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Mathematics · Textbook solutions

Linear Programming

Every solved example, exercise, and miscellaneous question, in the order the textbook teaches them. Plus every board question asked from this chapter. · 108 questions

7.1 Linear Inequations in Two Variables

41 q

Solved Examples

Worked · 9
  1. Show the solution sets for the following inequations graphically.
    a)
    x≤3x \le 3
  2. b)
    y≥−2y \ge -2
  3. c)
    x+2y≤0x + 2y \le 0
  4. d)
    2x+3y≥62x + 3y \ge 6
  5. e)
    2x−3y≥−62x - 3y \ge -6
  6. f)
    4x−5y≤204x - 5y \le 20
  7. 7.1 Ex.2
    Represent the solution set of the inequation 3x+2y≤63x + 2y \le 6 graphically.
  8. 7.1 Ex.3
    Find the common region of the solutions of the inequations x+2y≥4x + 2y \ge 4, 2x−y≤62x - y \le 6.
  9. 7.1 Ex.4
    Find the graphical solution of 3x+4y≤123x + 4y \le 12, and x−4y≤4x - 4y \le 4.

Exercise 7.1

Practice · 21
  1. Solve graphically:
    Q.1 i)
    x≥0x \ge 0
  2. Q.1 ii)
    x≤0x \le 0
  3. Q.1 iii)
    y≥0y \ge 0
  4. Q.1 iv)
    y≤0y \le 0
  5. Solve graphically:
    Q.2 i)
    x≥0x \ge 0 and y≥0y \ge 0
  6. Q.2 ii)
    x≤0x \le 0 and y≥0y \ge 0
  7. Q.2 iii)
    x≤0x \le 0 and y≤0y \le 0
  8. Q.2 iv)
    x≥0x \ge 0 and y≤0y \le 0
  9. Solve graphically:
    Q.3 i)
    2x−3≥02x - 3 \ge 0
  10. Q.3 ii)
    2y−5≥02y - 5 \ge 0
  11. Q.3 iii)
    3x+4≤03x + 4 \le 0
  12. Q.3 iv)
    5y+3≤05y + 3 \le 0
  13. Solve graphically:
    Q.4 i)
    x+2y≤6x + 2y \le 6
  14. Q.4 ii)
    2x−5y≥102x - 5y \ge 10
  15. Q.4 iii)
    3x+2y≥03x + 2y \ge 0
  16. Q.4 iv)
    5x−3y≤05x - 3y \le 0
  17. Solve graphically:
    Q.5 i)
    2x+y≥22x + y \ge 2 and x−y≤1x - y \le 1
  18. Q.5 ii)
    x−y≤2x - y \le 2 and x+2y≤8x + 2y \le 8
  19. Q.5 iii)
    x+y≥6x + y \ge 6 and x+2y≤10x + 2y \le 10
  20. Q.5 iv)
    2x+3y≤62x + 3y \le 6 and x+4y≥4x + 4y \ge 4
  21. Q.5 v)
    2x+y≥52x + y \ge 5 and x−y≤1x - y \le 1

Solved Examples (Feasible Region)

Worked · 3
  1. 7.1 Feasible Ex.1
    Find the graphical solution of the system of inequations 2x+y≤102x + y \le 10, 2x−y≤22x - y \le 2, x≥0x \ge 0, y≥0y \ge 0.
  2. 7.1 Feasible Ex.2
    Find the feasible solution of the system of inequations 3x+4y≥123x + 4y \ge 12, 2x+5y≥102x + 5y \ge 10, x≥0x \ge 0, y≥0y \ge 0.
  3. 7.1 Feasible Ex.3
    A manufacturer produces two items A and B. Both are processed on two machines I and II. A needs 2 hours on machine I and 2 hours on machine II. B needs 3 hours on machine I and 1 hour on machine II. If machine I can run maximum 12 hours per day and II for 8 hours per day, construct a problem in the form of inequations and find its feasible solution graphically.

Exercise 7.2

Practice · 8
  1. Q.1
    Find the feasible solution of the following inequations graphically: 3x+2y≤183x + 2y \le 18, 2x+y≤102x + y \le 10, x≥0x \ge 0, y≥0y \ge 0.
  2. Q.2
    Find the feasible solution of the following inequations graphically: 2x+3y≤62x + 3y \le 6, x+y≥2x + y \ge 2, x≥0x \ge 0, y≥0y \ge 0.
  3. Q.3
    Find the feasible solution of the following inequations graphically: 3x+4y≥123x + 4y \ge 12, 4x+7y≤284x + 7y \le 28, y≥1y \ge 1, x≥0x \ge 0.
  4. Q.4
    Find the feasible solution of the following inequations graphically: x+4y≤24x + 4y \le 24, 3x+y≤213x + y \le 21, x+y≤9x + y \le 9, x≥0x \ge 0, y≥0y \ge 0.
  5. Q.5
    Find the feasible solution of the following inequations graphically: 0≤x≤30 \le x \le 3, 0≤y≤30 \le y \le 3, x+y≤5x + y \le 5, 2x+y≥42x + y \ge 4.
  6. Q.6
    Find the feasible solution of the following inequations graphically: x−2y≤2x - 2y \le 2, x+y≥3x + y \ge 3, −2x+y≤4-2x + y \le 4, x≥0x \ge 0, y≥0y \ge 0.
  7. Q.7
    A company produces two types of articles A and B which requires silver and gold. Each unit of A requires 3 gm of silver and 1 gm of gold, while each unit of B requires 2 gm of silver and 2 gm of gold. The company has 6 gm of silver and 4 gm of gold. Construct the inequations and find the feasible solution graphically.
  8. Q.8
    A furniture dealer deals in tables and chairs. He has Rs.1,50,000 to invest and a space to store at most 60 pieces. A table costs him Rs.1500 and a chair Rs.750. Construct the inequations and find the feasible solution.

7.2 Linear Programming Problem (L.P.P.)

24 q

Solved Examples (Formulation)

Worked · 3
  1. 7.2 Ex.1
    A Toy manufacturer produces bicycles and tricycles, each of which must be processed through two machine A and B. Machine A has maximum of 120 hours available and machine B has a maximum of 180 hours available. Manufacturing a bicycle requires 4 hours on machine A and 10 hours on machine B. Manufacturing a tricycle required 6 hours on machine A and 3 hours on machine B. If profits are Rs.65 for a bicycle and Rs.45 for a tricycle, formulate L.P.P. to have maximum profit.
  2. 7.2 Ex.2
    A company manufactures two types of toys A and B. Each toy of type A requires 2 minutes for cutting and 1 minute for assembling. Each toy of type B requires 3 minutes for cutting and 4 minutes for assembling. There are 3 hours available for cutting and 2 hours are available for assembling. On selling a toy of type A the company gets a profit of Rs.10 and that on toy of type B is Rs.20. Formulate the L.P.P. to maximize profit.
  3. 7.2 Ex.3
    A horticulturist wishes to mix two brands of fertilizers that will provide a minimum of 15 units of potash, 20 units of nitrate and 24 units of phosphate. One unit of brand I provides 3 units of potash, 1 unit of nitrate, 3 units of phosphate. One unit of brand II provides 1 unit of potash, 5 units of nitrate and 2 units of phosphates. One unit of brand I costs Rs.120 and one unit of brand II costs Rs.60 per unit. Formulate this problems as L.P.P. to minimize the cost.

Exercise 7.3

Practice · 9
  1. Q.1
    A manufacturing firm produces two types of gadgets A and B, which are first processed in the foundry and then sent to machine shop for finishing. The number of man hours of labour required in each shop for production of A and B per unit and the number of man hours available for the firm are as follows:
    GadgetsFoundryMachine Shop
    A105
    B64
    Time available (hour)6035
    Profit on the sale of A is Rs. 30 and B is Rs. 20 per units. Formulate the L.P.P. to have maximum profit.
  2. Q.2
    In a cattle breading firm, it is prescribed that the food ration for one animal must contain 14, 22 and 1 units of nutrients A, B and C respectively. Two different kinds of fodder are available. Each unit of these two contains the following amounts of these three nutrients:
    NutrientFodder 1Fodder 2
    Nutrients A21
    Nutrients B23
    Nutrients C11
    The cost of fodder 1 is Rs.3 per unit and that of fodder Rs. 2. Formulate the L.P.P. to minimize the cost.
  3. Q.3
    A company manufactures two types of chemicals A and B. Each chemical requires two types of raw material P and Q. The table below shows number of units of P and Q required to manufacture one unit of A and one unit of B and the total availability of P and Q.
    Raw MaterialABAvailability
    P32120
    Q25160
    The company gets profits of Rs.350 and Rs.400 by selling one unit of A and one unit of B respectively. (Assume that the entire production of A and B can be sold). How many units of the chemicals A and B should be manufactured so that the company get maximum profit? Formulate the problem as L.P.P. to maximize the profit.
  4. Q.4
    A printing company prints two types of magazines A and B. The company earns Rs. 10 and Rs. 15 on magazines A and B per copy. These are processed on three machines I, II, III. Magazine A requires 2 hours on Machine I, 5 hours on Machine II and 2 hours on Machine III. Magazine B requires 3 hours on Machine I, 2 hours on Machine II and 6 hours on Machine III. Machines I, II, III are available for 36, 50, 60 hours per week respectively. Formulate the L.P.P. to determine weekly production of A and B, so that the total profit is maximum.
  5. Q.5
    A manufacture produces bulbs and tubes. Each of these must be processed through two machines M1M_1 and M2M_2. A package of bulbs require 1 hour of work on Machine M1M_1 and 3 hours of work on M2M_2. A package of tubes require 2 hours on Machine M1M_1 and 4 hours on Machine M2M_2. He earns a profit of Rs. 13.5 per package of bulbs and Rs. 55 per package of tubes. Formulate the LLP to maximize the profit, if he operates the machine M1M_1 for atmost 10 hours a day and machine M2M_2 for atmost 12 hours a day.
  6. Q.6
    A company manufactures two types of fertilizers F1F_1 and F2F_2. Each type of fertilizer requires two raw materials A and B. The number of units of A and B required to manufacture one unit of fertilizer F1F_1 and F2F_2 and availability of the raw materials A and B per day are given in the table below:
    Raw MaterialF1F2Availability
    A2340
    B1470
    By selling one unit of F1F_1 and one unit of F2F_2, company gets a profit of Rs. 500 and Rs. 750 respectively. Formulate the problem as L.P.P. to maximize the profit.
  7. Q.7
    A doctor has prescribed two different units of foods A and B to form a weekly diet for a sick person. The minimum requirements of fats, carbohydrates and proteins are 18, 28, 14 units respectively. One unit of food A has 4 units of fats, 14 units of carbohydrates and 8 units of protein. One unit of food B has 6 units of fat, 12 units of carbohydrates and 8 units of protein. The price of food A is 4.5 per unit and that of food B is 3.5 per unit. Form the L.P.P. so that the sick person's diet meets the requirements at a minimum cost.
  8. Q.8
    If John drives a car at a speed of 60 kms/hour he has to spend Rs. 5 per km on petrol. If he drives at a faster speed of 90 kms/hour, the cost of petrol increases to 8 per km. He has Rs. 600 to spend on petrol and wishes to travel the maximum distance within an hour. Formulate the above problem as L.P.P.
  9. Q.9
    The company makes concrete bricks made up of cement and sand. The weight of a concrete brick has to be least 5 kg. Cement costs Rs.20 per kg. and sand costs of Rs.6 per kg. strength consideration dictate that a concrete brick should contain minimum 4 kg. of cement and not more than 2 kg. of sand. Form the L.P.P. for the cost to be minimum.

Solved Examples (Graphical Method)

Worked · 4
  1. 1
    Solve graphically: Maximize z=9x+13yz = 9x + 13y subject to 2x+3y≤182x + 3y \le 18, 2x+y≤102x + y \le 10, x≥0x \ge 0, y≥0y \ge 0.
  2. 2
    Solve graphically: Maximize z=5x+2yz = 5x + 2y subject to 5x+y≥105x + y \ge 10, x+y≥6x + y \ge 6, x≥0x \ge 0, y≥0y \ge 0.
  3. 3
    Solve graphically: Maximize z=3x+4yz = 3x + 4y subject to x−y≥0x - y \ge 0, −x+3y≤3-x + 3y \le 3, x≥0x \ge 0, y≥0y \ge 0.
  4. 4
    Solve graphically: Maximize z=5x+2yz = 5x + 2y subject to 3x+5y≤153x + 5y \le 15, 5x+2y≤105x + 2y \le 10, x≥0x \ge 0, y≥0y \ge 0.

Exercise 7.4

Practice · 8
  1. Q.1
    Solve the following L.P.P. by graphical method: Maximize z=11x+8yz = 11x + 8y subject to x≤4x \le 4, y≤6y \le 6, x+y≤6x + y \le 6, x≥0x \ge 0, y≥0y \ge 0.
  2. Q.2
    Solve the following L.P.P. by graphical method: Maximize z=4x+6yz = 4x + 6y subject to 3x+2y≤123x + 2y \le 12, x+y≥4x + y \ge 4, x≥0x \ge 0, y≥0y \ge 0.
  3. Q.3
    Solve the following L.P.P. by graphical method: Maximize z=7x+11yz = 7x + 11y subject to 3x+5y≤263x + 5y \le 26, 5x+3y≤305x + 3y \le 30, x≥0x \ge 0, y≥0y \ge 0.
  4. Q.4
    Solve the following L.P.P. by graphical method: Maximize z=10x+25yz = 10x + 25y subject to 0≤x≤30 \le x \le 3, 0≤y≤30 \le y \le 3, x+y≤5x + y \le 5; also find maximum value of zz.
  5. Q.5
    Solve the following L.P.P. by graphical method: Maximize z=3x+5yz = 3x + 5y subject to x+4y≤24x + 4y \le 24, 3x+y≤213x + y \le 21, x+y≤9x + y \le 9, x≥0x \ge 0, y≥0y \ge 0; also find maximum value of zz.
  6. Q.6
    Solve the following L.P.P. by graphical method: Minimize z=7x+yz = 7x + y subject to 5x+y≥55x + y \ge 5, x+y≥3x + y \ge 3, x≥0x \ge 0, y≥0y \ge 0.
  7. Q.7
    Solve the following L.P.P. by graphical method: Minimize z=8x+10yz = 8x + 10y subject to 2x+y≥72x + y \ge 7, 2x+3y≥152x + 3y \ge 15, y≥2y \ge 2, x≥0x \ge 0, y≥0y \ge 0.
  8. Q.8
    Solve the following L.P.P. by graphical method: Minimize z=6x+2yz = 6x + 2y subject to x+2y≥3x + 2y \ge 3, x+4y≥4x + 4y \ge 4, 3x+y≥33x + y \ge 3, x≥0x \ge 0, y≥0y \ge 0.

Miscellaneous Exercise 7

43 q

Choose the correct alternative

Practice · 15
  1. Misc I (1)
    The value of objective function is maximum under linear constraints _______.

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  2. Misc I (2)
    Which of the following is correct _______.

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  3. Misc I (3)
    Objective function of L.P.P. is _______.

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  4. Misc I (4)
    The maximum value of z=5x+3yz = 5x + 3y subjected to the constraints 3x+5y≤153x + 5y \le 15, 5x+2y≤105x + 2y \le 10, x,y≥0x, y \ge 0 is _______.

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  5. Misc I (5)
    The maximum value of z=10x+6yz = 10x + 6y subjected to the constraints 3x+y≤123x + y \le 12, 2x+5y≤342x + 5y \le 34, x≥0x \ge 0, y≥0y \ge 0. _______.

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  6. Misc I (6)
    The point at which the maximum value of x+yx + y subject to the constraints x+2y≤70x + 2y \le 70, 2x+y≤952x + y \le 95, x≥0x \ge 0, y≥0y \ge 0 is obtained at _______.

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  7. Misc I (7)
    Of all the points of the feasible region, the optimal value of zz obtained at the point lies _______.

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  8. Misc I (8)
    Feasible region is the set of points which satisfy _______.

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  9. Misc I (9)
    Solution of L.P.P. to minimize z=2x+3yz = 2x + 3y s.t. x≥0x \ge 0, y≥0y \ge 0, 1≤x+2y≤101 \le x + 2y \le 10 is _______.

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  10. Misc I (10)
    The corner points of the feasible solution given by the inequation x+y≤4x + y \le 4, 2x+y≤72x + y \le 7, x≥0x \ge 0, y≥0y \ge 0 are _______.

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  11. Misc I (11)
    The corner points of the feasible solution are (0,0)(0, 0), (2,0)(2, 0), (127,37)\left(\tfrac{12}{7}, \tfrac{3}{7}\right), (0,1)(0, 1). Then Z=7x+yZ = 7x + y is maximum at _______.

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  12. Misc I (12)
    If the corner points of the feasible solution are (0,0)(0, 0), (3,0)(3, 0), (2,1)(2, 1) and (0,73)\left(0, \tfrac{7}{3}\right) the maximum value of z=4x+5yz = 4x + 5y is _______.

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  13. Misc I (13)
    If the corner points of the feasible solution are (0,10)(0, 10), (2,2)(2, 2) and (4,0)(4, 0) then the point of minimum z=3x+2yz = 3x + 2y is _______.

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  14. Misc I (14)
    The half plane represented by 3x+2y<83x + 2y < 8 contains the point _______.

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  15. Misc I (15)
    The half plane represented by 4x+3y>144x + 3y > 14 contains the point _______.

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Solve the following

Practice · 28
  1. Solve each of the following inequations graphically using X Y plane.
    Misc II Q.1 i)
    4x−18≥04x - 18 \ge 0
  2. Misc II Q.1 ii)
    −11x−55≤0-11x - 55 \le 0
  3. Misc II Q.1 iii)
    5y−12≥05y - 12 \ge 0
  4. Misc II Q.1 iv)
    y≤−3.5y \le -3.5
  5. Sketch the graph of each of following inequations in XOY co-ordinate system.
    Misc II Q.2 i)
    x≥5yx \ge 5y
  6. Misc II Q.2 ii)
    x+y≤0x + y \le 0
  7. Misc II Q.2 iii)
    2y−5x≥02y - 5x \ge 0
  8. Misc II Q.2 iv)
    1x+51≤y1x + 51 \le y
  9. Find graphical solution for each of the following system of linear inequation.
    Misc II Q.3 i)
    2x+y≥22x + y \ge 2, x−y≤1x - y \le 1
  10. Misc II Q.3 ii)
    x+2y≥4x + 2y \ge 4, 2x−y≤62x - y \le 6
  11. Misc II Q.3 iii)
    3x+4y≤123x + 4y \le 12, x−2y≥2x - 2y \ge 2, y≥−1y \ge -1
  12. Find feasible solution for each of the following system of linear inequations graphically.
    Misc II Q.4 i)
    2x+3y≤122x + 3y \le 12, 2x+y≤82x + y \le 8, x≥0x \ge 0, y≥0y \ge 0
  13. Misc II Q.4 ii)
    3x+4y≥123x + 4y \ge 12, 4x+7y≤284x + 7y \le 28, x≥0x \ge 0, y≥0y \ge 0
  14. Solve each of the following L.P.P.
    Misc II Q.5 i)
    Maximize z=5x1+6x2z = 5x_1 + 6x_2 subject to 2x1+3x2≤182x_1 + 3x_2 \le 18, 2x1+x2≤122x_1 + x_2 \le 12, x1≥0x_1 \ge 0, x2≥0x_2 \ge 0
  15. Misc II Q.5 ii)
    Maximize z=4x+2yz = 4x + 2y subject to 3x+y≥273x + y \ge 27, x+y≥21x + y \ge 21
  16. Misc II Q.5 iii)
    Maximize z=6x+10yz = 6x + 10y subject to 3x+5y≤103x + 5y \le 10, 5x+3y≤155x + 3y \le 15, x≥0x \ge 0, y≥0y \ge 0
  17. Misc II Q.5 iv)
    Maximize z=2x+3yz = 2x + 3y subject to x−y≥3x - y \ge 3, x≥0x \ge 0, y≥0y \ge 0
  18. Solve each of the following L.P.P.
    Misc II Q.6 i)
    Maximize z=4x1+3x2z = 4x_1 + 3x_2 subject to 3x1+x2≤153x_1 + x_2 \le 15, 3x1+4x2≤243x_1 + 4x_2 \le 24, x1≥0x_1 \ge 0, x2≥0x_2 \ge 0
  19. Misc II Q.6 ii)
    Maximize z=60x+50yz = 60x + 50y subject to x+2y≤40x + 2y \le 40, 3x+2y≤603x + 2y \le 60, x≥0x \ge 0, y≥0y \ge 0
  20. Misc II Q.6 iii)
    Maximize z=4x+2yz = 4x + 2y subject to 3x+y≥273x + y \ge 27, x+y≥21x + y \ge 21, x+2y≥30x + 2y \ge 30, x≥0x \ge 0, y≥0y \ge 0
  21. Misc II Q.7
    A carpenter makes chairs and tables. Profits are Rs.140/- per chair and Rs. 210/- per table. Both products are processed on three machines : Assembling, Finishing and Polishing. The time required for each product in hours and availability of each machine is given by following table: | Machine
    ProductChair (x)Table (y)Available time (hours)
    Assembling3336
    Finishing5250
    Polishing2660
    Formulate the above problem as L.P.P. Solve it graphically to get maximum profit.
  22. Misc II Q.8
    A company manufactures bicycles and tricycles, each of which must be processed through two machines A and B. Maximum availability of Machine A and B is respectively 120 and 180 hours. Manufacturing a bicycle requires 6 hours on Machine A and 3 hours on Machine B. Manufacturing a tricycles requires 4 hours on Machine A and 10 hours on Machine B. If profits are Rs.180/- for a bicycle and Rs.220/- for a tricycle. Determine the number of bicycles and tricycles that should be manufactured in order to maximize the profit.
  23. Misc II Q.9
    A factory produced two types of chemicals A and B. The following table gives the units of ingredients P and Q (per kg) of chemicals A and B as well as minimum requirements of P and Q and also cost per kg. chemicals A and B : | Ingredients per kg.
    Chemicals in unitsA (x)B (y)Minimum requirements in units
    P1280
    Q3175
    Cost (in Rs.)46--
    Find the number of units of chemicals A and B should be produced so as to minimize the cost.
  24. Misc II Q.10
    A company produces mixers and food processors. Profit on selling one mixer and one food processor is Rs. 2,000/- and Rs. 3,000/- respectively. Both the products are processed through three Machines A, B, C. The time required in hours by each product and total time available in hours per week on each machine are as follows : | Machine
    ProductMixer (per unit)Food Processor (per unit)Available time
    A3336
    B5250
    C2660
    How many mixers and food processors should be produced to maximize the profit?
  25. Misc II Q.11
    A chemical company produces a chemical containing three basic elements A, B, C so that it has at least 16 liters of A, 24 liters of B and 18 liters of C. This chemical is made by mixing two compounds I and II. Each unit of compound I has 4 liters of A, 12 liters of B, 2 liters of C. Each unit of compound II has 2 liters of A, 2 liters of B and 6 liters of C. The cost per unit of compound I is Rs.800/- and that of compound II is Rs.640/-. Formulate the problem as L.P.P. and solve it to minimize the cost.
  26. Misc II Q.12
    A person makes two types of gift items A and B requires the services of a cutter and a finisher. Gift item A requires 4 hours of cutter's time and 2 hours of finisher's time. B requires 2 hours of cutter's time and 4 hours of finisher's time. The cutter and finisher have 208 hours and 152 hours available times respectively every month. The profit of one gift item of type A is Rs.75/- and on gift item B is Rs.125/-. Assuming that the person can sell all the gift items produced, determine how many gift items of each type should he make every month to obtain the best returns?
  27. Misc II Q.13
    A firm manufactures two products A and B on which profit earned per unit Rs.3/- and Rs.4/- respectively. Each product is processed on two machines M1M_1 and M2M_2. The product A requires one minute of processing time on M1M_1 and two minute of processing time on M2M_2, B requires one minute of processing time on M1M_1 and one minute of processing time on M2M_2. Machine M1M_1 is available for use for 450 minutes while M2M_2 is available for 600 minutes during any working day. Find the number of units of product A and B to be manufactured to get the maximum profit.
  28. Misc II Q.14
    A firm manufacturing two types of electrical items A and B, can make a profit of Rs.20/- per unit of A and Rs.30/- per unit of B. Both A and B make use of two essential components a motor and a transformer. Each unit of A requires 3 motors and 2 transformers and each units of B requires 2 motors and 4 transformers. The total supply of components per month is restricted to 210 motors and 300 transformers. How many units of A and B should the manufacture per month to maximize profit? How much is the maximum profit?