In linear programming, both the objective function and constraints must be linear in the decision variables. Linearity means variables appear only to the first power without products like xy.
A linear constraint in two variables graphs as a
A Circle arc
B Parabola curve
C Straight line
D Spiral curve
A linear equation in x and y represents a straight line. For an inequality, that line forms a boundary, and the solution is one side of the line (a half-plane).
For an LPP, constraints must be
A Linear relations
B Random choices
C Nonlinear equations
D Trig identities
Constraints describe limitations like resources or requirements. In LPP, they must be linear so the feasible region becomes an intersection of half-planes, supporting graphical and corner-point methods.
A feasible solution must satisfy
A Objective only
B One constraint
C All constraints
D No constraint
A point is feasible only if it satisfies every constraint along with non-negativity conditions. Feasible solutions can be many, but only those are allowed candidates for the optimal solution.
Non-negativity condition means variables are
A Always integers
B Always equal
C Always fractions
D Not negative
Non-negativity requires decision variables to be zero or positive, matching real-life quantities like units, hours, or kilograms. It restricts solutions to the first quadrant in graphs.
In graphical method, the common shaded area is the
A Feasible region
B Objective region
C Random region
D Profit region
After shading each constraint’s satisfying half-plane, the overlap of all shaded parts gives the feasible region. Any point outside it violates at least one constraint.
A bounded feasible region means it is
A Infinite area
B Empty area
C Finite area
D Curved area
Bounded means the feasible set is enclosed and does not extend infinitely. Variables have upper limits due to constraints, and an optimum (if feasible) is easier to locate.
A solution is infeasible when it
A Maximizes objective
B Violates constraint
C Lies on vertex
D Has zero slack
If a point breaks even one constraint or non-negativity condition, it is not allowed and is called infeasible. Only feasible points are considered when searching for the optimum.
For maximization, iso-profit lines move toward
A Increasing Z
B Decreasing Z
C Zero slope
D Same point
Iso-profit lines represent Z = constant. For maximization, shifting the line parallel in the direction that increases Z until it last touches the feasible region gives the optimum location.
For minimization, iso-cost lines move toward
A Increasing Z
B Random Z
C Fixed Z
D Decreasing Z
In minimization, we shift the objective line parallel toward smaller Z values. The first point where the line touches the feasible region gives the minimum objective value.
The boundary of a ≤ constraint is drawn using
A Parallel lines
B Curved line
C Equality line
D Dotted circle
To graph an inequality like ax + by ≤ c, first draw the line ax + by = c. Then decide which side satisfies the inequality using a test point.
A point on the boundary line of a constraint
A May be feasible
B Always infeasible
C Always optimal
D Always excluded
Points on the constraint line satisfy it as equality. They are feasible if they also satisfy all other constraints and non-negativity. Many optimal points lie on boundaries.
The half-plane method is used for
A Differentiation
B Integration rules
C Matrix inverse
D Inequality plotting
Half-plane method means drawing each constraint line and selecting the correct side that satisfies the inequality. The intersection of all such half-planes gives the feasible region.
When a constraint is redundant, removing it
A Empties region
B Makes unbounded
C Changes nothing
D Flips shading
A redundant constraint is already implied by other constraints. Removing it does not alter the feasible region or the optimum because all feasible points already satisfy it automatically.
If constraints produce no common overlap, the feasible set is
A Triangular
B Empty
C Rectangular
D Circular
No overlap means no point satisfies all constraints together. This creates an infeasible LPP. Graphically, shaded half-planes do not share any common region.
In a two-variable LPP, the feasible region is generally
A Spherical region
B Elliptic region
C Polygonal region
D Random region
Each linear inequality gives a half-plane bounded by a line. Their intersection forms a polygon (possibly unbounded). Corner points of this polygon are key candidates for optimality.
Corner points are also called
A Vertices
B Radii
C Foci
D Tangents
Corner points are intersection points of boundary lines that form the feasible region’s “corners.” In linear programming, optimal solutions (if they exist) occur at one or more vertices.
A vertex is found by solving
A Two inequalities
B Two derivatives
C Two integrals
D Two equalities
To locate an intersection point, we take two boundary lines and solve them as equations. Then we verify whether that intersection satisfies all inequalities to be feasible.
In corner-point method, you must
A Ignore feasibility
B Skip vertices
C Check feasibility
D Use midpoints
Not every intersection point is feasible. After finding vertices, each must be tested against all constraints. Only feasible vertices are evaluated in the objective function.
If two vertices give same best Z, the LPP has
A Alternate optimum
B No optimum
C Infeasible region
D Redundant objective
When two different feasible vertices yield the same optimal objective value, the entire line segment between them (if feasible) also becomes optimal, giving multiple optimal solutions.
A common sign for “at most” in constraints is
A ≥ sign
B ≤ sign
C = sign
D ± sign
“At most” means cannot exceed a limit, so it becomes a ≤ constraint. Example: hours used ≤ available hours. This type often represents resource capacity.
A common sign for “at least” in constraints is
A ≤ sign
B = sign
C ≠ sign
D ≥ sign
“At least” means a minimum requirement, so it becomes a ≥ constraint. Example: protein intake ≥ required amount. Such constraints ensure minimum satisfaction levels.
Objective coefficients are often interpreted as
A Vertex per unit
B Slope per unit
C Profit per unit
D Constraint per unit
In many word problems, coefficients in the objective function represent profit or cost per unit of each activity. The objective calculates total profit/cost from chosen quantities.
For Z = 3x + 2y, coefficient of y is
A 2
B 3
C 5
D 1
The coefficient multiplying y is 2. It shows how much Z changes when y increases by one unit, keeping other variables fixed, assuming linear relation in objective.
A feasible solution in graphical method is a point
A On any line
B In shaded overlap
C Outside all lines
D On axes only
The feasible set is exactly the region that satisfies all inequalities, shown by the common overlapping shaded area. Any point in this region represents an allowed solution.
The objective line touches feasible region at optimum in
A Random contact point
B First quadrant only
C Last contact point
D Any interior point
For maximization, we move the objective line in the improving direction until it can’t move further while still touching the feasible region. That final contact gives the optimum.
When objective keeps improving without limit, maximum is
A Zero always
B At origin
C At midpoint
D Not defined
If Z can increase indefinitely while remaining feasible, then no finite maximum exists. This is an unbounded optimum situation and indicates missing limiting constraints.
If all constraints are satisfied but objective not best, the point is
A Optimal only
B Infeasible
C Feasible only
D Redundant
Satisfying constraints means the point is feasible. Optimality requires best objective value among feasible points. Many feasible points exist, but only one or more are optimal.
The feasible region of linear inequalities is always
A Convex set
B Concave set
C Circular set
D Disconnected always
Intersection of half-planes forms a convex region. For any two feasible points, every point on the segment joining them is also feasible, which supports the vertex-based optimization.
A line segment between two feasible points lies
A Outside region
B Inside region
C On circle only
D On axes only
Because the feasible region is convex, the entire segment joining any two feasible points remains feasible. This property is fundamental in linear programming geometry.
In standardization, converting ≥ to ≤ often involves
A Add same number
B Take square root
C Multiply by -1
D Add intercept
To change ax + by ≥ c into a ≤ form, multiply both sides by -1, reversing inequality: -ax – by ≤ -c. This helps align constraints for standard forms.
The main purpose of plotting constraints is to
A Identify feasible set
B Find derivatives
C Solve triangles
D Compute averages
Plotting shows where all constraints are satisfied simultaneously. Once the feasible region is identified, we list feasible corner points and evaluate the objective function there.
A typical production LPP decides
A Angles to measure
B Units to produce
C Curves to draw
D Roots to find
Production LPPs choose quantities of products (decision variables) to maximize profit or minimize cost under constraints like labor hours, machine time, and raw material availability.
The graphical method needs constraints in
A Four variables
B Many variables
C Two variables
D Complex variables
Graphical solution is mainly for two variables because we can draw constraints on a plane. With more variables, the feasible region becomes higher-dimensional and hard to visualize.
A point is checked for feasibility by
A Differentiating function
B Integrating function
C Factoring polynomial
D Substituting values
To verify whether (x, y) is feasible, substitute into every inequality and non-negativity condition. If all are satisfied, the point lies in feasible region.
A constraint line divides the plane into
A Many circles
B Four triangles
C Two half-planes
D One region
A line forms a boundary splitting the plane into two sides. A linear inequality chooses one side (including the boundary). The feasible region is the common side for all constraints.
If Z is constant, the objective graph is a
A Family of lines
B Family of circles
C Family of parabolas
D Family of ellipses
Z = ax + by gives parallel straight lines for different constant values of Z. These are iso-profit/iso-cost lines, moved to locate the best feasible touch point.
When a constraint is binding, its slack is
A Maximum
B Negative
C Infinite
D Zero
A binding constraint holds as equality at the solution. For a ≤ constraint, slack becomes zero, meaning the resource is fully used. Binding constraints often shape the optimal corner.
A non-binding ≤ constraint has slack
A Exactly zero
B Positive
C Always negative
D Always infinite
If a ≤ constraint is not tight, the left side is less than the right side, so slack is positive. It indicates unused resource and the constraint is not limiting at that point.
“Feasible corner points” are important because
A Curves occur there
B Derivatives vanish
C Optimum occurs there
D Integrals vanish
In linear programming with a convex polygonal feasible region, an optimal solution (if it exists) occurs at a feasible vertex. Hence we test only feasible corner points.
In a graph, non-negativity constraints restrict region to
A First quadrant
B Second quadrant
C Third quadrant
D Fourth quadrant
x ≥ 0 and y ≥ 0 keep the solution region where both coordinates are nonnegative, i.e., first quadrant. This matches real-life quantities which cannot be negative.
If the feasible region is nonempty and bounded, and the objective function is linear, where does an optimal value occur (if it exists)?
A At origin only
B At any interior
C Always on axis
D At a vertex
If there is no touching point because the feasible region is empty, no optimum exists. If the region exists, objective lines will touch; failure suggests infeasibility.
A constraint with “≤” generally means
A Lower limit
B Upper limit
C Equal demand
D Random demand
A ≤ constraint places a maximum on the left-side expression, like resource usage cannot exceed availability. It commonly models capacities, budgets, or limited hours.
A constraint with “≥” generally means
A Maximum capacity
B No requirement
C Minimum requirement
D Random selection
A ≥ constraint ensures at least some level of achievement, such as minimum production, minimum nutrition, or minimum service. It is a lower bound constraint.
In an LPP, all coefficients are assumed
A Known constants
B Changing daily
C Random variables
D Unknown slopes
Linear programming assumes coefficients in objective and constraints are fixed and known. If they keep changing, the model becomes unstable. This is why LPP avoids current-affairs type data.
The corner point method is mainly used to
A Find derivative zero
B Find curve center
C Find optimal vertex
D Find area only
Corner point method lists all feasible vertices and computes objective value at each. The vertex giving best value is optimal. This works well for two-variable problems.
A “feasible polygon” is formed by
A Adding circles
B Rotating axes
C Factoring lines
D Intersecting half-planes
Each inequality gives a half-plane. The feasible region is their intersection, producing a polygonal shape. Its edges are parts of constraint lines, and vertices guide optimization.
If a constraint line is wrongly shaded, it mainly affects
A Objective slope
B Feasible region
C Variable names
D Coefficients size
Incorrect shading selects the wrong half-plane, giving a wrong feasible region. Then the chosen vertices and computed optimum become incorrect, even if calculations afterward are done correctly.
Infeasible LPP means
A Many feasible points
B Unique optimum
C No feasible point
D Infinite optimum
Infeasible means there is no point satisfying all constraints at the same time. Graphically, half-planes have no common intersection. Hence optimization cannot be performed.
The simplex method is used mainly for
A More variables
B Two variables
C One variable
D Zero variables
Simplex is an algebraic method designed for LPPs with many variables and constraints where graphical method is impossible. It systematically moves along vertices to find the optimum.