The Additivity assumption states that the total contribution of all decision variables is obtained by simply adding their individual contributions. This means there are no interaction effects among the variables, and each variable contributes independently to the objective function and constraints. For example, if the objective function is: Z = ax + by then the total contribution is the sum of ax and by, without any additional interaction terms such as xy. Option C is correct because it correctly defines the assumption of additivity. Option A is incorrect because proportionality means each variable contributes directly in proportion to its value. Option B is incorrect because divisibility means decision variables may take fractional values. Option D is incorrect because certainty means all coefficients and parameters are known and fixed.

In constrained optimization, the objective function is optimized while satisfying restrictions imposed by limited resources. These resources commonly include labour, machine time, raw materials, storage space, and budget. Such limitations are expressed as linear constraints in an LPP. Option D is correct because constraints are formulated using the availability of resources such as labour, space, and materials. Option A is incorrect because derivatives and integrals belong to calculus and are not used in formulating LPP constraints. Option B is incorrect because constraints are not based solely on independent vectors. Option C is incorrect because non-linear moving averages belong to statistics and have no role in Linear Programming.

The Assertion is false because constraints in Linear Programming are represented by linear equations or linear inequalities, not quadratic equations. The Reason is also false because constraints exist precisely due to the limited availability of resources. If resources were infinite, optimization constraints would not be required. Therefore, both the Assertion and the Reason are false. Hence, Option A is the correct answer.

The objective function in a Linear Programming Problem is a real-valued linear function that represents the quantity to be optimized. Depending on the problem, it may represent profit, cost, production, distance, time, or any measurable objective. A general objective function is: Z = ax + by where a and b are constants. Option B is the incorrect statement because the objective function is not limited to time; it frequently represents profit, cost, production, or other measurable quantities. Option A is correct because the objective function is a real-valued function. Option C is correct because Z = ax + by is the standard linear objective function. Option D is correct because the objective function is always maximized or minimized.

Each concept has a standard meaning in Linear Programming. Corner Point → The intersection of two boundary lines. Z = ax + by → Linear objective function. x ≥ 0 → Represents the non-negative restriction, listed here as a trivial constraint element. Unbounded → A feasible region extending infinitely without complete enclosure. Thus, the correct matching is: 1 → d 2 → c 3 → a 4 → b Therefore, Option C is the correct answer.

To determine which side of a boundary line satisfies a linear inequality, the point (0,0) is usually tested. If (0,0) satisfies the inequality, then the half-plane containing the origin represents the solution region. If it does not satisfy the inequality, the opposite half-plane is selected. Therefore, if the inequality is true at (0,0), the feasible side contains the origin. Hence, Option D is correct.

The feasible region determined by the constraints is unbounded. Although the objective function has values of 3 and 7 at the given corner points, the feasible region extends infinitely in a direction where the value of Z = 6x + y continues to increase. Since larger values of Z can always be obtained by moving farther within the feasible region, no finite maximum value exists. Therefore, Option A is correct.

For an unbounded feasible region, the minimum value is verified using the following procedure: 1. Evaluate Z at all corner points to obtain the minimum candidate value m. 2. Draw the objective function line ax + by = m. 3. Check whether the half-plane ax + by < m contains any feasible point. 4. If no feasible point exists, then m is the minimum value of the objective function. Hence, the correct order is: 3 → 2 → 1 → 4 Therefore, Option B is correct.

The statement "at least 1500 pieces per day" specifies a minimum production requirement. Such conditions restrict the possible values of the decision variables and are therefore represented as linear constraints. For example, if x and y denote the working hours of Grade I and Grade II inspectors respectively, a possible constraint is: 20x + 14y ≥ 1500 This is a linear inequality that restricts the feasible region. Option C correctly identifies this as a linear constraint. Option A is incorrect because the objective function represents the quantity to be optimized. Option B is incorrect because non-negative constraints are of the form x ≥ 0, y ≥ 0. Option D is incorrect because an iso-profit line represents the objective function, not a production requirement.

The objective function represents the quantity to be optimized, such as minimizing total inspection cost. Since each inspection error costs ₹20, this cost contributes directly to the coefficients of the objective function. The objective function combines all relevant costs to determine the total cost for optimization. Option D is correct because cost information appears in the coefficients of the objective function. Option A is incorrect because corner points are determined only by the constraints. Option B is incorrect because intersection equations define corner points, not costs. Option C is incorrect because labour constraints specify available working time rather than inspection costs.

The given equations are: 4x + y = 20 2x + 3y = 30 From the first equation: y = 20 − 4x Substitute this into the second equation: 2x + 3(20 − 4x) = 30 2x + 60 − 12x = 30 −10x = −30 x = 3 Substitute x = 3 into the first equation: y = 20 − 12 y = 8 Therefore, the point of intersection is: (3, 8) Hence, Option A is the correct answer.

Evaluate the objective function: At A(6,0) Z = 6 + 2(0) = 6 At B(0,3) Z = 0 + 2(3) = 6 Since the minimum value is the same at both adjacent corner points, the objective function line is parallel to the line segment joining them. Therefore, every point on the line segment AB also gives Z = 6. This is the condition for multiple optimal solutions. Hence, Option B is the correct answer.

The variables x and y represent working hours of inspectors. Since working hours cannot be negative, they must satisfy the non-negative restrictions: x ≥ 0 y ≥ 0 These conditions ensure that the solution is physically meaningful and lies in the first quadrant. Therefore, Option C is the correct answer.

The objective function in a Linear Programming Problem is written as: Z = ax + by where x and y are the decision variables, and a and b are constants. The value of Z changes whenever x or y changes. Hence, the objective function is dependent on the decision variables. Therefore: Assertion (A) is false because Z depends directly on the values of x and y. Reason (R) is true because Z = ax + by is a linear function of the decision variables. Thus, Option D is the correct answer.

The feasible region is the common region obtained after considering all the constraints and the non-negative restrictions. Statement (i) is correct because every point in the feasible region satisfies all constraints simultaneously. Statement (ii) is correct because a feasible region may be either bounded or unbounded depending on the given constraints. Statement (iii) is correct because every point inside or on the boundary of the feasible region represents a feasible solution. Since all three statements are correct, Option A is the correct answer.

An infeasible solution is a point that fails to satisfy one or more constraints, including the non-negative restrictions. Option B is the incorrect statement because an infeasible solution does not satisfy all constraints. Option A is correct because an optimal solution must always be feasible. Option C is correct because violating x ≥ 0 makes a solution infeasible. Option D is correct because the feasible region contains only feasible solutions. Therefore, Option B is the correct answer.

According to the Fundamental Theorem of Linear Programming, if an optimal solution exists, then at least one corner (extreme) point of the feasible region will provide the optimal value of the objective function. This principle forms the basis of the Corner-Point Method, where the objective function is evaluated at every corner point, and the maximum or minimum value is selected. Option C is correct because an optimal solution occurs at an extreme point whenever it exists. Option A is incorrect because the center of the feasible region does not necessarily provide the optimum. Option B is incorrect because the origin may not even belong to the feasible region. Option D is incorrect because an optimal solution must always be feasible and therefore lies within or on the boundary of the feasible region.

In the Iso-profit Method, the objective function is represented by a family of parallel lines. For a maximization problem, these lines are shifted parallel to themselves in the direction of increasing objective-function values. The last line that still touches the feasible region gives the maximum value of the objective function. Thus, the optimal solution lies on the line farthest from the origin while still intersecting the feasible region. Therefore, Option D is correct.

The boundary line is: 3x + 5y = 15 To find the x-intercept, substitute y = 0: 3x = 15 x = 5 So, the x-intercept is (5,0). To find the y-intercept, substitute x = 0: 5y = 15 y = 3 So, the y-intercept is (0,3). Therefore, the boundary line intersects the coordinate axes at (5,0) and (0,3). Hence, Option A is the correct answer.

The factory has a total capacity of 8 units. If: x units are supplied to depot A, and y units are supplied to depot B, then the remaining units supplied to depot C are: Units to depot C = Total capacity − Units to A − Units to B Therefore, Units to depot C = 8 − x − y Hence, Option B is the correct answer.