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UNIT-V : LINEAR PROGRAMMING, , m=) dnote, , LINEAR, PROGRAMMING, , , , > Introduction, related terminology such as constraints, objective function, optimization, different types, of linear programming (L.P.) problems, graphical method of solution for problems in two variables,, feasible and infeasible region (hounded), feasible and infeasible solutions, optimal feasible solutions, (up to three non-trivial constraints)., , STAND ALONE MCQs (1 Mark each), , , , , , Q.1. The corner points of the feasible region determined Q. 2. The feasible solution for a LPP is shown in given, , by the system of linear constraints are (0, 0), (0, 40), figure. Let Z = 3x — 4y be the objective function, (20, 40), (60, 20), (60, 0). The objective function is Minimum of Z occurs at, Z=4x + By., , , , Compare the quantity in Column A and Column B, , Column A Column B, Maximum of Z 325 (0,8), , , , , , , , , , , , , , , , , , (A) The quantily in column A is greater., (B) The quantity in column Bis greater., (C) The two quantities are equal., (D) The relationship cannot be determined on the, basis of the information supplied., Ans, Option (B) is correct., , , , , , , , , , , , , , , , , , , , , , , , , , , , Explanation :, m Corresponding value 50, Corner points of Z = 4x + 3y (0, 0) , 0), (0, 0) 0 (A) 0,0) (B) (0,8), ; (C) 6,0) (D) (4, 10), (0, 40) a Ans. Option (B) is correct., (20, 40) 200, Explanation:, (0, 20) 300 — Maximum 5 ;, Sie Corresponding value, (60, 0) 240 Corner points af YZ = a2 Ay, Hence, maximum value of Z = 300 < 325 © 0) 0 , , , , , , , So, the quantity in column Bis greater.

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(5, 9) 15 © Maximum, (6, 5) 4, , (6, 8) -14, , (4, 10) =28, , (0, 8) -32 + Minimum, , , , , , , , , , Hence, the minimum of Z occurs at (0, 8) and its, minimum value is (-32)., , Q. 3. Refer to Q.2 of multiple choice questions, maximum, of Z occurs at, (A) 6,0), (C) 8), , Ans. Option (A) is correct., , (B) 5), (D) @, 10), , | Explanation: Maximum of Z occurs at (6,0). |, , Q. 4. Refer to Q.2 of multiple choice questions, (Maximum, value of Z + Minimum value of Z) is equal to, (A) 13 (B) 1, © 13 (D) 17, , Ans. Option (D) is correct., , Explanation: Maximum value of Z + Minimum, value of Z = 15 — 32= —17, , Q. 5. The feasible region for an LPP is shown in the given, Figure. Let F = 3x - 4y be the objective function., Maximum value of F is, , (12, 6), (0, 4), (6,0), (A) 0 (B) 8, (©) 12 (D) -18, , Ans. Option (C) is correct., , Explanation: The feasible region as shown in the, figure, has objective function F = 3x - 4y., , , , Corresponding value, , Corner points ae ae ay, , | Explanation: Minimum value of F is —16 at (0,4). |, , , , Corner points of the feasible region for an LPP are, (0, 2), (3, 0), (6,0), (6,8) and (0, 5). Let F = 4x + 6y be, the objective function., , The minimum value of F occurs at, , (A) 0,2) only, , (B) (3,0) only, , (C) the mid-point of the line segment joining the, points (0, 2) and (3, 0) only, , (D) any point on the line segment joining the points, (0, 2) and (3, 0), , Ans. Option (D) is correct., , , , , , , , , , , , , , Explanation :, Comer points | Corresponding value, (0, 2) 12 + Minimum, (3, 0) 12 — Minimum, (6, 0) 24, (6, 8) 72 + Maximum, (0, 5) 30, , , , , , , , , , Hence, minimum value of F occurs at any points, on the line segment joining the points (0, 2) and, , (3,0)., Q.8. Refer to Q. 7 above, Maximum of F — Minimum of, PS, (A) 60 (B) 48, (C) 42 (D) 18, , Ans. Option (A) is correct., , Explanation: Maximum of F — Minimum of, F=72-12=60, , Q. 9. Corner points of the feasible region determined by, the system of linear constraints are (0, 3), (1, 1) and, (3, 0). Let Z = px + gy, where p, q > 0. Condition on, pand qo that the minimum of Z occurs at (3, 0) and, , (L, 1) is, (A) p= 29 (B) p= qi, (C) p= 39 (D) p=q, , Ans. Option (B) is correct., , Explanation :, , , , Corresponding value, , Corner points of Z = px + qy;, , , , , , (0, 0) 0, (12, 6), @, 4), , Hence, the maximum value of F is 12., , , , 12 — Maximum, , , , , , , , , , 16 — Minimum, , , , Q. 6. Refer to Q.5 of multiple choice questions, minimum, value of F is, , , , , , , , , , , , peg > 0, (0, 3) 3q, (1,1) pre, (3, 0) 3p, , , , So, condition of p and q, so that the minimum of, Z occurs at (3, 0) and (1, 1) is, , (A) 0, (¢ 12, Ans. Option (B) is correct., , (B) -16, (D) does not exist, , p+q = 3p, , => ap=4q, =f, , ees

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ASSERTION AND REASON BASED MCQs (1 Mark each), , , , Directions: In the following questions, A statement, of Assertion (A) is followed by a statement of, Reason (R). Mark the correct choice as, , (A) Both Aand Rare true and Ris the correct explanation, of A, , (B) Both A and R are true but R is NOT the correct, explanation of A, , (C) Ais true but R is false, , (D) Ais false but Ris True, , Q.1. Assertion (A): Feasible region is the set of points, , which satisfy all of the given constraints and, objective function too., , , , Reason (R) is also correct., , , , , , , , , , , , , , , , , , , , Reason (R): The optimal value of the objective , =, function is attained at the points on X-axis only. Comer Folate: A HeSy, Ans. Option (C) is correct (3,0) 21, Explanation: The optimal value of the objective ( °) é, function is attained at the corner points of 2°2, feasible region., 7,0) 49 maximum, Q,2. Assertion (A): The intermediate solutions of (0,5) 5, constraints must be checked by substituting them, back into objective function. Q.4. Assertion (A): Z = 20x, + 20x,, subject to x,2 0,, Reason (R): Hy 22, x, + 2x, 2 8, 3x, + xy 2 15, 5x, + xy 220., , Out of the corner points of feasible region (8, 0),, , (é B22) and (0,10), the minimum value of, 2'4, , , , , , , , , , 2x+3y=6 a 2, 79, 2'4, Reason (R) :, Here (0, 2); (0, 0) and (3, 0) all are vertices of feasible Corner Points Z = 2x, | 20%,, region. 160, Ans. Option (D) is correct., 125, , Explanation: he intermediate solutions of, constraints must be checked by substituting,, them back into constraint equations., , , , 115 minimum, , , , Q.3. Assertion (A) : For the constraints of linear, optimizing function Z = x, + x, given by x, + x, $1, (0, 10) 200, 3x, + x,2 1, there is no feasible region., , Reason (R): Z = 7x + y, subject to 5x + y <5,, x + y23,x20, y 20. Out of the corner points of, , , , , , , , , , , , Ans. Option (A) is correct., , Explanation: Assertion (A) and Reason (R), both are correct and Reason (R) is the correct, , 1 mi, feasible region (3, 0), (24) (7, 0) and (0,5), the explanation of Assertion (A), maximum value of Z occurs at (7, 0). Q. 5. Assertion (A): For the constraints of a LPP problem, Ans. Option (B) is correct. given by, , x, + 2x, $ 2000, x, + x, $1500, x, $600 and x, x,20,, the points (1000, 0), (0, 500), (2, 0) lie in the positive, bounded region, but point (2000, 0) does not lie in, the positive bounded region., , Explanation: Assertion (A) is correct., Clearly from the graph below that there is no, feasible region.

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Reason (R):, X2, xytxy=1500, (0, 1500), (0,100), , x= 600, , , , 1, , of, , , , ¥1+2x7=2000, From the graph, it is clear that the point (2000, 0) is, outside., Ans. Option (A) is correct., Explanation: Assertion (A) and Reason (R) both, , are correct, Reason (I) is the correct explanation, of Assertion (A)., , CASE-BASED MCQs, , Q. 6. Assertion (A) : The graph of x <2.and y2 2 will be, situated in the first and second quadrants., Reason (R):, x, , , , y=2, , , , , , , , oO, x=2, , Ans. Option (A) is correct., , Explanation: It is clear from the graph given in, the Reason (R) that Assertion (A) is true., , , , Attempt any four sub-parts from each question., Each sub-part carries 1 mark., , . Read the following text and answer the following, questions on the basis af the same:, An aeroplane can carry a maximum of 200, passengers. A profit of % 1000 is made on each, executive class ticket and a profit of ® 600 is made, on each economy class tickel. The airline reserves, al least 20 seats for the executive class. However, al, least 4 times as many passengers prefer to travel by, economy class, than by executive class. It is given, that the number of executive class tickets is x and, , that of economy class tickets is y., , ea, , Q.1. The maximum value of x + vis,, (A) 100 (B) 200, (© 20 (D) 80, , , , Ans. Option (B) is correct., Q. 2. The relation between x and y is, (A) x<y (B) y > 80, (© x24y (D) y24e, Ans. Option (D) is correct., Q. 3. Which among these is not a constraint for this LPP?, , (A) x20 (B) x +200, (©) «280 (D) 4x-y<0, , Ans. Option (C) is correct., , Q. 4. The profit when x = 20 and y = 80is, , (A) 260,000 (B) 268,000, (©) 264,000 (D) 21,36,000, , Ans. Option (B) is correct., , Q.5. The maximum profit is, , (A) 136,000, (©) 68,000, ‘Ans. Option (A) is correct., , (B) 128,000, (D) 120,000, , Explanation:, Objective function :, Maximise Z = 1000x + 600y, Constraints:, xty 2200, y 2 20,x20, y 24x

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The corner points are A(20, 180), B(40, 160),, C(20, 80), Evaluating the objective function, Z= 1,000x + 600y at A, Band C, Z= 1,000 x 20 + 600 x 180, 0,000 + 1,08,000, % 1,28,000, 000 x 40 + 600 x 160, 0,000 + 96,000, % 1,36,000 (max.), 000 x 20 + 600 x 80, , = 20,000 + 48,000, , = % 68,000, or Zis maximum, when x = 40, y = 160., or 40 tickets of executive class and 160 tickets, of economy class should be sold to get the, maximum profit of & 1,36,000., , , , At A(20, 180),, , , , At B(40, 160), Z:, , , , , , , , ALC(20, 80),, , IL. Read the following text and answer the following, questions on the basis of the same:, A dealer in rural area wishes to purchase a number, of sewing machines. He has only 25,760 to invest, and has space for at most 20 items for storage. An, electronic sewing machine cost him 2360 and a, manually operated sewing machine 7240. He can, sell an electronic sewing machine at a profit of €22, and a manually operated machine at a profit of 218., Assume that the electronic sewing machines he can, , , , is x and that of manually operated machines is y., , , , . The objective function is, (A) Maximise Z = 360x + 240y, (B) Maximise Z = 22x + 18y, , (©) Minimise Z = 360x + 240y, (D) Minimise Z = 22x + 18y, Ans. Option (B) is correct., Q.2. The maximum value of x + is, (A) 5760 (B) 18, (C) 22, Ans. Option (D) is correct., Q. 3. Which of the following is not a constraint?, , (D) 20, , (A) x+y 220, (B) 360x + 240y < 5,760, (© x20, (D) y20, , ‘Ans. Option (A) is correct., , Q. 4. The profit is maximum when (3, y) =, , (A) G, 15) (B) (8, 12), (©) (12,8) (D) (15,5), Ans. Option (B) is correct., Q.5. The maximum profit is, (A) 5,760 (B) 392, (C) 362 (D) 290, Ans. Option (B) is correct., Explanation:, Objective function :, Maximise Z = 22x + 18y, Constraints, xty < 2, 360x + 240y < 5,760, Or 3x +2y < 48, x2 0, y20, Y, D (0,24), (0, 20), B(8, 12), , , , Vertices of feasible region are :, , A(0, 20), BB, 12), C(16, 0) & OC, 0), , P(A) = 360, P(B) = 392, P(C) = 352, , -. For Maximum P, Electronic machines = 8,, and Manual machines = 12. Max. profit ¥392