≥ 0, y ≥ 0

_{at the}

point

a) (6,0) b) (6,2)

c) (4.8,2.8) d) (0,4)

202. A printing company prints two types of magazines A and B. The company earns Rs. 10 and Rs.15 on each magazine A and B respectively. These are processed on three machines I, II, and III and total times in hour available per week on each machine is as follows

a) 5 b) 3

c) 4 d) 6

203. Minimum value of

z=4 x+8 y

under the constraints

200 x+100 y ≥2000, x+2 y ≥ 50,40 x +40 y ≥ 1400, x ≥ 0, y ≥0

is

a) 225 b) 175

c) 200 d) 280

204. If the constrains in an linear programming problems are changed:

a) Solution is not defined

.

b) The objective function has to be modified. c) The problem is to be revaluated.

d) The change in constraints is ignored. 205. The common region determined by

15 y+12 x ≤ 80, 3 x−4 y ≤ 12, andx ≥ 4, x ≥ 0, y ≥ 0

is a) (4,0) (10,-4) (-9,-4) b) (-4,0) (10,-4) (9,-4) c) (-4,0) (10,4) (9,4) d) (4,0) (4,10) (9,4)

206. The convex polygon given by

x ≤ 4, x− y ≥0, 3 x+ y ≥ 3 _{the vertices of convex} polygon

a) (4,9)(4,-4)(-1,0) b) (4,9)(4,-4)(-1,0) c) (4,-9)(4,4)(1,0) d) (4,9)(4,4)(-1,0)

207. The common region represented by the inequalities.

5 x+4 y ≥20, x ≤ 6 andy ≤ 4, x ≥ 0, y ≥ 0

_is a) (1,4) (-4,0) (6,0) b) (4,0)(-6,0)(1,-4) c) (4,0) (6,0) (6,4)(1,4) d) (4,0)(-6,0)(6,4) 208. Maximize z=x + y subject to x+2 y ≤ 8 _and 3 x+2 y ≤12, x ≥ 0, y ≥ 0 _is a)

Zmax=7

b) Zmax=5 c)

Zmax=6

d) Zmax=9 209. Minimize

z=x + y ,

subject to

5 x+2 y ≥10,2 x+3 y ≥ 6, x ≥1, x ≥ 0 andy ≥0

_is a) Z min¿20/11 b)

Z min

¿

27/11

c) Z min¿28/11 d)

Z min

¿

25/11

210. The following inequalities and indicate the solution set

x+2 y ≥ 2,3 x+ y ≥ 3, 4 x +3 y ≥ 6, x ≥ 0, y ≥ 0 a) (2,0)(0,3) b) (2,1)(2,3) c) (-2,0)(0,3) d) (2,6)(0,-3)

211. The Maximum value of the objective function z=x + y subject to constraints

x+2 y ≤ 8,3 x +2 y ≤ 12andx ≥ 0, y ≥ 0

a) 10 b) 20