CBSE Class 12 Mathematics Linear Programming Worksheet Set 01

CBSE Class 12 Mathematics Linear Programming (1). CBSE issues sample papers every year for students for class 12 board exams. Students should solve the CBSE issued sample papers to understand the pattern of the question paper which will come in class 12 board exams this year. The sample papers have been provided with marking scheme. It’s always recommended to practice as many CBSE sample papers as possible before the board examinations. Sample papers should be always practiced in examination condition at home or school and the student should show the answers to teachers for checking or compare with the answers provided. Students can download the sample papers in pdf format free and score better marks in examinations. Refer to other links too for latest sample papers.

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
Answer : A

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
Answer : B

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
Answer : C

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
Answer : D

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

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
Answer : B

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
Answer : B

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.
Answer : C

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
Answer : C

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.
Answer : C

CASE STUDY QUESTIONS

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

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

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

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

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

Short Answer Questions

Question. A small firm manufactures necklaces and bracelets. The total number of necklaces and bracelets that it can handle per day is at most 24. It takes one hour to make a bracelet and half an hour to make a necklace. The maximum number of hours available per day is 16. If the profit on a necklace is Rs. 100 and that on a bracelet is Rs. 300. Formulate an LPP. for finding how many of each should be produced daily to maximise the profit? It is being given that at least one of each must be produced. 
Answer: Let \( x \) and \( y \) be the number of necklaces and bracelets manufactured by small firm per day. If \( P \) be the profit, then objective function is given by
\( P = 100x + 300y \) which is to be maximised under the constrains
\( x + y \leq 24 \)
\( \frac{1}{2}x + y \leq 16 \)
\( x \geq 1, y \geq 1 \)

Question. Two tailors, A and B, earn Rs. 300 and Rs. 400 per day respectively. A can stitch 6 shirts and 4 pairs of trousers while B can stitch 10 shirts and 4 pairs of trousers per day. To find how many days should each of them work and if it is desired to produce at least 60 shirts and 32 pairs of trousers at a minimum labour cost, formulate this as an LPP. 
Answer: Let A and B work for \( x \) and \( y \) days respectively.
Let \( Z \) be the labour cost.
\( Z = 300x + 400y \)
Subject to constraints
\( 6x + 10y \geq 60 \)
\( 4x + 4y \geq 32 \)
\( x, y \geq 0 \)

Question. A company produces two types of goods A and B, that require gold and silver. Each unit of type A requires 3 g of silver and 1 g of gold while that of type B requires 1 g of silver and 2 g of gold. The company can produce a maximum of 9 g of silver and 8 g of gold. If each unit of type A brings a profit of Rs. 40 and that of type B Rs. 50, formulate LPP to maximize profit. 
Answer: Let \( x \) and \( y \) be the number of goods A and goods B respectively. If \( P \) be the profit then
\( P = 40x + 50y \) which is to be maximised under constraints
\( 3x + y \leq 9 \)
\( x + 2y \leq 8 \)
\( x \geq 0, y \geq 0 \)

Question. A firm has to transport atleast 1200 packages daily using large vans which carry 200 packages each and small vans which can take 80 packages each. The cost for engaging each large van is Rs. 400 and each small van is Rs. 200. Not more than Rs. 3,000 is to be spent daily on the job and the number of large vans cannot exceed the number of small vans. Formulate this problem as a LPP given that the objective is to minimize cost. 
Answer: Let the number of large vans and small vans be \( x \) and \( y \) respectively.
Here transportation cost \( Z \) be objective function, then
\( Z = 400x + 200y \), which is to be minimized under constraints
\( 200x + 80y \geq 1200 \)
\( \implies \) \( 5x + 2y \geq 30 \)
\( 400x + 200y \leq 3000 \)
\( \implies \) \( 2x + y \leq 15 \)
\( x \leq y, x \geq 0, y \geq 0 \)