Linear Programming VBQs Class 12 Mathematics

VBQs Linear Programming Class 12 Mathematics with solutions has been provided below for standard students. We have provided chapter wise VBQ for Class 12 Mathematics with solutions. The following Linear Programming Class 12 Mathematics value based questions with answers will come in your exams. Students should understand the concepts and learn the solved cased based VBQs provided below. This will help you to get better marks in class 12 examinations.

Linear Programming VBQs Class 12 Mathematics

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

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

Question. A Linear function, which is minimized or maximized is called
(a) an objective function
(b) an optimal function
(c) A feasible function
(d) None of these

Question. The maximum value of Z = 3x + 4y subject to the constraints : x+ y ≤ 4, x ≥ 0 , y ≥ 0 is :
(a) 0
(b) 12
(c) 16
(d) 18

Question. Any feasible solution which maximizes or minimizes the objective function is Called:
(a) A regional feasible solution
(b)An optimal feasible solution
(c)An objective feasible solution
(d) None of these

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)

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

Question. The solution set of the in equation 2𝑥 + 𝑦 > 5 is
(a) Half plane that contains the origin
(b) Open half plane not containing the origin
(c) Whole 𝑥𝑦 −plane except the points lying on the line 2𝑥 + 𝑦 = 5
(d) None of these

Question. The optimal value of the objective function is attained at the points :
(a) Given the intersection of inequations with the axes only
(b) Given by intersection of inequations with X-axis only
(c) Given by corner points of the feasible region
(d) None of these.

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

Question. Which of the following statements is correct?
(a) Every L P P admits an optimal solution
(b)A L P P admits unique optimal solution
(c) If a L P P admits two optimal solution solutions, it has aninfinite number of optimal solutions
(d) The set of all feasible solutions of a LPP is a finite set.

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.

Question. Objective function of a LPP is
(a) a constraint
(b) a function to be optimized
(c)a relation between the variables
(d) none of these

Question. The maximum value of 𝑍 = 4𝑥 + 2𝑦 subjected to the Constraints2𝑥 + 3𝑦 ≤ 18 ,𝑥 + 𝑦 ≥ 10 ;𝑥, 𝑦 ≥ 0 is
(a) 320
(b) 300
(c) 230
(d) none of these

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

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

Question. The maximum value of Z = 2x +3y subjectto the constraints :
𝑥 + 𝑦 ≤ 1 , 3𝑥 + 𝑦 ≤ 4 , 𝑥, 𝑦 ≥ 0is
(a) 2
(b) 4
(c) 5
(d) 3

Question. The point in the half plane 2𝑥 + 3𝑦 − 12 ≥ 0 is :
(a) (- 7,8 )
(b) ( 7 , – 8 )
(c) ( -7 , – 8 )
(d) (7, 8 )

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

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)

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

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

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

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

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

II. A company started airlines business and for running business it bought aeroplanes . Now an aeroplane can carry maximum of 200 passengers . A profit of Rs.400 is made on each first class ticket and a profit of Rs.300 is made on each second class ticket . The airline reserves at least 20 seats for first class .However , at least four times as many passengers prefer to travel by second class then by first class . Company wants to make maximum profit by selling tickets of first class (𝑥) and second class (𝑦) .
Using the above information give the answer of the following questions.

Question. Minimum profit is equal to
(a) 8000
(b) 6000
(c) 64000
(d) none of the above

Question. The objective function is
(a) 400𝑥 + 300𝑦
(b) 300𝑥 + 400𝑦
(c) 𝑥 + 𝑦
(d) none of the above

Question. To get maximum profit how many first class tickets should be sold –
(a) 20
(b) 180
(c) 160
(d) 40

Question. Corner points of feasible region are
(a) (20,180)
(b) (20,0)
(c) (40,0)
(d) all the above

Question. Difference between the maximum profit and minimum profit is equal to
(a) 8000
(b) 56000
(c) 64000
(d) none of the above

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) 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.

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 .

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 .

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 .

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.