Class 12 Mathematics Chapter 28 Linear Programming

This quiz on ICSE Class 12 Mathematics Chapter 28 – Linear Programming is designed to assess students’ understanding of fundamental concepts, methods, and applications of linear programming in solving real-life optimization problems. It covers key topics such as formulation of linear programming problems, graphical method of solution for two-variable problems, identification of feasible and infeasible regions, and determination of optimal solutions for maximizing or minimizing linear objective functions under given constraints. Students will also apply mathematical reasoning to interpret graphical results and analyze conditions for bounded and unbounded solutions. This quiz aims to strengthen problem-solving and analytical skills essential for higher-level mathematics and practical decision-making scenarios.

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Category: A method for optimizing (maximizing/minimizing) a linear function subject to linear constraints.

1. (A) The optimal solution of a linear programming problem always occurs at a corner point of the feasible region.
(R) The objective function in a linear programming problem is linear and the feasible region is convex.

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.

b) Both Assertion and Reason are true, but Reason is NOT the correct explanation of Assertion.

c) Assertion is true, but Reason is false.

d) Assertion is false, but Reason is true.

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Category: Identify common overlapping region (solution region).

2. (A) The solution region of the system $2x + y \leq 6$ and $x - y \geq 1$ is bounded by the intersection of two half-planes.
(R) The common solution set of a system of linear inequalities is always an unbounded region.

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.

b) Both Assertion and Reason are true, but Reason is NOT the correct explanation of Assertion.

c) Assertion is true, but Reason is false.

d) Assertion is false, but Reason is true.

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Category: Convex region.

3. Consider the feasible region defined by the constraints $3x + y \leq 12$, $x + 2y \leq 10$, $x \geq 0$, $y \geq 0$. Which of the following statements about this region is true?

a) The feasible region is non-convex.

b) The feasible region is unbounded.

c) The feasible region is convex but not closed.

d) The feasible region is convex and closed.

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Category: Infeasible solution.

4. Consider the LPP: Minimize $Z = x + y$ subject to $3x + 4y \geq 12$, $2x + y \geq 4$, $x \geq 0$, $y \geq 0$. Which of the following solutions is infeasible?

a) $x = 1$, $y = 1$

b) $x = 2$, $y = 2$

c) $x = 0$, $y = 3$

d) $x = 4$, $y = 0$

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Category: Transportation Problems – Minimize cost of shipment.

5. For a transportation problem with two factories (M and N) and three warehouses (U, V, W), the demand constraints must ensure that the total shipments to each warehouse meet or exceed its demand. If the demand at U is 150 units, V is 200 units, and W is 250 units, which of the following correctly represents the demand constraint for warehouse V?

a) $x_{MU} + x_{NU} \geq 200$

b) $x_{MV} + x_{NV} \geq 200$

c) $x_{MW} + x_{NW} \geq 200$

d) $x_{MV} + x_{NW} \geq 200$

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Category: CORNER POINT METHOD.

6. Consider a linear programming problem with the following constraints:
$x + y \leq 6$
$2x + y \leq 8$
$x \geq 0, y \geq 0$
The objective is to maximize $Z = 3x + 5y$. What is the optimal value of $Z$?

a) 26

b) 30

c) 12

d) 0

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Category: LINEAR PROGRAMMING

7. A nutritionist wants to create a diet plan with two types of food, A and B, that meets the minimum daily requirements for vitamins X and Y. Each unit of food A contains 5 units of vitamin X and 3 units of vitamin Y, while each unit of food B contains 4 units of vitamin X and 6 units of vitamin Y. The minimum daily requirement is at least 200 units of vitamin X and 180 units of vitamin Y. If food A costs \$20 per unit and food B costs \$30 per unit, how many units of each food should be used to minimize the cost?

a) 15 units of A and 25 units of B

b) 20 units of A and 20 units of B

c) 10 units of A and 35 units of B

d) 25 units of A and 15 units of B

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Category: Theorem

8. Maximize $P = 10x + 15y$ under the constraints $5x + 4y \leq 80$, $3x + 5y \leq 75$, $x \geq 0$, $y \geq 0$. Which of the following is the maximum value of $P$?

a) 225

b) 160

c) 232.69

d) 180

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Category: Optimal feasible solution.

9. Consider the LPP:
Maximize $Z = 3x + 5y$
subject to constraints:
$x + y \leq 6$,
$2x + y \leq 8$,
$x \geq 0$, $y \geq 0$.
Which of the following points is NOT a corner point of the feasible region?

a) $(0,6)$

b) $(2,4)$

c) $(3,2)$

d) $(4,0)$

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Category: GRAPHICAL SOLUTION OF LINEAR INEQUATIONS

10. A company produces two products with constraints $3x + 2y \leq 18$, $x \leq 4$, $y \leq 6$, and $x, y \geq 0$. Which corner point maximizes the profit function $P = 5x + 4y$?

a) (0,6)

b) (4,0)

c) (4,3)

d) (2,6)

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Category: Apply non-negativity restrictions.

11. In an LP problem with constraints $3x + 2y \leq 12$, $x - y \geq 1$, $x \geq 0$, $y \geq 0$, which point lies outside the feasible region?

a) (2,1)

b) (1,0)

c) (3,1.5)

d) (0.5,-0.6)

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Category: Infeasible solution.

12. Which of the following is an example of an infeasible solution for the LPP: Maximize $Z = 5x + 3y$ subject to $x + y \leq 4$, $x \geq 0$, $y \geq 0$?

a) $x = 1, y = 3$

b) $x = 2, y = 2$

c) $x = -1, y = 2$

d) $x = 0, y = 4$

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Category: Optimal feasible solution.

13. For the LPP:
Minimize $Z = 4x + 3y$
subject to constraints:
$5x + 2y \geq 15$,
$3x + 4y \leq 24$,
$x \geq 0$, $y \geq 0$,
what is the optimal value of Z?

a) 12

b) 15.5

c) 18

d) 20

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Category: Represent graphically and find feasible region.

14. What do the constraints $x \geq 0$ and $y \geq 0$ represent in a linear programming problem?

a) The minimum limit for decision variables.

b) The non-negativity conditions for decision variables.

c) The equality constraints for decision variables.

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Category: Apply non-negativity restrictions.

15. (A) In a linear programming problem, the non-negativity constraints ensure that the solution is feasible and realistic.

(R) Decision variables in real-world problems often represent quantities like production units or resource allocations, which cannot be negative.

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.

b) Both Assertion and Reason are true, but Reason is NOT the correct explanation of Assertion.

c) Assertion is true, but Reason is false.

d) Assertion is false, but Reason is true.

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Category: Non-negativity restriction.

16. What happens if non-negativity restrictions are violated in an LPP?

a) The solution becomes optimal

b) The feasible region expands

c) The solution becomes infeasible

d) No impact on the solution

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Category: LINAR PROGRAMMING-DEFINITION

17. What is a feasible region in linear programming?

a) Region with maximum value of the objective function

b) Common area satisfying all constraints

c) Area where no solutions exist

d) Region outside the constraints

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Category: SOME DEFINITIONS AND CONCEPTS RELATED TO LPP

18. For the LPP: Minimize $Z = 5x + 7y$ subject to $x + y \leq 8$, $2x + 3y \geq 18$, $x \geq 0$, $y \geq 0$, which of the following is an optimal feasible solution?

a) (0, 6)

b) (9, 0)

c) (6, 2)

d) (8, 0)

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Category: AREAS OF APPLICATIONS OF LINEAR PROGRAMMING

19. A logistics firm ships goods from plants P$_1$ and P$_2$ to markets M$_1$, M$_2$, and M$_3$. The supply capacities are 100 tons at P$_1\$ and 120 tons at P$_2\$. The demands at M$_1\$, M$_2\$, and M$_3\$ are 70, 80, and 50 tons respectively. The transportation costs per ton are \$5 (P$_1$-M$_1\$), \$4 (P$_1$-M$_2\$), \$3 (P$_1$-M$_3\$), \$7 (P$_2$-M$_1\$), \$6 (P$_2$-M$_2\$), and \$9 (P$_2$-M$_3\$). Additionally, blending requirements mandate that at least 20\% of M$_1\$'s demand must come from P$_2$. What is the minimum transportation cost while satisfying all constraints?

a) \$920

b) \$940

c) \$948

d) \$958

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Category: FORMULATION OF A LINEAR PROGRAMMING PROBLEM

20. (A) In the diet problem formulation, the objective function represents the total cost of different foods.
(R) The decision variables in a diet problem represent the quantities of different constituents/nutrients to be consumed.

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.

b) Both Assertion and Reason are true, but Reason is NOT the correct explanation of Assertion.

c) Assertion is true, but Reason is false.

d) Assertion is false, but Reason is true.

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Category: Linear Inequations in Two Variables

21. (A) The feasible region for the system of inequalities $x + y \geq 4$, $2x - y \leq 6$, and $x \geq 0$ must always be bounded.
(R) A feasible region formed by a system of linear inequalities can only be unbounded if at least one of the constraints is non-strict ($\geq$ or $\leq$) with an open boundary.

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.

b) Both Assertion and Reason are true, but Reason is NOT the correct explanation of Assertion.

c) Assertion is true, but Reason is false.

d) Assertion is false, but Reason is true.

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Category: Apply non-negativity restrictions.

22. Which of the following correctly represents the non-negativity constraint for decision variables $x_1$ and $x_2$ in an LPP?

a) $x_1 + x_2 \geq 0$

b) $x_1 \leq 0$ and $x_2 \leq 0$

c) $x_1 \geq 0$ and $x_2 \geq 0$

d) $x_1 = 0$ and $x_2 = 0$

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Category: Developed by Kantorovich and George Dantzig.

23. For the LPP:
Maximize $Z = 50x + 40y$
subject to $5x + 6y \leq 38$, $3x + 2y \leq 18$, $x \geq 0$, $y \geq 0$, which corner point yields the maximum value of $Z$?

a) $(0, 6.3)$

b) $(6, 0)$

c) $(4, 3)$

d) $(0, 0)$

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Category: Non-negativity restriction.

24. Consider a linear programming problem with the objective to maximize $Z = 4x + 6y$ subject to $x + y \leq 5$, $2x + y \leq 8$, and non-negativity restrictions $x \geq 0$, $y \geq 0$. Which of the following points lies in the feasible region but violates an implied condition for optimality due to non-negativity?

a) (0, 5)

b) (4, 0)

c) (2, 3)

d) (3, -1)

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Category: Applications: maximize profit, minimize cost, optimal resource utilization.

25. What is the objective function in a Linear Programming Problem (LPP)?

a) A system of equations representing constraints

b) A linear equation representing the quantity to be maximized or minimized

c) The variable whose values are restricted by inequalities

d) The region defined by all constraints

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Category: ADVANTAGES OF LINEAR PROGRAMMING PROBLEMS

26. (A) Linear programming ensures optimal resource allocation by maximizing profit or minimizing cost under given constraints.
(R) Linear programming uses an objective function and constraints to mathematically model and solve optimization problems.

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.

b) Both Assertion and Reason are true, but Reason is NOT the correct explanation of Assertion.

c) Assertion is true, but Reason is false.

d) Assertion is false, but Reason is true.

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Category: Graph each inequality on the same coordinate axes.

27. (A) The solution set of the system of inequalities $y \leq x + 3$, $y \geq 2x - 1$, and $x \geq 0$ is represented by the intersection of the shaded regions of each inequality on the same coordinate axes.
(R) The intersection of the half-planes corresponding to each inequality gives the common feasible region satisfying all the constraints.

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.

b) Both Assertion and Reason are true, but Reason is NOT the correct explanation of Assertion.

c) Assertion is true, but Reason is false.

d) Assertion is false, but Reason is true.

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Category: Applications: maximize profit, minimize cost, optimal resource utilization.

28. Why are non-negativity restrictions important in LPP?

a) To ensure the objective function is always positive

b) To restrict variables from taking negative values

c) To simplify the mathematical model

d) To increase the number of feasible solutions

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Category: Developed by Kantorovich and George Dantzig.

29. Which of the following represents the general form of the objective function in a linear programming problem?

a) $Z = x^2 + y^2$

b) $Z = c_1x_1 + c_2x_2 + \dots + c_nx_n$

c) $Z = \sin(x) + \cos(y)$

d) $Z = e^{x} + \log(y)$

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Category: Graph each inequality on the same coordinate axes.

30. For the system of inequalities $y \leq x + 2$, $y \geq -x + 1$, $y \leq 3$, which of the following points lies within the feasible region?

a) $(4, 2)$

b) $(0, 4)$

c) $(1, 1)$

d) $(-1, -1)$

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Category: CORNER POINT METHOD.

31. Given the feasibility region bounded by the constraints $x \geq 0$, $y \geq 0$, $x + y \leq 4$, and $2x + y \leq 6$, what are the corner points of this region?

a) $(0, 0), (4, 0), (2, 2), (0, 4)$

b) $(0, 0), (6, 0), (2, 2), (0, 4)$

c) $(0, 0), (4, 0), (2, 2), (0, 6)$

d) $(0, 0), (6, 0), (3, 0), (0, 6)$

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Category: Diet Problems – Minimize cost subject to nutritional constraints.

32. A nutritionist needs to plan a diet using foods X and Y. Food X costs \$5 per unit and provides 2 units of carbohydrates and 4 units of fats. Food Y costs \$3 per unit and provides 3 units of carbohydrates and 1 unit of fats. The diet must have at least 12 units of carbohydrates and 6 units of fats. What is the minimum cost solution?

a) \$12

b) \$15

c) \$16

d) \$18

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Category: Graph each inequality on the same coordinate axes.

33. Which of the following inequalities corresponds to the region bounded by $2x + y \geq 6$ and $x \geq 0$, $y \geq 0$?

a) The area below $2x + y = 6$ in the first quadrant

b) The area above $2x + y = 6$ in the first quadrant

c) The area to the left of $x = 0$ and above $y = 0$

d) The entire xy-plane except the first quadrant

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Category: FORMULATION OF A LINEAR PROGRAMMING PROBLEM

34. Which of the following represents the non-negativity constraints in a linear programming problem involving decision variables $x$ and $y$?

a) $x > 0, y > 0$

b) $x \leq 0, y \leq 0$

c) $x \geq 0, y \geq 0$

d) $x < 0, y < 0$

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Category: Non-negativity restriction.

35. (A) The non-negativity constraints in a linear programming problem ensure that all variables are positive or zero.

(R) Negative values for decision variables would make the solution infeasible in real-world applications.

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.

b) Both Assertion and Reason are true, but Reason is NOT the correct explanation of Assertion.

c) Assertion is true, but Reason is false.

d) Assertion is false, but Reason is true.

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Category: Graphical Method of Solution

36. Given the constraints $y \geq 0$, $x \geq 0$, and $3x + 2y \leq 12$, which of the following points lies in the feasible region?

a) $(3, 2)$

b) $(1, 4)$

c) $(0, 7)$

d) $(4, 0)$

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Category: Infeasible solution.

37. (A) The solution $x = -2, y = 7$ is an infeasible solution for the LPP with constraints $3x + 2y \leq 30$, $x + 2y \leq 22$, $x \geq 0$, $y \geq 0$.
(R) An infeasible solution does not satisfy all the constraints and non-negativity conditions of an LPP.

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.

b) Both Assertion and Reason are true, but Reason is NOT the correct explanation of Assertion.

c) Assertion is true, but Reason is false.

d) Assertion is false, but Reason is true.

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Category: Areas of Application

38. A nutritionist is designing a diet with two food items, A and B, to meet minimum daily requirements of nutrients P, Q, and R. The cost and nutrient content per unit of each food item are given below:

Food A: P = 2, Q = 3, R = 4, Cost = \$5

Food B: P = 4, Q = 2, R = 1, Cost = \$3

The minimum daily requirements are: P $\geq$ 12, Q $\geq$ 12, R $\geq$ 8. What is the cheapest combination of food items that meets all the requirements?

a) 0 units of A and 6 units of B

b) 3 units of A and 1.5 units of B

c) 4 units of A and 0 units of B

d) 2 units of A and 3 units of B

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Category: GENERAL LINAR PROGRAMMING PROBLEM

39. A nutritionist is designing a diet plan with two types of food, $F_1$ and $F_2$, to meet minimum daily requirements of vitamins A, B, and C. Food $F_1$ costs \$3 per unit and provides 4 units of vitamin A, 1 unit of vitamin B, and 2 units of vitamin C per unit. Food $F_2$ costs \$5 per unit and provides 2 units of vitamin A, 3 units of vitamin B, and 4 units of vitamin C per unit. The daily requirement is at least 20 units of vitamin A, 15 units of vitamin B, and 16 units of vitamin C. Formulate the LPP to minimize the cost.

a) Minimize $Z = 3x_1 + 5x_2$ subject to $4x_1 + 2x_2 \leq 20$, $x_1 + 3x_2 \leq 15$, $2x_1 + 4x_2 \leq 16$, $x_1 \geq 0$, $x_2 \geq 0$

b) Minimize $Z = 3x_1 + 5x_2$ subject to $4x_1 + 2x_2 \geq 20$, $x_1 + 3x_2 \geq 15$, $2x_1 + 4x_2 \geq 16$, $x_1 \geq 0$, $x_2 \geq 0$

c) Maximize $Z = 3x_1 + 5x_2$ subject to $4x_1 + 2x_2 \geq 20$, $x_1 + 3x_2 \geq 15$, $2x_1 + 4x_2 \geq 16$, $x_1 \geq 0$, $x_2 \geq 0$

d) Minimize $Z = 3x_1 + 5x_2$ subject to $4x_1 + 2x_2 \geq 20$, $x_1 + 3x_2 \geq 15$, $2x_1 + 4x_2 \geq 16$, $x_1 \leq 0$, $x_2 \leq 0$

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Category: The common region represents the solution set (infinite ordered pairs).

40. A system of inequalities is given by $2x + y \geq 4$, $x - y \leq 1$, and $x \geq 0$. Which of the following points lies in the common solution region?

a) $(1, 3)$

b) $(2, 1)$

c) $(0, 4)$

d) $(-1, 6)$

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Category: A method for optimizing (maximizing/minimizing) a linear function subject to linear constraints.

41. (A) The optimal solution of a linear programming problem always lies at one of the corner points of the feasible region.
(R) The feasible region of a linear programming problem is a convex polygon formed by the intersection of constraints.

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.

b) Both Assertion and Reason are true, but Reason is NOT the correct explanation of Assertion.

c) Assertion is true, but Reason is false.

d) Assertion is false, but Reason is true.

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Category: Identify decision variables (e.g., x, y).

42. A farmer wants to allocate land to grow wheat and barley. Each acre of wheat yields a profit of \$200, and each acre of barley yields \$300. The farmer has 50 acres available and cannot plant more than 40 acres of wheat or 30 acres of barley. What is the objective function for maximizing profit?

a) $Z = 50x + 40y$

b) $Z = 300x + 200y$

c) $Z = 200x + 300y$

d) $Z = x + y$

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Category: GRAPHICAL SOLUTION OF LINEAR INEQUATIONS

43. For the system $x + y \leq 5$, $x \geq 1$, $y \geq 0$, which point lies in the feasible region?

a) (0,0)

b) (6,-1)

c) (2,3)

d) (1,6)

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Category: Linear Programming Problem (LPP) Example

44. A candidate is preparing for an exam where they need to solve problems worth 5, 4, and 6 points respectively. The total time available is 3.5 hours (210 minutes), and they can attempt up to 100 problems. The objective function to maximize the total points is:

a) $Z = 3x_1 + 2x_2 + 4x_3$

b) $Z = x_1 + x_2 + x_3$

c) $Z = 5x_1 + 4x_2 + 6x_3$

d) $Z = 2x_1 + 3x_2 + 5x_3$

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Category: Define objective function (profit, cost, etc.).

45. A company produces two products, P and Q, with profit per unit of \$20 and \$30 respectively. The total production must not exceed 500 units due to resource constraints. If $x$ and $y$ represent the number of units of P and Q produced, which of the following correctly represents the objective function for maximizing profit?

a) $Z = 20x - 30y$

b) $Z = 30x + 20y$

c) $Z = 20x + 30y$

d) $Z = 30x - 20y$

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Category: Linear Programming Problem (LPP) Example

46. Which of the following represents a valid non-negativity constraint for the variables in an LPP?

a) $x_1 \leq 0$, $x_2 \leq 0$, $x_3 \leq 0$

b) $x_1 \geq 0$, $x_2 \geq 0$, $x_3 \geq 0$

c) $x_1 + x_2 + x_3 \geq 0$

d) $x_1x_2x_3 \geq 0$

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Category: Advertising Problems – Allocate funds for maximum exposure.

47. (A) Linear programming is useful for allocating a fixed advertising budget to maximize total effective audience.
(R) The objective function in such problems is to maximize $Z = c_1x_1 + c_2x_2 + \ldots + c_nx_n$, where $c_i$ represents the effectiveness of each media and $x_i$ is the amount allocated.

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.

b) Both Assertion and Reason are true, but Reason is NOT the correct explanation of Assertion.

c) Assertion is true, but Reason is false.

d) Assertion is false, but Reason is true.

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Category: Write down constraints in terms of inequalities/equations.

48. A furniture manufacturer produces chairs and tables. The company has 500 labor hours and 400 units of wood available per day. Each chair requires 3 labor hours and 2 units of wood, while each table requires 5 labor hours and 1 unit of wood. Which of the following represents a correct constraint for labor hours?

a) $3x + 5y \geq 500$

b) $5x + 3y \leq 500$

c) $3x + 5y \leq 500$

d) $2x + y \leq 500$

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Category: Corner points (vertices) can be found by solving boundary equations in pairs.

49. A feasible region is defined by the inequations $x + y \geq 4$, $2x - y \leq 6$, and $x \geq 1$. What is one of the corner points of this region?

a) (\$\frac{10}{3}, \frac{2}{3}\$)

b) (\$5, -1\$)

c) (\$1, -4\$)

d) (\$2, 2\$)

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Category: Investment Problems – Maximize return on capital allocation.

50. A company has \$120,000 to invest in two securities with returns of 4\% and 7\% respectively. The investment in the first security must be at least \$30,000 but not more than \$70,000. What is the maximum possible return on the total investment?

a) \$6,300

b) \$7,500

c) \$5,800

d) \$8,100

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Category: THE GRAPHICAL METHOD OF SOLUTION OF LINEAR PROGRAMMING PROBLEMS

51. Given the constraints for a linear programming problem:
$4x + y \leq 20$,
$2x + 3y \leq 24$,
$x \geq 0$, $y \geq 0$.
Which of the following points lies outside the feasible region?

a) $(2,4)$

b) $(3,6)$

c) $(4,5)$

d) $(1,3)$

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Category: Diet Problems – Minimize cost subject to nutritional constraints.

52. A health planner uses foods P and Q to meet dietary requirements. Food P costs \$6 per unit and provides 5 units of iron and 2 units of calcium. Food Q costs \$4 per unit and provides 2 units of iron and 3 units of calcium. The diet requires at least 20 units of iron and 12 units of calcium. Find the minimum cost solution.

a) \$24

b) \$26

c) \$28

d) \$30

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Category: Non-negativity restriction.

53. In a transportation problem, the supply at source A is 50 units and at source B is 60 units. The demand at destination C is 40 units and at D is 70 units. Let $x_{AC}$ and $x_{AD}$ be the quantities transported from A to C and D respectively, while $x_{BC}$ and $x_{BD}$ are from B to C and D. If the non-negativity restrictions are applied, which of the following combinations is invalid?

a) (20, 30, 20, 40)

b) (0, 50, 40, 20)

c) (25, 25, 15, 45)

d) (10, 20, -10, 50)

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Category: Check feasible side using a test point (often origin).

54. (A) For the line $x + 2y = 8$, the origin $(0, 0)$ satisfies $x + 2y \leq 8$ because substituting $(0,0)$ gives $0 \leq 8$.
(R) The feasible region and the origin lie on the same side of the line $x + 2y = 8$.

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.

b) Both Assertion and Reason are true, but Reason is NOT the correct explanation of Assertion.

c) Assertion is true, but Reason is false.

d) Assertion is false, but Reason is true.

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Category: Identify common overlapping region (solution region).

55. Given the system of inequalities: $2x + 3y \geq 6$, $x - y \leq 2$, and $y \geq 0$, which point lies within the common solution region?

a) $(1, -1)$

b) $(2, 1)$

c) $(0, 0)$

d) $(3, -1)$

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Category: The common region represents the solution set (infinite ordered pairs).

56. Which of the following ordered pairs lies in the solution set of the system of inequalities: $x + y \leq 4$ and $y \geq x - 1$?

a) $(0, 5)$

b) $(1, 2)$

c) $(3, 3)$

d) $(4, 0)$

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Category: Theorem

57. A company produces two products $X$ and $Y$. The profit per unit is \$5 for $X$ and \$3 for $Y$. The constraints are:
$2x + 3y \leq 120$
$4x + y \leq 80$
with $x \geq 0$, $y \geq 0$. What is the maximum profit?

a) \$120

b) \$100

c) \$156

d) \$200

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Category: A method for optimizing (maximizing/minimizing) a linear function subject to linear constraints.

58. A company produces two products A and B. The profit per unit for A is \$5 and for B is \$8. The production requires 2 hours of labor per unit of A and 3 hours per unit of B. The total labor hours available are 120. Additionally, product A requires 1 kg of raw material and product B requires 2 kg, with a total of 60 kg available. What is the maximum profit the company can achieve?

a) 280

b) 312

c) 300

d) 240

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Category: THE GRAPHICAL METHOD OF SOLUTION OF LINEAR PROGRAMMING PROBLEMS

59. In the graphical method of solving an LPP, how is the feasible region determined?

a) By solving only the objective function for different values of variables.

b) By considering only the non-negativity constraints.

c) By plotting each constraint as an equation and finding the common region satisfying all constraints.

d) By ignoring inequality constraints and focusing only on equality constraints.

60 / 576

Category: System of Inequations

60. What is the feasibility region in linear programming problems?

a) The area outside all constraint boundaries

b) Only the lines representing the constraints

c) Intersection of all constraint inequalities

d) Any arbitrary region satisfying some constraints

61 / 576

Category: Graphical solution of linear inequalities forms the basis of linear programming.

61. For the inequality $y > x + 2$, what is the correct description of the solution region?

a) All points below the line $y = x + 2$

b) All points above the line $y = x + 2$

c) Only the points on the line $y = x + 2$

d) None of the above

62 / 576

Category: Blending Problems – Optimize combination of raw materials.

62. Why are non-negativity restrictions included in blending problems?

a) To ensure all variables are positive for mathematical simplicity

b) To guarantee the solution meets quality requirements

c) Because raw material quantities cannot be negative

d) To maximize the objective function

63 / 576

Category: Blending Problems – Optimize combination of raw materials.

63. In a fuel blending problem, a company needs at least 1000 liters of fuel with a minimum octane rating of 90. The octane ratings of raw fuels X and Y are 85 and 95, respectively. If $x$ and $y$ represent the quantities of X and Y used, which of the following represents the octane constraint?

a) $85x + 95y \geq 90000$

b) $85x + 95y \leq 90000$

c) $x + y \geq 1000$

d) $85x + 95y = 90000$

64 / 576

Category: Constraints.

64. Which of the following constraints is essential to ensure that the solution to an LPP makes practical sense?

a) $x + y \leq 100$

b) $2x - y \geq 50$

c) $x \geq 0$

d) $3x + 4y = 120$

65 / 576

Category: LINEAR PROGRAMMING

65. What do non-negativity restrictions signify in an LPP?

a) Constraints where variables must be positive

b) Conditions ensuring all decision variables are non-negative

c) The requirement to maximize the objective function

d) The total number of constraints in the problem

66 / 576

Category: Steps in Formulating a Linear Programming Model

66. (A) In a linear programming problem, the decision variables must always have non-negative values.
(R) Negative values of decision variables have no valid interpretation in physical problems.

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.

b) Both Assertion and Reason are true, but Reason is NOT the correct explanation of Assertion.

c) Assertion is true, but Reason is false.

d) Assertion is false, but Reason is true.

67 / 576

Category: Linear Programming Problem (LPP) Example

67. Which of the following constraints represents the total time limit for solving numerical problems (first two sets)?

a) $3x_1 + 2x_2 \leq 210$

b) $3x_1 + 4x_3 \leq 150$

c) $3x_1 + 2x_2 \leq 150$

d) $x_1 + x_2 + x_3 \leq 100$

68 / 576

Category: GRAPHICAL SOLUTION OF LINEAR INEQUATIONS

68. What is the equivalent inequality of $4x - 3y < 12$ when solved for $y$?

a) $y > -\frac{4}{3}x + 4$

b) $y < \frac{4}{3}x - 4$

c) $y > \frac{4}{3}x - 4$

d) $y < -\frac{4}{3}x + 4$

69 / 576

Category: A method for optimizing (maximizing/minimizing) a linear function subject to linear constraints.

69. Which of the following best describes the feasible region in linear programming?

a) The area where the objective function is maximized

b) The set of points satisfying all constraints and non-negativity conditions

c) Only the corner points of the solution space

d) The region where at least one constraint is violated

70 / 576

Category: Identify decision variables (e.g., x, y).

70. (A) In a linear programming problem, the objective function $Z = 3x + 4y$ implies that the profit per unit of product $x$ is \$3 and per unit of product $y$ is \$4.
(R) The decision variables $x$ and $y$ represent the quantities of products to be produced, and their coefficients in the objective function directly indicate their respective profit contributions.

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.

b) Both Assertion and Reason are true, but Reason is NOT the correct explanation of Assertion.

c) Assertion is true, but Reason is false.

d) Assertion is false, but Reason is true.

71 / 576

Category: Graphical solution of linear inequalities forms the basis of linear programming.

71. Which of the following systems of inequalities defines a feasible region where $x \geq 0$, $y \geq 0$, and satisfies both $x + y \leq 5$ and $3x - 2y \geq 6$?

a) $x \geq 0$, $y \geq 0$, $x + y \leq 5$, $3x - 2y \leq 6$

b) $x \geq 0$, $y \geq 0$, $x + y \geq 5$, $3x - 2y \geq 6$

c) $x \geq 0$, $y \geq 0$, $x + y \leq 5$, $3x - 2y \geq 6$

d) $x \leq 0$, $y \leq 0$, $x + y \leq 5$, $3x - 2y \geq 6$

72 / 576

Category: Advertising Problems – Allocate funds for maximum exposure.

72. What is the objective function in an advertising media selection problem where $x_1$ and $x_2$ represent the number of TV and radio ads respectively, with audience reach per ad being 3000 and 1500?

a) Maximize $Z = 3000x_1 - 1500x_2$

b) Minimize $Z = 3000x_1 + 1500x_2$

c) Maximize $Z = 1500x_1 + 3000x_2$

d) Maximize $Z = 3000x_1 + 1500x_2$

73 / 576

Category: Optimal feasible solution.

73. What is an optimal feasible solution in the context of Linear Programming Problems (LPP)?

a) A solution that satisfies some constraints

b) A solution that gives the minimum or maximum value of the objective function without considering constraints

c) A solution that satisfies all constraints and optimizes the objective function

d) A solution that only considers non-negativity constraints

74 / 576

Category: LINAR PROGRAMMING-DEFINITION

74. Which of the following correctly defines an objective function in Linear Programming?

a) A linear function representing constraints

b) A linear inequality describing non-negativity restrictions

c) A linear function to be optimized (maximized or minimized)

d) A convex set defining feasible solutions

75 / 576

Category: Identify common overlapping region (solution region).

75. Which inequation must be added to $x \geq 0$ and $y \geq 0$ so that the solution region is bounded by the first quadrant and the line $2x + y = 4$?

a) $2x + y \geq 4$

b) $2x - y \leq 4$

c) $2x + y \leq 4$

d) $x + 2y \leq 4$

76 / 576

Category: System of Inequations

76. (A) The feasible region determined by the inequalities $x + y \leq 5$, $3x + 4y \geq 12$, $x \geq 0$, and $y \geq 0$ is bounded.
(R) A feasible region is always bounded if all the constraints are linear inequalities.

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.

b) Both Assertion and Reason are true, but Reason is NOT the correct explanation of Assertion.

c) Assertion is true, but Reason is false.

d) Assertion is false, but Reason is true.

77 / 576

Category: Graphical solution of linear inequalities forms the basis of linear programming.

77. What is the feasibility region in a linear programming problem?

a) The set of points satisfying the objective function

b) The set of points satisfying any one of the constraints

c) The intersection of all constraint half-planes

d) The union of all constraint half-planes

78 / 576

Category: The common region represents the solution set (infinite ordered pairs).

78. The corner points of the common solution region for the system $$ 3x + y \geq 6$$, $$ x + 2y \geq 8$$, $$ x \geq 0$$, and $$ y \geq 0$$ are:

a) $(2, 0), (0, 6), (8, 0)$

b) $(0, 4), (2, 0), \left(\frac{4}{5}, \frac{18}{5}\right)$

c) $(1, 3), (4, 2), (0, 0)$

d) $(3, 0), (0, 8), (2, 2)$

79 / 576

Category: Constraints.

79. Which of the following is NOT a valid constraint sign in an LPP?

a) $<$

b) $=$

c) $>$

d) $\neq$

80 / 576

Category: GRAPHICAL SOLUTION OF LINEAR INEQUATIONS

80. (A) The solution set of the inequality $x + y \geq 1$ includes the line $x + y = 1$.
(R) For inequalities with $\geq$ or $\leq$, the associated line is included in the solution set.

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.

b) Both Assertion and Reason are true, but Reason is NOT the correct explanation of Assertion.

c) Assertion is true, but Reason is false.

d) Assertion is false, but Reason is true.

81 / 576

Category: FORMULATION OF A LINEAR PROGRAMMING PROBLEM

81. A farmer has 100 acres of land and can grow wheat or barley. The cost of cultivation per acre is \$40 for wheat and \$60 for barley. The farmer has a budget of \$4800. The net profit per acre is \$100 for wheat and \$120 for barley. Formulate the LPP to maximize the total profit.

a) Maximize $Z = 100x_1 + 120x_2$; Subject to: $x_1 + x_2 \leq 100$, $40x_1 + 60x_2 \leq 4800$, $x_1, x_2 \geq 0$

b) Maximize $Z = 120x_1 + 100x_2$; Subject to: $x_1 + x_2 \leq 100$, $40x_1 + 60x_2 \leq 4800$, $x_1, x_2 \geq 0$

c) Maximize $Z = 100x_1 + 120x_2$; Subject to: $x_1 + x_2 \leq 100$, $60x_1 + 40x_2 \leq 4800$, $x_1, x_2 \geq 0$

d) Maximize $Z = 120x_1 + 100x_2$; Subject to: $x_1 + x_2 \leq 100$, $60x_1 + 40x_2 \leq 4800$, $x_1, x_2 \geq 0$

82 / 576

Category: Feasible region : i region and Feasible solution.

82. For the LPP: Minimize $Z = 4x + 6y$ subject to $x + y \geq 5$, $2x + y \geq 8$, $x \geq 0$, $y \geq 0$, which of the following is the optimal solution?

a) $(0,5)$

b) $(3,2)$

c) $(4,1)$

d) $(2,4)$

83 / 576

Category: Check feasible side using a test point (often origin).

83. Given the feasible region bounded by $x + y \leq 6$, $2x - y \geq 0$, $x \geq 0$, $y \geq 0$, what is the maximum value of the objective function $Z = 4x + 3y$?

a) 20

b) 18

c) 24

d) 30

84 / 576

Category: Identify common overlapping region (solution region).

84. For the system of inequalities $3x + 4y \leq 12$, $x - y \leq 1$, and $x, y \geq 0$, what are the corner points of the solution region?

a) $(0,0)$, $(4,0)$, $(0,3)$

b) $(1,0)$, $(4,0)$, $(\frac{16}{7}, \frac{9}{7})$

c) $(0,1)$, $(3,0)$, $(\frac{9}{7}, \frac{16}{7})$

d) $(0,0)$, $(1,0)$, $(0,1)$

85 / 576

Category: SOLVED EXAMPLES

85. (A) The objective function in an LPP represents the quantity to be maximized or minimized.
(R) In Example 1, the objective function $Z = 5x_1 + 4x_2 + 6x_3$ represents the total points earned by the candidate.

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.

b) Both Assertion and Reason are true, but Reason is NOT the correct explanation of Assertion.

c) Assertion is true, but Reason is false.

d) Assertion is false, but Reason is true.

86 / 576

Category: SOLVED EXAMPLES

86. For the problem: Minimize $Z = x - 7y + 190$ under constraints $x + y \leq 8$, $x \leq 5$, $y \leq 5$, $x + y \geq 4$, $x, y \geq 0$, what is the minimum value of $Z$ obtained at which corner point?

a) 162 at $(0, 4)$

b) 155 at $(0, 5)$

c) 158 at $(3, 5)$

d) 174 at $(5, 3)$

87 / 576

Category: Objective function.

87. (A) The objective function in a linear programming problem is always a linear function of the decision variables.
(R) The objective function represents the quantity to be optimized, either maximized or minimized, subject to given constraints.

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.

b) Both Assertion and Reason are true, but Reason is NOT the correct explanation of Assertion.

c) Assertion is true, but Reason is false.

d) Assertion is false, but Reason is true.

88 / 576

Category: Applications: maximize profit, minimize cost, optimal resource utilization.

88. Which of the following best describes constraints in an LPP?

a) Conditions that must be satisfied for the problem to be solved

b) The final solution to the optimization problem

c) The range of possible values for the objective function

d) The set of feasible solutions

89 / 576

Category: Advertising Problems – Allocate funds for maximum exposure.

89. A company wants to allocate its \$50,000 advertising budget across TV and online ads. TV ads cost \$5,000 per slot with an effectiveness of 400 audience units per dollar, while online ads cost \$2,000 per campaign with 250 audience units per dollar. The company wants at least 4 TV ad slots and no more than 12 online campaigns. What is the optimal allocation that maximizes total audience exposure?

a) 4 TV slots and 12 online campaigns

b) 6 TV slots and 10 online campaigns

c) 5 TV slots and 12 online campaigns

d) 7 TV slots and 7 online campaigns

90 / 576

Category: Areas of Application

90. Which of the following is a limitation of linear programming?

a) It requires nonlinear equations for accurate results

b) It can incorporate qualitative factors like human behavior

c) It cannot handle problems involving quantitative constraints

d) It does not consider qualitative factors such as human relations

91 / 576

Category: Fundamental Theorem (Extreme Point Theorem)

91. According to the Extreme Point Theorem, where does the optimal solution of a linear programming problem occur if it exists?

a) At any point inside the feasible region

b) At any boundary point of the feasible region

c) At one of the corner points of the feasible region

d) Only at the origin

92 / 576

Category: AREAS OF APPLICATIONS OF LINEAR PROGRAMMING

92. Why is linear programming useful in transportation problems?

a) To increase transportation time

b) To maximize fuel consumption

c) To determine the least expensive transportation routes

d) To complicate logistics

93 / 576

Category: CORNER POINT METHOD.

93. Given the feasible region with corner points $(0, 0)$, $(4, 0)$, $(3, 2)$, and $(0, 3)$, what is the maximum value of the objective function $Z = 5x + 4y$?

a) At $(0, 0)$: $Z = 5(0) + 4(0) = 0$

b) At $(4, 0)$: $Z = 5(4) + 4(0) = 20$

c) At $(3, 2)$: $Z = 5(3) + 4(2) = 15 + 8 = 23$

d) At $(0, 3)$: $Z = 5(0) + 4(3) = 12$ \end{itemize} The maximum value of $Z$ is $23$ at the point $(3, 2)$.

e) 20

f) 23

g) 12

h) 0

94 / 576

Category: Steps in Formulating a Linear Programming Model

94. (A) Decision variables must be assigned symbols such as $x$ and $y$.
(R) Decision variables represent quantities whose values need to be determined.

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.

b) Both Assertion and Reason are true, but Reason is NOT the correct explanation of Assertion.

c) Assertion is true, but Reason is false.

d) Assertion is false, but Reason is true.

95 / 576

Category: Represent graphically and find feasible region.

95. A company produces two products, P and Q. The constraints are: $x + y \leq 6$, $x \geq 0$, $y \geq 0$, and $2x + y \leq 8$. Which of the following points lies in the feasible region?

a) $(2, 3)$

b) $(4, 2)$

c) $(0, 8)$

d) $(3, -1)$

96 / 576

Category: Graphical Method of Solution

96. What is the first step in solving a linear programming problem using the corner point method?

a) Solve the objective function for maximum value

b) Find the co-ordinates of each corner point

c) Plot the constraints and identify the feasible region

d) Evaluate the objective function at each corner point

97 / 576

Category: Represent graphically and find feasible region.

97. Maximize $Z = 3x + 4y$ subject to $x + y \leq 8$, $2x + y \leq 12$, $x \geq 0$, $y \geq 0$. At which corner point does $Z$ attain its maximum?

a) (0, 0)

b) (0, 8)

c) (6, 0)

d) (4, 4)

98 / 576

Category: Graphical Method of Solution

98. Consider the following constraints for a linear programming problem: $x \geq 0$, $y \geq 0$, $x + y \leq 4$, and $2x + y \leq 6$. What is the maximum value of the objective function $Z = 3x + 4y$?

a) 12

b) 14

c) 16

d) 18

99 / 576

Category: ADVANTAGES OF LINEAR PROGRAMMING PROBLEMS

99. (A) Linear programming ensures optimal allocation of limited resources like manpower and materials.
(R) Linear programming models can be adjusted to accommodate changes in constraints or objectives without restarting the analysis.

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.

b) Both Assertion and Reason are true, but Reason is NOT the correct explanation of Assertion.

c) Assertion is true, but Reason is false.

d) Assertion is false, but Reason is true.

100 / 576

Category: Apply non-negativity restrictions.

100. Consider the LP problem: Maximize $Z = 5x + 3y$ subject to $x + y \leq 6$, $2x + y \leq 8$, with only $y \geq 0$. What is the optimal value of Z if x can be negative?

a) 22

b) 18

c) 20

d) Unbounded

101 / 576

Category: GRAPHICAL SOLUTION OF LINEAR INEQUATIONS

101. Which of the following represents the feasible region for the system of inequalities $x + y \leq 4$, $2x - y \geq 2$, and $x \geq 0$, $y \geq 0$?

a) A triangle with vertices at (0,0), (1,0), and (0,4)

b) A quadrilateral with vertices at (0,0), (1,0), (2,2), and (0,4)

c) An unbounded region extending infinitely to the right

d) A single point at (2,2)

102 / 576

Category: A method for optimizing (maximizing/minimizing) a linear function subject to linear constraints.

102. A company manufactures two types of products, A and B. The profit per unit for product A is \$10 and for product B is \$15. The constraints are $2x + y \leq 100$, $x + y \leq 80$, $x \geq 0$, $y \geq 0$, where $x$ and $y$ represent the number of units of A and B, respectively. What is the optimal production mix to maximize profit?

a) 100 units of A and 0 units of B

b) 20 units of A and 60 units of B

c) 0 units of A and 80 units of B

d) 40 units of A and 40 units of B

103 / 576

Category: LINEAR PROGRAMMING

103. Which condition must hold true for the feasible region in linear programming?

a) It must be a straight line.

b) It must have no boundary points.

c) It must satisfy all constraints and non-negativity conditions.

d) It must include only integer solutions.

104 / 576

Category: LINAR PROGRAMMING-DEFINITION

104. In a Linear Programming Problem (LPP), what is a feasible region?

a) The set of all infeasible solutions

b) The region where all constraints are violated

c) The common region satisfying all constraints and non-negativity restrictions

d) The area outside the solution space

105 / 576

Category: A method for optimizing (maximizing/minimizing) a linear function subject to linear constraints.

105. Consider a feasibility region defined by the inequalities
$x + y \leq 10, \quad 2x + y \leq 15, \quad x \geq 0, \quad y \geq 0.$
Which of the following points is NOT a corner point of this region?

a) (0, 0)

b) (5, 5)

c) (6, 4)

d) (7.5, 0)

106 / 576

Category: Graphical solution of linear inequalities forms the basis of linear programming.

106. (A) The feasible region for the constraints $x + y \leq 5$, $x \geq 0$, $y \geq 0$ is a convex region.
(R) A region formed by the intersection of half-planes corresponding to linear inequalities is always convex.

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.

b) Both Assertion and Reason are true, but Reason is NOT the correct explanation of Assertion.

c) Assertion is true, but Reason is false.

d) Assertion is false, but Reason is true.

107 / 576

Category: Feasible region : i region and Feasible solution.

107. For the LPP with constraints $x + 2y \leq 8$, $x - y \leq 1$, $2x + y \geq 2$, $x \geq 0$, $y \geq 0$, at which corner point does the objective function $Z = 3x + 4y$ attain its maximum value?

a) $(1, 0)$

b) $\left(\frac{10}{3}, \frac{7}{3}\right)$

c) $(8, 0)$

d) $(0, 1)$

108 / 576

Category: Constraints.

108. (A) The feasible region of a linear programming problem is always a convex set.
(R) The intersection of any number of convex sets is also a convex set.

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.

b) Both Assertion and Reason are true, but Reason is NOT the correct explanation of Assertion.

c) Assertion is true, but Reason is false.

d) Assertion is false, but Reason is true.

109 / 576

Category: Understanding regions of inequality is essential before tackling LPP problems.

109. Which region represents the solution to the inequality $y \leq 4$ in the $x-y$ plane?

a) The region above the line $y = 4$

b) The region to the right of the line $y = 4$

c) The region below the line $y = 4$, including the line

d) The region to the left of the line $y = 4$

110 / 576

Category: Check feasible side using a test point (often origin).

110. Consider the following system of inequalities:
$3x + 4y \leq 12$
$5x - 2y \geq 10$
$x \geq 0, y \geq 0$.

Which of the following points lies in the feasible region?

a) $(2,1)$

b) $(1,1)$

c) $(3,0)$

d) $(0,3)$

111 / 576

Category: Convex region.

111. Which of the following best describes the feasible region in an LPP?

a) A region where the objective function is minimized or maximized.

b) The area outside the constraint boundaries.

c) The intersection of all constraint regions, forming a convex set.

d) The unbounded solution space of the problem.

112 / 576

Category: Fundamental Theorem (Extreme Point Theorem)

112. If a linear programming problem has an unbounded feasibility region, which of the following statements about its optimal solution is correct?

a) It always exists and occurs at a corner point.

b) It never exists because the region is unbounded.

c) It may or may not exist; if it exists, it occurs at a corner point.

d) It exists only when the objective function is minimized.

113 / 576

Category: Diet Problems – Minimize cost subject to nutritional constraints.

113. When can the graphical method be used to solve a linear programming problem?

a) When there are more than three variables

b) When the problem has non-linear constraints

c) When there are exactly two variables

d) When the objective function is quadratic

114 / 576

Category: Manufacturing Problems – Optimize production for profit.

114. In a linear programming problem, what is the feasible region defined by?

a) All points that satisfy the objective function

b) All points that lie outside the constraints

c) All points that satisfy all the constraints

d) Only the origin point $(0, 0)$

115 / 576

Category: LINAR PROGRAMMING-DEFINITION

115. Consider an LPP with the constraints $2x + 3y \leq 12$, $4x + y \leq 8$, $x \geq 0$, $y \geq 0$. Which of the following points lies in the feasible region?

a) (0, 4)

b) (2, 0)

c) (3, 1)

d) (-1, 5)

116 / 576

Category: LINEAR PROGRAMMING

116. A factory has constraints such that the production of product X requires 2 hours of labor and product Y requires 3 hours. The total labor available is 120 hours. Which of the following represents this constraint?

a) $2x + 3y \geq 120$

b) $3x + 2y \leq 120$

c) $2x + 3y \leq 120$

d) $3x + 2y \geq 120$

117 / 576

Category: Understanding regions of inequality is essential before tackling LPP problems.

117. Consider the system of linear inequalities: $2x + y \leq 10$, $x + 3y \leq 15$, $x \geq 0$, $y \geq 0$. Which of the following points lies in the feasible region?

a) $(4, 2)$

b) $(5, 1)$

c) $(1, 5)$

d) $(-1, 4)$

118 / 576

Category: SOME DEFINITIONS AND CONCEPTS RELATED TO LPP

118. (A) The feasible region of a linear programming problem is always a convex set.
(R) The intersection of any two convex sets is also a convex set.

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.

b) Both Assertion and Reason are true, but Reason is NOT the correct explanation of Assertion.

c) Assertion is true, but Reason is false.

d) Assertion is false, but Reason is true.

119 / 576

Category: LINAR PROGRAMMING-DEFINITION

119. (A) For a given Linear Programming Problem (LPP), the solution set of all points that satisfy all constraints including non-negativity restrictions is called the feasible region.
(R) The optimal solution to a LPP must always lie on the boundary of the feasible region.

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.

b) Both Assertion and Reason are true, but Reason is NOT the correct explanation of Assertion.

c) Assertion is true, but Reason is false.

d) Assertion is false, but Reason is true.

120 / 576

Category: Areas of Application

120. A manufacturing company has two machines, A and B, producing two types of products, X and Y. Machine A can produce 5 units of X or 3 units of Y per hour, while machine B can produce 4 units of X or 6 units of Y per hour. The profit per unit for X is \$8 and for Y is \$12. If the company has 40 hours available on both machines, how should the production be allocated to maximize profit?

a) Allocate all hours to producing product X on both machines

b) Allocate all hours to producing product Y on both machines

c) Allocate all hours to producing product X on machine A and product Y on machine B

d) Allocate half the hours to each product on both machines

121 / 576

Category: Understanding regions of inequality is essential before tackling LPP problems.

121. (A) The feasible region in a linear programming problem is the intersection of all regions defined by the constraints.
(R) The feasible region must satisfy all the given inequalities simultaneously to be considered valid.

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.

b) Both Assertion and Reason are true, but Reason is NOT the correct explanation of Assertion.

c) Assertion is true, but Reason is false.

d) Assertion is false, but Reason is true.

122 / 576

Category: Theorem

122. (A) The maximum value of the linear objective function $Z = ax + by$ over a bounded feasibility region always occurs at a corner point.
(R) The feasibility region is formed by the intersection of closed half-planes and has finite corner points.

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.

b) Both Assertion and Reason are true, but Reason is NOT the correct explanation of Assertion.

c) Assertion is true, but Reason is false.

d) Assertion is false, but Reason is true.

123 / 576

Category: CORNER POINT METHOD.

123. A company produces two products, P and Q, with the following constraints:
$4P + 3Q \leq 24$
$P + 2Q \leq 10$
$P \geq 0, Q \geq 0$
If the profit function is $Z = 5P + 4Q$, what is the maximum profit?

a) 20

b) 29

c) 30

d) 0

124 / 576

Category: The common region represents the solution set (infinite ordered pairs).

124. The common region of the inequalities $2x + y \leq 6$ and $x \geq 0$, $y \geq 0$ is a:

a) Rectangle

b) Triangle

c) Circle

d) Parabola

125 / 576

Category: A method for optimizing (maximizing/minimizing) a linear function subject to linear constraints.

125. What is the objective function in a linear programming problem?

a) A nonlinear equation representing constraints

b) The linear function whose maximum or minimum value is determined

c) A set of inequalities describing feasible solutions

d) The region where all constraints are satisfied

126 / 576

Category: SOME DEFINITIONS AND CONCEPTS RELATED TO LPP

126. In an LPP with objective function $Z = 3x_1 + 5x_2$, if we have constraints $x_1 + x_2 \leq 4$ and $x_1 \geq 1, x_2 \geq 0$, what is the maximum value of Z?

a) 12

b) 15

c) 18

d) 20

127 / 576

Category: Feasible region : i region and Feasible solution.

127. (A) The feasible region of an LPP is always a convex set.
(R) The intersection of any two convex sets is convex, and the feasible region is formed by the intersection of multiple half-spaces (constraints), which are convex.

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.

b) Both Assertion and Reason are true, but Reason is NOT the correct explanation of Assertion.

c) Assertion is true, but Reason is false.

d) Assertion is false, but Reason is true.

128 / 576

Category: Manufacturing Problems – Optimize production for profit.

128. (A) In a manufacturing problem, the optimal solution for profit maximization always lies at one of the corner points of the feasible region.
(R) The feasible region in linear programming is a convex polygon, and the maximum or minimum value of the objective function occurs at its vertices.

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.

b) Both Assertion and Reason are true, but Reason is NOT the correct explanation of Assertion.

c) Assertion is true, but Reason is false.

d) Assertion is false, but Reason is true.

129 / 576

Category: Check feasible side using a test point (often origin).

129. For the LPP with constraints $x + y \leq 6$, $x \geq 0$, $y \geq 0$, which corner point maximizes $Z = 4x + 3y$?

a) $(0, 0)$

b) $(6, 0)$

c) $(0, 6)$

d) $(3, 3)$

130 / 576

Category: Linear Inequations in Two Variables

130. Which of the following represents the solution region for the inequality $2x - y \geq 4$?

a) Shaded area above the line $y = 2x - 4$

b) Shaded area below the line $y = 2x - 4$

c) Shaded area to the right of the line $y = 2x - 4$

d) Shaded area to the left of the line $y = 2x - 4$

131 / 576

Category: Transportation Problems – Minimize cost of shipment.

131. (A) In a transportation problem, if the total supply equals the total demand, then it is called a balanced transportation problem.
(R) A balanced transportation problem ensures that all demand constraints are satisfied without exceeding any supply constraints.

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.

b) Both Assertion and Reason are true, but Reason is NOT the correct explanation of Assertion.

c) Assertion is true, but Reason is false.

d) Assertion is false, but Reason is true.

132 / 576

Category: Investment Problems – Maximize return on capital allocation.

132. (A) The objective function $Z = 0.05x_1 + 0.07x_2 + 0.10x_3$ in an investment problem must always be maximized to achieve optimal capital allocation.
(R) The coefficients $0.05$, $0.07$, and $0.10$ represent the return rates of the securities, which are directly proportional to the total return.

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.

b) Both Assertion and Reason are true, but Reason is NOT the correct explanation of Assertion.

c) Assertion is true, but Reason is false.

d) Assertion is false, but Reason is true.

133 / 576

Category: Constraints.

133. If a company produces two products $x$ and $y$, and has a labor constraint of $4x + 3y \leq 120$ hours per day, what does this constraint represent?

a) Total profit from producing $x$ and $y$

b) Minimum labor required for production

c) Maximum labor available for production

d) Cost of raw materials

134 / 576

Category: Define objective function (profit, cost, etc.).

134. A logistics company operates vans (cost \$80/day) and trucks (cost \$120/day) to meet daily delivery requirements (at least 30 vehicles total) with driver limitations (max 20 trucks available). If they must minimize costs while ensuring at least 15 vans are used for urban deliveries, which represents proper cost minimization objective with all constraints?

a) Minimize 120V + 80T with V + T ≥ 30, T ≤ 20

b) Minimize 80V + 120T with V + T ≥ 30, T ≤ 20, V ≥ 15

c) Minimize 80V + 120T with only vehicle total constraint

d) Minimize 80V + 120T with V + T ≥ 30, T ≤ 20, V ≥ 15, V ≥ 0, T ≥ 0

135 / 576

Category: Check feasible side using a test point (often origin).

135. For the system $2x + 5y \leq 10$, $x \geq 0$, $y \geq 0$, what is the feasible side of the first inequality when tested with the origin $(0,0)$?

a) The side excluding the origin

b) The side including the origin

c) Only the boundary line

d) None of these

136 / 576

Category: Graph each inequality on the same coordinate axes.

136. Which of the following represents the solution region for the inequality $x + y \leq 4$?

a) The area above the line $x + y = 4$

b) The area below the line $x + y = 4$ including the line

c) Only the line $x + y = 4$

d) The area to the right of the line $x + y = 4$

137 / 576

Category: Applications: maximize profit, minimize cost, optimal resource utilization.

137. A company produces two types of food supplements: Type A and Type B. Each unit of Type A provides 3 units of protein and 1 unit of carbohydrates at a cost of \$4. Each unit of Type B provides 1 unit of protein and 2 units of carbohydrates at a cost of \$3. The minimum daily requirement is 10 units of protein and 7 units of carbohydrates. What is the minimum cost to meet the daily requirements?

a) \$15

b) \$20

c) \$25

d) \$30

138 / 576

Category: ADVANTAGES OF LINEAR PROGRAMMING PROBLEMS

138. During the simplex method iterations, if a basic variable in the optimal tableau has a value of zero, which phenomenon occurs?

a) The problem is infeasible.

b) The problem has multiple optimal solutions.

c) Degeneracy occurs.

d) The dual problem becomes unbounded.

139 / 576

Category: SOME DEFINITIONS AND CONCEPTS RELATED TO LPP

139. Consider the LPP: Maximize $Z = 3x + 4y$ subject to $2x + y \leq 10$, $x + 3y \leq 12$, $x \geq 0$, $y \geq 0$. Which of the following points lies in the feasible region?

a) (1, 5)

b) (3, 2)

c) (4, 6)

d) (-1, 8)

140 / 576

Category: Developed by Kantorovich and George Dantzig.

140. (A) In a linear programming problem, the optimal solution always occurs at a corner point of the feasible region.
(R) The Extreme Point Theorem states that an optimal solution to an LPP, if it exists, occurs at one of the corner points of the feasible region.

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.

b) Both Assertion and Reason are true, but Reason is NOT the correct explanation of Assertion.

c) Assertion is true, but Reason is false.

d) Assertion is false, but Reason is true.

141 / 576

Category: LINEAR PROGRAMMING

141. (A) The feasible region for a linear programming problem is always a convex set.
(R) Any point within the feasible region satisfying all constraints and non-negativity conditions can be an optimal solution.

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.

b) Both Assertion and Reason are true, but Reason is NOT the correct explanation of Assertion.

c) Assertion is true, but Reason is false.

d) Assertion is false, but Reason is true.

142 / 576

Category: Advertising Problems – Allocate funds for maximum exposure.

142. (A) In an advertising problem, allocating more funds to the medium with the highest effectiveness coefficient will always maximize the total audience reach.
(R) The effectiveness coefficient represents the audience per unit cost for each medium, and higher coefficients indicate better efficiency in reaching the audience.

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.

b) Both Assertion and Reason are true, but Reason is NOT the correct explanation of Assertion.

c) Assertion is true, but Reason is false.

d) Assertion is false, but Reason is true.

143 / 576

Category: Identify common overlapping region (solution region).

143. Find the common solution region for the system of inequalities: $x + y \leq 4$, $2x - y \geq 1$, and $x \geq 0$.

a) A triangle with vertices $(0,4)$, $(5/3,7/3)$, and $(0.5,0)$

b) A quadrilateral with vertices $(0,4)$, $(5/3,7/3)$, $(4,0)$, and $(0.5,0)$

c) An unbounded region extending infinitely upwards

d) No common solution exists

144 / 576

Category: SYSTEM OF THREE OR MORE INEQULITIES

144. (A) The feasible region of a system of linear inequalities is always a convex set.
(R) The intersection of any number of convex sets is always a convex set.

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.

b) Both Assertion and Reason are true, but Reason is NOT the correct explanation of Assertion.

c) Assertion is true, but Reason is false.

d) Assertion is false, but Reason is true.

145 / 576

Category: The common region represents the solution set (infinite ordered pairs).

145. Which of the following systems of inequalities represents the shaded region bounded by the lines $x = 0$, $y = 0$, and $x + y = 4$?

a) $x + y \geq 4$, $x \geq 0$, $y \geq 0$

b) $x + y \leq 4$, $x \leq 0$, $y \leq 0$

c) $x + y \leq 4$, $x \geq 0$, $y \geq 0$

d) $x + y \geq 4$, $x \leq 0$, $y \leq 0$

146 / 576

Category: Identify decision variables (e.g., x, y).

146. Why are non-negativity constraints included in a linear programming model?

a) To make the problem easier to solve

b) To restrict the feasible region

c) Because negative values have no valid interpretation

d) To balance the objective function

147 / 576

Category: Understanding regions of inequality is essential before tackling LPP problems.

147. For the LPP: Minimize $Z = 3x + 5y$ subject to $x + 2y \geq 10$, $2x + y \geq 12$, $x \geq 0$, $y \geq 0$, which of the following points is a feasible solution?

a) $(4, 3)$

b) $(2, 6)$

c) $(6, 2)$

d) $(0, 5)$

148 / 576

Category: Non-negativity restriction.

148. (A) In a Linear Programming Problem (LPP), non-negativity restrictions are applied to decision variables.
(R) Non-negativity restrictions ensure that the solutions obtained are feasible and practical in real-world scenarios.

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.

b) Both Assertion and Reason are true, but Reason is NOT the correct explanation of Assertion.

c) Assertion is true, but Reason is false.

d) Assertion is false, but Reason is true.

149 / 576

Category: Investment Problems – Maximize return on capital allocation.

149. Which of the following represents the objective function in an investment problem aiming to maximize returns?

a) Minimize $Z = r_1 x + r_2 y$

b) Maximize $Z = k_1 x + k_2 y$

c) Maximize $Z = r_1 x + r_2 y$

d) Minimize $Z = k_1 x + k_2 y$

150 / 576

Category: Advertising Problems – Allocate funds for maximum exposure.

150. What are the non-negativity constraints for variables $x_1$ and $x_2$ representing the number of TV and radio ads respectively?

a) $x_1 > 0$, $x_2 > 0$

b) $x_1 \leq 0$, $x_2 \leq 0$

c) $x_1 \geq 0$, $x_2 \geq 0$

d) $x_1 < 0$, $x_2 < 0$

151 / 576

Category: FORMULATION OF A LINEAR PROGRAMMING PROBLEM

151. A farmer has 200 acres of land and grows wheat and rice. The profit per acre of wheat is \$500 and for rice is \$800. Wheat requires 2 workers per acre and rice requires 3 workers per acre. The farmer has 600 workers available. Also, water consumption for wheat is 4000 liters/acre and for rice is 5000 liters/acre, with a total of 800,000 liters available. Formulate the LPP to maximize profit.

a) Maximize $Z = 500x_1 + 800x_2$ subject to $x_1 + x_2 \leq 200$, $2x_1 + 3x_2 \leq 600$, $4x_1 + 5x_2 \leq 800$, $x_1, x_2 \geq 0$

b) Maximize $Z = 500x_1 + 800x_2$ subject to $x_1 + x_2 \leq 200$, $2x_1 + 3x_2 \geq 600$, $4x_1 + 5x_2 \leq 800$, $x_1, x_2 \geq 0$

c) Maximize $Z = 500x_1 + 800x_2$ subject to $x_1 + x_2 \leq 200$, $2x_1 + 3x_2 \leq 600$, $4x_1 + 5x_2 \geq 800$, $x_1, x_2 \geq 0$

d) Maximize $Z = 500x_1 + 800x_2$ subject to $x_1 + x_2 \geq 200$, $2x_1 + 3x_2 \leq 600$, $4x_1 + 5x_2 \leq 800$, $x_1, x_2 \geq 0$

152 / 576

Category: Theorem

152. What does the given theorem guarantee about the solution to a linear programming problem?

a) The solution is always at the origin

b) The solution can be anywhere inside the region

c) The optimal value occurs at a corner point of the feasibility region

d) No solution exists for such problems

153 / 576

Category: SOLVED EXAMPLES

153. A candidate is required to answer 5-point, 4-point, and 6-point questions in a competitive examination. The time taken to solve each question type is 1 minute, 2 minutes, and 4 minutes respectively. The total available time is 210 minutes, with at most 150 minutes allocated for numerical calculations (first two sets of problems). What is the correct formulation to maximize the total points earned?

a) Maximize $Z = 5x_1 + 4x_2 + 6x_3$ subject to $x_1 + 2x_2 + 4x_3 \leq 210$, $x_1 + 2x_2 \leq 150$, $x_1, x_2, x_3 \geq 0$

b) Maximize $Z = 5x_1 + 4x_2$ subject to $x_1 + 2x_2 + 4x_3 \leq 210$, $x_1 + 2x_2 \leq 150$, $x_1, x_2, x_3 \geq 0$

c) Maximize $Z = 5x_1 + 6x_3$ subject to $x_1 + 4x_3 \leq 210$, $x_1 + 2x_2 \leq 150$, $x_1, x_2, x_3 \geq 0$

d) Maximize $Z = 4x_2 + 6x_3$ subject to $2x_2 + 4x_3 \leq 210$, $x_1 + 2x_2 \leq 150$, $x_1, x_2, x_3 \geq 0$

154 / 576

Category: Feasible region : i region and Feasible solution.

154. Which of the following is an example of an infeasible solution?

a) $x = 2$, $y = 3$

b) $x = 5$, $y = 0$

c) $x = -2$, $y = 7$

d) $x = 0$, $y = 10$

155 / 576

Category: Diet Problems – Minimize cost subject to nutritional constraints.

155. A nutritionist wants to minimize the cost of a diet consisting of two foods, X and Y. Food X costs \$4 per unit and provides 3 units of protein and 1 unit of carbohydrates. Food Y costs \$5 per unit and provides 1 unit of protein and 2 units of carbohydrates. The diet must provide at least 9 units of protein and 8 units of carbohydrates. What is the correct formulation of this problem?

a) Minimize $Z = 4x_1 + 5x_2$ subject to $3x_1 + x_2 \leq 9$, $x_1 + 2x_2 \leq 8$, $x_1, x_2 \geq 0$

b) Minimize $Z = 4x_1 + 5x_2$ subject to $3x_1 + x_2 \geq 9$, $x_1 + 2x_2 \geq 8$, $x_1, x_2 \geq 0$

c) Minimize $Z = 3x_1 + x_2$ subject to $4x_1 + 5x_2 \geq 9$, $x_1 + 2x_2 \geq 8$, $x_1, x_2 \geq 0$

d) Minimize $Z = x_1 + 2x_2$ subject to $3x_1 + x_2 \geq 9$, $4x_1 + 5x_2 \geq 8$, $x_1, x_2 \geq 0$

156 / 576

Category: Corner points (vertices) can be found by solving boundary equations in pairs.

156. (A) Solving the pair of equations $x + y = 5$ and $2x - y = 1$ gives a corner point of the feasible region.
(R) The corner points of a feasible region formed by linear inequations can be found by solving the boundary equations in pairs.

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.

b) Both Assertion and Reason are true, but Reason is NOT the correct explanation of Assertion.

c) Assertion is true, but Reason is false.

d) Assertion is false, but Reason is true.

157 / 576

Category: Define objective function (profit, cost, etc.).

157. Which condition ensures that the decision variables in a linear programming problem are physically meaningful?

a) $x + y \leq 100$

b) $x \geq 0$ and $y \geq 0$

c) $x - y = 0$

d) $x^2 + y^2 \leq 1$

158 / 576

Category: Constraints.

158. (A) The constraints $3x + 2y \leq 30$ and $x + 2y \leq 22$ define the feasible region for a given LPP.
(R) The feasible region represents all possible solutions that satisfy the given constraints.

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.

b) Both Assertion and Reason are true, but Reason is NOT the correct explanation of Assertion.

c) Assertion is true, but Reason is false.

d) Assertion is false, but Reason is true.

159 / 576

Category: Areas of Application

159. During World War II, the British military used linear programming to allocate resources (e.g., bombs, fuel) across multiple missions to maximize enemy damage while minimizing costs. Which of the following scenarios would render linear programming LEAST applicable?

a) Damage caused by each bomb type is proportional to the quantity used.

b) Pilot fatigue affects mission success non-linearly.

c) Fuel consumption varies linearly with distance traveled.

d) Enemy defenses are static and known in advance.

160 / 576

Category: Write down constraints in terms of inequalities/equations.

160. (A) The constraint $2x + 3y \leq 60$ represents a limitation on labor hours in a linear programming problem.
(R) In linear programming, constraints are always written as linear inequalities or equations to represent resource limitations.

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.

b) Both Assertion and Reason are true, but Reason is NOT the correct explanation of Assertion.

c) Assertion is true, but Reason is false.

d) Assertion is false, but Reason is true.

161 / 576

Category: Understanding regions of inequality is essential before tackling LPP problems.

161. Which of the following represents the feasible region for the system of inequalities $2x + y \leq 8$, $x + y \geq 4$, $x \geq 0$, $y \geq 0$?

a) Only the area below $2x + y = 8$ and above $x + y = 4$

b) Only the area above $2x + y = 8$ and below $x + y = 4$

c) Only the area between $x = 0$ and $y = 0$

d) The triangular region bounded by $(4, 0)$, $(0, 4)$, and $(0, 8)$

162 / 576

Category: Non-negativity restriction.

162. A company produces two products, X and Y. The profit per unit of X is \$5 and per unit of Y is \$3. The constraints are given by:
$x + 2y \leq 10$
$3x + y \leq 15$
with non-negativity restrictions $x \geq 0$, $y \geq 0$. Which of the following points violates the non-negativity restrictions?

a) (2, 2)

b) (0, 5)

c) (-1, 4)

d) (3, 0)

163 / 576

Category: Manufacturing Problems – Optimize production for profit.

163. (A) The optimal solution to a linear programming problem always lies at a corner point of the feasible region.
(R) The objective function in a linear programming problem is linear, and the maximum or minimum value of a linear function over a convex feasible region occurs at an extreme point.

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.

b) Both Assertion and Reason are true, but Reason is NOT the correct explanation of Assertion.

c) Assertion is true, but Reason is false.

d) Assertion is false, but Reason is true.

164 / 576

Category: GRAPHICAL SOLUTION OF LINEAR INEQUATIONS

164. What are the corner points of the feasible region bounded by $x \geq 0$, $y \geq 0$, and $2x + 3y \leq 6$?

a) (0,0), (0,3), (2,0)

b) (0,0), (0,2), (3,0)

c) (0,0), (3,0), (2,3)

d) (0,2), (2,0), (3,0)

165 / 576

Category: Graphical solution of linear inequalities forms the basis of linear programming.

165. A company produces two products, P and Q. The production constraints are given by $3x + 4y \leq 120$, $x + 2y \leq 50$, $x \geq 0$, $y \geq 0$, where $x$ and $y$ represent the quantities of P and Q respectively. The profit function is $Z = 5x + 6y$. Determine the coordinates of the corner point that maximizes the profit.

a) $(0, 0)$

b) $(40, 0)$

c) $(20, 15)$

d) $(0, 25)$

166 / 576

Category: Theorem

166. (A) The maximum value of the objective function $Z = 5x + 3y$ over the feasibility region defined by $x \geq 0$, $y \geq 0$, $x + y \leq 10$, and $2x + y \leq 16$ will always occur at one of the corner points of the region.
(R) For any linear programming problem with a bounded feasibility region formed by the intersection of closed half-planes, the optimal solution occurs at a corner point.

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.

b) Both Assertion and Reason are true, but Reason is NOT the correct explanation of Assertion.

c) Assertion is true, but Reason is false.

d) Assertion is false, but Reason is true.

167 / 576

Category: Areas of Application

167. In which of the following fields is linear programming commonly applied?

a) Industrial production optimization

b) Weather forecasting

c) Human psychology research

d) Historical data analysis

168 / 576

Category: Linear Programming Problem (LPP) Example

168. What is the objective function for maximizing the total credit in the given LPP problem?

a) Maximize $Z = 5x_1 + 4x_2 + 6x_3$

b) Minimize $Z = 5x_1 + 4x_2 + 6x_3$

c) Maximize $Z = 3x_1 + 2x_2 + 4x_3$

d) Minimize $Z = 3x_1 + 2x_2 + 4x_3$

169 / 576

Category: SOLVED EXAMPLES

169. Minimize $Z = x - 7y + 190$ under the constraints: $x + y \leq 8$, $x \leq 5$, $y \leq 5$, $x + y \geq 4$, $x \geq 0$, $y \geq 0$.

a) The minimum value of $Z$ is 156

b) The minimum value of $Z$ is 160

c) The minimum value of $Z$ is 165

d) The minimum value of $Z$ is 170

170 / 576

Category: SOME DEFINITIONS AND CONCEPTS RELATED TO LPP

170. (A) The feasible region in a Linear Programming Problem (LPP) is always convex.
(R) The intersection of two convex sets is also a convex set.

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.

b) Both Assertion and Reason are true, but Reason is NOT the correct explanation of Assertion.

c) Assertion is true, but Reason is false.

d) Assertion is false, but Reason is true.

171 / 576

Category: CORNER POINT METHOD.

171. Consider the objective function $Z = 3x + 7y$ over the feasible region with corner points $(1, 5)$, $(1, 2)$, $(4, 1)$, and $(6, 0)$. What is the minimum value of $Z$?

a) At $(1, 5)$: $Z = 3(1) + 7(5) = 3 + 35 = 38$

b) At $(1, 2)$: $Z = 3(1) + 7(2) = 3 + 14 = 17$

c) At $(4, 1)$: $Z = 3(4) + 7(1) = 12 + 7 = 19$

d) At $(6, 0)$: $Z = 3(6) + 7(0) = 18 + 0 = 18$ \end{itemize} The minimum value of $Z$ is $17$ at the point $(1, 2)$.

e) 17

f) 38

g) 19

h) 18

172 / 576

Category: Identify decision variables (e.g., x, y).

172. In a linear programming model where $x$ represents the number of units of product P and $y$ represents the number of units of product Q, which set of constraints ensures non-negative production quantities?

a) $x \leq 0$, $y \leq 0$

b) $x \geq 0$, $y \leq 0$

c) $x \geq 0$, $y \geq 0$

d) $x < 0$, $y < 0$

173 / 576

Category: Feasible region : i region and Feasible solution.

173. What is an optimal feasible solution in an LPP?

a) A solution that satisfies only some constraints.

b) A solution that maximizes the constraints.

c) A feasible solution that also optimizes the objective function.

d) A solution that lies outside the feasible region but optimizes the objective function.

174 / 576

Category: Understanding regions of inequality is essential before tackling LPP problems.

174. In an LPP, what defines an optimal feasible solution?

a) Any solution that satisfies the constraints but not necessarily optimizes the objective

b) The first feasible solution found during calculations

c) The solution that lies at the center of the feasible region

d) The feasible solution that gives the best value for the objective function

175 / 576

Category: LINEAR PROGRAMMING

175. A company manufactures two products, P and Q, which require processing on machines M1 and M2. Each unit of P requires 3 hours on M1 and 2 hours on M2, while each unit of Q requires 2 hours on M1 and 4 hours on M2. The available time on M1 is 120 hours and on M2 is 160 hours. The profit on each unit of P is \$50 and on each unit of Q is \$60. How many units of each product should be produced to maximize profit?

a) 10 units of P and 40 units of Q

b) 20 units of P and 30 units of Q

c) 30 units of P and 20 units of Q

d) 40 units of P and 10 units of Q

176 / 576

Category: Developed by Kantorovich and George Dantzig.

176. (A) George B. Dantzig developed the technique of linear programming.
(R) Linear programming originated during World War II for military strategies.

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.

b) Both Assertion and Reason are true, but Reason is NOT the correct explanation of Assertion.

c) Assertion is true, but Reason is false.

d) Assertion is false, but Reason is true.

177 / 576

Category: SOME DEFINITIONS AND CONCEPTS RELATED TO LPP

177. Which of the following describes a feasible solution in an LPP?

a) A solution that violates the constraints

b) A point that lies outside the feasible region

c) A point that satisfies all the constraints and non-negativity conditions

d) A solution that optimizes the objective function

178 / 576

Category: Linear Inequations in Two Variables

178. A system of inequalities is given by $2x + y \geq 4$, $x - y \leq 1$, and $x \geq 0$. Which point lies in the feasible region satisfying all three inequalities?

a) $(1, 2)$

b) $(0, 5)$

c) $(2, -1)$

d) $(-1, 6)$

179 / 576

Category: Convex region.

179. (A) The feasible region in a Linear Programming Problem (LPP) is always convex.
(R) Any two points in the feasible region of an LPP will have the line segment joining them entirely lying within the region.

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.

b) Both Assertion and Reason are true, but Reason is NOT the correct explanation of Assertion.

c) Assertion is true, but Reason is false.

d) Assertion is false, but Reason is true.

180 / 576

Category: SOLVED EXAMPLES

180. (A) In the given example of a candidate solving problems for maximum credits, the constraint $3x_1 + 2x_2 \leq 150$ ensures that the candidate does not spend more than $2 \frac{1}{2}$ hours on the first two sets of problems.
(R) The time required to solve each problem type is proportional to the points allocated to it.

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.

b) Both Assertion and Reason are true, but Reason is NOT the correct explanation of Assertion.

c) Assertion is true, but Reason is false.

d) Assertion is false, but Reason is true.

181 / 576

Category: Linear Inequations in Two Variables

181. A company produces two products with constraints $x + y \leq 10$, $2x + y \leq 16$, $x \geq 0$, $y \geq 0$. The profit function is $P = 3x + 4y$. What is the maximum profit achievable?

a) \$34

b) \$40

c) \$24

d) \$0

182 / 576

Category: Theorem

182. Which of the following points is an infeasible solution for the LPP:
Minimize $Z = x + 2y$
Subject to:
$x + y \geq 3$,
$2x + y \leq 4$,
$x, y \geq 0$?

a) $(1, 2)$

b) $(0, 4)$

c) $(1, 1)$

d) $(2, 0)$

183 / 576

Category: Linear Programming Problem (LPP) Example

183. (A) In the given LPP formulation for maximizing credits, the constraint $3x_1 + 2x_2 \leq 150$ ensures that the candidate does not exceed the time limit for solving 5-point and 4-point problems combined.
(R) The time spent on solving 6-point problems is not considered in this constraint because it is part of a separate constraint $3x_1 + 2x_2 + 4x_3 \leq 210$.

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.

b) Both Assertion and Reason are true, but Reason is NOT the correct explanation of Assertion.

c) Assertion is true, but Reason is false.

d) Assertion is false, but Reason is true.

184 / 576

Category: AREAS OF APPLICATIONS OF LINEAR PROGRAMMING

184. A company has two production plants P1 and P2 supplying goods to three markets M1, M2, and M3. Plant P1 can produce 50 units and P2 can produce 70 units. Markets M1, M2, and M3 demand 40, 30, and 50 units respectively. The goal is to minimize transportation cost. Which of the following conditions ensures that the total supply meets the total demand?

a) $x_{11} + x_{12} + x_{13} \leq 50$

b) $x_{21} + x_{22} + x_{23} \leq 70$

c) $x_{11} + x_{21} = 40$, $x_{12} + x_{22} = 30$, $x_{13} + x_{23} = 50$

d) No additional condition is needed as total supply equals total demand

185 / 576

Category: Blending Problems – Optimize combination of raw materials.

185. (A) In a blending problem, the objective function is always to minimize cost or maximize profit.
(R) The objective function is formulated as a linear combination of decision variables representing quantities of raw materials.

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.

b) Both Assertion and Reason are true, but Reason is NOT the correct explanation of Assertion.

c) Assertion is true, but Reason is false.

d) Assertion is false, but Reason is true.

186 / 576

Category: AREAS OF APPLICATIONS OF LINEAR PROGRAMMING

186. (A) Linear programming can be used to determine the optimal diet plan for an individual by minimizing cost while meeting nutritional requirements.

(R) Linear programming can handle qualitative factors such as taste preferences and cultural dietary restrictions in diet optimization problems.

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.

b) Both Assertion and Reason are true, but Reason is NOT the correct explanation of Assertion.

c) Assertion is true, but Reason is false.

d) Assertion is false, but Reason is true.

187 / 576

Category: Blending Problems – Optimize combination of raw materials.

187. A company blends two types of juices, A and B, using three different fruits: Orange, Mango, and Pineapple. Each liter of juice A requires 4 units of Orange, 2 units of Mango, and 1 unit of Pineapple. Each liter of juice B requires 2 units of Orange, 3 units of Mango, and 2 units of Pineapple. The company has 120 units of Orange, 90 units of Mango, and 60 units of Pineapple available daily. If the profit per liter is \$8 for juice A and \$7 for juice B, how many liters of each juice should be produced to maximize profit while not exceeding available resources?

a) 10 liters of A and 25 liters of B

b) 15 liters of A and 20 liters of B

c) 20 liters of A and 20 liters of B

d) 25 liters of A and 15 liters of B

188 / 576

Category: System of Inequations

188. Which region represents the solution set of the system of inequalities $2x + y \leq 6$ and $x \geq 0, y \geq 0$?

a) Region above the line $2x + y = 6$ and outside the first quadrant

b) Entire first quadrant

c) Triangular region in the first quadrant bounded by $2x + y = 6$

d) Only the line $2x + y = 6$

189 / 576

Category: Identify decision variables (e.g., x, y).

189. A company produces two products, A and B. Let $x$ represent the number of units of product A and $y$ represent the number of units of product B produced per day. The profit per unit for product A is \$8 and for product B is \$12. What is the correct objective function to maximize the total daily profit?

a) $8x - 12y$

b) $12x + 8y$

c) $8x + 12y$

d) $20(x + y)$

190 / 576

Category: Objective function.

190. What distinguishes an optimal feasible solution from other feasible solutions in an LPP?

a) It lies on the boundary of the feasible region.

b) It satisfies only the non-negativity constraints.

c) It optimizes the objective function while satisfying all constraints.

d) It has the highest number of variables.

191 / 576

Category: ADVANTAGES OF LINEAR PROGRAMMING PROBLEMS

191. How does linear programming contribute to business profitability?

a) By increasing labor costs

b) By reducing resource allocation efficiency

c) By minimizing operational costs

d) By ignoring constraints

192 / 576

Category: SOME DEFINITIONS AND CONCEPTS RELATED TO LPP

192. Which of the following statements about the feasible region in a linear programming problem is correct?

a) Feasible region is always bounded

b) Non-negativity restrictions are not part of feasible region

c) Feasible region is always a convex set

d) Optimal solution must lie at one of the vertices of feasible region

193 / 576

Category: Write down constraints in terms of inequalities/equations.

193. A factory has 200 hours of machine time available. Producing one unit of product P requires 5 hours, and one unit of product Q requires 4 hours. Which inequality represents the resource constraint for machine time?

a) $4p + 5q \leq 200$

b) $5p + 4q \geq 200$

c) $5p + 4q \leq 200$

d) $p + q \leq 200$

194 / 576

Category: System of Inequations

194. For the LPP with objective function $Z = 5x + 10y$ subject to $x + y \leq 10$, $x \geq 2$, $y \geq 3$, which corner point gives the maximum value of Z?

a) $(2, 3)$

b) $(7, 3)$

c) $(2, 8)$

d) $(5, 5)$

195 / 576

Category: Applications: maximize profit, minimize cost, optimal resource utilization.

195. (A) Linear programming can be used to determine the optimal allocation of a fixed advertising budget across different media channels to maximize total effective audience reach.
(R) The objective in linear programming is always to maximize profit by minimizing costs under given constraints.

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.

b) Both Assertion and Reason are true, but Reason is NOT the correct explanation of Assertion.

c) Assertion is true, but Reason is false.

d) Assertion is false, but Reason is true.

196 / 576

Category: Manufacturing Problems – Optimize production for profit.

196. A company manufactures two products X and Y. Each unit of X requires 3 hours of labor and 1 hour of machine time, while each unit of Y requires 2 hours of labor and 2 hours of machine time. The profit per unit of X is \$8 and for Y is \$10. The total available labor hours are 120 and machine hours are 80. What is the maximum achievable profit?

a) \$320

b) \$400

c) \$460

d) \$500

197 / 576

Category: Developed by Kantorovich and George Dantzig.

197. Consider an LPP with constraints $x \geq 0$, $y \geq 0$, $3x + 4y \leq 60$, and $5x + 2y \leq 50$. The objective is to maximize $Z = 6x + 8y$. Which of the following points gives the optimal solution?

a) $(0,0)$

b) $(40/7, 60/7)$

c) $(0,15)$

d) $(10,0)$

198 / 576

Category: GENERAL LINAR PROGRAMMING PROBLEM

198. (A) In a linear programming problem, if the feasible region is unbounded, the optimal solution may not exist.
(R) The corner point method requires the feasible region to be bounded for determining the optimal solution.

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.

b) Both Assertion and Reason are true, but Reason is NOT the correct explanation of Assertion.

c) Assertion is true, but Reason is false.

d) Assertion is false, but Reason is true.

199 / 576

Category: Diet Problems – Minimize cost subject to nutritional constraints.

199. For the diet problem with the following formulation:
$\text{Minimize } Z = 2x_1 + 3x_2 \quad \text{subject to }$
$x_1 + 2x_2 \geq 6$
$2x_1 + x_2 \geq 6$
$x_1, x_2 \geq 0$
Which of the following is a feasible solution that satisfies all constraints?

a) $(2, 1)$

b) $(1, 4)$

c) $(0, 6)$

d) $(3, 0)$

200 / 576

Category: Advertising Problems – Allocate funds for maximum exposure.

200. A company can advertise in Magazine A (\$2000 per ad, reaches 8000 people) or Magazine B (\$3500 per ad, reaches 14000 people). Their budget is \$28,000 and they want the difference between total ads placed in both magazines to not exceed 3. Additionally, Magazine A ads must be at least twice as many as Magazine B ads. What's the optimal ad combination?

a) 6 Magazine A ads and 3 Magazine B ads

b) 4 Magazine A ads and 2 Magazine B ads

c) 5 Magazine A ads and 2 Magazine B ads

d) 3 Magazine A ads and 1 Magazine B ads

201 / 576

Category: Convex region.

201. In an LPP with a convex feasible region, the optimal solution must lie:

a) At an interior point of the feasible region.

b) At a non-vertex boundary point.

c) At a vertex of the feasible region.

d) Outside the feasible region.

202 / 576

Category: Understanding regions of inequality is essential before tackling LPP problems.

202. (A) The inequality $y \geq 3$ includes the line $y = 3$ in its graphical representation.
(R) Non-strict inequalities like $\geq$ or $\leq$ have solid boundary lines to indicate inclusion of the boundary points.

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.

b) Both Assertion and Reason are true, but Reason is NOT the correct explanation of Assertion.

c) Assertion is true, but Reason is false.

d) Assertion is false, but Reason is true.

203 / 576

Category: Transportation Problems – Minimize cost of shipment.

203. (A) The goal of a transportation problem is to minimize the total cost of shipping goods from suppliers to markets.
(R) The objective function $Z = \sum_{i=1}^{m} \sum_{j=1}^{n} c_{ij} x_{ij}$ represents the total transportation cost.

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.

b) Both Assertion and Reason are true, but Reason is NOT the correct explanation of Assertion.

c) Assertion is true, but Reason is false.

d) Assertion is false, but Reason is true.

204 / 576

Category: Identify decision variables (e.g., x, y).

204. A company produces two products, P and Q. The profit per unit of P is \$15 and per unit of Q is \$20. Production requires 3 hours of labor for P and 4 hours for Q weekly, with total labor not exceeding 120 hours per week. The company can produce at most 30 units of P and 25 units of Q weekly. What are the decision variables in this problem?

a) Number of labor hours and profit per unit

b) Number of units of P and Q produced weekly

c) Total labor hours and production limits

d) Profit per unit of P and Q

205 / 576

Category: Graphical Method of Solution

205. Consider the following linear programming problem:
Maximize $Z = 15x + 20y$
Subject to:
$x \geq 0$, $y \geq 0$,
$4x + 3y \leq 120$,
$2x + 5y \leq 100$.
Which of the following points gives the maximum value of $Z$?

a) $(0, 20)$

b) $(30, 0)$

c) $(\frac{150}{7}, \frac{80}{7})$

d) $(0, 0)$

206 / 576

Category: Define objective function (profit, cost, etc.).

206. A manufacturing company produces two types of products X and Y with profits of \$120 and \$80 per unit respectively. The production is constrained by: machine time (4x + 2y ≤ 200 hours), labor (2x + 3y ≤ 150 hours), and storage (x + y ≤ 60 units). What should be the correct objective function if the company wants to maximize its profit?

a) $Z = 120x + 80y$

b) $Z = 80x + 120y$

c) $Z = 200x + 150y$

d) $Z = 4x + 2y$

207 / 576

Category: Developed by Kantorovich and George Dantzig.

207. What type of constraints are used in linear programming problems?

a) Quadratic equations

b) Trigonometric functions

c) Non-linear inequalities

d) Linear equations or inequations

208 / 576

Category: Understanding regions of inequality is essential before tackling LPP problems.

208. Consider the LPP: Maximize $Z = 4x + 6y$ subject to $x + y \leq 5$, $2x + 3y \leq 12$, $x \geq 0$, $y \geq 0$. Which of the following is the optimal feasible solution?

a) $(5, 0)$ with $Z = 20$

b) $(0, 4)$ with $Z = 24$

c) $(3, 2)$ with $Z = 24$

d) Both $(3, 2)$ and $(0, 4)$ with $Z = 24$

209 / 576

Category: Write down constraints in terms of inequalities/equations.

209. A factory makes chairs and tables. Each chair requires 2 hours of assembly and 3 hours of finishing, while each table requires 4 hours of assembly and 2 hours of finishing. The assembly department has 80 hours available, and finishing has 60 hours. The factory must produce at least 5 tables due to orders. If c = number of chairs and t = number of tables, which represents ALL constraints correctly?

a) $2c + 4t \leq 80$, $3c + 2t \leq 60$, $t \geq 5$

b) $2c + 4t \leq 80$, $3c + 2t \leq 60$, $t \geq 5$, $c \geq 0$

c) $2c + 4t \leq 80$, $3c + 2t \leq 60$, $t \geq 5$, $c \geq 0$, $t \geq 0$

d) $4c + 2t \leq 80$, $2c + 3t \leq 60$, $t \geq 5$, $c \geq 0$, $t \geq 0$

210 / 576

Category: Check feasible side using a test point (often origin).

210. Which side of the inequality $x + 3y \leq 6$ contains the feasible region when tested with the origin $(0,0)$?

a) The side opposite to the origin

b) The side including the origin

c) Both sides

d) Neither side

211 / 576

Category: Linear Programming Problem (LPP) Example

211. A toy company produces dolls and cars. Each doll requires 2 hours of molding and 3 hours of painting, while each car requires 4 hours of molding and 1 hour of painting. The total molding time available is 80 hours, and painting time is 60 hours. If the profit per doll is \$10 and per car is \$15, what is the optimal production mix for maximum profit?

a) 300

b) 320

c) 340

d) 360

212 / 576

Category: Represent graphically and find feasible region.

212. Given the feasible region bounded by $x \geq 0$, $y \geq 0$, $x + 2y \leq 8$, and $3x + y \leq 9$, which corner point maximizes $Z = 5x + 10y$?

a) $(0, 0)$

b) $(0, 4)$

c) $(3, 0)$

d) $(2, 3)$

213 / 576

Category: Corner points (vertices) can be found by solving boundary equations in pairs.

213. If the objective function $Z = 3x + 4y$ is evaluated at the corner points $(2, 1)$, $(4, 0)$, and $(1, 3)$, what is its maximum value?

a) 10

b) 12

c) 15

d) 18

214 / 576

Category: Theorem

214. For a linear programming problem with objective function $Z = 5x + 3y$, where should you look first to find its maximum value?

a) At any point inside the feasibility region

b) At the center of the feasibility region

c) At one of the corner points of the feasibility region

d) Outside the feasibility region

215 / 576

Category: Advertising Problems – Allocate funds for maximum exposure.

215. (A) Linear programming can be used to optimally allocate a fixed advertising budget across media channels to maximize audience reach.
(R) The objective function in such problems maximizes the total effective audience while satisfying budget and other constraints.

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.

b) Both Assertion and Reason are true, but Reason is NOT the correct explanation of Assertion.

c) Assertion is true, but Reason is false.

d) Assertion is false, but Reason is true.

216 / 576

Category: Feasible region : i region and Feasible solution.

216. In an LPP, the system of constraints is given by $3x + 4y \leq 24$, $x + 2y \geq 6$, $x \geq 0$, $y \geq 0$. Which statement about the feasible region is true?

a) The feasible region is empty

b) The feasible region is unbounded

c) The feasible region is non-empty and convex

d) The feasible region is non-convex

217 / 576

Category: Graphical solution of linear inequalities forms the basis of linear programming.

217. What is optimized in a linear programming problem?

a) The constraints

b) The feasibility region

c) The linear function to be maximized or minimized

d) The corner points of the feasible region

218 / 576

Category: AREAS OF APPLICATIONS OF LINEAR PROGRAMMING

218. (A) Linear programming can be used to determine the least expensive way of transporting a commodity from multiple factories to different markets.
(R) Transportation problems involve minimizing costs while satisfying supply and demand constraints, which can be represented as linear equations.

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.

b) Both Assertion and Reason are true, but Reason is NOT the correct explanation of Assertion.

c) Assertion is true, but Reason is false.

d) Assertion is false, but Reason is true.

219 / 576

Category: Objective function.

219. (A) The objective function $5x + 3y$ will always have a unique maximum value in a feasible region bounded by linear constraints.
(R) A bounded feasible region in an LPP ensures that the objective function attains its maximum or minimum value at one of the corner points.

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.

b) Both Assertion and Reason are true, but Reason is NOT the correct explanation of Assertion.

c) Assertion is true, but Reason is false.

d) Assertion is false, but Reason is true.

220 / 576

Category: Linear Inequations in Two Variables

220. The inequalities $x + 2y \leq 8$, $x - y \leq 1$, and $y \geq 0$ define a feasible region. Which vertex of this region maximizes $3x + 5y$?

a) $(1, 0)$

b) $(8, 0)$

c) $(\frac{10}{3}, \frac{7}{3})$

d) $(0, 4)$

221 / 576

Category: Optimal feasible solution.

221. (A) The point (2, 3) is an optimal feasible solution for the LPP Maximize $Z = 6x + 8y$ subject to constraints $3x + 2y \leq 30$, $x + 2y \leq 22$, $x \geq 0$, $y \geq 0$.
(R) It yields the maximum value of the objective function among all feasible solutions.

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.

b) Both Assertion and Reason are true, but Reason is NOT the correct explanation of Assertion.

c) Assertion is true, but Reason is false.

d) Assertion is false, but Reason is true.

222 / 576

Category: Investment Problems – Maximize return on capital allocation.

222. (A) In a linear programming problem, the objective function must always be to maximize profit or minimize cost.
(R) The objective function represents the goal of the optimization problem, which is typically to maximize returns or minimize expenses.

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.

b) Both Assertion and Reason are true, but Reason is NOT the correct explanation of Assertion.

c) Assertion is true, but Reason is false.

d) Assertion is false, but Reason is true.

223 / 576

Category: Identify common overlapping region (solution region).

223. The common solution region for the inequations $y \geq 2$ and $x \leq 3$ includes which of the following points?

a) $(2,1)$

b) $(4,2)$

c) $(2,3)$

d) $(3,1)$

224 / 576

Category: Objective function.

224. (A) In a linear programming problem, the objective function can be either maximized or minimized.
(R) The word 'optimize' means to maximize or minimize.

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.

b) Both Assertion and Reason are true, but Reason is NOT the correct explanation of Assertion.

c) Assertion is true, but Reason is false.

d) Assertion is false, but Reason is true.

225 / 576

Category: Check feasible side using a test point (often origin).

225. (A) The point $(0, 0)$ lies in the feasible region for the system of inequations: $x + y \leq 5$, $2x + y \leq 8$, $x \geq 0$, $y \geq 0$.
(R) The origin $(0, 0)$ satisfies all the given constraints when substituted into each inequality.

a) For $x + y \leq 5$: $0 + 0 \leq 5$ → $0 \leq 5$ (True).

b) For $2x + y \leq 8$: $0 + 0 \leq 8$ → $0 \leq 8$ (True).

c) For $x \geq 0$ and $y \geq 0$: Both are satisfied as $0 \geq 0$ (True). \end{itemize} Since all constraints are satisfied, the Assertion (A) is true. The Reason (R) correctly explains why the origin is part of the feasible region by showing that it satisfies all constraints. Hence, both are true, and Reason is the correct explanation of Assertion.

d) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.

e) Both Assertion and Reason are true, but Reason is NOT the correct explanation of Assertion.

f) Assertion is true, but Reason is false.

g) Assertion is false, but Reason is true.

226 / 576

Category: Steps in Formulating a Linear Programming Model

226. Which of the following best describes the objective function in a linear programming problem?

a) It defines the constraints for the problem

b) It limits the range of decision variables

c) It specifies the non-negativity conditions

d) It represents the goal (maximize/minimize) as a linear function of decision variables

227 / 576

Category: CORNER POINT METHOD.

227. (A) The optimal solution of a linear programming problem always occurs at a corner point of the feasible region.
(R) According to the Extreme Point Theorem, if an optimum solution exists for an LPP, it must occur at one of the corner points of the feasible region.

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.

b) Both Assertion and Reason are true, but Reason is NOT the correct explanation of Assertion.

c) Assertion is true, but Reason is false.

d) Assertion is false, but Reason is true.

228 / 576

Category: Areas of Application

228. (A) Linear programming is not suitable for solving problems where qualitative factors like human behavior and credibility are significant.
(R) Linear programming techniques can only be applied when all elements related to a problem can be quantified.

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.

b) Both Assertion and Reason are true, but Reason is NOT the correct explanation of Assertion.

c) Assertion is true, but Reason is false.

d) Assertion is false, but Reason is true.

229 / 576

Category: Check feasible side using a test point (often origin).

229. Consider the inequality $3x + 4y \leq 12$. Which of the following test points lies on the feasible side of this inequality?

a) $(0, 0)$

b) $(1, 1)$

c) $(4, 0)$

d) $(5, 1)$

230 / 576

Category: ADVANTAGES OF LINEAR PROGRAMMING PROBLEMS

230. What is one key advantage of linear programming in resource allocation?

a) Ensures subjectivity in decision-making

b) Leads to wastage of resources

c) Helps maximize output or minimize cost efficiently

d) Reduces flexibility in problem-solving

231 / 576

Category: The common region represents the solution set (infinite ordered pairs).

231. Which of the following systems represents the feasible region bounded by the lines $x = 0$, $y = 0$, $x + y = 4$, and $2x + y = 6$?

a) $x \geq 0$, $y \geq 0$, $x + y \geq 4$, $2x + y \geq 6$

b) $x \geq 0$, $y \geq 0$, $x + y \leq 4$, $2x + y \leq 6$

c) $x \leq 0$, $y \geq 0$, $x + y \leq 4$, $2x + y \leq 6$

d) $x \geq 0$, $y \leq 0$, $x + y \leq 4$, $2x + y \leq 6$

232 / 576

Category: SYSTEM OF THREE OR MORE INEQULITIES

232. (A) The solution set of the system of inequalities $y \geq 2$, $x - y \leq 1$, and $2x + y \leq 10$ includes the point $(3, 2)$.
(R) The point $(3, 2)$ satisfies all three given inequalities.

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.

b) Both Assertion and Reason are true, but Reason is NOT the correct explanation of Assertion.

c) Assertion is true, but Reason is false.

d) Assertion is false, but Reason is true.

233 / 576

Category: THE GRAPHICAL METHOD OF SOLUTION OF LINEAR PROGRAMMING PROBLEMS

233. Which of the following is the correct statement regarding the corner point method in linear programming problems?

a) The optimal solution always lies at the midpoint of the feasible region.

b) The optimal solution can lie anywhere inside the feasible region.

c) The optimal solution must satisfy all constraints but need not be a corner point.

d) The optimal solution occurs at one of the corner points of the feasible region.

234 / 576

Category: Corner points (vertices) can be found by solving boundary equations in pairs.

234. A solution set is bounded by the lines $3x + 2y = 6$, $x - y = 1$, and $y = 0$. Which of the following is NOT a corner point of this region?

a) \$(\frac{8}{5}, \frac{3}{5})\$

b) \$(1, 0)\$

c) \$(2, 0)\$

d) \$(0, 3)\$

235 / 576

Category: Optimal feasible solution.

235. For the LPP:
Minimize $Z = 2x + 5y$
subject to:
$x + y \geq 4$, $x + 3y \geq 6$, $x \geq 0$, $y \geq 0$,
which corner point minimizes the objective function?

a) $(0, 2)$

b) $(4, 0)$

c) $(3, 1)$

d) $(6, 0)$

236 / 576

Category: Manufacturing Problems – Optimize production for profit.

236. A company manufactures two products X and Y. Product X gives \$60 profit per unit while Y gives \$80. Production is constrained by: 4X + 5Y ≤ 200 machine hours, 3X + 2Y ≤ 120 labor hours, and Y ≤ 25 units. What is the optimal production combination to maximize profit?

a) 15 units of X and 30 units of Y

b) 18.75 units of X and 25 units of Y

c) 20 units of X and 24 units of Y

d) 25 units of X and 20 units of Y

237 / 576

Category: ADVANTAGES OF LINEAR PROGRAMMING PROBLEMS

237. (A) Linear programming helps in optimizing resource allocation.
(R) It provides a systematic method to determine the best possible outcome under given constraints.

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.

b) Both Assertion and Reason are true, but Reason is NOT the correct explanation of Assertion.

c) Assertion is true, but Reason is false.

d) Assertion is false, but Reason is true.

238 / 576

Category: Corner points (vertices) can be found by solving boundary equations in pairs.

238. (A) The corner points of the feasible region formed by the intersection of the inequalities $x + 2y \leq 8$, $2x + y \geq 2$, $x - y \leq 1$, $x \geq 0$, and $y \geq 0$ are $(0, 2)$, $(3, 2)$, and $(5, 4)$.
(R) The corner points of a feasible region can always be found by solving the boundary equations in pairs.

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.

b) Both Assertion and Reason are true, but Reason is NOT the correct explanation of Assertion.

c) Assertion is true, but Reason is false.

d) Assertion is false, but Reason is true.

239 / 576

Category: Understanding regions of inequality is essential before tackling LPP problems.

239. For the LPP: Maximize $Z = 3x + 4y$ subject to $x + y \leq 8$, $2x + y \leq 12$, $x \geq 0$, $y \geq 0$, what is the optimal solution using the corner point method?

a) $(0, 0)$ with $Z = 0$

b) $(6, 0)$ with $Z = 18$

c) $(4, 4)$ with $Z = 28$

d) $(0, 8)$ with $Z = 32$

240 / 576

Category: Corner points (vertices) can be found by solving boundary equations in pairs.

240. Which of the following constraints represents a half-plane that includes the origin?

a) $3x - 2y \geq 6$

b) $5x + 2y \leq 10$

c) $-x + y > 3$

d) $4x + 5y < -8$

241 / 576

Category: Corner points (vertices) can be found by solving boundary equations in pairs.

241. Given the system of inequations $2x + y \leq 8$, $x + 3y \leq 9$, $x \geq 0$, $y \geq 0$, which corner point maximizes the objective function $P = 3x + 4y$?

a) \$(4, 0)\$

b) \$(0, 3)\$

c) \$(3, 2)\$

d) \$(0, 0)\$

242 / 576

Category: CORNER POINT METHOD.

242. What is the fundamental theorem upon which the Corner Point Method is based?

a) Linear Programming Theorem

b) Extreme Point Theorem

c) Optimization Theorem

d) Feasibility Theorem

243 / 576

Category: Apply non-negativity restrictions.

243. In a production problem, what does the non-negativity restriction $x \geq 0$ imply if $x$ represents the number of units produced?

a) The production quantity can be any real number.

b) The production quantity must be exactly zero.

c) The production quantity cannot be negative.

d) The production quantity must be a fraction.

244 / 576

Category: Apply non-negativity restrictions.

244. What do non-negativity restrictions ensure in a linear programming problem?

a) Decision variables can be negative or positive.

b) Decision variables must be integers.

c) Decision variables cannot be negative.

d) Decision variables must be zero.

245 / 576

Category: Write down constraints in terms of inequalities/equations.

245. In a linear programming model, which of the following constraints ensures non-negativity for decision variables $x$ and $y$?

a) $x \leq 0$, $y \leq 0$

b) $x > 0$, $y > 0$

c) $x \geq 0$, $y \geq 0$

d) $x = y \geq 0$

246 / 576

Category: AREAS OF APPLICATIONS OF LINEAR PROGRAMMING

246. Which of the following is an objective of using linear programming in diet problems?

a) Minimizing cost while meeting nutritional requirements

b) Maximizing taste preferences

c) Ignoring the cost factor completely

d) Prioritizing expensive food items only

247 / 576

Category: Applications: maximize profit, minimize cost, optimal resource utilization.

247. A company produces two products, A and B. Each unit of product A requires 3 hours of labor and 2 hours of machine time, while each unit of product B requires 1 hour of labor and 4 hours of machine time. The company has 120 hours of labor and 160 hours of machine time available. If the profit per unit of A is \$50 and per unit of B is \$60, how many units of each should be produced to maximize profit?

a) 20 units of A and 30 units of B

b) 32 units of A and 24 units of B

c) 40 units of A and 20 units of B

d) 28 units of A and 26 units of B

248 / 576

Category: SOME DEFINITIONS AND CONCEPTS RELATED TO LPP

248. What do non-negativity restrictions imply in an LPP?

a) Variables can take only integer values

b) Variables must be positive or zero

c) Constraints must be linear

d) Objective function must be quadratic

249 / 576

Category: Identify common overlapping region (solution region).

249. Which of the following systems represents the feasible region bounded by the lines $y = x + 1$, $y = -2x + 4$, and $y = 0$?

a) $y \geq x + 1$, $y \geq -2x + 4$, $y \leq 0$

b) $y \leq x + 1$, $y \geq -2x + 4$, $y \geq 0$

c) $y \leq x + 1$, $y \leq -2x + 4$, $y \geq 0$

d) $y \geq x + 1$, $y \leq -2x + 4$, $y \leq 0$

250 / 576

Category: THE GRAPHICAL METHOD OF SOLUTION OF LINEAR PROGRAMMING PROBLEMS

250. What is the purpose of evaluating the objective function at each corner point of the feasible region?

a) To find the average value of the objective function over the feasible region.

b) To eliminate infeasible solutions from consideration.

c) To determine the maximum or minimum value of the objective function.

d) To check if the constraints are consistent.

251 / 576

Category: Diet Problems – Minimize cost subject to nutritional constraints.

251. Using the same problem formulation as in Q2, what is the minimum cost achievable for the diet?

a) \$18

b) \$10

c) \$12

d) \$14

252 / 576

Category: Constraints.

252. Given the LPP:
Minimize $Z = 2x + 5y$
subject to:
$x + y = 7$
$3x + y \geq 15$
$x \geq 0$, $y \geq 0$.
Which of the following points is a feasible solution that minimizes $Z$?

a) (4, 3)

b) (5, 2)

c) (6, 1)

d) (7, 0)

253 / 576

Category: SOLVED EXAMPLES

253. A manufacturer produces products A, B, and C using rubber and other materials. The profits are \$40, \$45, and \$24 per unit respectively. The daily material limits are 100 units of rubber and 120 units of other materials. The material requirements are: A (3 rubber, 3 others), B (3 rubber, 2 others), C (2 rubber, 1 other). Formulate the LPP to maximize profit.

a) Maximize $Z = 40x_1 + 45x_2 + 24x_3$ subject to $3x_1 + 3x_2 + 2x_3 \leq 100$, $3x_1 + 2x_2 + x_3 \leq 120$, $x_1, x_2, x_3 \geq 0$

b) Maximize $Z = 40x_1 + 45x_2 + 24x_3$ subject to $3x_1 + 2x_2 + x_3 \leq 120$, $x_1 + x_2 + x_3 \leq 100$, $x_1, x_2, x_3 \geq 0$

c) Maximize $Z = 40x_1 + 45x_2 + 24x_3$ subject to $3x_1 + 3x_2 + 2x_3 \leq 120$, $3x_1 + 2x_2 + x_3 \leq 100$, $x_1, x_2, x_3 \geq 0$

d) Maximize $Z = 40x_1 + 45x_2 + 24x_3$ subject to $3x_1 + 2x_2 + x_3 \leq 100$, $x_1, x_2, x_3 \geq 0$

254 / 576

Category: Steps in Formulating a Linear Programming Model

254. Why are non-negativity constraints added in a linear programming model?

a) To maximize the objective function

b) To minimize the number of decision variables

c) Because negative values of decision variables are invalid in real-world problems

d) To simplify the mathematical model

255 / 576

Category: FORMULATION OF A LINEAR PROGRAMMING PROBLEM

255. (A) In a linear programming problem, the decision variables must always be non-negative.
(R) Negative values of decision variables have no valid interpretation in physical problems.

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.

b) Both Assertion and Reason are true, but Reason is NOT the correct explanation of Assertion.

c) Assertion is true, but Reason is false.

d) Assertion is false, but Reason is true.

256 / 576

Category: Steps in Formulating a Linear Programming Model

256. A farmer has 50 acres of land and can grow either wheat or barley. Wheat yields \$200 profit per acre and requires 3 days of labor per acre, while barley yields \$150 profit per acre and requires 2 days of labor. If only 120 days of labor are available, what is the correct objective function to maximize profit?

a) $150x + 200y$

b) $200x + 150y$

c) $200x - 150y$

d) $150x - 200y$

257 / 576

Category: Steps in Formulating a Linear Programming Model

257. (A) In the given example problem, the objective function $Z = 5x_1 + 4x_2 + 6x_3$ is correctly formulated to maximize the candidate's total credits.
(R) The objective function must always represent the quantity to be optimized, which, in this case, is the total points earned by the candidate.

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.

b) Both Assertion and Reason are true, but Reason is NOT the correct explanation of Assertion.

c) Assertion is true, but Reason is false.

d) Assertion is false, but Reason is true.

258 / 576

Category: Infeasible solution.

258. (A) The solution $x = 5$, $y = -3$ is an infeasible solution for the LPP Maximize $Z = 6x + 8y$ subject to the constraints $3x + 2y \leq 30$, $x + 2y \leq 22$, $x \geq 0$, $y \geq 0$.
(R) An infeasible solution violates at least one of the constraints or non-negativity conditions of the LPP.

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.

b) Both Assertion and Reason are true, but Reason is NOT the correct explanation of Assertion.

c) Assertion is true, but Reason is false.

d) Assertion is false, but Reason is true.

259 / 576

Category: AREAS OF APPLICATIONS OF LINEAR PROGRAMMING

259. A nutritionist wants to minimize the cost of a diet consisting of two foods, F1 and F2. Food F1 costs \$10 per unit and provides 200 calories and 8 grams of protein per unit. Food F2 costs \$15 per unit and provides 300 calories and 5 grams of protein per unit. The diet must provide at least 2000 calories and at least 100 grams of protein. Which of the following represents the correct constraint for the protein requirement?

a) $8x_1 + 5x_2 \leq 100$

b) $8x_1 + 5x_2 = 100$

c) $8x_1 + 5x_2 \geq 100$

d) $5x_1 + 8x_2 \geq 100$

260 / 576

Category: FORMULATION OF A LINEAR PROGRAMMING PROBLEM

260. A company produces two products, X and Y, using two machines, M1 and M2. Each unit of X requires 2 hours on M1 and 1 hour on M2. Each unit of Y requires 1 hour on M1 and 3 hours on M2. Machine M1 is available for at most 70 hours and M2 is available for at most 90 hours per week. The profit per unit of X is \$30 and for Y is \$50. Formulate the LPP to maximize the weekly profit.

a) Maximize $Z = 30x_1 + 50x_2$; Subject to: $2x_1 + x_2 \leq 70$, $x_1 + 3x_2 \leq 90$, $x_1, x_2 \geq 0$

b) Maximize $Z = 50x_1 + 30x_2$; Subject to: $2x_1 + x_2 \leq 70$, $x_1 + 3x_2 \leq 90$, $x_1, x_2 \geq 0$

c) Maximize $Z = 30x_1 + 50x_2$; Subject to: $x_1 + 2x_2 \leq 70$, $3x_1 + x_2 \leq 90$, $x_1, x_2 \geq 0$

d) Maximize $Z = 50x_1 + 30x_2$; Subject to: $x_1 + 2x_2 \leq 70$, $3x_1 + x_2 \leq 90$, $x_1, x_2 \geq 0$

261 / 576

Category: Objective function.

261. In a Linear Programming Problem (LPP), what is the objective function?

a) A system of inequalities defining the solution space

b) The linear function whose maximum or minimum value is determined

c) The region where all constraints are satisfied

d) The point that optimizes the given constraints

262 / 576

Category: Define objective function (profit, cost, etc.).

262. What is the purpose of an objective function in a linear programming model?

a) To define the constraints of the problem

b) To identify the decision variables

c) To specify non-negativity conditions

d) To represent the goal to maximize or minimize a quantity

263 / 576

Category: Fundamental Theorem (Extreme Point Theorem)

263. Consider the following linear programming problem: Maximize $Z = 5x + 3y$ subject to the constraints $2x + y \leq 10$, $x + 2y \leq 8$, $x \geq 0$, $y \geq 0$. Which of the following points is not a corner point of the feasible region?

a) $(0, 0)$

b) $(4, 2)$

c) $(0, 5)$

d) $(8, 0)$

264 / 576

Category: Graph each inequality on the same coordinate axes.

264. The feasible region for a linear programming problem has vertices at $(0, 0)$, $(0, 4)$, $(2, 3)$, and $(3, 0)$. The objective function is to maximize $Z = 2x + 3y$. What is the maximum value of $Z$?

a) 10

b) 12

c) 13

d) 15

265 / 576

Category: SYSTEM OF THREE OR MORE INEQULITIES

265. Identify the system of inequalities whose feasible region is represented by the intersection of the following half-planes: $y \leq 3$, $x + y \geq 1$, and $2x - y \leq 4$.

a) $y \geq 3$, $x + y \leq 1$, $2x - y \geq 4$

b) $y \leq 3$, $x + y \geq 1$, $2x - y \leq 4$

c) $y \geq 3$, $x + y \geq 1$, $2x - y \leq 4$

d) $y \leq 3$, $x + y \leq 1$, $2x - y \geq 4$

266 / 576

Category: GENERAL LINAR PROGRAMMING PROBLEM

266. A company produces two products, P and Q, using machines M1 and M2. Each unit of P requires 3 hours on M1 and 1 hour on M2. Each unit of Q requires 2 hours on M1 and 2 hours on M2. Machine M1 is available for at most 18 hours, and M2 for at most 10 hours. The profit per unit of P is \$8 and Q is \$6. Formulate the LPP to maximize profit.

a) Maximize $Z = 8x_1 + 6x_2$ subject to $3x_1 + 2x_2 \geq 18$, $x_1 + 2x_2 \geq 10$, $x_1 \geq 0$, $x_2 \geq 0$

b) Maximize $Z = 8x_1 + 6x_2$ subject to $3x_1 + 2x_2 \leq 18$, $x_1 + 2x_2 \leq 10$, $x_1 \geq 0$, $x_2 \geq 0$

c) Minimize $Z = 8x_1 + 6x_2$ subject to $3x_1 + 2x_2 \leq 18$, $x_1 + 2x_2 \leq 10$, $x_1 \geq 0$, $x_2 \geq 0$

d) Maximize $Z = 8x_1 + 6x_2$ subject to $3x_1 + 2x_2 \leq 18$, $x_1 + 2x_2 \leq 10$, $x_1 \leq 0$, $x_2 \leq 0$

267 / 576

Category: Linear Inequations in Two Variables

267. (A) The inequality $3x + 4y \leq 12$ represents a closed half-plane including the line $3x + 4y = 12$.
(R) For any linear inequality in two variables, if the inequality includes equality ($\leq$ or $\geq$), the associated boundary line is part of the solution region.

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.

b) Both Assertion and Reason are true, but Reason is NOT the correct explanation of Assertion.

c) Assertion is true, but Reason is false.

d) Assertion is false, but Reason is true.

268 / 576

Category: Transportation Problems – Minimize cost of shipment.

268. Which condition correctly represents the demand constraint in a transportation problem?

a) $\sum_{j=1}^{n} x_{ij} \leq b_j$

b) $\sum_{i=1}^{m} x_{ij} \geq b_j$

c) $x_{ij} \leq c_{ij}$

d) $a_i \geq b_j$

269 / 576

Category: The common region represents the solution set (infinite ordered pairs).

269. For the given system of inequalities $$ x + y \leq 5$$, $$ 2x - y \leq 4$$, and $$ y \geq 0$$, which point does NOT lie in the common solution region?

a) $(1, 2)$

b) $(3, 1)$

c) $(0, 0)$

d) $(2, 2)$

270 / 576

Category: Fundamental Theorem (Extreme Point Theorem)

270. For the objective function $Z = 4x + 5y$ subject to the constraints $x \geq 0$, $y \geq 0$, $x + 2y \leq 8$, $3x + y \leq 9$, which corner point gives the maximum value of $Z$?

a) $(0, 0)$

b) $(0, 4)$

c) $(3, 0)$

d) $(2, 3)$

271 / 576

Category: Areas of Application

271. (A) Linear programming techniques cannot account for qualitative factors such as human behavior and credibility.
(R) Linear programming requires all elements related to a problem to be quantifiable.

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.

b) Both Assertion and Reason are true, but Reason is NOT the correct explanation of Assertion.

c) Assertion is true, but Reason is false.

d) Assertion is false, but Reason is true.

272 / 576

Category: GENERAL LINAR PROGRAMMING PROBLEM

272. A factory produces two types of widgets: X and Y. The profit from each unit of X is \$3 and from each unit of Y is \$4. The production requires 2 hours of labor for X and 3 hours for Y per unit. The total labor hours available are 120. Also, the maximum number of units that can be produced in total is 50. What is the correct formulation of this LPP if the goal is to maximize the profit?

a) Maximize $Z = 3x + 4y$; Subject to: $2x + 3y \leq 120$, $x + y \leq 50$, $x, y \geq 0$

b) Maximize $Z = 4x + 3y$; Subject to: $3x + 2y \leq 120$, $x + y \leq 50$, $x, y \geq 0$

c) Maximize $Z = 3x + 4y$; Subject to: $2x + 3y \geq 120$, $x + y \leq 50$, $x, y \geq 0$

d) Maximize $Z = 4x + 3y$; Subject to: $2x + 3y \leq 120$, $x + y \geq 50$, $x, y \geq 0$

273 / 576

Category: AREAS OF APPLICATIONS OF LINEAR PROGRAMMING

273. A factory produces two products, A and B. Each unit of A requires 2 hours of machining and 1 hour of assembly. Each unit of B requires 1 hour of machining and 3 hours of assembly. The factory has 60 hours of machining time and 90 hours of assembly time available per week. If the profit per unit of A is \$20 and B is \$30, what is the constraint representing the assembly time limitation?

a) $2x_A + x_B \leq 60$

b) $x_A + 3x_B \leq 90$

c) $x_A + x_B \leq 150$

d) $3x_A + x_B \leq 90$

274 / 576

Category: Linear Programming Problem (LPP) Example

274. A farmer wants to cultivate wheat and barley on his farm. Wheat gives a profit of \$50 per acre and requires 2 hours of labor per acre. Barley gives a profit of \$70 per acre and requires 3 hours of labor per acre. The farmer has 60 acres of land and 150 hours of labor available. How should he allocate the land to maximize profit?

a) 3000

b) 3400

c) 3600

d) 4000

275 / 576

Category: Non-negativity restriction.

275. Consider the following LPP: Maximize $Z = 4x + 5y$ subject to $3x + y \leq 12$, $x + 2y \leq 10$, and $x \geq -1$, $y \geq 0$. Which statement is true about the non-negativity restrictions here?

a) The LPP follows non-negativity restrictions as $y \geq 0$.

b) The LPP does not follow non-negativity restrictions because $x \geq -1$ allows negative values for $x$.

c) Non-negativity restrictions are not required in this LPP.

d) Both $x$ and $y$ must be strictly positive.

276 / 576

Category: CORNER POINT METHOD.

276. Given the constraints:
$2x + y \geq 8$
$x + 2y \geq 10$
$x \geq 0, y \geq 0$
Find the minimum value of $Z = x + 3y$.

a) 24

b) 14

c) 10

d) 8

277 / 576

Category: Linear Programming Problem (LPP) Example

277. What is one of the conditions required for a solution to lie within the feasible region of the given LPP?

a) $x_1 + x_2 + x_3 \leq 100$

b) $3x_1 + 2x_2 + 4x_3 \leq 210$

c) $x_1 \geq 0, x_2 \geq 0, x_3 \geq 0$

d) $5x_1 + 4x_2 + 6x_3 \leq 500$

278 / 576

Category: Constraints.

278. Consider the following LPP:
Maximize $Z = 5x + 3y$
subject to:
$2x + y \leq 20$
$x + 2y \leq 16$
$x \geq 0$, $y \geq 0$.
What is the maximum value of $Z$ in the feasible region?

a) 24

b) 40

c) 50

d) 52

279 / 576

Category: AREAS OF APPLICATIONS OF LINEAR PROGRAMMING

279. What is the primary goal of applying linear programming in manufacturing problems?

a) Minimizing product quality

b) Maximizing profit within constraints

c) Reducing customer satisfaction

d) Increasing waste production

280 / 576

Category: Represent graphically and find feasible region.

280. What is the feasible region in a linear programming problem?

a) The area where the objective function is maximized.

b) The area where at least one constraint is satisfied.

c) The area outside the constraints.

281 / 576

Category: LINEAR PROGRAMMING

281. (A) In a Linear Programming Problem, the decision variables must always satisfy non-negativity constraints.
(R) Negative values of decision variables have no valid interpretation in physical problems involving resources or quantities.

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.

b) Both Assertion and Reason are true, but Reason is NOT the correct explanation of Assertion.

c) Assertion is true, but Reason is false.

d) Assertion is false, but Reason is true.

282 / 576

Category: Convex region.

282. (A) The feasible region in a linear programming problem is always convex.
(R) A convex region ensures that any line segment joining two points within the region lies entirely within the region.

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.

b) Both Assertion and Reason are true, but Reason is NOT the correct explanation of Assertion.

c) Assertion is true, but Reason is false.

d) Assertion is false, but Reason is true.

283 / 576

Category: ADVANTAGES OF LINEAR PROGRAMMING PROBLEMS

283. In a primal LPP where the objective is maximized, if the shadow price for a constraint is \$5, what does this indicate about the dual problem?

a) The dual problem has no feasible solution.

b) The optimal value of the corresponding dual variable is \$5.

c) The dual problem must be unbounded.

d) The shadow price is unrelated to the dual problem.

284 / 576

Category: Manufacturing Problems – Optimize production for profit.

284. In an optimal solution producing P and Q, the shadow price for material constraint is \$15 per kg. If we increase material availability by 2 kg, what happens to maximum profit?

a) Increases by exactly \$30

b) Increases by more than \$30

c) Decreases by \$30

d) Remains unchanged

285 / 576

Category: Blending Problems – Optimize combination of raw materials.

285. Consider a blending problem where raw materials P and Q are mixed. The constraints are $2p + q \leq 10$, $p + 2q \leq 8$, $p \geq 0$, $q \geq 0$. Which of the following points lies within the feasible region?

a) $(1, 5)$

b) $(2, 2)$

c) $(3, 4)$

d) $(0, 6)$

286 / 576

Category: Constraints.

286. What are constraints in a Linear Programming Problem (LPP)?

a) Non-linear equations describing profit maximization

b) System of inequations or equations defining optimization conditions

c) Objective function of the LPP

d) Graphical representation of feasible region

287 / 576

Category: Manufacturing Problems – Optimize production for profit.

287. A company makes two models M and N. Model M requires 2 units of resource R and 3 units of resource S, while model N requires 4 units of R and 1 unit of S. The profit per unit of M is \$12 and for N is \$8. Total resources available are 100 units of R and 90 units of S. Which point lies in the feasible region for this problem?

a) $(10, 20)$

b) $(30, 10)$

c) $(20, 20)$

d) $(15, 25)$

288 / 576

Category: The common region represents the solution set (infinite ordered pairs).

288. (A) The common region formed by the intersection of graphs of inequalities represents the solution set of the system.
(R) The solution set of a system of inequalities consists of ordered pairs that satisfy all the given inequalities simultaneously.

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.

b) Both Assertion and Reason are true, but Reason is NOT the correct explanation of Assertion.

c) Assertion is true, but Reason is false.

d) Assertion is false, but Reason is true.

289 / 576

Category: Identify common overlapping region (solution region).

289. Which point lies in the common solution region for the system of inequalities $x + y \leq 4$ and $x - y \geq 1$?

a) $(1,1)$

b) $(3,1)$

c) $(0,4)$

d) $(5,0)$

290 / 576

Category: Developed by Kantorovich and George Dantzig.

290. A company produces two products P and Q using two machines M1 and M2. Product P requires 1 hour on M1 and 2 hours on M2; product Q requires 3 hours on M1 and 1 hour on M2. Available machine time is 60 hours for M1 and 40 hours for M2. The profit per unit is \$300 for P and \$400 for Q. Assuming production quantities are non-negative, what is the maximum possible profit?

a) \$8000

b) \$10000

c) \$6000

d) \$9000

291 / 576

Category: Investment Problems – Maximize return on capital allocation.

291. An investor has \$80,000 to allocate between two bonds with annual returns of 6\% and 9\%. No more than 50\% of the capital can be invested in the higher-return bond due to risk constraints. What is the optimal allocation to maximize the return?

a) \$5,500

b) \$6,000

c) \$6,500

d) \$7,000

292 / 576

Category: Developed by Kantorovich and George Dantzig.

292. Which of the following represents the standard form of a linear programming problem to maximize $Z$?

a) Minimize $Z = c_1x_1 + c_2x_2 + \dots + c_nx_n$ with equality constraints.

b) Maximize $Z = c_1x_1 + c_2x_2 + \dots + c_nx_n$ with inequality constraints and non-negativity.

c) Minimize $Z = c_1x_1 + c_2x_2 + \dots + c_nx_n$ with non-negativity only.

d) Maximize $Z = c_1x_1 + c_2x_2 + \dots + c_nx_n$ without constraints.

293 / 576

Category: System of Inequations

293. Consider the constraints $2x + 3y \leq 6$ and $x \geq 0, y \geq 0$. Which point lies within the feasible region?

a) $(0, 3)$

b) $(1, 1)$

c) $(3, 0)$

d) $(-1, 1)$

294 / 576

Category: SOLVED EXAMPLES

294. A dietitian wants to minimize the cost of a diet with foods F1 (\$2/unit) and F2 (\$3/unit). The diet must contain at least 5 units of carbohydrates and 7 units of proteins. F1 provides 1 unit of carbohydrates and 2 units of proteins per unit, while F2 provides 2 units of carbohydrates and 1 unit of proteins per unit. What is the minimum cost?

a) \$8

b) \$9

c) \$10

d) \$11

295 / 576

Category: Linear Programming Problem (LPP) Example

295. The candidate must not spend more than 150 minutes on 5-point and 4-point problems combined. If $x_1$ and $x_2$ are the number of 5-point and 4-point problems respectively, the constraint is:

a) $2x_1 + 3x_2 \leq 150$

b) $3x_1 + 2x_2 \geq 150$

c) $3x_1 + 2x_2 \leq 150$

d) $5x_1 + 4x_2 \leq 150$

296 / 576

Category: Fundamental Theorem (Extreme Point Theorem)

296. Which of the following best describes a feasible region in linear programming?

a) The region where the objective function is maximized

b) The set of all points satisfying all constraints of the LPP

c) The area outside the given constraints

d) The union of all infeasible solutions

297 / 576

Category: Graphical solution of linear inequalities forms the basis of linear programming.

297. Which region represents the solution to the inequality $2x - y \leq 4$?

a) Above the line $2x - y = 4$

b) Below or on the line $2x - y = 4$

c) Only on the line $2x - y = 4$

d) None of the above

298 / 576

Category: Identify decision variables (e.g., x, y).

298. In a linear programming problem, what do decision variables represent?

a) The final solution to the problem

b) The constraints of the problem

c) The objective function coefficients

d) Quantities whose values we wish to determine

299 / 576

Category: LINAR PROGRAMMING-DEFINITION

299. (A) In a linear programming problem, the objective function must always be a linear expression in terms of the decision variables.
(R) The non-negativity restrictions ensure that all feasible solutions lie in the first quadrant or on its axes.

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.

b) Both Assertion and Reason are true, but Reason is NOT the correct explanation of Assertion.

c) Assertion is true, but Reason is false.

d) Assertion is false, but Reason is true.

300 / 576

Category: LINAR PROGRAMMING-DEFINITION

300. What do non-negativity restrictions enforce in a Linear Programming Problem (LPP)?

a) Decision variables must be integers

b) Decision variables must be non-negative

c) Objective function must be minimized

d) Constraints must be linear equations

301 / 576

Category: SYSTEM OF THREE OR MORE INEQULITIES

301. (A) The feasible region for the system of inequalities $x + y \geq 4$, $3x - y \leq 6$, and $x - y \leq 2$ is unbounded.
(R) An unbounded feasible region implies that the objective function in a linear programming problem has no finite optimal solution.

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.

b) Both Assertion and Reason are true, but Reason is NOT the correct explanation of Assertion.

c) Assertion is true, but Reason is false.

d) Assertion is false, but Reason is true.

302 / 576

Category: Transportation Problems – Minimize cost of shipment.

302. In a transportation problem with three sources (A, B, C) and two destinations (X, Y), the supply constraints for sources A, B, and C are 200, 300, and 400 units respectively. If $x_{AX}$, $x_{AY}$, $x_{BX}$, $x_{BY}$, $x_{CX}$, $x_{CY}$ represent the quantities transported, which of the following correctly represents the supply constraint for source B?

a) $x_{BX} + x_{BY} \geq 300$

b) $x_{AX} + x_{AY} \leq 300$

c) $x_{BX} + x_{BY} \leq 300$

d) $x_{CX} + x_{CY} \leq 300$

303 / 576

Category: Infeasible solution.

303. A solution to an LPP with constraints $2x + 3y \leq 12$, $x \geq 0$, $y \geq 0$ is infeasible if:

a) $x = 0, y = 0$

b) $x = 3, y = 2$

c) $x = 10, y = -5$

d) $x = 6, y = 0$

304 / 576

Category: System of Inequations

304. In an LPP, what is the purpose of the objective function $Z = 5x + 3y$?

a) Defines the constraints of the problem

b) Determines the feasibility region boundary

c) Represents the function to be maximized or minimized

d) Calculates the area of the feasible solutions

305 / 576

Category: Identify common overlapping region (solution region).

305. Given a graph where the common solution region satisfies $y \leq x + 1$, $y \geq -x + 1$, and $y \geq 0$, which additional constraint would make the solution set a triangle?

a) $x \leq 1$

b) $y \leq 2$

c) $x + y \leq 3$

d) $x \geq 0$

306 / 576

Category: Blending Problems – Optimize combination of raw materials.

306. Which of the following represents a constraint in a blending problem where $x_1$ and $x_2$ are quantities of two raw materials, and $b_1$ is the available quantity limit?

a) $Z = c_1x_1 + c_2x_2$

b) $x_1 + x_2 \geq 0$

c) $a_{11}x_1 + a_{12}x_2 \leq b_1$

d) $x_1 - x_2 = 0$

307 / 576

Category: Graphical Method of Solution

307. A farmer wants to allocate land between wheat and corn. The profit per acre is \$100 for wheat and \$150 for corn. The constraints are:
$2x + y \leq 80$ (land),
$x + 2y \leq 100$ (labor),
where $x$ and $y$ are acres of wheat and corn, respectively. How many acres of each crop should be planted to maximize profit?

a) $(40, 0)$

b) $(0, 50)$

c) $(20, 40)$

d) $(0, 0)$

308 / 576

Category: Constraints.

308. In a linear programming problem, the constraints are given as $2x + y \leq 50$, $x + 2y \leq 70$, $x \geq 0$, and $y \geq 0$. Which of the following is NOT part of the feasible region?

a) (10, 20)

b) (15, 25)

c) (20, 15)

d) (5, 10)

309 / 576

Category: SYSTEM OF THREE OR MORE INEQULITIES

309. For the system $x + y \leq 4$, $x \geq 1$, $y \geq 0$, what is one of the corner points of the feasible region?

a) $(0, 0)$

b) $(1, 3)$

c) $(2, 2)$

d) $(4, 4)$

310 / 576

Category: Steps in Formulating a Linear Programming Model

310. What is the first step in formulating a linear programming model?

a) Identify the constraints and express them as linear equations/inequations

b) Identify the objective function and express it as a linear function

c) Add non-negativity restrictions on the decision variables

d) Identify the decision variables and assign symbols to them

311 / 576

Category: Applications: maximize profit, minimize cost, optimal resource utilization.

311. (A) In a transportation problem, the objective is to minimize the total cost of transporting goods from sources to destinations while satisfying supply and demand constraints.
(R) The constraints in a transportation problem must always be formulated as equations because supply must exactly match demand.

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.

b) Both Assertion and Reason are true, but Reason is NOT the correct explanation of Assertion.

c) Assertion is true, but Reason is false.

d) Assertion is false, but Reason is true.

312 / 576

Category: Infeasible solution.

312. Which of the following is an infeasible solution to the LPP: Maximize $Z = 6x + 8y$ subject to $3x + 2y \leq 30$, $x + 2y \leq 22$, $x \geq 0$, $y \geq 0$?

a) $(x, y) = (-2, 7)$

b) $(x, y) = (0, -2)$

c) $(x, y) = (2, 3)$

d) $(x, y) = (5, 10)$

313 / 576

Category: SOLVED EXAMPLES

313. (A) In the given problem of maximizing credits with time constraints, the candidate should solve all 6-point problems first to maximize his score.
(R) The 6-point problems give the highest credit per unit time spent compared to 5-point and 4-point problems.

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.

b) Both Assertion and Reason are true, but Reason is NOT the correct explanation of Assertion.

c) Assertion is true, but Reason is false.

d) Assertion is false, but Reason is true.

314 / 576

Category: Developed by Kantorovich and George Dantzig.

314. (A) The technique of linear programming was developed by Kantorovich and George Dantzig to optimize an objective function under given constraints.
(R) Linear programming is suitable only for problems involving two variables.

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.

b) Both Assertion and Reason are true, but Reason is NOT the correct explanation of Assertion.

c) Assertion is true, but Reason is false.

d) Assertion is false, but Reason is true.

315 / 576

Category: Graphical Method of Solution

315. (A) The optimal solution to a linear programming problem with two variables will always lie at a corner point of the feasible region.
(R) According to the Extreme Point Theorem, if an optimal solution exists for a linear programming problem, it occurs at one of the extreme (corner) points of the feasible region.

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.

b) Both Assertion and Reason are true, but Reason is NOT the correct explanation of Assertion.

c) Assertion is true, but Reason is false.

d) Assertion is false, but Reason is true.

316 / 576

Category: Write down constraints in terms of inequalities/equations.

316. A company wants to maximize profit (\$) by producing two products, A and B. The profit per unit of A is \$20 and per unit of B is \$30. If $x$ and $y$ represent the quantities of A and B respectively, which of the following correctly models the objective function?

a) $Z = 30x + 20y$

b) $Z = 20x - 30y$

c) $Z = 20x + 30y$

d) $Z = 50xy$

317 / 576

Category: Write down constraints in terms of inequalities/equations.

317. A farmer wants to maximize profit by growing wheat and barley. Wheat gives \$120/acre profit and requires 2 tons of fertilizer per acre. Barley gives \$90/acre profit and needs 1 ton of fertilizer per acre. The farm has 100 acres and can obtain up to 120 tons of fertilizer. Let w = wheat acres and b = barley acres. Which is NOT a valid constraint for this problem?

a) $w + b \leq 100$

b) $2w + b \leq 120$

c) $w \leq 40$

d) $120w + 90b \leq 10,000$

318 / 576

Category: The common region represents the solution set (infinite ordered pairs).

318. (A) The solution set of the system of inequations $ 2x + 3y \leq 6 \end{dmatrix*}, $ x \geq 0 \end{dmatrix*}, $ y \geq 0 \end{dmatrix*} is represented by an infinite set of ordered pairs in the first quadrant.

(R) The common region formed by the intersection of the half-planes of these inequations includes all points that satisfy all three conditions simultaneously.

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.

b) Both Assertion and Reason are true, but Reason is NOT the correct explanation of Assertion.

c) Assertion is true, but Reason is false.

d) Assertion is false, but Reason is true.

319 / 576

Category: Graphical solution of linear inequalities forms the basis of linear programming.

319. (A) The feasible region of a linear programming problem is always a convex set.
(R) The intersection of any number of half-planes results in a convex region.

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.

b) Both Assertion and Reason are true, but Reason is NOT the correct explanation of Assertion.

c) Assertion is true, but Reason is false.

d) Assertion is false, but Reason is true.

320 / 576

Category: Identify common overlapping region (solution region).

320. (A) The solution region for the system of inequalities $x + y \leq 5$ and $x - y \geq 1$ includes the point $(3, 2)$.
(R) A point that satisfies all given inequalities lies in the common overlapping region.

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.

b) Both Assertion and Reason are true, but Reason is NOT the correct explanation of Assertion.

c) Assertion is true, but Reason is false.

d) Assertion is false, but Reason is true.

321 / 576

Category: Blending Problems – Optimize combination of raw materials.

321. A fertilizer company produces two blends, X and Y, using three nutrients N, P, and K. Blend X requires 2kg of N, 1kg of P, and 3kg of K per bag. Blend Y requires 1kg of N, 2kg of P, and 2kg of K per bag. The company has 100kg of N, 80kg of P, and 120kg of K available. What is the maximum number of bags that can be produced if both blends must be made in equal quantities?

a) 40 bags

b) 44 bags

c) 48 bags

d) 52 bags

322 / 576

Category: Blending Problems – Optimize combination of raw materials.

322. A paint manufacturer makes two types of paint: Interior (I) and Exterior (E). Each gallon of I uses 2 units of pigment and 1 unit of binder, yielding \$30 profit. Each gallon of E uses 1 unit of pigment and 2 units of binder, yielding \$40 profit. If there are 60 units of pigment and 75 units of binder available, and demand limits E to at most 30 gallons, what is the optimal production mix?

a) 20 gallons I, 20 gallons E

b) 10 gallons I, 30 gallons E

c) 15 gallons I, 30 gallons E

d) 30 gallons I, 15 gallons E

323 / 576

Category: Manufacturing Problems – Optimize production for profit.

323. (A) In a manufacturing problem, if the feasible region is unbounded, the optimal solution for profit maximization may not exist.
(R) An unbounded feasible region implies that the objective function can increase indefinitely without violating any constraints.

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.

b) Both Assertion and Reason are true, but Reason is NOT the correct explanation of Assertion.

c) Assertion is true, but Reason is false.

d) Assertion is false, but Reason is true.

324 / 576

Category: SOME DEFINITIONS AND CONCEPTS RELATED TO LPP

324. (A) The feasible region of a Linear Programming Problem (LPP) is always a convex set.
(R) A convex region is defined as a region where the line segment joining any two points in the region lies entirely within the region.

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.

b) Both Assertion and Reason are true, but Reason is NOT the correct explanation of Assertion.

c) Assertion is true, but Reason is false.

d) Assertion is false, but Reason is true.

325 / 576

Category: SOME DEFINITIONS AND CONCEPTS RELATED TO LPP

325. Given the LPP: Maximize $Z = 2x + 3y$ subject to $x - y \leq 2$, $x + y \leq 4$, $x \geq 0$, $y \geq 0$. Which of the following solutions is infeasible?

a) (1, 1)

b) (2, 2)

c) (3, 1)

d) (5, 1)

326 / 576

Category: ADVANTAGES OF LINEAR PROGRAMMING PROBLEMS

326. In which of the following industries can linear programming be applied due to its versatility?

a) Only in manufacturing

b) Only in finance

c) Only in agriculture

d) Manufacturing, transportation, finance, and agriculture

327 / 576

Category: Diet Problems – Minimize cost subject to nutritional constraints.

327. (A) In a linear programming diet problem, the constraints always include non-negativity conditions for the decision variables.
(R) Non-negativity constraints ensure that the quantities of foods included in the diet cannot be negative.

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.

b) Both Assertion and Reason are true, but Reason is NOT the correct explanation of Assertion.

c) Assertion is true, but Reason is false.

d) Assertion is false, but Reason is true.

328 / 576

Category: System of Inequations

328. A company manufactures two products A and B with profits of \$4 and \$5 per unit, respectively. The production constraints are:
$2x + 3y \leq 24$
$x + y \leq 10$
$x \geq 0, y \geq 0$
where $x$ and $y$ are units of A and B produced. What is the maximum profit achievable under these constraints?

a) \$40

b) \$42

c) \$44

d) \$50

329 / 576

Category: Advertising Problems – Allocate funds for maximum exposure.

329. An advertiser has \$100,000 to spend on three media: Radio (\$3,000 per spot, 150 audience/unit), Print (\$1,500 per ad, 80 audience/unit), and Digital (\$4,000 per campaign, 300 audience/unit). They must buy at least 10 radio spots, print ads cannot exceed 25\% of total units purchased, and digital campaigns must be at least half of radio spots. What combination maximizes audience reach?

a) 10 radio, 6 print, 10 digital

b) 12 radio, 6 print, 6 digital

c) 10 radio, 5 print, 5 digital

d) 15 radio, 3 print, 8 digital

330 / 576

Category: Manufacturing Problems – Optimize production for profit.

330. For a manufacturing problem with constraints: x + 2y ≤ 100, 3x + y ≤ 90, x ≥ 0, y ≥ 0, which point lies outside the feasible region?

a) (10,45)

b) (20,30)

c) (16,42)

d) (30,40)

331 / 576

Category: Represent graphically and find feasible region.

331. For the inequalities $2x + y \geq 4$ and $x - y \leq 2$, which region represents the feasible solution?

a) Above $2x + y = 4$ only

b) Below $x - y = 2$ only

c) Intersection of regions from both inequalities

d) Outside both regions

332 / 576

Category: LINEAR PROGRAMMING

332. A company produces two products, P and Q, which yield a profit of \$5 and \$3 per unit respectively. The objective function to maximize the profit is:

a) $Z = 3x + 5y$

b) $Z = 5x - 3y$

c) $Z = 5x + 3y$

d) $Z = 3x - 5y$

333 / 576

Category: Non-negativity restriction.

333. What is the purpose of non-negativity restrictions in a Linear Programming Problem (LPP)?

a) To maximize the objective function

b) To minimize computational complexity

c) To ensure variables take non-negative values only

d) To eliminate all constraints from the problem

334 / 576

Category: GENERAL LINAR PROGRAMMING PROBLEM

334. A company manufactures two products A and B. Profit per unit of A is \$20 and of B is \$30. Each product requires processing on machines M1 and M2. Product A needs 2 hours on M1 and 1 hour on M2; B needs 1 hour on M1 and 2 hours on M2. The total time available on M1 is 60 hours and on M2 is 48 hours. Which of the following correctly represents the constraints?

a) $2x + y \leq 60$, $x + 2y \leq 48$, $x, y \geq 0$

b) $x + 2y \leq 60$, $2x + y \leq 48$, $x, y \geq 0$

c) $2x + y \leq 48$, $x + 2y \leq 60$, $x, y \geq 0$

d) $x + y \leq 60$, $x + y \leq 48$, $x, y \geq 0$

335 / 576

Category: Transportation Problems – Minimize cost of shipment.

335. A company has two factories (P and Q) and two warehouses (R and S). The supply capacities of P and Q are 100 and 150 units respectively. The demands at R and S are 120 and 130 units respectively. The transportation costs per unit are: from P to R is \$5, from P to S is \$8, from Q to R is \$6, and from Q to S is \$4. Which of the following correctly represents the objective function to minimize total transportation cost?

a) $Z = 5x_{PR} + 8x_{PS} + 6x_{QR}$

b) $Z = 5x_{PR} + 8x_{PS} + 6x_{QR} + 4x_{QS}$

c) $Z = 5x_{PR} + 6x_{QR} + 4x_{QS}$

d) $Z = 8x_{PS} + 6x_{QR} + 4x_{QS}$

336 / 576

Category: Define objective function (profit, cost, etc.).

336. A factory manufactures chairs and tables. Each chair requires 2 hours of labor and each table requires 3 hours. The total available labor hours are 120. If $x$ is the number of chairs and $y$ is the number of tables, what is the constraint representing the labor limitation?

a) $2x + 3y \geq 120$

b) $3x + 2y \leq 120$

c) $2x + 3y \leq 120$

d) $x + y \leq 120$

337 / 576

Category: Non-negativity restriction.

337. Which of the following constraints represents a non-negativity restriction?

a) $3x + 2y \leq 30$

b) $x \geq 0$

c) $x + y = 10$

d) $4x - 5y > 12$

338 / 576

Category: Objective function.

338. Which of the following correctly defines the objective function in a Linear Programming Problem (LPP)?

a) A nonlinear function involving quadratic terms.

b) The linear function whose maximum or minimum value is determined.

c) A system of inequalities representing constraints.

d) The region containing all feasible solutions.

339 / 576

Category: SYSTEM OF THREE OR MORE INEQULITIES

339. Given the system of inequalities:
$x + 2y \leq 6$
$x - y \geq 1$
$x, y \geq 0$
What are the corner points of the feasible region?

a) $(1, 0)$, $(0, 0)$, $(6, 0)$

b) $(2.67, 1.67)$, $(1, 0)$, $(6, 0)$

c) $(1, 0)$, $(6, 0)$, $(0, 3)$

d) $(0, 0)$, $(0, 3)$, $(1, 0)$

340 / 576

Category: SYSTEM OF THREE OR MORE INEQULITIES

340. The system of inequalities $x^2 + y^2 \leq 16$, $x + y \geq 2$, and $y \leq x + 2$ defines a feasible region. Which point lies within this region?

a) $(0, 0)$

b) $(4, 0)$

c) $(2, 2)$

d) $(1, -2)$

341 / 576

Category: AREAS OF APPLICATIONS OF LINEAR PROGRAMMING

341. A company produces two types of health supplements, A and B. Each unit of supplement A costs \$3 to produce and provides 10 mg of protein and 5 mg of vitamins. Each unit of supplement B costs \$4 to produce and provides 8 mg of protein and 10 mg of vitamins. The daily requirement is at least 200 mg of protein and 150 mg of vitamins. If the company also wants to invest in a fixed-income security yielding a return of 5\% per annum while minimizing production costs, what is the optimal number of units of each supplement to be produced daily?

a) 10 units of A and 15 units of B

b) 15 units of A and 10 units of B

c) 20 units of A and 0 units of B

d) 0 units of A and 20 units of B

342 / 576

Category: Infeasible solution.

342. Which of the following is an infeasible solution for the LPP: Maximize $Z = 5x + 3y$ subject to $2x + y \leq 10$, $x + 3y \leq 15$, $x \geq 0$, $y \geq 0$?

a) $x = 4$, $y = 1$

b) $x = 6$, $y = 2$

c) $x = 0$, $y = 5$

d) $x = 3$, $y = 2$

343 / 576

Category: Define objective function (profit, cost, etc.).

343. An agricultural firm grows crops A and B on limited land (5A + 3B ≤ 300 acres) with water restriction (2A + 4B ≤ 200 units). If crop A yields \$150/acre profit but requires special fertilizer (A ≤ 40 acres), while crop B gives \$100/acre profit, what is the complete optimization problem formulation?

a) Maximize 150A + 100B subject to 5A + 3B ≤ 300, 2A + 4B ≤ 200, A ≥ 0

b) Maximize 150A + 100B subject to 5A + 3B ≤ 300, 2A + 4B ≤ 200, A ≤ 40, A ≥ 0, B ≥ 0

c) Maximize 100A + 150B subject to all given constraints

d) Maximize 150A + 100B with only land and water constraints

344 / 576

Category: GRAPHICAL SOLUTION OF LINEAR INEQUATIONS

344. (A) The solution region of the system of inequations $3x + 5y \leq 15$ and $x \geq 1$ includes all points on the line $x = 1$.
(R) The boundary line $x = 1$ is part of the solution set for the inequation $x \geq 1$.

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.

b) Both Assertion and Reason are true, but Reason is NOT the correct explanation of Assertion.

c) Assertion is true, but Reason is false.

d) Assertion is false, but Reason is true.

345 / 576

Category: Represent graphically and find feasible region.

345. (A) The feasible region for a linear programming problem is always a convex polygon.
(R) The intersection of multiple linear inequalities results in a convex set.

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.

b) Both Assertion and Reason are true, but Reason is NOT the correct explanation of Assertion.

c) Assertion is true, but Reason is false.

d) Assertion is false, but Reason is true.

346 / 576

Category: Graphical solution of linear inequalities forms the basis of linear programming.

346. Find the system of linear inequalities representing the feasible region with corner points at $(0, 0)$, $(4, 0)$, $(3, 2)$, and $(0, 3)$.

a) $x \geq 0$, $y \geq 0$, $2x + y \leq 8$, $x + 3y \leq 9$

b) $x \geq 0$, $y \geq 0$, $x + y \leq 5$, $2x + y \leq 8$

c) $x \geq 0$, $y \geq 0$, $x + 2y \leq 6$, $3x + y \leq 12$

d) $x \geq 0$, $y \geq 0$, $x + y \leq 7$, $x + 2y \leq 10$

347 / 576

Category: Understanding regions of inequality is essential before tackling LPP problems.

347. Which of the following would make a solution infeasible for an LPP with constraints $3x + 2y \leq 18$, $x + 4y \leq 16$, and non-negativity restrictions?

a) $(4, 2)$

b) $(0, 4)$

c) $(5, 1)$

d) $(2, 6)$

348 / 576

Category: Define objective function (profit, cost, etc.).

348. (A) The objective function in a linear programming problem represents the quantity to be maximized or minimized.
(R) The objective function is typically expressed as a linear combination of decision variables, such as $Z = c_1x + c_2y$.

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.

b) Both Assertion and Reason are true, but Reason is NOT the correct explanation of Assertion.

c) Assertion is true, but Reason is false.

d) Assertion is false, but Reason is true.

349 / 576

Category: System of Inequations

349. Which of the following systems of inequalities has no feasible solution?

a) $x + y \leq 4$, $2x + y \geq 6$, $x \geq 0$, $y \geq 0$

b) $3x + 2y \leq 6$, $-x + y \geq 1$, $x \geq 0$, $y \geq 0$

c) $x + y \geq 5$, $x + y \leq 1$, $x \geq 0$, $y \geq 0$

d) $x \geq 0$, $y \geq 0$, $x + y \leq 3$

350 / 576

Category: Convex region.

350. What is a convex region in the context of Linear Programming Problems (LPP)?

a) A region with only straight edges.

b) A region where any two points can be connected by a straight line lying entirely within the region.

c) A region bounded by linear inequalities.

d) A region containing all its boundary points.

351 / 576

Category: Linear Inequations in Two Variables

351. For the system of inequalities: $x + y \leq 5$ and $2x - y \geq 2$, which point lies in the feasible region?

a) $(2, 3)$

b) $(3, 1)$

c) $(4, 0)$

d) $(1, 4)$

352 / 576

Category: A method for optimizing (maximizing/minimizing) a linear function subject to linear constraints.

352. For an LPP, the feasible region has corner points at $(0, 0)$, $(0, 5)$, $(4, 3)$, and $(6, 0)$. If the objective function is $Z = 3x + 4y$, which corner point minimizes $Z$?

a) (0, 5)

b) (4, 3)

c) (6, 0)

d) (0, 0)

353 / 576

Category: THE GRAPHICAL METHOD OF SOLUTION OF LINEAR PROGRAMMING PROBLEMS

353. (A) The graphical method can be used to solve linear programming problems with two variables.
(R) The graphical method involves plotting the constraints and finding the feasible region.

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.

b) Both Assertion and Reason are true, but Reason is NOT the correct explanation of Assertion.

c) Assertion is true, but Reason is false.

d) Assertion is false, but Reason is true.

354 / 576

Category: Feasible region : i region and Feasible solution.

354. What is a feasible solution in Linear Programming Problems (LPP)?

a) A solution that violates at least one constraint or non-negativity condition.

b) A solution that optimizes the objective function.

c) Any point within the feasible region that satisfies all constraints and non-negativity conditions.

d) A solution that lies outside the feasible region.

355 / 576

Category: ADVANTAGES OF LINEAR PROGRAMMING PROBLEMS

355. A company wants to maximize profit by producing two products, X and Y, with constraints given by $2x + y \leq 8$ and $x + 3y \leq 9$, where $x \geq 0$ and $y \geq 0$. If the objective function is $z = 5x + 4y$, what is the maximum value of $z$ at an optimal feasible solution?

a) 20

b) 23

c) 12

d) 18

356 / 576

Category: Linear Inequations in Two Variables

356. What is the associated equation of the inequality $5x + 4y < 20$?

a) $5x + 4y > 20$

b) $5x + 4y = 20$

c) $5x - 4y = 20$

d) $5x + 4y \leq 20$

357 / 576

Category: Check feasible side using a test point (often origin).

357. Given the feasible region lies on the opposite side of the origin for the line $2x - 5y = 10$, what is the correct constraint?

a) $2x - 5y \leq 10$

b) $2x - 5y < 10$

c) $2x - 5y \geq 10$

d) $2x - 5y > 10$

358 / 576

Category: Developed by Kantorovich and George Dantzig.

358. Who were the key developers of linear programming?

a) Isaac Newton and Albert Einstein

b) Leonhard Euler and Carl Friedrich Gauss

c) Kantorovich and George Dantzig

d) John von Neumann and Alan Turing

359 / 576

Category: Transportation Problems – Minimize cost of shipment.

359. (A) The objective function in a transportation problem is to maximize the total shipping cost.

(R) In transportation problems, we aim to minimize costs while meeting supply and demand constraints.

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.

b) Both Assertion and Reason are true, but Reason is NOT the correct explanation of Assertion.

c) Assertion is true, but Reason is false.

d) Assertion is false, but Reason is true.

360 / 576

Category: Fundamental Theorem (Extreme Point Theorem)

360. Consider the following constraints for a linear programming problem:
$x \geq 0, y \geq 0, 2x + 3y \leq 12, x + y \leq 5$
Which of the following points is NOT a corner point of the feasibility region?

a) $(0, 0)$

b) $(0, 4)$

c) $(5, 0)$

d) $(6, 0)$

361 / 576

Category: Advertising Problems – Allocate funds for maximum exposure.

361. A company can spend a maximum of \$100,000 on advertising across three media: TV (\$x_1$), radio (\$x_2$), and newspapers (\$x_3$). The minimum spending on TV is \$20,000 and on radio is \$10,000. The non-negativity constraint applies to all variables. Which of the following correctly represents the budget constraint?

a) $x_1 + x_2 + x_3 \leq 100,000$ with $x_1 \geq 20,000$, $x_2 \geq 10,000$

b) $x_1 + x_2 + x_3 \geq 100,000$

c) $x_1 + x_2 \leq 100,000$

d) $x_1 + x_2 + x_3 = 100,000$

362 / 576

Category: Check feasible side using a test point (often origin).

362. For the inequality $4x - y \geq 8$, which side is feasible when tested with the origin $(0,0)$?

a) The side including the origin

b) The side opposite to the origin

c) Both sides

d) Neither side

363 / 576

Category: LINAR PROGRAMMING-DEFINITION

363. Which of the following best describes constraints in a linear programming problem?

a) Equations defining the optimal solution

b) System of equations without inequalities

c) Conditions imposed on decision variables

d) Only non-negative restrictions

364 / 576

Category: Applications: maximize profit, minimize cost, optimal resource utilization.

364. An investor wants to allocate \$100,000 between two investments: Investment X yields 5\% return and Investment Y yields 8\% return. Due to risk constraints, no more than \$60,000 can be invested in Y. Additionally, at least \$20,000 must be invested in X. What is the maximum possible return on investment?

a) \$6,800

b) \$7,200

c) \$7,500

d) \$8,000

365 / 576

Category: Apply non-negativity restrictions.

365. Which of the following correctly represents the non-negativity constraint for a decision variable $x_j$?

a) $x_j > 0$

b) $x_j \leq 0$

c) $x_j \geq 0$

d) $x_j = 0$

366 / 576

Category: Graph each inequality on the same coordinate axes.

366. For the system of inequalities $x + y \leq 4$, $x - y \leq 2$, and $y \geq 0$, one of the corner points of the solution region is:

a) $(0, 0)$

b) $(3, 1)$

c) $(1, 3)$

d) $(-2, 0)$

367 / 576

Category: Transportation Problems – Minimize cost of shipment.

367. What is the primary objective of a transportation problem in linear programming?

a) Maximize profit

b) Balance supply and demand only

c) Minimize the number of shipments

d) Minimize the cost of shipment

368 / 576

Category: FORMULATION OF A LINEAR PROGRAMMING PROBLEM

368. A factory produces two types of products, A and B. Each unit of product A requires 3 hours of labor and 4 hours of machine time, while each unit of product B requires 5 hours of labor and 2 hours of machine time. The factory has a maximum of 150 hours of labor and 120 hours of machine time available per day. The profit per unit for product A is \$10 and for product B is \$8. Formulate the LPP to maximize the daily profit.

a) Maximize $Z = 10x_1 + 8x_2$; Subject to: $3x_1 + 5x_2 \leq 150$, $4x_1 + 2x_2 \leq 120$, $x_1, x_2 \geq 0$

b) Maximize $Z = 8x_1 + 10x_2$; Subject to: $3x_1 + 5x_2 \leq 150$, $4x_1 + 2x_2 \leq 120$, $x_1, x_2 \geq 0$

c) Maximize $Z = 10x_1 + 8x_2$; Subject to: $5x_1 + 3x_2 \leq 150$, $2x_1 + 4x_2 \leq 120$, $x_1, x_2 \geq 0$

d) Maximize $Z = 8x_1 + 10x_2$; Subject to: $5x_1 + 3x_2 \leq 150$, $2x_1 + 4x_2 \leq 120$, $x_1, x_2 \geq 0$

369 / 576

Category: GRAPHICAL SOLUTION OF LINEAR INEQUATIONS

369. What does the intersection of two shaded regions represent in the graphical solution of a system of inequalities?

a) The union of the two inequalities

b) The difference between the two inequalities

c) The common solution set satisfying both inequalities

d) The complement of one inequality with respect to the other

370 / 576

Category: Corner points (vertices) can be found by solving boundary equations in pairs.

370. What is the corner point obtained by solving the boundary equations $x + y = 5$ and $x - y = 1$?

a) $(2, 3)$

b) $(3, 2)$

c) $(4, 1)$

d) $(1, 4)$

371 / 576

Category: Investment Problems – Maximize return on capital allocation.

371. A portfolio manager has \$150,000 to invest in three securities with returns of 3\%, 5\%, and 8\%. The investments in the first and third securities must each be at least 20\% of the total capital, and the investment in the second security cannot exceed 40\% of the total capital. If only two securities can be chosen, which combination yields the highest return?

a) 3\% and 5\%

b) 5\% and 8\%

c) 3\% and 8\%

d) None of the above

372 / 576

Category: Write down constraints in terms of inequalities/equations.

372. (A) In a linear programming problem, if the constraint $5x_1 + 3x_2 \leq 15$ represents a resource limitation, then increasing the right-hand side value from 15 to 20 will always expand the feasible region.
(R) Increasing the right-hand side of a less-than-or-equal-to constraint relaxes the condition, thereby allowing more combinations of decision variables to satisfy it.

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.

b) Both Assertion and Reason are true, but Reason is NOT the correct explanation of Assertion.

c) Assertion is true, but Reason is false.

d) Assertion is false, but Reason is true.

373 / 576

Category: Advertising Problems – Allocate funds for maximum exposure.

373. If a company has a budget of \$8000, with TV ads costing \$400 each and radio ads costing \$100 each, what is the budget constraint for allocating funds between these two media?

a) $400x_1 + 100x_2 \geq 8000$

b) $400x_1 - 100x_2 \leq 8000$

c) $100x_1 + 400x_2 \leq 8000$

d) $400x_1 + 100x_2 \leq 8000$

374 / 576

Category: GRAPHICAL SOLUTION OF LINEAR INEQUATIONS

374. Which inequality represents the area to the right of the $y$-axis?

a) $x < 0$

b) $y > 0$

c) $x > 0$

d) $y < 0$

375 / 576

Category: Objective function.

375. Consider the LPP: Maximize $Z = 5x + 4y$ subject to the constraints $3x + 2y \leq 18$, $x + y \leq 6$, $x \geq 0$, $y \geq 0$. Which of the following points is a feasible solution but does NOT optimize the objective function?

a) (2, 2)

b) (6, 0)

c) (0, 6)

d) (3, 3)

376 / 576

Category: Blending Problems – Optimize combination of raw materials.

376. What is the purpose of the objective function in a blending problem?

a) To define the constraints on raw materials

b) To specify the non-negativity conditions

c) To minimize cost or maximize profit based on given conditions

d) To list all available raw materials

377 / 576

Category: SYSTEM OF THREE OR MORE INEQULITIES

377. Consider the linear programming problem:
Maximize $Z = 3x + 5y$ subject to:
$x + y \leq 8$
$2x + y \leq 12$
$x, y \geq 0$
Which point provides the optimal solution?

a) $(4, 4)$

b) $(0, 8)$

c) $(6, 0)$

d) $(0, 0)$

378 / 576

Category: Advertising Problems – Allocate funds for maximum exposure.

378. A company has a total advertising budget of \$50,000 to spend on TV and online ads. The effectiveness per unit cost for TV is 500 viewers/\$100 and for online ads is 300 viewers/\$100. The minimum spending requirement for TV is \$10,000 and the maximum is \$30,000. For online ads, the minimum is \$5,000 and the maximum is \$25,000. What is the optimal allocation to maximize the total audience reach?

a) \$10,000 on TV and \$25,000 on online ads

b) \$30,000 on TV and \$5,000 on online ads

c) \$20,000 on TV and \$20,000 on online ads

d) \$15,000 on TV and \$30,000 on online ads

379 / 576

Category: Graphical Method of Solution

379. (A) The maximum value of the objective function $Z = 3x + 4y$ subject to the constraints $x \geq 0, y \geq 0, x + y \leq 6, 2x + y \leq 8$ is 24.
(R) The feasible region for these constraints is a polygon with corner points at (0,0), (0,6), (4,0), and (2,4).

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.

b) Both Assertion and Reason are true, but Reason is NOT the correct explanation of Assertion.

c) Assertion is true, but Reason is false.

d) Assertion is false, but Reason is true.

380 / 576

Category: FORMULATION OF A LINEAR PROGRAMMING PROBLEM

380. In a manufacturing problem, if a factory has a maximum of 200 hours of labor available and producing one unit of Product X takes 2 hours while Product Y takes 3 hours, what is the correct constraint?

a) $3x + 2y \leq 200$

b) $2x + 3y \geq 200$

c) $2x + 3y \leq 200$

d) $x + y \leq 200$

381 / 576

Category: Define objective function (profit, cost, etc.).

381. In a linear programming problem, why is the non-negativity constraint $x \geq 0$ and $y \geq 0$ necessary?

a) To make the problem mathematically solvable

b) To ensure the solution is feasible in real-world contexts

c) To simplify the objective function

d) To align with graphical methods

382 / 576

Category: Graphical Method of Solution

382. (A) The optimal solution of a linear programming problem with two variables always occurs at one of the corner points of the feasible region.
(R) According to the Extreme Point Theorem, an optimum solution to a linear programming problem, if it exists, occurs at one of the corner points of the feasibility region.

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.

b) Both Assertion and Reason are true, but Reason is NOT the correct explanation of Assertion.

c) Assertion is true, but Reason is false.

d) Assertion is false, but Reason is true.

383 / 576

Category: CORNER POINT METHOD.

383. Which of the following represents a valid feasibility region for the constraints $x + y \leq 6$, $2x + y \leq 8$, and $x, y \geq 0$?

a) Intersection of $x + y = 6$ and $2x + y = 8$ gives $(2, 4)$

b) Intersection of $x + y = 6$ and $y = 0$ gives $(6, 0)$

c) Intersection of $2x + y = 8$ and $x = 0$ gives $(0, 8)$

d) Origin $(0, 0)$ is also a corner point \end{itemize} The feasible region includes all these corner points and the area bounded by them.

e) A triangle with vertices $(0, 0)$, $(0, 6)$, and $(6, 0)$

f) A polygon with vertices $(0, 0)$, $(0, 8)$, $(2, 4)$, and $(6, 0)$

g) A quadrilateral with vertices $(0, 0)$, $(0, 6)$, $(4, 0)$, and $(2, 4)$

h) A rectangle with vertices $(0, 0)$, $(0, 6)$, $(4, 0)$, and $(4, 6)$

384 / 576

Category: LINAR PROGRAMMING-DEFINITION

384. An LPP aims to minimize $Z = 5x + 10y$ subject to $x + 2y \geq 100$, $2x + y \geq 120$, $x \geq 0$, $y \geq 0$. What is the optimal feasible solution?

a) (50, 25)

b) (40, 30)

c) ($\frac{140}{3}, \frac{80}{3}$)

d) (60, 20)

385 / 576

Category: Investment Problems – Maximize return on capital allocation.

385. (A) In a linear programming problem for investment optimization, the objective function must always represent the total return on investments.
(R) The objective function in an LPP is formulated to maximize or minimize a linear function of decision variables, which can represent returns, costs, or other measurable quantities.

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.

b) Both Assertion and Reason are true, but Reason is NOT the correct explanation of Assertion.

c) Assertion is true, but Reason is false.

d) Assertion is false, but Reason is true.

386 / 576

Category: GRAPHICAL SOLUTION OF LINEAR INEQUATIONS

386. For the inequality $y \leq 3$, which region should be shaded on the graph?

a) Above the line $y = 3$

b) Below the line $y = 3$ (including the line)

c) Above and including the line $x = 3$

d) Below the line $x = 3$

387 / 576

Category: LINEAR PROGRAMMING

387. In an LPP, what are constraints?

a) The values of decision variables

b) The linear functions used to define the objective

c) The system of inequations or equations involving decision variables

d) The optimum solution of the problem

388 / 576

Category: Graph each inequality on the same coordinate axes.

388. Consider the system of inequalities: $x + y \geq 4$, $2x - y \leq 6$, $3x + 2y \geq 6$, $x \geq 0$, $y \geq 0$. Find the number of corner points of the feasible region.

a) 4

b) 5

c) 3

d) 6

389 / 576

Category: SYSTEM OF THREE OR MORE INEQULITIES

389. Which of the following represents the system of inequalities for the shaded feasible region bounded by lines passing through points $(0, 2)$, $(2, 3)$, and $(4, 0)$?

a) $y \geq \frac{1}{2}x + 2$, $y \geq -\frac{3}{2}x + 6$, $y \leq -\frac{1}{2}x + 2$

b) $y \leq \frac{1}{2}x + 2$, $y \leq -\frac{3}{2}x + 6$, $y \geq -\frac{1}{2}x + 2$

c) $y \geq \frac{1}{2}x + 2$, $y \leq -\frac{3}{2}x + 6$, $y \leq -\frac{1}{2}x + 2$

d) $y \leq \frac{1}{2}x + 2$, $y \geq -\frac{3}{2}x + 6$, $y \geq -\frac{1}{2}x + 2$

390 / 576

Category: LINAR PROGRAMMING-DEFINITION

390. (A) The objective function in a linear programming problem is always maximized.
(R) The purpose of the objective function is to optimize a quantity, which can be either maximizing profit or minimizing cost.

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.

b) Both Assertion and Reason are true, but Reason is NOT the correct explanation of Assertion.

c) Assertion is true, but Reason is false.

d) Assertion is false, but Reason is true.

391 / 576

Category: A method for optimizing (maximizing/minimizing) a linear function subject to linear constraints.

391. In an LPP, the objective function is given by $Z = 4x + 6y$. If the feasible region has corner points at $(0, 0)$, $(5, 0)$, $(3, 2)$, and $(0, 4)$, what is the minimum value of $Z$?

a) 0

b) 20

c) 24

d) 30

392 / 576

Category: THE GRAPHICAL METHOD OF SOLUTION OF LINEAR PROGRAMMING PROBLEMS

392. A linear programming problem has the objective function $Z = 2x + 5y$ and the following constraints:
$x + y \leq 9$,
$3x + y \leq 15$,
$x \geq 0$, $y \geq 0$.
At which corner point does the minimum value of $Z$ occur?

a) $(0,9)$

b) $(5,0)$

c) $(3,6)$

d) $(0,0)$

393 / 576

Category: Infeasible solution.

393. Consider the LPP: Maximize $Z = 4x + 5y$ subject to $2x + 3y \leq 12$, $x \geq 4$, $y \geq 0$. Which of the following points is infeasible?

a) $(x, y) = (4, 1)$

b) $(x, y) = (5, 0)$

c) $(x, y) = (6, 1)$

d) $(x, y) = (4, -1)$

394 / 576

Category: Understanding regions of inequality is essential before tackling LPP problems.

394. What is the feasible region in a Linear Programming Problem (LPP)?

a) A region where the objective function has multiple solutions

b) The intersection of all constraints, including non-negativity

c) The area where at least one constraint is satisfied

d) The region outside the constraints' boundary

395 / 576

Category: Represent graphically and find feasible region.

395. (A) The feasible region for the system of inequalities $3x + 4y \leq 12$, $5x - 2y \geq 10$, $x \geq 0$, $y \geq 0$ is unbounded.
(R) A feasible region formed by a set of linear inequalities is bounded if it can be enclosed within a finite area.

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.

b) Both Assertion and Reason are true, but Reason is NOT the correct explanation of Assertion.

c) Assertion is true, but Reason is false.

d) Assertion is false, but Reason is true.

396 / 576

Category: Theorem

396. Which of the following best describes a feasibility region in linear programming?

a) The union of open half-planes

b) The intersection of a finite number of closed half-planes

c) A single point satisfying all constraints

d) The area outside the constraint boundaries

397 / 576

Category: Objective function.

397. Which of the following is true about a feasible solution in an LPP?

a) It must always optimize the objective function

b) It lies outside the feasible region

c) It violates at least one constraint

d) It satisfies all constraints and non-negativity conditions

398 / 576

Category: Convex region.

398. Consider a region defined by $x + y \leq 4$, $x \geq 0$, $y \geq 0$. Is this region convex?

a) No, because it has sharp corners

b) Yes, because it is formed by linear inequalities

c) No, because it is a triangular region

d) Yes, but only if $x$ and $y$ are integers

399 / 576

Category: Diet Problems – Minimize cost subject to nutritional constraints.

399. A dietitian wants to minimize the cost of a meal consisting of two foods, A and B. Food A costs \$4 per unit and provides 3 units of protein and 1 unit of vitamins. Food B costs \$2 per unit and provides 1 unit of protein and 2 units of vitamins. The meal must contain at least 9 units of protein and 8 units of vitamins. What is the minimum cost of the meal?

a) \$12

b) \$14

c) \$16

d) \$18

400 / 576

Category: Linear Inequations in Two Variables

400. Which of the following regions represents the solution to $y \leq 3x - 1$?

a) Above the line $y = 3x - 1$

b) Below the line $y = 3x - 1$

c) Left of the line $y = 3x - 1$

d) Right of the line $y = 3x - 1$

401 / 576

Category: Infeasible solution.

401. What defines an infeasible solution in a Linear Programming Problem (LPP)?

a) A solution that maximizes the objective function

b) A solution that violates at least one constraint or non-negativity condition

c) A solution that lies on the boundary of the feasible region

d) A solution that satisfies all constraints except the objective function

402 / 576

Category: Apply non-negativity restrictions.

402. (A) In a linear programming problem, if non-negativity constraints are relaxed for all decision variables, the feasible region may become unbounded.
(R) Non-negativity restrictions ensure that the feasible region remains bounded in all linear programming problems.

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.

b) Both Assertion and Reason are true, but Reason is NOT the correct explanation of Assertion.

c) Assertion is true, but Reason is false.

d) Assertion is false, but Reason is true.

403 / 576

Category: Fundamental Theorem (Extreme Point Theorem)

403. (A) If the feasibility region of a linear programming problem is unbounded, the Extreme Point Theorem guarantees that an optimal solution exists at one of the corner points.
(R) The feasibility region being unbounded implies that the objective function $Z = ax + by$ can take infinitely large values, making it impossible to find a finite maximum or minimum.

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.

b) Both Assertion and Reason are true, but Reason is NOT the correct explanation of Assertion.

c) Assertion is true, but Reason is false.

d) Assertion is false, but Reason is true.

404 / 576

Category: GRAPHICAL SOLUTION OF LINEAR INEQUATIONS

404. (A) The maximum value of the linear expression $5x + 7y$ over the feasibility region defined by the constraints $x \geq 0$, $y \geq 0$, $3x + 2y \leq 18$, $x + y \leq 8$ will occur at one of the corner points of the region.
(R) According to the theorem on linear programming, if a linear expression has a maximum or minimum value over a feasibility region that is bounded and has corner points, then the extremum occurs at one of these corner points.

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.

b) Both Assertion and Reason are true, but Reason is NOT the correct explanation of Assertion.

c) Assertion is true, but Reason is false.

d) Assertion is false, but Reason is true.

405 / 576

Category: Write down constraints in terms of inequalities/equations.

405. A company produces two products X and Y, with profits of \$4 per unit for X and \$5 per unit for Y. What is the correct objective function to maximize the profit?

a) $Z = 5x + 4y$

b) $Z = 4x - 5y$

c) $Z = x + y$

d) $Z = 4x + 5y$

406 / 576

Category: Transportation Problems – Minimize cost of shipment.

406. What is the primary objective of a transportation problem in linear programming?

a) Maximize the profit from transportation

b) Minimize the number of shipments

c) Minimize the time taken for transportation

d) Minimize the cost of shipment

407 / 576

Category: Fundamental Theorem (Extreme Point Theorem)

407. (A) The maximum value of the linear expression $Z = 5x + 10y$ over a bounded feasibility region must occur at one of its corner points.
(R) A linear function achieves its maximum and minimum values at corner points when optimized over a bounded convex feasibility region.

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.

b) Both Assertion and Reason are true, but Reason is NOT the correct explanation of Assertion.

c) Assertion is true, but Reason is false.

d) Assertion is false, but Reason is true.

408 / 576

Category: Corner points (vertices) can be found by solving boundary equations in pairs.

408. (A) The optimal solution of a linear programming problem always occurs at a corner point of the feasible region.
(R) According to the Extreme Point Theorem, if the objective function has a maximum or minimum value over the feasibility region, it must occur at some corner point.

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.

b) Both Assertion and Reason are true, but Reason is NOT the correct explanation of Assertion.

c) Assertion is true, but Reason is false.

d) Assertion is false, but Reason is true.

409 / 576

Category: Represent graphically and find feasible region.

409. Where does the optimal solution lie in a linear programming problem with a bounded feasible region?

a) Only at the origin $(0, 0)$.

b) At one of the corner points of the feasible region.

c) Outside the feasible region.

410 / 576

Category: Apply non-negativity restrictions.

410. A linear programming problem has the following constraints: $x + y \leq 10$, $2x - y \geq 4$, $y \geq 0$. If $x$ is not explicitly restricted to be non-negative, which of the following points is a feasible solution?

a) (3,4)

b) (-1,5)

c) (5,5)

d) (2,-1)

411 / 576

Category: Developed by Kantorovich and George Dantzig.

411. What is guaranteed about the optimal solution in a bounded feasible region of an LPP?

a) It lies at the center of the feasible region.

b) It lies at any arbitrary point within the feasible region.

c) It lies at one of the corner points of the feasible region.

d) It does not exist if the region is bounded.

412 / 576

Category: Areas of Application

412. A logistics company needs to transport goods from three warehouses (W1, W2, W3) to four retailers (R1, R2, R3, R4). The transportation costs per unit (\$) are given by the matrix below. The supply at each warehouse and demand at each retailer are also provided:

$\text{Cost Matrix} = \\ \begin{bmatrix} \\ 10 & 12 & 15 & 20 \\ \\ 8 & 11 & 14 & 18 \\ \\ 9 & 10 & 13 & 16 \\ \end{bmatrix}$

Supply: W1 = 50, W2 = 60, W3 = 70

Demand: R1 = 40, R2 = 30, R3 = 60, R4 = 50

Which of the following is the least cost transportation plan that meets all demands without exceeding supplies?

a) W1-R1: 40, W2-R2: 30, W3-R3: 60, W3-R4: 50

b) W2-R1: 40, W3-R2: 30, W3-R3: 40, W1-R3: 20, W1-R4: 30, W2-R4: 20

c) W1-R2: 30, W2-R1: 40, W3-R3: 60, W3-R4: 50

d) W3-R1: 40, W1-R2: 30, W2-R3: 60, W3-R4: 50

413 / 576

Category: Identify decision variables (e.g., x, y).

413. (A) In a linear programming problem, the decision variables represent the quantities that need to be determined.
(R) Decision variables are assigned symbols like $x$ and $y$ to denote the unknowns in the problem.

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.

b) Both Assertion and Reason are true, but Reason is NOT the correct explanation of Assertion.

c) Assertion is true, but Reason is false.

d) Assertion is false, but Reason is true.

414 / 576

Category: Investment Problems – Maximize return on capital allocation.

414. What do the variables $x$ and $y$ represent in the investment problem formulation?

a) Amounts invested in risk-free assets only

b) Returns on investments in different securities

c) Amounts invested in different fixed income securities

d) Percentages of total budget allocated to investments

415 / 576

Category: SOLVED EXAMPLES

415. Which of the following represents the correct graph for the inequality $3x - y \geq 1$?

a) A solid line passing through $(0, -1)$ and $(1, 2)$, with shading above the line.

b) A dashed line passing through $(0, -1)$ and $(1, 2)$, with shading below the line.

c) A solid line passing through $(0, -1)$ and $(1, 2)$, with shading below the line.

d) A dashed line passing through $(0, -1)$ and $(1, 2)$, with shading above the line.

416 / 576

Category: Corner points (vertices) can be found by solving boundary equations in pairs.

416. For the system of inequalities:
$2x + y \leq 10, \, x + 2y \leq 8, \, x \geq 0, \, y \geq 0$
determine the number of corner points of the feasible region.

a) 3

b) 4

c) 5

d) 6

417 / 576

Category: AREAS OF APPLICATIONS OF LINEAR PROGRAMMING

417. (A) Linear programming can be used to find the least expensive routes for transporting goods from multiple factories to various markets.
(R) Transportation problems involve minimizing costs while meeting demand and supply constraints, which is a linear optimization problem.

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.

b) Both Assertion and Reason are true, but Reason is NOT the correct explanation of Assertion.

c) Assertion is true, but Reason is false.

d) Assertion is false, but Reason is true.

418 / 576

Category: THE GRAPHICAL METHOD OF SOLUTION OF LINEAR PROGRAMMING PROBLEMS

418. Consider the following linear programming problem:
Maximize $Z = 3x + 4y$
Subject to:
$x + y \leq 5$,
$2x + y \leq 8$,
$x \geq 0$, $y \geq 0$.
What is the maximum value of $Z$?

a) 10

b) 14

c) 16

d) 17

419 / 576

Category: System of Inequations

419. (A) The feasible region for the system of inequalities $x + 2y \leq 10$, $3x - y \geq 0$, $x \geq 0$, $y \geq 0$ is a convex set.
(R) The intersection of any number of closed half-planes always results in a convex set.

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.

b) Both Assertion and Reason are true, but Reason is NOT the correct explanation of Assertion.

c) Assertion is true, but Reason is false.

d) Assertion is false, but Reason is true.

420 / 576

Category: Graph each inequality on the same coordinate axes.

420. (A) The maximum value of the objective function $P = 5x + 3y$ for the feasible region defined by the inequalities $x + y \leq 6, 2x + 3y \geq 12, x \geq 0, y \geq 0$ occurs at the point $(6, 0)$.
(R) The corner points of a feasibility region are obtained by solving the boundary equations in pairs and the maximum or minimum value of an objective function over the feasibility region occurs at one of these corner points.

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.

b) Both Assertion and Reason are true, but Reason is NOT the correct explanation of Assertion.

c) Assertion is true, but Reason is false.

d) Assertion is false, but Reason is true.

421 / 576

Category: SYSTEM OF THREE OR MORE INEQULITIES

421. Find the coordinates of the vertices of the feasible region formed by the following system of inequalities:
$y \geq 2x - 4$, $y \leq -x + 5$, and $x + y \geq 1$.

a) $(2, 1), (3, 2), (1, 3)$

b) $(3, 2), \left(\frac{5}{3}, -\frac{2}{3}\right), (-1, 6)$

c) $(0, 5), (1, 0), (3, 2)$

d) $(3, 2), \left(\frac{5}{3}, -\frac{2}{3}\right), (0, 1)$

422 / 576

Category: Convex region.

422. Why is the convexity of the feasible region important in LPP?

a) It guarantees multiple optimal solutions.

b) It allows nonlinear constraints to be used.

c) It ensures the solution is always bounded.

d) It limits the optimal solution to the vertices of the region.

423 / 576

Category: The common region represents the solution set (infinite ordered pairs).

423. Which of the following points lies in the common solution region for the system of inequalities $x + y \geq 4$, $2x - y \leq 6$, and $y \geq 1$?

a) $(2, 3)$

b) $(1, 1)$

c) $(0, 4)$

d) $(5, 0)$

424 / 576

Category: Graph each inequality on the same coordinate axes.

424. Consider the system of inequalities $2x + 3y \leq 6$, $x \geq 0$, and $y \geq 1$. Which of the following points lies within the common solution region?

a) $(0, 0)$

b) $(1, 1.5)$

c) $(1, 1.2)$

d) $(2, 2)$

425 / 576

Category: SOLVED EXAMPLES

425. A candidate is preparing for a competitive examination where he needs to solve problems worth 5, 4, and 6 points each. He can spend a maximum of 3$\frac{1}{2}$ hours on all problems, including at most 2$\frac{1}{2}$ hours on 5-point and 4-point problems. He can submit at most 100 solutions. Formulate the LPP to maximize his total score.

a) Maximize $Z = 5x_1 + 4x_2 + 6x_3$ subject to $x_1 + x_2 \leq 100$, $3x_1 + 2x_2 + 4x_3 \leq 210$, $3x_1 + 2x_2 \leq 150$

b) Maximize $Z = 5x_1 + 4x_2 + 6x_3$ subject to $x_1 + x_2 + x_3 \leq 100$, $3x_1 + 2x_2 + 4x_3 \leq 210$, $3x_1 + 2x_2 \leq 150$, $x_1, x_2, x_3 \geq 0$

c) Maximize $Z = 5x_1 + 4x_2 + 6x_3$ subject to $x_1 + x_2 + x_3 \leq 210$, $3x_1 + 2x_2 \leq 150$, $x_1, x_2, x_3 \geq 0$

d) Maximize $Z = 5x_1 + 4x_2 + 6x_3$ subject to $x_1 + x_2 + x_3 \leq 100$, $3x_1 + 2x_2 + 4x_3 \leq 150$, $x_1, x_2, x_3 \geq 0$

426 / 576

Category: Linear Programming Problem (LPP) Example

426. (A) In a Linear Programming Problem, the objective function must always be linear.
(R) The constraints in an LPP can be non-linear.

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.

b) Both Assertion and Reason are true, but Reason is NOT the correct explanation of Assertion.

c) Assertion is true, but Reason is false.

d) Assertion is false, but Reason is true.

427 / 576

Category: Investment Problems – Maximize return on capital allocation.

427. A company has \$10,000 to invest in two types of securities: bonds (\$100 per unit) and stocks (\$200 per unit). The expected return per unit for bonds is \$8, and for stocks is \$15. The company wants to maximize its return while ensuring that the investment in stocks does not exceed 40\% of the total investment. What is the optimal allocation?

a) Invest 60 units in bonds and 20 units in stocks

b) Invest 50 units in bonds and 25 units in stocks

c) Invest 70 units in bonds and 15 units in stocks

d) Invest 80 units in bonds and 10 units in stocks

428 / 576

Category: Optimal feasible solution.

428. (A) The optimal feasible solution of an LPP must always occur at a corner point of the feasibility region.
(R) The extreme point theorem guarantees that any linear function defined over a closed and bounded convex set attains its maximum or minimum value at one of the extreme points.

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.

b) Both Assertion and Reason are true, but Reason is NOT the correct explanation of Assertion.

c) Assertion is true, but Reason is false.

d) Assertion is false, but Reason is true.

429 / 576

Category: Applications: maximize profit, minimize cost, optimal resource utilization.

429. (A) In manufacturing problems, the objective function in LPP is always to maximize profit.
(R) The constraints in manufacturing problems ensure that resources like manpower and raw materials are utilized optimally.

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.

b) Both Assertion and Reason are true, but Reason is NOT the correct explanation of Assertion.

c) Assertion is true, but Reason is false.

d) Assertion is false, but Reason is true.

430 / 576

Category: Identify decision variables (e.g., x, y).

430. (A) Decision variables in a linear programming problem must always represent non-negative quantities.

(R) Negative values for decision variables are not meaningful in most practical scenarios.

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.

b) Both Assertion and Reason are true, but Reason is NOT the correct explanation of Assertion.

c) Assertion is true, but Reason is false.

d) Assertion is false, but Reason is true.

431 / 576

Category: Linear Inequations in Two Variables

431. (A) The graph of the inequation $3x - y \geq 1$ includes the line $y = 3x - 1$ as part of the solution set.
(R) For any inequation of the form $ax + by \leq c$, the line $ax + by = c$ is always included in the solution set.

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.

b) Both Assertion and Reason are true, but Reason is NOT the correct explanation of Assertion.

c) Assertion is true, but Reason is false.

d) Assertion is false, but Reason is true.

432 / 576

Category: SOME DEFINITIONS AND CONCEPTS RELATED TO LPP

432. For an LPP with constraints $x + y \leq 8$, $2x + 3y \leq 18$, $x \geq 0$, $y \geq 0$, which point represents an optimal solution when maximizing $Z = 5x + 7y$?

a) (0,6)

b) (8,0)

c) (6,2)

d) (4,4)

433 / 576

Category: Areas of Application

433. A manufacturing company wants to maximize its profit by producing two products, A and B. Product A yields a profit of \$5 per unit, and Product B yields \$7 per unit. The production process has constraints: the total labor hours cannot exceed 300, and the total raw material cannot exceed 200 units. Each unit of A requires 1 labor hour and 2 units of raw material, while each unit of B requires 3 labor hours and 1 unit of raw material. Which of the following is NOT a limitation of using linear programming for this problem?

a) It assumes linear relationships between variables.

b) It ignores qualitative factors like worker morale.

c) It requires all constraints to be linear.

d) It guarantees global optimal solutions.

434 / 576

Category: Manufacturing Problems – Optimize production for profit.

434. A factory produces two items P and Q. Each unit of P requires 4 kg of raw material and 5 hours of labor, while each unit of Q requires 6 kg of raw material and 4 hours of labor. The profit per unit of P is \$7 and for Q is \$9. If 240 kg of raw material and 200 labor hours are available, how many units of Q should be produced to maximize profit?

a) 20

b) 25

c) 30

d) 40

435 / 576

Category: Apply non-negativity restrictions.

435. (A) Non-negativity restrictions in LPP ensure that all decision variables are greater than or equal to zero.
(R) Negative values for decision variables do not represent realistic solutions in practical scenarios.

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.

b) Both Assertion and Reason are true, but Reason is NOT the correct explanation of Assertion.

c) Assertion is true, but Reason is false.

d) Assertion is false, but Reason is true.

436 / 576

Category: A method for optimizing (maximizing/minimizing) a linear function subject to linear constraints.

436. According to the Corner Point Method, where does the optimal solution for a linear programming problem lie?

a) At any random point inside the feasible region

b) Only at the center of the feasible region

c) At one of the corner points of the feasible region

d) Outside the feasible region

437 / 576

Category: Steps in Formulating a Linear Programming Model

437. A company produces two items P and Q using machines M1 and M2. If machine M1 has a maximum availability of 120 hours and item P takes 3 hours while item Q takes 4 hours on M1, identify the correct constraint.

a) $3x + 4y \geq 120$

b) $3x + 4y = 120$

c) $3x + 4y \leq 120$

d) $4x + 3y \leq 120$

438 / 576

Category: ADVANTAGES OF LINEAR PROGRAMMING PROBLEMS

438. Which advantage of linear programming helps in efficiently utilizing scarce resources like manpower and materials to achieve the best outcome?

a) Improved Decision Making

b) Cost Reduction

c) Optimal Resource Allocation

d) Time Efficiency

439 / 576

Category: Optimal feasible solution.

439. Consider the LPP:
Maximize $Z = 3x + 4y$
subject to the constraints:
$x + y \leq 5$, $2x + y \leq 8$, $x \geq 0$, $y \geq 0$.
Which of the following is an optimal feasible solution for this problem?

a) $(1, 4)$

b) $(5, 0)$

c) $(3, 2)$

d) $(0, 5)$

440 / 576

Category: Apply non-negativity restrictions.

440. How do non-negativity restrictions affect the feasible region in a linear programming problem?

a) They expand the feasible region

b) They eliminate the feasible region

c) They restrict the feasible region to non-negative values

d) They convert the feasible region into an unbounded one

441 / 576

Category: Graph each inequality on the same coordinate axes.

441. Given the inequalities $y < 4 - x$ and $y \geq 2x - 2$, determine which quadrant contains part of the common solution region.

a) Quadrant I

b) Quadrant II

c) Quadrant III

d) Quadrant IV

442 / 576

Category: Feasible region : i region and Feasible solution.

442. Which of the following sets of constraints defines a feasible region with corner points $(0, 0)$, $(0, 4)$, $(2, 3)$, and $(3, 0)$?

a) $2x + y \leq 6$, $x + 2y \leq 8$, $x \geq 0$, $y \geq 0$

b) $x + y \leq 5$, $2x + y \leq 6$, $x \geq 0$, $y \geq 0$

c) $x + 2y \leq 8$, $3x + y \leq 9$, $x \geq 0$, $y \geq 0$

d) $3x + 2y \leq 12$, $x + y \leq 5$, $x \geq 0$, $y \geq 0$

443 / 576

Category: GRAPHICAL SOLUTION OF LINEAR INEQUATIONS

443. Which region represents the solution set of the inequality $2x - y < 4$?

a) Below the line $y = 2x - 4$

b) Above the line $y = 2x - 4$

c) On the line $y = 2x - 4$

d) To the left of the line $y = 2x - 4$

444 / 576

Category: Advertising Problems – Allocate funds for maximum exposure.

444. If the effectiveness ratio of radio ads is 4 viewers/\$100 and print ads is 2 viewers/\$100, what is the objective function to maximize total audience reach if $x_1$ is the amount spent on radio and $x_2$ on print?

a) $Z = 4x_1 + 2x_2$

b) $Z = 0.04x_1 + 0.02x_2$

c) $Z = 2x_1 + 4x_2$

d) $Z = 0.02x_1 + 0.04x_2$

445 / 576

Category: Applications: maximize profit, minimize cost, optimal resource utilization.

445. A beverage company blends two ingredients, I and II, to produce drinks A and B. Drink A requires 2 units of I and 1 unit of II per liter. Drink B requires 1 unit of I and 2 units of II per liter. The company has 80 units of I and 70 units of II. Profit is \$3 per liter for A and \$2 per liter for B. How should resources be allocated to maximize profit?

a) \$110

b) \$120

c) \$130

d) \$140

446 / 576

Category: SOLVED EXAMPLES

446. A company manufactures two products, X and Y, with profits \$10 and \$8 per unit respectively. Product X requires 2 hours of labor and 1 unit of material, while Y requires 1 hour of labor and 2 units of material. The company has 100 hours of labor and 80 units of material. What is the optimal production mix to maximize profit?

a) 15 units of X and 35 units of Y

b) 20 units of X and 30 units of Y

c) 25 units of X and 25 units of Y

d) 30 units of X and 20 units of Y

447 / 576

Category: THE GRAPHICAL METHOD OF SOLUTION OF LINEAR PROGRAMMING PROBLEMS

447. (A) The optimal solution to a linear programming problem always occurs at one of the extreme points of the feasible region.
(R) According to the Extreme Point Theorem, if an optimum solution exists to an LPP, it must lie at one of the corner points of the feasible region.

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.

b) Both Assertion and Reason are true, but Reason is NOT the correct explanation of Assertion.

c) Assertion is true, but Reason is false.

d) Assertion is false, but Reason is true.

448 / 576

Category: Transportation Problems – Minimize cost of shipment.

448. In a balanced transportation problem, what must be true about total supply and total demand?

a) Total supply must be greater than total demand

b) Total demand must be greater than total supply

c) Total supply must equal total demand

d) Total supply and demand can be any value

449 / 576

Category: Represent graphically and find feasible region.

449. Consider the following system of inequalities:
$x + y \leq 6$, $x - y \geq -2$, $x \geq 0$, $y \geq 0$. Which of the following points lies in the feasible region?

a) (2, 3)

b) (4, 3)

c) (1, -1)

d) (0, 7)

450 / 576

Category: Convex region.

450. Which of the following regions is NOT convex?

a) The region bounded by a rectangle.

b) The region bounded by a circle.

c) The feasible region of a standard LPP.

d) A star-shaped region with an indentation.

451 / 576

Category: AREAS OF APPLICATIONS OF LINEAR PROGRAMMING

451. A factory can manufacture products X and Y using machines M$_1\$ and M$_2\$. Producing one unit of X requires 2 hours on M$_1\$ and 1 hour on M$_2\$, while Y requires 1 hour on M$_1\$ and 3 hours on M$_2\$. The factory has 16 hours available on M$_1\$ and 18 hours on M$_2\$ per day. Profit is \$4 per unit of X and \$5 per unit of Y. Additionally, the marketing team allocates \$1000 per day among TV, radio, and print ads, where each \$100 spent increases sales potential by 3 units for X and 2 units for Y. How should the factory allocate production and ad spending to maximize profit?

a) 4 units of X, 5 units of Y, \$400 on TV, \$600 on radio

b) 6 units of X, 4 units of Y, \$500 on TV, \$500 on radio

c) 5 units of X, 6 units of Y, \$300 on TV, \$700 on print

d) 7 units of X, 3 units of Y, \$200 on radio, \$800 on print

452 / 576

Category: Investment Problems – Maximize return on capital allocation.

452. Which constraint ensures that the total investment does not exceed the available budget $B$?

a) $x - y \geq B$

b) $x + y \geq B$

c) $x + y \leq B$

d) $xy \leq B$

453 / 576

Category: Steps in Formulating a Linear Programming Model

453. In a manufacturing problem, the profit from producing product A is \$10 per unit and product B is \$15 per unit. The objective function to maximize the total profit can be represented as:

a) $Z = 10x_1 + 15x_2$

b) $Z = 15x_1 + 10x_2$

c) $Z = x_1 + x_2$

d) $Z = 25(x_1 + x_2)$

454 / 576

Category: Define objective function (profit, cost, etc.).

454. (A) The objective function in a linear programming problem must always be linear.
(R) A linear relationship ensures that the optimization problem remains solvable using standard linear programming techniques.

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.

b) Both Assertion and Reason are true, but Reason is NOT the correct explanation of Assertion.

c) Assertion is true, but Reason is false.

d) Assertion is false, but Reason is true.

455 / 576

Category: Fundamental Theorem (Extreme Point Theorem)

455. In the context of linear programming, what is the purpose of evaluating the objective function at each corner point of the feasible region?

a) To determine the infeasible solutions

b) To find the slope of the objective function

c) To identify the optimal solution

d) To plot the feasible region graphically

456 / 576

Category: Constraints.

456. (A) The system of inequalities $3x + 2y \leq 30$ and $x + 2y \leq 22$ are constraints in a Linear Programming Problem.
(R) Constraints in a LPP describe the conditions under which the optimization is to be attained.

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.

b) Both Assertion and Reason are true, but Reason is NOT the correct explanation of Assertion.

c) Assertion is true, but Reason is false.

d) Assertion is false, but Reason is true.

457 / 576

Category: Constraints.

457. A company produces two products, A and B. The production is constrained by:
$2x + y \leq 10$
$x + 3y \leq 12$
$x \geq 0$, $y \geq 0$
where $x$ and $y$ are units of A and B produced respectively. Which of the following points lies outside the feasible region?

a) (0, 0)

b) (3, 2)

c) (4, 1)

d) (5, 2)

458 / 576

Category: Feasible region : i region and Feasible solution.

458. (A) The point $x = 3$, $y = 4$ is a feasible solution to the LPP with constraints $2x + y \leq 10$, $x + 3y \leq 15$, $x \geq 0$, $y \geq 0$.
(R) A feasible solution satisfies all the constraints of an LPP, including non-negativity conditions.

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.

b) Both Assertion and Reason are true, but Reason is NOT the correct explanation of Assertion.

c) Assertion is true, but Reason is false.

d) Assertion is false, but Reason is true.

459 / 576

Category: GENERAL LINAR PROGRAMMING PROBLEM

459. Which of the following signs can be used in constraints of a Linear Programming Problem?

a) Only $\leq$

b) Only $\geq$

c) Only $=$

d) $\leq$, $\geq$, and $=$

460 / 576

Category: ADVANTAGES OF LINEAR PROGRAMMING PROBLEMS

460. How does linear programming aid decision-making?

a) Adds complexity to problems

b) Provides a subjective approach

c) Eliminates all constraints

d) Offers a systematic and scientific approach

461 / 576

Category: Investment Problems – Maximize return on capital allocation.

461. An investor wants to allocate \$50,000 between two funds: Fund X yields 6\% annually and Fund Y yields 9\% annually. The investor wants at least 30\% of the total investment in Fund X and no more than \$20,000 in Fund Y. How should the investor distribute the amount to maximize annual returns?

a) Invest \$30,000 in Fund X and \$20,000 in Fund Y

b) Invest \$20,000 in Fund X and \$30,000 in Fund Y

c) Invest \$25,000 in Fund X and \$25,000 in Fund Y

d) Invest \$35,000 in Fund X and \$15,000 in Fund Y

462 / 576

Category: CORNER POINT METHOD.

462. According to the Corner Point Method, how do we determine the optimal solution for a minimization problem?

a) Identify the largest value of the objective function among all corner points

b) Identify the smallest value of the objective function among all corner points

c) Calculate the average value of the objective function across all corner points

d) Select any random corner point as the solution

463 / 576

Category: Graph each inequality on the same coordinate axes.

463. (A) The solution set of a system of inequalities represents the intersection of all individual inequality graphs.
(R) When graphing a system of inequalities, the common region satisfying all inequalities appears as the overlapping (intersection) area of their respective half-planes.

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.

b) Both Assertion and Reason are true, but Reason is NOT the correct explanation of Assertion.

c) Assertion is true, but Reason is false.

d) Assertion is false, but Reason is true.

464 / 576

Category: Define objective function (profit, cost, etc.).

464. (A) In a linear programming problem, the objective function $Z = c_1x + c_2y$ must always represent either profit maximization or cost minimization.
(R) The coefficients $c_1$ and $c_2$ in the objective function always denote the profit per unit of decision variables $x$ and $y$ respectively.

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.

b) Both Assertion and Reason are true, but Reason is NOT the correct explanation of Assertion.

c) Assertion is true, but Reason is false.

d) Assertion is false, but Reason is true.

465 / 576

Category: LINEAR PROGRAMMING

465. A company has two warehouses W1 and W2 and two retail stores R1 and R2. The capacities of W1 and W2 are 100 and 150 units respectively. The demand at R1 is 80 units and at R2 is 120 units. The transportation costs per unit are \$10 from W1 to R1, \$15 from W1 to R2, \$12 from W2 to R1, and \$20 from W2 to R2. What should be the optimal transportation plan to minimize the total cost?

a) 70 units from W1 to R1, 30 units from W1 to R2, 10 units from W2 to R1, 90 units from W2 to R2

b) 80 units from W1 to R1, 20 units from W1 to R2, 0 units from W2 to R1, 100 units from W2 to R2

c) 60 units from W1 to R1, 40 units from W1 to R2, 20 units from W2 to R1, 110 units from W2 to R2

d) 50 units from W1 to R1, 50 units from W1 to R2, 30 units from W2 to R1, 120 units from W2 to R2

466 / 576

Category: Transportation Problems – Minimize cost of shipment.

466. Which of the following represents the supply constraint in a transportation problem?

a) $\sum_{i=1}^{m} x_{ij} \geq b_j$

b) $\sum_{j=1}^{n} x_{ij} \leq a_i$

c) $x_{ij} \leq 0$

d) $\sum_{i=1}^{m} x_{ij} = b_j$

467 / 576

Category: THE GRAPHICAL METHOD OF SOLUTION OF LINEAR PROGRAMMING PROBLEMS

467. (A) In the graphical method of solving a linear programming problem, the optimal solution always lies at one of the corner points of the feasible region.
(R) The feasible region of an LPP is always a convex polygon with finitely many corner points.

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.

b) Both Assertion and Reason are true, but Reason is NOT the correct explanation of Assertion.

c) Assertion is true, but Reason is false.

d) Assertion is false, but Reason is true.

468 / 576

Category: SOLVED EXAMPLES

468. A candidate has to answer questions worth 5 points, 4 points, and 6 points, taking 1 minute, 2 minutes, and 4 minutes respectively per question. The total time available is 210 minutes, with a maximum of 150 minutes for the 5-point and 4-point questions. Which of the following correctly represents the objective function and constraints?

a) Maximize $Z = 5x_1 + 4x_2 + 6x_3$; Constraints: $x_1 + 2x_2 + 4x_3 \leq 210$, $x_1 + 2x_2 \leq 150$, $x_1, x_2, x_3 \geq 0$

b) Minimize $Z = 5x_1 + 4x_2 + 6x_3$; Constraints: $x_1 + 2x_2 + 4x_3 \geq 210$, $x_1 + 2x_2 \leq 150$, $x_1, x_2, x_3 \geq 0$

c) Maximize $Z = x_1 + x_2 + x_3$; Constraints: $x_1 + 2x_2 + 4x_3 \leq 210$, $x_1 + 2x_2 \leq 150$, $x_1, x_2, x_3 \geq 0$

d) Minimize $Z = x_1 + x_2 + x_3$; Constraints: $x_1 + 2x_2 + 4x_3 \geq 210$, $x_1 + 2x_2 \leq 150$, $x_1, x_2, x_3 \geq 0$

469 / 576

Category: GENERAL LINAR PROGRAMMING PROBLEM

469. What is the general form of an objective function in a Linear Programming Problem?

a) $ax^2 + by + c$

b) $ax + by + c$

c) $ax + by^2 + c$

d) $ax^2 + by^2 + c$

470 / 576

Category: CORNER POINT METHOD.

470. (A) The optimal solution to a linear programming problem always occurs at a corner point of the feasible region.
(R) According to the extreme point theorem, if an optimal solution exists for a linear programming problem with a bounded feasible region, it must occur at one of the corner points.

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.

b) Both Assertion and Reason are true, but Reason is NOT the correct explanation of Assertion.

c) Assertion is true, but Reason is false.

d) Assertion is false, but Reason is true.

471 / 576

Category: FORMULATION OF A LINEAR PROGRAMMING PROBLEM

471. (A) In a linear programming problem, the objective function must always be linear.
(R) The constraints in a linear programming problem can be non-linear.

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.

b) Both Assertion and Reason are true, but Reason is NOT the correct explanation of Assertion.

c) Assertion is true, but Reason is false.

d) Assertion is false, but Reason is true.

472 / 576

Category: Infeasible solution.

472. (A) The solution $x = -3$, $y = 5$ is an infeasible solution for the LPP: Maximize $Z = 4x + 6y$ subject to $2x + 3y \leq 12$, $x + y \geq 1$, $x \geq 0$, $y \geq 0$.
(R) An infeasible solution violates at least one of the constraints or non-negativity conditions of the LPP.

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.

b) Both Assertion and Reason are true, but Reason is NOT the correct explanation of Assertion.

c) Assertion is true, but Reason is false.

d) Assertion is false, but Reason is true.

473 / 576

Category: Write down constraints in terms of inequalities/equations.

473. A company produces two products X and Y using three resources: labor, material, and machine time. The production process requires 4 hours of labor and 3 units of material per unit of X, and 5 hours of labor and 6 units of material per unit of Y. The total available labor is 200 hours, material is 180 units, and machine time is at most 150 hours (each product takes 1 hour). If x represents units of X and y represents units of Y, which set correctly represents all constraints?

a) $4x + 5y \leq 200$, $3x + 6y \leq 180$, $x \geq 0$, $y \geq 0$

b) $4x + 5y \leq 200$, $3x + 6y \leq 180$, $x + y \leq 150$, $x \geq 0$

c) $4x + 5y \leq 200$, $3x + 6y \leq 180$, $x + y \leq 150$, $x \geq 0$, $y \geq 0$

d) $5x + 4y \leq 200$, $6x + 3y \leq 180$, $x + y \leq 150$, $x \geq 0$, $y \geq 0$

474 / 576

Category: Theorem

474. (A) The maximum value of a linear expression $ax + by$ over a feasibility region occurs at one of its corner points.
(R) A feasibility region is the intersection of closed half-planes and has corner points.

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.

b) Both Assertion and Reason are true, but Reason is NOT the correct explanation of Assertion.

c) Assertion is true, but Reason is false.

d) Assertion is false, but Reason is true.

475 / 576

Category: Applications: maximize profit, minimize cost, optimal resource utilization.

475. A company has two factories located at cities X and Y. Factory X can produce 100 units and factory Y can produce 150 units. The demand at markets A, B, and C is 80, 90, and 80 units respectively. The transportation cost per unit from X to A, B, C is \$5, \$6, and \$7 respectively, and from Y to A, B, C is \$8, \$6, and \$5 respectively. What is the optimal transportation plan that minimizes total cost?

a) X to A=80, X to C=20, Y to B=90, Y to C=60

b) X to A=80, X to B=20, Y to B=70, Y to C=80

c) X to B=80, X to C=20, Y to A=80, Y to B=70

d) X to A=70, X to B=30, Y to A=10, Y to B=60, Y to C=80

476 / 576

Category: Steps in Formulating a Linear Programming Model

476. A company makes two models of chairs: Standard and Deluxe. Each Standard chair requires 2 hours of assembly and 1 hour of finishing, while each Deluxe requires 3 hours of assembly and 4 hours of finishing. There are 100 assembly hours and 120 finishing hours available daily. Profit is \$25 per Standard and \$40 per Deluxe chair. What is the complete LPP formulation?

a) Maximize $25x + 40y$ s.t. $2x + 3y \leq 100$, $x + 4y \leq 120$, $x,y \geq 0$

b) Maximize $40x + 25y$ s.t. $2x + 3y \leq 100$, $x + 4y \leq 120$, $x,y \geq 0$

c) Maximize $25x + 40y$ s.t. $3x + 2y \leq 100$, $4x + y \leq 120$, $x,y \geq 0$

d) Maximize $25x + 40y$ s.t. $2x + 3y \geq 100$, $x + 4y \geq 120$, $x,y \geq 0$

477 / 576

Category: Non-negativity restriction.

477. Which of the following is an example of a valid non-negativity restriction in an LPP?

a) $x \leq 0$

b) $x > 0$

c) $x \geq 0$

d) $y \leq -5$

478 / 576

Category: Identify decision variables (e.g., x, y).

478. A factory manufactures chairs and tables. Each chair requires 3 hours of labor and each table requires 5 hours. The total available labor hours per day are 150. If $x$ represents chairs and $y$ represents tables, which inequality correctly represents the labor constraint?

a) $3x + 5y \geq 150$

b) $5x + 3y \leq 150$

c) $3x + 5y \leq 150$

d) $3x - 5y \leq 150$

479 / 576

Category: Graphical solution of linear inequalities forms the basis of linear programming.

479. (A) The solution set of the inequality $2x + 3y \leq 6$ includes all points on the line $2x + 3y = 6$.
(R) The boundary line of a linear inequality is always part of the solution set when the inequality is non-strict ($\leq$ or $\geq$).

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.

b) Both Assertion and Reason are true, but Reason is NOT the correct explanation of Assertion.

c) Assertion is true, but Reason is false.

d) Assertion is false, but Reason is true.

480 / 576

Category: Apply non-negativity restrictions.

480. In a linear programming problem, what is the purpose of non-negativity restrictions?

a) To maximize the objective function

b) To ensure decision variables are non-negative

c) To reduce the feasible region size

d) To convert inequalities into equations

481 / 576

Category: Check feasible side using a test point (often origin).

481. (A) The origin (0, 0) is always a valid test point to determine the feasible side of an inequality in linear programming problems.
(R) The origin satisfies all non-negativity constraints by default.

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.

b) Both Assertion and Reason are true, but Reason is NOT the correct explanation of Assertion.

c) Assertion is true, but Reason is false.

d) Assertion is false, but Reason is true.

482 / 576

Category: Steps in Formulating a Linear Programming Model

482. A factory produces two types of products A and B. Product A gives a profit of \$8 per unit and product B gives \$5 per unit. The production requires 4 hours of machine time and 2 hours of labor for each unit of A, while B requires 2 hours of machine time and 3 hours of labor. The total available machine time is 100 hours and labor time is 90 hours. How should the problem be formulated to maximize profit?

a) Maximize $8x + 5y$ subject to: $4x + 2y \leq 100$, $2x + 3y \leq 80$, $x,y \geq 0$

b) Maximize $8x + 5y$ subject to: $4x + 2y \leq 100$, $2x + 3y \leq 90$, $x,y \geq 0$

c) Maximize $8x + 5y$ subject to: $4x + 3y \leq 100$, $2x + 3y \leq 90$, $x,y \geq 0$

d) Maximize $5x + 8y$ subject to: $4x + 2y \leq 100$, $2x + 3y \leq 90$, $x,y \geq 0$

483 / 576

Category: Optimal feasible solution.

483. (A) The point (4, 3) is an optimal feasible solution for the LPP: Maximize $Z = 5x + 4y$ subject to $2x + y \leq 12$, $x + 3y \leq 15$, $x \geq 0$, $y \geq 0$.
(R) The point (4, 3) satisfies all constraints and gives the highest value of $Z$ among all feasible solutions.

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.

b) Both Assertion and Reason are true, but Reason is NOT the correct explanation of Assertion.

c) Assertion is true, but Reason is false.

d) Assertion is false, but Reason is true.

484 / 576

Category: Developed by Kantorovich and George Dantzig.

484. In a linear programming problem, the constraints include $2x + y \leq 80$, $x + 2y \leq 70$, $x \leq 25$, $x \geq 0$, $y \geq 0$. If the objective is to minimize $Z = 3x + 5y$, at which point does this minimum occur?

a) $(25,0)$

b) $(0,35)$

c) $(0,0)$

d) $(25,22.5)$

485 / 576

Category: Feasible region : i region and Feasible solution.

485. A company produces two products, X and Y, with profits of \$10 and \$5 per unit respectively. The production is constrained by $2x + y \leq 100$, $x + 3y \leq 120$, $x \geq 0$, $y \geq 0$. Which of the following points is a corner point of the feasible region for this problem?

a) $(30, 20)$

b) $(36, 28)$

c) $(25, 35)$

d) $(40, 10)$

486 / 576

Category: Objective function.

486. What defines a convex region in the context of an LPP?

a) A polygon with more than four sides

b) A region where the line segment between any two points is entirely within the region

c) A region with only straight edges

d) A region that intersects itself

487 / 576

Category: Steps in Formulating a Linear Programming Model

487. Which of the following represents non-negativity constraints in a linear programming problem?

a) $x_1 \leq 0, x_2 \leq 0$

b) $x_1 \geq 0, x_2 \geq 0$

c) $x_1 > 0, x_2 > 0$

d) $x_1 + x_2 \geq 0$

488 / 576

Category: Graphical Method of Solution

488. A company produces two products, A and B. The profit per unit of A is \$8 and for B is \$12. The production constraints are:
$x + 2y \leq 30$ (labor hours),
$3x + y \leq 45$ (machine hours),
where $x$ and $y$ are the quantities of A and B, respectively. What is the optimal production mix to maximize profit?

a) $(15, 0)$

b) $(0, 15)$

c) $(12, 9)$

d) $(0, 0)$

489 / 576

Category: Diet Problems – Minimize cost subject to nutritional constraints.

489. What is the primary objective of a diet problem in linear programming?

a) Maximize the variety of food items

b) Minimize the cost of the diet

c) Maximize the nutritional content

d) Minimize the number of food items

490 / 576

Category: A method for optimizing (maximizing/minimizing) a linear function subject to linear constraints.

490. Consider the following LPP: Maximize $Z = 3x + 5y$ subject to the constraints $x + y \leq 6$, $x \geq 0$, $y \geq 0$, and $2x + y \leq 8$. What is the maximum value of $Z$?

a) 12

b) 18

c) 26

d) 30

491 / 576

Category: Applications: maximize profit, minimize cost, optimal resource utilization.

491. An investor has \$100,000 to invest in three funds: Fund X offers 6\% return, Fund Y offers 8\% return, and Fund Z offers 10\% return. The investor wants at least 30\% of the investment in Fund Z and no more than 20\% in Fund X. How should the money be allocated to maximize return?

a) \$25,000 in X, \$35,000 in Y, \$40,000 in Z

b) \$15,000 in X, \$45,000 in Y, \$40,000 in Z

c) \$20,000 in X, \$50,000 in Y, \$30,000 in Z

d) \$10,000 in X, \$55,000 in Y, \$35,000 in Z

492 / 576

Category: Linear Inequations in Two Variables

492. Given the lines $2x + y = 6$, $x - y = 1$, and $y = 0$, which set of constraints defines the feasible region where the origin $(0,0)$ lies on the opposite side of all shaded regions?

a) $2x + y \leq 6$, $x - y \geq 1$, $y \geq 0$

b) $2x + y \geq 6$, $x - y \geq 1$, $y \leq 0$

c) $2x + y \geq 6$, $x - y \leq 1$, $y \geq 0$

d) $2x + y \geq 6$, $x - y \geq 1$, $y \geq 0$

493 / 576

Category: Feasible region : i region and Feasible solution.

493. Consider the LPP:
Maximize $Z = 5x + 3y$ subject to $2x + y \leq 10$, $x + 3y \leq 15$, $x \geq 0$, $y \geq 0$. Which of the following points is NOT a feasible solution?

a) $(4,2)$

b) $(0,5)$

c) $(6,0)$

d) $(3,3)$

494 / 576

Category: Blending Problems – Optimize combination of raw materials.

494. A factory blends two types of raw materials, A and B, to produce a product. The cost per unit of material A is \$3 and material B is \$5. The objective function representing the total cost (Z) for using $x_1$ units of A and $x_2$ units of B is:

a) $Z = 3x_1 + 5x_2$

b) $Z = 5x_1 + 3x_2$

c) $Z = x_1 + x_2$

d) $Z = 8(x_1 + x_2)$

495 / 576

Category: The common region represents the solution set (infinite ordered pairs).

495. Given the system of inequalities $x + y \leq 6$, $2x - y \geq 0$, $y \geq 0$, what are the corner points of the feasible region?

a) $(0, 0)$, $(3, 3)$, $(6, 0)$

b) $(0, 0)$, $(2, 4)$, $(6, 0)$

c) $(1, 5)$, $(2, 4)$, $(6, 0)$

d) $(0, 0)$, $(2, 4)$, $(4, 2)$

496 / 576

Category: Optimal feasible solution.

496. Which of the following is a feasible solution for the LPP: Maximize $Z = 6x + 8y$ subject to $3x + 2y \leq 30$, $x + 2y \leq 22$, $x \geq 0$, $y \geq 0$?

a) $x = 2$, $y = 3$

b) $x = -2$, $y = 7$

c) $x = 5$, $y = 0$

d) $x = 0$, $y = -2$

497 / 576

Category: Graph each inequality on the same coordinate axes.

497. For the system of inequalities $y \geq x - 2$ and $y \leq 3$, which region represents the solution set?

a) The area above both lines $y = x - 2$ and $y = 3$

b) The area below both lines $y = x - 2$ and $y = 3$

c) The area between the lines $y = x - 2$ and $y = 3$

d) The area outside both lines $y = x - 2$ and $y = 3$

498 / 576

Category: SYSTEM OF THREE OR MORE INEQULITIES

498. Which of the following systems represents the solution set where $x \geq 0$, $y \geq 0$, and $2x + y \leq 6$?

a) $x \geq 0$, $y \geq 0$, $2x - y \leq 6$

b) $x > 0$, $y > 0$, $2x + y < 6$

c) $x \geq 0$, $y \geq 0$, $2x + y \leq 6$

d) $x \leq 0$, $y \leq 0$, $2x + y \geq 6$

499 / 576

Category: Identify decision variables (e.g., x, y).

499. A factory produces chairs and tables. Each chair requires 2 kg of wood and 1 hour of labor; each table requires 5 kg of wood and 3 hours of labor. Available resources are 100 kg of wood and 60 hours of labor daily. If $x$ is chairs and $y$ is tables, which point lies within the feasibility region?

a) $(30, 10)$

b) $(20, 10)$

c) $(10, 20)$

d) $(25, 15)$

500 / 576

Category: FORMULATION OF A LINEAR PROGRAMMING PROBLEM

500. What is the correct objective function for a linear programming problem where a company wants to maximize profit (\$P) from selling two products, A and B, with profits of \$10 per unit of A and \$15 per unit of B?

a) Maximize $P = 10x + 15y$

b) Minimize $P = 15x + 10y$

c) Maximize $P = 5x + 15y$

d) Minimize $P = 10x + 15y$

501 / 576

Category: ADVANTAGES OF LINEAR PROGRAMMING PROBLEMS

501. In which sectors can linear programming be applied?

a) Only in manufacturing

b) Exclusively in finance

c) Across multiple sectors like manufacturing, transportation, and agriculture

d) Only in theoretical research

502 / 576

Category: Represent graphically and find feasible region.

502. (A) The feasible region in a linear programming problem is always bounded.
(R) The feasible region is the intersection of all constraint inequalities and non-negativity restrictions.

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.

b) Both Assertion and Reason are true, but Reason is NOT the correct explanation of Assertion.

c) Assertion is true, but Reason is false.

d) Assertion is false, but Reason is true.

503 / 576

Category: Optimal feasible solution.

503. A feasible region in an LPP is bounded by the lines:
$2x + y = 6$, $x + 2y = 8$, $x = 0$, $y = 0$.
If the objective is to maximize $Z = 5x + 3y$, which point is optimal?

a) $(3, 0)$

b) $(0, 4)$

c) $(\frac{4}{3}, \frac{10}{3})$

d) $(2, 2)$

504 / 576

Category: Convex region.

504. Which of the following statements best defines a convex region?

a) A region with at least one "dent"

b) A region where any two points can be connected by a curve within the region

c) A region where any two points can be connected by a straight line segment within the region

d) A region formed by non-linear constraints

505 / 576

Category: Write down constraints in terms of inequalities/equations.

505. In a linear programming problem, which of the following constraints ensures that decision variables cannot be negative?

a) $x + y \leq 10$

b) $x \geq 0$, $y \geq 0$

c) $x \leq 100$

d) $3x + 2y \leq 50$

506 / 576

Category: Check feasible side using a test point (often origin).

506. For the system of inequalities $\begin{cases} x^2 + y^2 \leq 25 \\ x + y \leq 5 \\ x \geq 0, y \geq 0 \end{cases}$, which of the following points lies outside the feasible region?

a) $(4,1)$

b) $(1,3)$

c) $(0,5)$

d) $(3,4)$

507 / 576

Category: FORMULATION OF A LINEAR PROGRAMMING PROBLEM

507. A company produces two products, A and B. The profit per unit of product A is \$10 and for product B is \$15. The production process requires 2 hours of labor for A and 3 hours for B. The total available labor hours are 120. Also, the raw material required for A is 4 kg and for B is 5 kg, with a total of 200 kg available. Additionally, the company cannot produce more than 50 units of A due to market constraints. Formulate the LPP to maximize profit.

a) Maximize $Z = 10x_1 + 15x_2$ subject to $x_1 + x_2 \leq 100$, $2x_1 + 3x_2 \leq 120$, $x_1 \leq 50$, $x_1, x_2 \geq 0$

b) Maximize $Z = 10x_1 + 15x_2$ subject to $2x_1 + 3x_2 \leq 120$, $4x_1 + 5x_2 \leq 200$, $x_1 \leq 50$, $x_1, x_2 \geq 0$

c) Maximize $Z = 10x_1 + 15x_2$ subject to $2x_1 + 3x_2 \geq 120$, $4x_1 + 5x_2 \leq 200$, $x_1 \leq 50$, $x_1, x_2 \geq 0$

d) Maximize $Z = 10x_1 + 15x_2$ subject to $2x_1 + 3x_2 \leq 120$, $4x_1 + 5x_2 \geq 200$, $x_1 \leq 50$, $x_1, x_2 \geq 0$

508 / 576

Category: System of Inequations

508. (A) The solution set of the system of inequalities $x + y \leq 4$ and $x \geq 0$, $y \geq 0$ is bounded.
(R) The feasible region of a system of linear inequalities is always bounded if all constraints are non-negative.

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.

b) Both Assertion and Reason are true, but Reason is NOT the correct explanation of Assertion.

c) Assertion is true, but Reason is false.

d) Assertion is false, but Reason is true.

509 / 576

Category: Identify common overlapping region (solution region).

509. (A) The solution region for the system of inequalities $x + y \geq 4$, $2x - y \leq 6$, and $x \geq 0$ is bounded and has three corner points.
(R) A bounded solution region in a system of linear inequalities must have at least three corner points formed by the intersection of the boundary lines.

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.

b) Both Assertion and Reason are true, but Reason is NOT the correct explanation of Assertion.

c) Assertion is true, but Reason is false.

d) Assertion is false, but Reason is true.

510 / 576

Category: Objective function.

510. Identify the infeasible solution for the LPP: Maximize $Z = 2x + 3y$ subject to $x + y \leq 8$, $2x + y \leq 10$, $x \geq 0$, $y \geq 0$.

a) (3, 4)

b) (5, 0)

c) (-1, 9)

d) (4, 3)

511 / 576

Category: Blending Problems – Optimize combination of raw materials.

511. (A) In a blending problem, the objective function $Z = 3x_1 + 7x_2$ represents the total cost to be minimized for raw materials A and B, where $x_1$ and $x_2$ are quantities of A and B respectively.
(R) The coefficients 3 and 7 in the objective function represent the per-unit cost of raw materials A and B.

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.

b) Both Assertion and Reason are true, but Reason is NOT the correct explanation of Assertion.

c) Assertion is true, but Reason is false.

d) Assertion is false, but Reason is true.

512 / 576

Category: Blending Problems – Optimize combination of raw materials.

512. (A) In a blending problem, if the cost per unit of raw materials $A$ and $B$ are \$5 and \$8 respectively, the objective function to minimize cost would be $Z = 5x_1 + 8x_2$.
(R) The objective function in blending problems always represents either cost minimization or profit maximization under given constraints.

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.

b) Both Assertion and Reason are true, but Reason is NOT the correct explanation of Assertion.

c) Assertion is true, but Reason is false.

d) Assertion is false, but Reason is true.

513 / 576

Category: Areas of Application

513. A logistics company needs to transport goods from three warehouses (W1, W2, W3) to four retail stores (R1, R2, R3, R4). The supply capacities are 50, 100, and 150 units, respectively, and the demands are 80, 60, 70, and 90 units, respectively. The transportation cost matrix (in \$ per unit) is given below:

$\begin{bmatrix} \\ 5 & 8 & 6 & 10 \\ \\ 7 & 6 & 9 & 8 \\ \\ 6 & 5 & 7 & 4 \\ \\ \end{bmatrix}$

Which of the following statements about solving this problem using linear programming is FALSE?

a) The problem can be modeled as a minimization of total transportation cost.

b) The constraints ensure that total supply equals total demand.

c) The solution will always involve shipping goods from the cheapest routes exclusively.

d) The problem can have multiple optimal solutions.

514 / 576

Category: Diet Problems – Minimize cost subject to nutritional constraints.

514. Which of the following represents a typical constraint in a diet problem?

a) $x_1 + x_2 \leq 100$

b) $a_{11}x_1 + a_{12}x_2 \geq b_1$

c) $c_1x_1 + c_2x_2 = Z$

d) $x_1 - x_2 > 0$

515 / 576

Category: LINEAR PROGRAMMING

515. What is the objective function in a Linear Programming Problem (LPP)?

a) A set of inequalities representing constraints

b) The linear function to be maximized or minimized

c) The non-negativity restrictions on variables

d) The feasible region of the problem

516 / 576

Category: Theorem

516. Consider the following linear programming problem:
Maximize $Z = 5x + 3y$
Subject to:
$2x + y \leq 10$,
$x + 3y \leq 12$,
$x, y \geq 0$.
Which of the following points is a feasible solution that yields the maximum value of $Z$?

a) $(0, 4)$

b) $(5, 0)$

c) $(3.6, 2.8)$

d) $(0, 0)$

517 / 576

Category: Diet Problems – Minimize cost subject to nutritional constraints.

517. (A) In a diet problem, the decision variables represent the quantities of different foods to be consumed.
(R) The objective function in a diet problem is always linear and represents the total cost to be minimized.

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.

b) Both Assertion and Reason are true, but Reason is NOT the correct explanation of Assertion.

c) Assertion is true, but Reason is false.

d) Assertion is false, but Reason is true.

518 / 576

Category: Fundamental Theorem (Extreme Point Theorem)

518. (A) The optimal solution to a linear programming problem always occurs at a corner point of the feasible region.
(R) The feasible region in a linear programming problem is always a convex set.

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.

b) Both Assertion and Reason are true, but Reason is NOT the correct explanation of Assertion.

c) Assertion is true, but Reason is false.

d) Assertion is false, but Reason is true.

519 / 576

Category: SYSTEM OF THREE OR MORE INEQULITIES

519. Given the inequalities $x - y \geq 1$, $y \geq 2$, and $5y - 2x \leq 10$, which point lies in the solution set?

a) $(0, 0)$

b) $(1, 1)$

c) $(4, 3)$

d) $(5, 5)$

520 / 576

Category: Manufacturing Problems – Optimize production for profit.

520. A company produces two products, $X$ and $Y$. The profit per unit of product $X$ is $\$4$ and for product $Y$ is $\$3$. What is the objective function to maximize the total profit if $x$ represents units of $X$ and $y$ represents units of $Y$?

a) $Z = 3x + 4y$

b) $Z = 4x - 3y$

c) $Z = 4x + 3y$

d) $Z = x + y$

521 / 576

Category: Transportation Problems – Minimize cost of shipment.

521. In the transportation problem, what does the supply constraint $\sum_{j=1}^{n} x_{ij} \leq a_i$ represent?

a) Total demand must be met

b) Transport cost per unit

c) Quantity shipped from source $i$ cannot exceed its supply

d) All shipments must be positive

522 / 576

Category: Theorem

522. A linear programming problem has the constraints:
$3x + 4y \leq 24$,
$x + y \leq 7$,
$x \geq 0$, $y \geq 0$.
If the objective function is $Z = 2x + 5y$, which point among the following gives the minimum value of $Z$?

a) $(4, 3)$

b) $(0, 7)$

c) $(8, 0)$

d) $(0, 0)$

523 / 576

Category: Areas of Application

523. What is the optimal solution point in the given example where the objective function $Z = 50x + 40y$ is maximized under the constraints $5x + 6y \leq 38$ and $3x + 2y \leq 18$?

a) $(3, 4)$

b) $(4, 3)$

c) $(5, 2)$

d) $(2, 5)$

524 / 576

Category: GENERAL LINAR PROGRAMMING PROBLEM

524. Why are non-negativity restrictions applied to decision variables in Linear Programming Problems?

a) To make the problem more complex

b) To ensure the solution is always positive

c) Because negative values have no valid interpretation in physical problems

d) To limit the number of solutions

525 / 576

Category: Investment Problems – Maximize return on capital allocation.

525. A firm invests in two projects A and B. Project A requires an initial investment of \$5,000 and yields \$700 annually, while Project B requires \$8,000 and yields \$1,000 annually. The total budget is \$40,000, and the firm must invest at least \$10,000 in Project A. What is the optimal investment strategy to maximize annual returns?

a) Invest in 2 units of Project A and 3 units of Project B

b) Invest in 3 units of Project A and 4 units of Project B

c) Invest in 4 units of Project A and 1 unit of Project B

d) Invest in 5 units of Project A and 2 units of Project B

526 / 576

Category: Manufacturing Problems – Optimize production for profit.

526. A factory produces two goods $P$ and $Q$. Each unit of $P$ requires 2 hours of labor, and each unit of $Q$ requires 3 hours of labor. If the factory has a maximum of 90 labor hours available, which inequality represents the labor constraint?

a) $2x + 3y \geq 90$

b) $2x + 3y \leq 90$

c) $3x + 2y \leq 90$

d) $x + y \leq 90$

527 / 576

Category: A method for optimizing (maximizing/minimizing) a linear function subject to linear constraints.

527. (A) The corner point method can be used to find the optimal solution of a linear programming problem with two variables.
(R) The extreme point theorem states that the optimal solution of an LPP, if it exists, occurs at one of the corner points of the feasible region.

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.

b) Both Assertion and Reason are true, but Reason is NOT the correct explanation of Assertion.

c) Assertion is true, but Reason is false.

d) Assertion is false, but Reason is true.

528 / 576

Category: Convex region.

528. Why is the feasible region in a Linear Programming Problem always convex?

a) It has multiple optimal solutions

b) It is formed by non-linear constraints

c) It is the intersection of convex sets (half-planes)

d) It includes all boundary points

529 / 576

Category: LINAR PROGRAMMING-DEFINITION

529. What is the objective function in a linear programming problem?

a) A function that represents constraints

b) A function that defines non-negativity restrictions

c) The linear function to be optimized

d) A function used for graphical representation

530 / 576

Category: Understanding regions of inequality is essential before tackling LPP problems.

530. (A) The feasible region for the system of inequalities $2x + 3y \leq 6$ and $4x - y \geq 2$ includes the origin $(0, 0)$.

(R) The origin $(0, 0)$ satisfies both inequalities $2x + 3y \leq 6$ and $4x - y \geq 2$.

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.

b) Both Assertion and Reason are true, but Reason is NOT the correct explanation of Assertion.

c) Assertion is true, but Reason is false.

d) Assertion is false, but Reason is true.

531 / 576

Category: Identify decision variables (e.g., x, y).

531. What is the purpose of the objective function in a linear programming model?

a) To define the constraints

b) To represent the goal of the optimization problem

c) To limit the feasible solutions

d) To assign values to decision variables

532 / 576

Category: THE GRAPHICAL METHOD OF SOLUTION OF LINEAR PROGRAMMING PROBLEMS

532. For the constraints $x + y \leq 5$, $2x + y \leq 8$, $x \geq 0$, $y \geq 0$, what is the minimum value of the objective function $Z = 3x + 4y$?

a) \$0

b) \$12

c) \$17

d) \$20

533 / 576

Category: Graphical solution of linear inequalities forms the basis of linear programming.

533. Which region represents the solution to the inequality $y \geq 2$ in the $x-y$ plane?

a) The region below the line $y = 2$

b) The region above the line $y = 2$, including the line

c) The region to the right of the line $y = 2$

d) The region to the left of the line $y = 2$

534 / 576

Category: Non-negativity restriction.

534. (A) The feasible region of a linear programming problem (LPP) cannot include points where any decision variable is negative.

(R) Non-negativity restrictions ensure that the solutions to an LPP are physically meaningful by preventing variables from taking negative values.

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.

b) Both Assertion and Reason are true, but Reason is NOT the correct explanation of Assertion.

c) Assertion is true, but Reason is false.

d) Assertion is false, but Reason is true.

535 / 576

Category: Fundamental Theorem (Extreme Point Theorem)

535. For the linear programming problem: Maximize $Z = 3x + 2y$ subject to $x + y \leq 5$, $2x + y \geq 12$, $x \geq 0$, $y \geq 0$, which statement is correct?

a) The optimal solution occurs at (6, -1)

b) The problem has multiple optimal solutions

c) The feasible region is unbounded

d) There is no feasible solution

536 / 576

Category: Areas of Application

536. (A) Linear programming is useful in determining the optimal mix of advertising media to maximize reach or minimize costs.
(R) Linear programming can handle both quantitative and qualitative factors in decision-making.

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.

b) Both Assertion and Reason are true, but Reason is NOT the correct explanation of Assertion.

c) Assertion is true, but Reason is false.

d) Assertion is false, but Reason is true.

537 / 576

Category: Optimal feasible solution.

537. According to the Extreme Point Theorem, where does the optimal solution of an LPP occur?

a) At any point inside the feasible region

b) At the intersection of two constraint lines (corner point)

c) At the center of the feasible region

d) On the boundary but not necessarily at a corner point

538 / 576

Category: Graphical Method of Solution

538. Why is the graphical method not suitable for problems with more than two variables?

a) It requires complex calculations

b) It cannot handle non-linear constraints

c) It is difficult to plot in more than two dimensions

d) It always gives incorrect results for multi-variable problems

539 / 576

Category: Fundamental Theorem (Extreme Point Theorem)

539. A company produces two products A and B using three machines M1, M2, and M3. The time required (in hours) per unit for each product on each machine is given by:
Product A: M1 - 2hrs, M2 - 1hr, M3 - 2hrs
Product B: M1 - 2hrs, M2 - 2hrs, M3 - 1hr
The available time on M1, M2, and M3 are 100, 120, and 80 hours respectively. The profit per unit for A is \$40 and for B is \$30. What is the maximum profit if at least 10 units of product B must be produced?

a) \$1900

b) \$2000

c) \$2133

d) \$2200

540 / 576

Category: Write down constraints in terms of inequalities/equations.

540. (A) In a linear programming problem, the constraints $x \geq 0$ and $y \geq 0$ are always included to ensure non-negativity of variables.
(R) Non-negativity constraints prevent solutions with negative quantities, which are unrealistic in real-world resource allocation problems.

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.

b) Both Assertion and Reason are true, but Reason is NOT the correct explanation of Assertion.

c) Assertion is true, but Reason is false.

d) Assertion is false, but Reason is true.

541 / 576

Category: GENERAL LINAR PROGRAMMING PROBLEM

541. (A) The objective function in a linear programming problem must always be maximized.
(R) The feasible region of a linear programming problem is formed by the intersection of all constraints.

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.

b) Both Assertion and Reason are true, but Reason is NOT the correct explanation of Assertion.

c) Assertion is true, but Reason is false.

d) Assertion is false, but Reason is true.

542 / 576

Category: Linear Programming Problem (LPP) Example

542. (A) The feasible region of a linear programming problem is always convex.
(R) The intersection of any number of convex sets is also a convex set.

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.

b) Both Assertion and Reason are true, but Reason is NOT the correct explanation of Assertion.

c) Assertion is true, but Reason is false.

d) Assertion is false, but Reason is true.

543 / 576

Category: Non-negativity restriction.

543. In a Linear Programming Problem (LPP), what does the non-negativity restriction ensure?

a) Decision variables can take any real value.

b) Only the objective function must be non-negative.

c) All decision variables must be non-negative.

d) Only constraints must be non-negative.

544 / 576

Category: Constraints.

544. A company produces two products, P and Q. The constraints for production are given as $x \geq 0$, $y \geq 0$, $3x + 4y \leq 120$, and $x + y \leq 40$. Which of the following points lies within the feasible region?

a) (10, 20)

b) (30, 15)

c) (25, 20)

d) (0, 35)

545 / 576

Category: Infeasible solution.

545. For the LPP: Maximize $Z = 2x + 5y$ subject to $x + y \leq 6$, $x - y \geq 2$, $x \geq 0$, $y \geq 0$, which of the following is an infeasible solution?

a) $x = 4$, $y = 2$

b) $x = 1$, $y = 5$

c) $x = 3$, $y = 1$

d) $x = 0$, $y = 0$

546 / 576

Category: Objective function.

546. Given the LPP: Minimize $Z = 3x + 5y$ subject to $x + 2y \geq 10$, $3x + y \geq 12$, $x \geq 0$, $y \geq 0$. Which point provides the optimal solution?

a) (0, 5)

b) (4, 0)

c) (2.8, 3.6)

d) (1, 4.5)

547 / 576

Category: Linear Inequations in Two Variables

547. What is the solution set for the system: $x + y \geq 2$ and $x \geq 0$?

a) On or below the line $x + y = 2$ and left of the $y$-axis

b) On or above the line $x + y = 2$ and right of the $y$-axis

c) Only on the line $x + y = 2$

d) Only the first quadrant

548 / 576

Category: GENERAL LINAR PROGRAMMING PROBLEM

548. A manufacturer has two warehouses W1 and W2, and three retail stores R1, R2, and R3. The supply at W1 is 50 units and at W2 is 60 units. The demand at R1 is 30 units, R2 is 40 units, and R3 is 40 units. The transportation costs per unit are \$2 from W1 to R1, \$3 from W1 to R2, \$4 from W1 to R3, \$3 from W2 to R1, \$2 from W2 to R2, and \$1 from W2 to R3. Formulate the LPP to minimize transportation cost.

a) Minimize $Z = 2x_{11} + 3x_{12} + 4x_{13} + 3x_{21} + 2x_{22} + 1x_{23}$ subject to $x_{11} + x_{12} + x_{13} \geq 50$, $x_{21} + x_{22} + x_{23} \geq 60$, $x_{11} + x_{21} \leq 30$, $x_{12} + x_{22} \leq 40$, $x_{13} + x_{23} \leq 40$, all $x_{ij} \geq 0$

b) Minimize $Z = 2x_{11} + 3x_{12} + 4x_{13} + 3x_{21} + 2x_{22} + 1x_{23}$ subject to $x_{11} + x_{12} + x_{13} \leq 50$, $x_{21} + x_{22} + x_{23} \leq 60$, $x_{11} + x_{21} = 30$, $x_{12} + x_{22} = 40$, $x_{13} + x_{23} = 40$, all $x_{ij} \geq 0$

c) Maximize $Z = 2x_{11} + 3x_{12} + 4x_{13} + 3x_{21} + 2x_{22} + 1x_{23}$ subject to $x_{11} + x_{12} + x_{13} \leq 50$, $x_{21} + x_{22} + x_{23} \leq 60$, $x_{11} + x_{21} = 30$, $x_{12} + x_{22} = 40$, $x_{13} + x_{23} = 40$, all $x_{ij} \geq 0$

d) Minimize $Z = 2x_{11} + 3x_{12} + 4x_{13} + 3x_{21} + 2x_{22} + 1x_{23}$ subject to $x_{11} + x_{12} + x_{13} \leq 50$, $x_{21} + x_{22} + x_{23} \leq 60$, $x_{11} + x_{21} = 30$, $x_{12} + x_{22} = 40$, $x_{13} + x_{23} = 40$, all $x_{ij} \leq 0$

549 / 576

Category: THE GRAPHICAL METHOD OF SOLUTION OF LINEAR PROGRAMMING PROBLEMS

549. Given the constraints $2x + y \leq 6$, $x + y \leq 4$, $x \geq 0$, $y \geq 0$, which of the following points is NOT a corner point of the feasible region?

a) $(0, 0)$

b) $(2, 2)$

c) $(1, 3)$

d) $(3, 0)$

550 / 576

Category: Identify common overlapping region (solution region).

550. Which of the following represents the common solution region for the system of inequations $x + y \geq 2$, $2x - y \leq 4$, and $x \geq 0$?

a) A triangle with vertices at $(0,2)$, $(2,0)$, and $(0,-4)$

b) A quadrilateral with vertices at $(0,2)$, $(2,0)$, $(3,2)$, and $(0,-4)$

c) An unbounded region extending infinitely upwards

d) A single point at $(1,1)$

551 / 576

Category: Graphical Method of Solution

551. A linear programming problem has the feasible region with corner points at $(0, 0)$, $(0, 5)$, $(4, 2)$, and $(6, 0)$. If the objective function is $Z = 2x + 5y$, which corner point minimizes $Z$?

a) $(0, 5)$

b) $(4, 2)$

c) $(6, 0)$

d) $(0, 0)$

552 / 576

Category: Linear Programming Problem (LPP) Example

552. A company manufactures two types of products A and B. Product A gives a profit of \$20 per unit and requires 3 hours of labor and 1 kg of material. Product B gives a profit of \$30 per unit and requires 4 hours of labor and 2 kg of material. The company has 120 hours of labor and 60 kg of material available daily. What is the maximum profit the company can achieve while satisfying all constraints?

a) 700

b) 800

c) 900

d) 1000

553 / 576

Category: Corner points (vertices) can be found by solving boundary equations in pairs.

553. For the given constraints:
$x + y \geq 4, \, 2x + y \geq 6, \, x \geq 0, \, y \geq 0$
identify which pair of equations gives the corner point $(2, 2)$.

a) $x + y = 4$ and $x = 0$

b) $2x + y = 6$ and $y = 0$

c) $x + y = 4$ and $2x + y = 6$

d) $x + y = 4$ and $x + y = 6$

554 / 576

Category: FORMULATION OF A LINEAR PROGRAMMING PROBLEM

554. A toy manufacturer makes two types of toys, cars and bikes. The profit per car is \$20 and per bike is \$30. Each car requires 1 hour of machine time and 2 hours of painting time. Each bike requires 3 hours of machine time and 1 hour of painting time. The total available machine time is 90 hours and painting time is 80 hours. The demand for cars is at most 40 units. Formulate the LPP to maximize profit.

a) Maximize $Z = 20x_1 + 30x_2$ subject to $x_1 + 3x_2 \leq 90$, $2x_1 + x_2 \leq 80$, $x_1 \leq 40$, $x_1, x_2 \geq 0$

b) Maximize $Z = 20x_1 + 30x_2$ subject to $x_1 + 3x_2 \geq 90$, $2x_1 + x_2 \leq 80$, $x_1 \leq 40$, $x_1, x_2 \geq 0$

c) Maximize $Z = 20x_1 + 30x_2$ subject to $x_1 + 3x_2 \leq 90$, $2x_1 + x_2 \geq 80$, $x_1 \leq 40$, $x_1, x_2 \geq 0$

d) Maximize $Z = 20x_1 + 30x_2$ subject to $x_1 + 3x_2 \leq 90$, $2x_1 + x_2 \leq 80$, $x_1 \geq 40$, $x_1, x_2 \geq 0$

555 / 576

Category: Graphical solution of linear inequalities forms the basis of linear programming.

555. A farmer wants to maximize his profit by growing wheat and barley. The constraints are $2x + y \leq 80$, $x + y \leq 50$, $x \geq 0$, $y \geq 0$, where $x$ and $y$ are acres of wheat and barley respectively. The profit per acre is \$100 for wheat and \$120 for barley. What is the maximum profit?

a) \$4000

b) \$5400

c) \$6000

d) \$6500

556 / 576

Category: Graphical Method of Solution

556. According to the extreme point theorem, where does the optimal solution of a linear programming problem lie?

a) At the center of the feasible region

b) Along the boundary of the feasible region

c) At one of the corner points of the feasible region

d) Outside the feasible region

557 / 576

Category: GENERAL LINAR PROGRAMMING PROBLEM

557. (A) In a linear programming problem, the maximum or minimum value of the objective function always occurs at a corner point of the feasibility region.
(R) The feasibility region in a linear programming problem is always a convex set formed by the intersection of half-planes.

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.

b) Both Assertion and Reason are true, but Reason is NOT the correct explanation of Assertion.

c) Assertion is true, but Reason is false.

d) Assertion is false, but Reason is true.

558 / 576

Category: Objective function.

558. Which of the following is an example of a feasible solution for the LPP: Maximize $Z = 6x + 8y$ subject to $3x + 2y \leq 30$, $x + 2y \leq 22$, $x \geq 0$, $y \geq 0$?

a) $x = -1$, $y = 10$

b) $x = 5$, $y = -3$

c) $x = 2$, $y = 3$

d) $x = 15$, $y = 5$

559 / 576

Category: Convex region.

559. (A) The feasible region defined by the constraints $x + y \leq 4$, $x \geq 0$, $y \geq 0$ is convex.
(R) For any two points $(x_1, y_1)$ and $(x_2, y_2)$ in the feasible region, the point $\left( \lambda x_1 + (1 - \lambda)x_2, \lambda y_1 + (1 - \lambda)y_2 \right)$ lies within the region for all $\lambda \in [0, 1]$.

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.

b) Both Assertion and Reason are true, but Reason is NOT the correct explanation of Assertion.

c) Assertion is true, but Reason is false.

d) Assertion is false, but Reason is true.

560 / 576

Category: Represent graphically and find feasible region.

560. The feasible region for the inequalities $2x + y \leq 10$, $x \geq 0$, $y \geq 0$ is bounded by which of the following points?

a) (0, 0), (5, 0), (0, 5)

b) (0, 0), (10, 0), (0, 10)

c) (0, 0), (5, 0), (0, 10)

d) (0, 0), (5, 5), (0, 10)

561 / 576

Category: System of Inequations

561. Which of the following represents the common solution region for the system of inequalities $x + y \leq 4$ and $x - y \geq 2$?

a) Region above both lines

b) Region below both lines

c) Region between the two lines

d) No common region exists

562 / 576

Category: THE GRAPHICAL METHOD OF SOLUTION OF LINEAR PROGRAMMING PROBLEMS

562. A company produces two products, P and Q. The profit per unit of P is \$8 and per unit of Q is \$12. The production constraints are given by:
$x + 2y \leq 10$, $3x + y \leq 15$, $x \geq 0$, $y \geq 0$. What is the maximum profit achievable?

a) \$60

b) \$40

c) \$68

d) \$0

563 / 576

Category: Optimal feasible solution.

563. A manufacturing problem has constraints forming an unbounded feasible region with vertices at $(0,4)$, $(2,2)$, and $(3,1)$. If the objective is to maximize $P = 2x + y$, which statement is true?

a) Maximum P is unbounded

b) Maximum P occurs at $(2,2)$

c) Maximum P occurs at $(3,1)$

d) No optimal solution exists

564 / 576

Category: Feasible region : i region and Feasible solution.

564. (A) The feasible region of an LPP is always a convex set.
(R) Every feasible solution lies on the boundary of the feasible region.

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.

b) Both Assertion and Reason are true, but Reason is NOT the correct explanation of Assertion.

c) Assertion is true, but Reason is false.

d) Assertion is false, but Reason is true.

565 / 576

Category: LINEAR PROGRAMMING

565. (A) In a linear programming problem, the feasible region is always a convex polygon.
(R) The feasible region is formed by the intersection of half-spaces defined by linear inequalities.

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.

b) Both Assertion and Reason are true, but Reason is NOT the correct explanation of Assertion.

c) Assertion is true, but Reason is false.

d) Assertion is false, but Reason is true.

566 / 576

Category: SOME DEFINITIONS AND CONCEPTS RELATED TO LPP

566. What is the objective function in a linear programming problem?

a) A non-linear equation representing constraints

b) The linear function whose maximum or minimum value is determined

c) The region where all constraints are satisfied

d) A solution that satisfies all constraints

567 / 576

Category: Diet Problems – Minimize cost subject to nutritional constraints.

567. (A) In a diet problem formulated as a linear programming model, the point $(15, 0)$ minimizes the cost function $Z = 5x_1 + 7x_2$ but is not a feasible solution.
(R) A feasible solution must satisfy all constraints, including nutritional requirements, and $(15, 0)$ violates the protein constraint $2x_1 + 3x_2 \geq 60$.

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.

b) Both Assertion and Reason are true, but Reason is NOT the correct explanation of Assertion.

c) Assertion is true, but Reason is false.

d) Assertion is false, but Reason is true.

568 / 576

Category: GENERAL LINAR PROGRAMMING PROBLEM

568. Consider the feasible region defined by the inequalities $x + 2y \leq 10$, $3x + y \leq 15$, $x \geq 0$, $y \geq 0$. At which corner point does the objective function $Z = 2x + 5y$ attain its maximum value?

a) (0, 0)

b) (0, 5)

c) (5, 0)

d) (4, 3)

569 / 576

Category: The common region represents the solution set (infinite ordered pairs).

569. (A) The common region formed by the intersection of graphs of $x + y \geq 4$ and $2x - y \leq 6$ represents an infinite set of ordered pairs as its solution set.
(R) The intersection of two or more half-planes always results in a region bounded by straight lines, which contains infinitely many points.

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.

b) Both Assertion and Reason are true, but Reason is NOT the correct explanation of Assertion.

c) Assertion is true, but Reason is false.

d) Assertion is false, but Reason is true.

570 / 576

Category: Infeasible solution.

570. Which statement correctly describes an infeasible solution in an LPP?

a) It satisfies all constraints and non-negativity conditions.

b) It violates at least one constraint or non-negativity condition.

c) It maximizes the objective function.

d) It has no effect on the feasible region.

571 / 576

Category: Theorem

571. Find the minimum value of $Z = 8x + 6y$ subject to $x + 2y \geq 4$, $3x + y \geq 6$, $x \geq 0$, $y \geq 0$.

a) 20

b) 12

c) 16

d) 18

572 / 576

Category: Corner points (vertices) can be found by solving boundary equations in pairs.

572. Given the system of inequalities:
$x + y \leq 6, \, x - y \geq 1, \, x \geq 0, \, y \geq 0$
Find the coordinates of the corner points of the feasible region.

a) $(\frac{7}{2}, \frac{5}{2}), (3, 1), (5, 1)$

b) $(1, 0), (6, 0), (0, 6)$

c) $(\frac{7}{2}, \frac{5}{2}), (1, 0), (6, 0)$

d) $(0, 6), (3, 3), (6, 0)$

573 / 576

Category: System of Inequations

573. Consider the system of inequalities:
$5x + 4y \leq 20$
$x + 2y \geq 6$
$x \geq 0, y \geq 0$
Which of the following points lies in the feasible region of this system?

a) (0, 5)

b) (2, 2)

c) (6, 0)

d) (3, 1)

574 / 576

Category: CORNER POINT METHOD.

574. (A) The optimal solution to a linear programming problem must always occur at a corner point of the feasible region.
(R) The Extreme Point Theorem states that if an optimal solution exists, it must lie at one of the corner points of the feasible region.

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.

b) Both Assertion and Reason are true, but Reason is NOT the correct explanation of Assertion.

c) Assertion is true, but Reason is false.

d) Assertion is false, but Reason is true.

575 / 576

Category: Define objective function (profit, cost, etc.).

575. In a production problem, if the profit per unit of product A is \$20 and per unit of product B is \$30, what is the correct objective function for maximizing profit where $x$ and $y$ represent the quantities of products A and B respectively?

a) $Z = 30x + 20y$

b) $Z = x + y$

c) $Z = 50xy$

d) $Z = 20x + 30y$

576 / 576

Category: LINAR PROGRAMMING-DEFINITION

576. A company manufactures two products, X and Y. The profit per unit is \$40 for X and \$30 for Y. The production process requires 3 hours of labor and 1 hour of machine time for product X, and 2 hours each of labor and machine time for product Y. If the total available labor is 60 hours and machine time is 36 hours, which of the following represents the optimal feasible solution to maximize profit?

a) (5, 10)

b) (15, 7.5)

c) (12, 12)

d) (8, 16)

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