Q.1 A linear programming problem (LPP) involves maximizing Z = 3x + 4y subject to constraints x + y ≤ 5, x ≥ 0, y ≥ 0. What are the corner points of the feasible region?
(0,0), (5,0), (0,5)
(0,0), (5,0), (5,5)
(0,0), (0,5), (5,5)
(0,0), (2,3), (5,0)
Explanation - The feasible region is bounded by x ≥ 0, y ≥ 0, and x + y ≤ 5. The intercepts on axes give the corner points: (0,0), (5,0), and (0,5).
Correct answer is: (0,0), (5,0), (0,5)
Q.2 For the LPP: Maximize Z = 2x + 3y subject to x + 2y ≤ 8, x ≥ 0, y ≥ 0, which method is most commonly used to find the optimal solution?
Graphical Method
Simplex Method
Dual Method
Monte Carlo Method
Explanation - For two-variable LPPs, the Graphical Method is commonly used to find the optimal solution by plotting constraints and evaluating Z at corner points.
Correct answer is: Graphical Method
Q.3 A feasible solution of an LPP is
Any solution satisfying all constraints
The solution that maximizes Z
The solution that minimizes Z
Any solution satisfying only non-negativity constraints
Explanation - A feasible solution satisfies all the constraints of the linear programming problem, including inequalities and non-negativity conditions.
Correct answer is: Any solution satisfying all constraints
Q.4 If Z = x + y and constraints are x + y ≤ 4, x ≥ 0, y ≥ 0, what is the maximum value of Z?
2
4
0
Infinity
Explanation - Maximum value occurs at the corner point (4,0) or (0,4), giving Z = 4.
Correct answer is: 4
Q.5 Which of the following is a necessary condition for a linear programming problem?
All equations are quadratic
Objective function is linear
Constraints are non-linear
Variables are unrestricted in sign
Explanation - In LPP, both the objective function and all constraints must be linear.
Correct answer is: Objective function is linear
Q.6 The feasible region of a linear programming problem is always
A convex polygon
A concave polygon
A circle
A parabola
Explanation - The feasible region formed by linear inequalities is always a convex set.
Correct answer is: A convex polygon
Q.7 If an LPP has three variables, x, y, z, the feasible region is a
3D polyhedron
2D polygon
Line
Circle
Explanation - In three-variable LPP, the feasible region is a polyhedron in 3D space formed by planes representing constraints.
Correct answer is: 3D polyhedron
Q.8 Maximize Z = 5x + 2y subject to x + y ≤ 6, x ≤ 4, y ≤ 3, x ≥ 0, y ≥ 0. The optimal solution occurs at
(4,2)
(2,3)
(0,3)
(4,0)
Explanation - Evaluate Z at all corner points: (0,0)=0, (4,0)=20, (0,3)=6, (4,2)=24. Maximum Z = 24 at (4,2).
Correct answer is: (4,2)
Q.9 In LPP, a constraint of type '≥' can be converted into standard form by
Adding slack variable
Subtracting slack variable
Adding surplus variable
Dividing by 2
Explanation - A '≥' inequality is converted by subtracting a surplus variable to transform it into an equality for simplex method.
Correct answer is: Adding surplus variable
Q.10 A linear programming problem with no feasible solution is called
Infeasible
Unbounded
Degenerate
Redundant
Explanation - If no point satisfies all constraints simultaneously, the LPP has no feasible solution and is called infeasible.
Correct answer is: Infeasible
Q.11 A linear programming problem with feasible solutions but the objective function can increase indefinitely is called
Unbounded
Infeasible
Optimal
Degenerate
Explanation - If the feasible region is open in the direction of optimization, the LPP is unbounded.
Correct answer is: Unbounded
Q.12 Graphical method of solving LPP can be applied when
Number of variables is 2
Number of variables is 3
Number of constraints is 2
Number of variables is 4 or more
Explanation - Graphical method is feasible only for 2-variable LPPs because we can represent constraints in a 2D plane.
Correct answer is: Number of variables is 2
Q.13 A basic feasible solution in simplex method corresponds to
A corner point of feasible region
The centroid of feasible region
Any interior point
The maximum value only
Explanation - In simplex method, each basic feasible solution corresponds to a vertex (corner point) of the feasible region.
Correct answer is: A corner point of feasible region
Q.14 If an LPP has redundant constraints, it means
Some constraints do not affect feasible region
Feasible region is empty
Objective function is zero
Feasible region is unbounded
Explanation - Redundant constraints do not limit or change the feasible region and can be removed without affecting the solution.
Correct answer is: Some constraints do not affect feasible region
Q.15 The optimal solution of an LPP always occurs at
One of the corner points
Midpoint of feasible region
Origin
Any interior point
Explanation - By the fundamental theorem of linear programming, the optimal value occurs at a corner point of the feasible region.
Correct answer is: One of the corner points
Q.16 If the objective function Z = 4x + 5y has the same value at two corner points, the LPP is said to have
Multiple optimal solutions
No solution
Unbounded solution
Degenerate solution
Explanation - When two or more corner points give the same maximum (or minimum) value, the LPP has multiple optimal solutions.
Correct answer is: Multiple optimal solutions
Q.17 In a two-variable LPP, if the line of objective function is parallel to a constraint boundary and gives maximum Z on that line, the solution is
Multiple optimal solutions
No solution
Unique solution
Unbounded solution
Explanation - Parallelism between objective function and constraint boundary indicates all points along that line segment are optimal.
Correct answer is: Multiple optimal solutions
Q.18 Slack variables are introduced in LPP to convert inequalities into
Equalities
Quadratic equations
Non-linear inequalities
Redundant constraints
Explanation - Slack variables are added to '≤' inequalities to convert them into equalities for applying simplex method.
Correct answer is: Equalities
Q.19 In LPP, the dual of a maximization problem is always
A minimization problem
Also a maximization problem
Quadratic problem
Non-linear problem
Explanation - The dual of a maximization LPP is a minimization problem with constraints formed from the primal variables and vice versa.
Correct answer is: A minimization problem
Q.20 If a feasible region of an LPP is a line segment, then the number of corner points is
2
1
0
Infinite
Explanation - A line segment has two endpoints, which act as corner points in the feasible region.
Correct answer is: 2
Q.21 The simplex method was developed by
George Dantzig
Isaac Newton
Leonhard Euler
Carl Gauss
Explanation - George Dantzig developed the simplex method in 1947 to solve linear programming problems efficiently.
Correct answer is: George Dantzig
Q.22 For an LPP with two variables, if feasible region is unbounded in the direction of maximization, the problem is
Unbounded
Infeasible
Multiple optimal
Degenerate
Explanation - If the objective function can increase indefinitely in the feasible region, the LPP is unbounded.
Correct answer is: Unbounded
Q.23 A degenerate basic feasible solution occurs when
At least one basic variable is zero
All basic variables are positive
Feasible region is empty
Objective function is zero
Explanation - Degeneracy occurs in simplex method when a basic variable takes value zero at a corner point.
Correct answer is: At least one basic variable is zero
Q.24 A two-variable LPP has constraints x ≥ 0, y ≥ 0, x + y ≤ 10, 2x + y ≤ 12. Which point is feasible?
(3,4)
(7,6)
(5,8)
(-1,2)
Explanation - Check constraints: x + y = 3+4=7 ≤10, 2x + y =6+4=10 ≤12, x ≥0, y ≥0. Feasible point is (3,4).
Correct answer is: (3,4)
Q.25 If an LPP has two constraints that intersect outside the first quadrant, the feasible region will be
Only in first quadrant
Unbounded in negative region
No feasible region in first quadrant
Always infinite
Explanation - Since all variables are non-negative, if intersection is outside the first quadrant, feasible region is empty in first quadrant.
Correct answer is: No feasible region in first quadrant