CBSE Class 12 Mathematics Linear Programming Problems Worksheet

Question. Which of the term is not used in a linear programming problem :
(a) Slack inequation
(b) Objective function
(c) Concave region
(d) Feasible Region
Answer : C

Question. The feasible solution of a LPP belongs to
(a) First and second quadrants
(b) First and third quadrants.
(c) Second quadrant
(d) Only first quadrant.
Answer : D

Question. The point at which the maximum value of 𝑥 + 𝑦 , subject to the Constraints 𝑥 + 2𝑦 ≤ 70 , 2𝑥 + 𝑦 ≤ 95 , 𝑥, 𝑦 ≥ 0is obtained, is
(a) (30, 25)
(b) (20, 35 )
(c) (35 ,20 )
(d) (40 , 15)
Answer : D

Question. The value of objective function is maximum under linear constraints
(a) At the centre of feasible region
(b) At (0,0)
(c) At any vertex of feasible region
(d) The vertex which is at maximum distance from (0,0)
Answer : C

Question. If the constraints in a linear programming problem are changed :
(a) The problem is to be re-evaluated
(b) Solution is not defined
(c) The objective function has to be modified
(d) The change in constraints is ignored
Answer : A

Question. A linear programming of linear functions deals with :
(a) Minimizing
(b) Optimizing
(c) Maximizing
(d) None
Answer : B

Question. Solution set of inequations 𝑥 − 2𝑦 ≥ 0, 2𝑥 − 𝑦 ≤ −2 , 𝑥 ≥ 0, 𝑦 ≥ 0 is
(a) First quadrant
(b) infinite
(c) Empty
(d) closed half plane
Answer : C

Question. Maximum value of the objective function 𝑍 = 4𝑥 + 3𝑦 subject to the constraints
3𝑥 + 2𝑦 ≤ 160, 5𝑥 + 2𝑦 ≥ 200, 𝑥 + 2𝑦 ≥ 80 , 𝑥, 𝑦 ≥ 0 is
(a) 320
(b) 300
(c) 230
(d) none of these
Answer : A

Question. The corner points of the feasible region determined by the following System Of linear inequalities: 2𝑥 + 𝑦 ≤ 10 , 𝑥 + 3𝑦 ≤ 15 ,
𝑥, 𝑦 ≥ 0 are (0,0),(5,0), (3,4) and (0, 5 ) .
Let 𝑍 = 𝑝𝑥 + 𝑞𝑦, where 𝑝 , 𝑞 > 0.Condition on 𝑝 and 𝑞so that the maximum of 𝑍 occurs at both ( 3, 4 ) and ( 0, 5) is
(a) 𝑝 = 𝑞
(b) 𝑝 = 2𝑞
(c) 𝑝 = 3𝑞
(d) 𝑞 = 3𝑝
Answer : D

Question. By graphical method, the solution of linear programming problem Maximize : Z= 3x + 5y
Subject to : 3x +2y ≤ 18 , x ≤ 4, y ≤ 6 and x, y ≥ 0 ,is
(a) x = 2 ,y = 0 ,Z = 6
(b) x = 2 , y = 6, z=36
(c) x=4, y = 3 , Z= 27
(d) X = 4, y = 6,Z = 42
Answer : B

CASE STUDY QUESTIONS

I. A small firm manufacturers gold rings and chains. The total number of rings and chains manufactured per day is atmost 24 . it takes 1 hour to make ring and 30 minutes to make a chain . The maximum number of hours available per day is 16 . If the profit on a ring is Rs.300 and that on a chain is Rs.190 . Firm is concerned about earning maximum profit on the number of rings(𝑥) and chains(𝑦) that have to be manufactured per day.
Using the above information give the answer of the following questions.

Question. For maximum profit firm has to make the number of rings and chains –
(a) 0,24
(b) 8,16
(c) 16,8
(d) 16,0
Answer : B

Question. Constraints of the above LPP are
(a) 𝑥 ≤ 0
(b) 2𝑥 + 𝑦 ≤ 32
(c) 𝑦 ≥ 1
(d) none of the above
Answer : B

Question. Corner points of feasible region are
(a) (0,24)
(b) (8,16)
(c) a &b both
(d) (12,0)
Answer : C

Question. The objective function is
(a) 190𝑥 + 300𝑦
(b) 300𝑥 + 190𝑦
(c) 𝑥 + 𝑦
(d) none of the above
Answer : B

Question. Maximum profit earned by the firm is equal to
(a) 6440
(b) 4560
(c) 5000
(d) 5440
Answer : D

ASSERTION AND REASON
Directions (Q. Nos. 1-5) Each of these questions contains two statements: Assertion (A) and Reason (R). Each of these questions also has four alternative choices, any one of which is the correct answer. You have to select one of the codes (a), (b). (c) and (d) given below.
(a) A is true, R is true: R is a correct explanation for A.
(b) A is true, R is true; R is not a correct explanation for A.
(c) A is true: R is false.
(d) A is false: R is true.

Question. Assertion (A) Objective function 𝑍 = 13𝑥 − 15𝑦 , is minimized subject to constraints 𝑥 + 𝑦 ≤ 7 , 2𝑥 − 3𝑦 + 6 ≥ 0 , 𝑥 ≥ 0 , 𝑦 ≥ 0 occur at corner point (0,2) .
Reason (R) If the feasible region of the given LPP is bounded , then the maximum or minimum values of an objective function occur at corner points .
Answer : A

Question. Assertion (A) Maximum value of 𝑍 = 11𝑥 + 7𝑦 , subject to constraints 2𝑥 + 𝑦 ≤ 6, 𝑥 ≤ 2 , 𝑥 ≥ 0 , 𝑦 ≥ 0 will be obtained at (0,6) .
Reason (R)In a bounded feasible region, it always exist a maximum and minimum value.
Answer : B

Question. Assertion (A) Maximise𝑍 = 3𝑥 + 4𝑦, subject to constraints : 𝑥 + 𝑦 ≤ 1 ,, 𝑥 ≥ 0 , 𝑦 ≥ 0 . Then maximum value of Z is 4 .
Reason (R) If the shaded region is not bounded then maximum value cannot be determined.
Answer : C

Question. Assertion (A) For an objective function 𝑍 = 4𝑥 + 3𝑦 , corner points are (0,0), (25,0) , (16,16) and (0,24) . Then optimal values are 112 and 0 respectively .
Reason (R) Themaximum or minimum values of an objective function is known as optimal value of LPP . These values are obtained at corner points .
Answer : A

Question. Assertion (A)The linear programming problem, maximize 𝑍 = 2𝑥 + 3𝑦
subject to constraints 𝑥 + 𝑦 ≤ 4 , 𝑥 ≥ 0 , 𝑦 ≥ 0
It gives the maximum value of Z as 8 .
Reason (R)To obtain maximum value of Z, we need to compare value of Z at all the corner points of the feasible region .
Answer : D

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