12th Math chapter 12: full solution, notes, MCQ, & free PDF | Bihar board

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12th Math chapter 12 Bihar board with NCERT-Details & Highlights

name of content	12th Math chapter 12 full solution
content	Math objective type MCQ question with answer
name of subjects	class 12 Math
important for exam	Bihar board & other state board
pattern & Level	NCERT & State level question

12th Math chapter 12 all objective solutions – download PDF

Question 1. Objective function of a L.P.P.is

• • (a) a constant
    (b) a function to be optimised
    (c) a relation between the variables
    (d) none of these

Answer: (b) a function to be optimised

Question 2. The optimal value of the objective function is attained at the points

• • (a) on X-axis
    (b) on Y-axis
    (c) which are comer points of the feascible region
    (d) none of these

Answer: (c) which are comer points of the feascible region

Question 3. In solving the LPP:
“minimize f = 6x + 10y subject to constraints x ≥ 6, y ≥ 2, 2x + y ≥ 10, x ≥ 0, y ≥ 0” redundant constraints are

• • (a) x ≥ 6, y ≥ 2
    (b) 2x + y ≥ 10, x ≥ 0, y ≥ 0
    (c) x ≥ 6
    (d) none of these

Answer: (b) 2x + y ≥ 10, x ≥ 0, y ≥ 0

Question 4. Region represented by x ≥ 0, y ≥ 0 is

• • (a) first quadrant
    (b) second quadrant
    (c) third quadrant
    (d) fourth quadrant

Answer: (a) first quadrant

Question 5. The region represented by the inequalities
x ≥ 6, y ≥ 2, 2x + y ≤ 0, x ≥ 0, y ≥ 0 is

• • (a) unbounded
    (b) a polygon
    (c) exterior of a triangle
    (d) None of these

Answer: (d) None of these

Question 6. Maximize Z = 3x + 5y, subject to x + 4y ≤ 24, 3x + y ≤ 21, x + y ≤ 9, x ≥ 0, y ≥ 0.

• • (a) 20 at (1, 0)
    (b) 30 at (0, 6)
    (c) 37 at (4, 5)
    (d) 33 at (6, 3)

Answer: (c) 37 at (4, 5)

Question 7. Maximize Z = 4x + 6y, subject to 3x + 2y ≤ 12, x + y ≥ 4, x, y ≥ 0.

• • (a) 16 at (4, 0)
    (b) 24 at (0, 4)
    (c) 24 at (6, 0)
    (d) 36 at (0, 6)

Answer: (d) 36 at (0, 6)

Question 8. Maximize Z = 6x + 4y, subject to x ≤ 2, x + y ≤ 3, -2x + y ≤ 1, x ≥ 0, y ≥ 0.

• • (a) 12 at (2, 0)
    (b) 1403 at (23, 13)
    (c) 16 at (2, 1)
    (d) 4 at (0, 1)

Answer: (c) 16 at (2, 1)

Question 9. Maximize Z = 10×1 + 25×2, subject to 0 ≤ x1 ≤ 3, 0 ≤ x2 ≤ 3, x1 + x2 ≤ 5.

• • (a) 80 at (3, 2)
    (b) 75 at (0, 3)
    (c) 30 at (3, 0)
    (d) 95 at (2, 3)

Answer: (d) 95 at (2, 3)

Bihar Board 12th math chapter 12 most mcq notes and solution

Question 10. Z = 20x₁ + 20x₂, subject to x₁ ≥ 0, x₂ ≥ 0, x₁ + 2x₂ ≥ 8, 3x₁ + 2x₂ ≥ 15, 5x₁ + 2x₂ ≥ 20. The minimum value of Z occurs at

• • (a) (8, 0)
    (b) (52,154)
    (c) (72,94)
    (d) (0, 10)

Answer: (c) (72,94)

Question 11. Z = 7x + y, subject to 5x + y ≥ 5, x + y ≥ 3, x ≥ 0, y ≥ 0. The minimum value of Z occurs at

• • (a) (3, 0)
    (b) (12,52)
    (c) (7, 0)
    (d) (0, 5)

Answer: (d) (0, 5)

Question 12. Minimize Z = 20x₁ + 9x₂, subject to x₁ ≥ 0, x₂ ≥ 0, 2x₁ + 2x₂ ≥ 36, 6x₁ + x₂ ≥ 60.

• • (a) 360 at (18, 0)
    (b) 336 at (6, 4)
    (c) 540 at (0, 60)
    (d) 0 at (0, 0)

Answer: (b) 336 at (6, 4)

Question 13. Z = 8x + 10y, subject to 2x + y ≥ 1, 2x + 3y ≥ 15, y ≥ 2, x ≥ 0, y ≥ 0. The minimum value of Z occurs at

• • (a) (4.5, 2)
    (b) (1.5, 4)
    (c) (0, 7)
    (d) (7, 0)

Answer: (b) (1.5, 4)

Question 14. Z = 4x₁ + 5x₂, subject to 2x₁ + x₂ ≥ 7, 2x₁ + 3x₂ ≤ 15, x₂ ≤ 3, x₁, x₂ ≥ 0. The minimum value of Z occurs at

• • (a) (3.5, 0)
    (b) (3, 3)
    (c) (7.5, 0)
    (d) (2, 3)

Answer: (a) (3.5, 0)

Question 15. The maximum value of f = 4x + 3y subject to constraints x ≥ 0, y ≥ 0, 2x + 3y ≤ 18; x + y ≥ 10 is

• • (a) 35
    (b) 36
    (c) 34
    (d) none of these

Answer: (d) none of these

Question 16. The minimum value of Z = 4x + 3y subjected to the constraints 3x + 2y ≥ 160, 5 + 2y ≥ 200, 2y ≥ 80; x, y ≥ 0 is

• • (a) 220
    (b) 300
    (c) 230
    (d) none of these

Answer: (a) 220

Question 17. The maximum value of Z = 3x + 2y, subjected to x + 2y ≤ 2, x + 2y ≥ 8; x, y ≥ 0 is

• • (a) 32
    (b) 24
    (c) 40
    (d) none of these

Answer: (d) none of these

Question 18. Maximize Z = 11x + 8y, subject to x ≤ 4, y ≤ 6, x ≥ 0, y ≥ 0.

• • (a) 44 at (4, 2)
    (b) 60 at (4, 2)
    (c) 62 at (4, 0)
    (d) 48 at (4, 2)

Answer: (b) 60 at (4, 2)

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SL.N.	Subject name with solutions
01	Physics
02	Chemistry
03	Biology
04	Math
05	Hindi
06	English