68 free MCQs on Linear Programming, each with its own worked answer and explanation. Optimizing a linear objective function subject to linear constraints using the graphical corner point method.
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The feasible region (shaded) is bounded by the constraint lines; the fundamental theorem of LPP guarantees the optimal value of the objective function occurs at one of the corner points (O, A, B, C), so only these need to be checked, not the entire region.
Q1.
In a linear programming problem, the function to be maximized or minimized is called the:
Q2.
The unknown quantities that a linear programming problem solves for are called:
Q3.
The linear inequalities that decision variables must satisfy in an LPP are called:
Q4.
The set of all points satisfying every constraint of an LPP simultaneously is called the:
Q5.
According to the fundamental theorem of linear programming, if an optimal value exists, where does it occur?
Q6.
A feasible region that can be enclosed within a circle of finite radius is called:
Q7.
Decision variables in a typical LPP are restricted to be:
Q8.
A diet problem in linear programming typically aims to:
Q9.
A manufacturing problem in linear programming typically aims to:
Q10.
The graphical method of solving an LPP by evaluating the objective function at every vertex of the feasible region is called the:
Q11.
Linear programming is a method for finding the ___ value of a linear function subject to constraints:
Q12.
The linear function to be maximised or minimised in an LPP is called the:
Q13.
The conditions written as linear inequalities in an LPP are called:
Q14.
The common region that satisfies all the constraints is the:
Q15.
The requirement that the variables be greater than or equal to zero is the ___ restriction:
Q16.
The optimal solution of an LPP occurs at a ___ of the feasible region:
Q17.
A feasible region that extends indefinitely is described as:
Q18.
With x ≥ 0 and y ≥ 0, the feasible region lies in the:
Q19.
Points that satisfy all the constraints of an LPP are called ___ solutions:
Q20.
A linear programming problem has a linear objective function together with linear:
Q21.
Maximize Z = 3x + 4y subject to x + y <= 4, x >= 0, y >= 0. What is the maximum value of Z?
Q22.
For the feasible region with corners (0,0), (4,0), (0,4) from x + y <= 4 (x,y >= 0), what is the maximum value of Z = 5x + 2y?
Q23.
A transportation problem in linear programming typically minimizes:
Q24.
Maximize Z = 50x + 60y subject to 2x + y <= 120, x + 2y <= 120, x >= 0, y >= 0. The corner points of the feasible region are (0,0), (60,0), (40,40), and (0,60). What is the maximum value of Z?
Q25.
If the feasible region of an LPP is unbounded, what is true about finding the maximum of the objective function?
Q26.
Minimize Z = 200x + 500y subject to x + 2y >= 10, 3x + 4y <= 24, x >= 0, y >= 0. The feasible region has corners (0,5), (4,3), and (0,6). What is the minimum value of Z?
Q27.
In an LPP, what does a corner point represent geometrically?
Q28.
Maximize Z = 7x + 4y subject to 3x + y <= 600, x + y <= 300, x >= 0, y >= 0. The relevant corner points are (0,0), (200,0), (150,150), and (0,300). What is the maximum Z?
Q29.
If two corner points of a feasible region give the same maximum value of the objective function, what can be concluded?
Q30.
For constraints x >= 0, y >= 0, and x <= 10 only (no upper bound on y), the feasible region is:
Q31.
By the corner-point theorem, the optimum of the objective function is attained at a:
Q32.
If the feasible region of an LPP is bounded, the objective function has:
Q33.
To solve an LPP graphically, the objective function is evaluated at each:
Q34.
Maximise Z = 3x + 2y over the corner points (0,0), (4,0) and (0,5). The maximum value is:
Q35.
The constraints of an LPP usually take a form such as ax + by ≤ c or:
Q36.
If the objective is Z = x + y and a corner point is (3, 4), the value of Z there is:
Q37.
For a minimisation LPP, we choose the corner point giving the ___ value of Z:
Q38.
The feasible region of a linear programming problem is always a ___ set:
Q39.
If two corner points give the same optimal value of Z, then the optimum occurs at:
Q40.
The boundary lines of the constraints are obtained by replacing each inequality with:
Q41.
Minimize Z = 3x + 9y subject to x + 3y <= 60, x + y >= 10, x <= y, x >= 0, y >= 0. The corner points of the feasible region are (0,10), (5,5), and (15,15). Which gives the minimum Z?
Q42.
A furniture maker produces chairs (x) and tables (y). Each chair needs 2 hours of carpentry and 1 hour of finishing; each table needs 1 hour of carpentry and 3 hours of finishing. Only 40 carpentry hours and 60 finishing hours are available. If profit is Rs 30 per chair and Rs 60 per table, which is the correct constraint pair?
Q43.
For the furniture problem with constraints 2x + y <= 40, x + 3y <= 60, x >= 0, y >= 0 and objective Z = 30x + 60y, the corner points are (0,0), (20,0), (0,20), and the intersection of the two lines. Find the intersection point.
Q44.
Continuing the furniture problem (Z = 30x + 60y, corners (0,0), (20,0), (12,16), (0,20)), what is the maximum profit?
Q45.
A diet problem requires at least 8 units of vitamin A and at least 11 units of vitamin B daily. Food 1 (x units) gives 2 units of A and 1 unit of B per item; Food 2 (y units) gives 1 unit of A and 2 units of B per item. Which constraints model the minimum requirements?
Q46.
For the diet problem with constraints 2x + y >= 8, x + 2y >= 11, x >= 0, y >= 0 and cost Z = 50x + 70y, the relevant corner points are (0,8), (5/3, 14/3), and (11,0). What is the minimum cost?
Q47.
In an LPP to minimize cost with an unbounded feasible region (extending infinitely upward and to the right), which statement is generally true?
Q48.
Two corner points (10, 0) and (0, 15) of a feasible region both give Z = 300 for the objective function Z = 30x + 20y. What does this indicate about the LPP?
Q49.
A factory produces two products A and B. Each unit of A needs 3 machine-hours and 1 labour-hour; each unit of B needs 1 machine-hour and 2 labour-hours. Only 12 machine-hours and 10 labour-hours are available daily. Which constraint pair models this?
Q50.
For the LPP Maximize Z = 5x + 4y subject to 3x + y <= 12 and x + 2y <= 10, x,y >= 0, the corner points include (0,0), (4,0), (0,5), and the intersection of the two boundary lines. Find Z at that intersection point.
Q51.
Maximise Z = 5x + 3y subject to x + y ≤ 4, x ≥ 0, y ≥ 0 (corners (0,0), (4,0), (0,4)). The maximum Z is:
Q52.
For an unbounded feasible region, the maximum value of the objective function may:
Q53.
In the LPP maximise Z = x + y subject to x + y ≤ 6, the maximum value of Z is:
Q54.
The set of all feasible solutions of a two-variable LPP forms a:
Q55.
Minimise Z = 2x + 3y over the corner points (0,0), (2,0) and (0,2). The minimum value is:
Q56.
For the constraint 2x + y ≤ 10, the boundary line meets the y-axis (x = 0) at:
Q57.
The point (2, 3) satisfies the constraint x + y ≤ 6 because:
Q58.
The number of decision variables in a standard two-variable LPP is:
Q59.
In a typical diet problem, the objective function usually represents the ___ to be minimised:
Q60.
Maximise Z = 4x + 3y over corners (0,0), (6,0), (0,4) and (3,3). The maximum value is:
Q61.
For an LPP with corner points (0,0), (4,0), (0,5) and (2,3), the maximum of Z = 3x + 4y is:
Q62.
The feasible region of any linear programming problem is always a:
Q63.
For corner points (0,0), (3,0) and (0,4), the maximum of Z = 2x + 3y is:
Q64.
For corner points (2,3), (4,1) and (1,5), the minimum of Z = x + y is:
Q65.
Subject to 2x + y ≤ 10 with x, y ≥ 0, the maximum value of x is:
Q66.
If the optimal value of an LPP occurs at two adjacent corner points, then the problem has:
Q67.
For corner points (0,0), (0,8), (4,10), (6,8) and (6,0), the maximum of Z = 4x + 3y is:
Q68.
Subject to x + y ≤ 4 with x, y ≥ 0, the maximum of Z = x + 2y is:
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