ECAT Mathematics
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ECAT Mathematics
Chapter 19: Linear Inequalities and Linear Programming
75 Questions
90 Minutes
Pass: 60%+
ECAT Prep
0
A linear inequality in two variables is of the form:
A
ax²+by²<c
B
ax+by<c (or ≤,>,≥)
C
ax+by=c
D
ax²+by=c
1
The solution of 2x+3<9:
A
x<2
B
x<3
C
x<6
D
x>2
2
The solution of −3x>12:
A
x>−4
B
x<−4
C
x>4
D
x<4
3
The feasible region in linear programming is:
A
The objective function
B
The region satisfying all constraints
C
The optimal point
D
The gradient
4
The objective function in LP is:
A
The constraint
B
What we maximize or minimize (e.g., z=ax+by)
C
The feasible region
D
A non-linear function
5
The optimal solution in LP occurs at:
A
Any interior point
B
A corner (vertex) of the feasible region
C
The midpoint of a constraint boundary
D
The origin always
6
The corner point method tests the objective function at:
A
All interior points
B
All vertices of the feasible region
C
Midpoints of edges
D
Origin only
7
The constraint 2x+y≤10, x≥0, y≥0 forms a feasible region that is:
A
Unbounded
B
A single point
C
A bounded polygon (triangle or quadrilateral)
D
Empty
8
For the LP problem: maximize z=3x+4y subject to x+y≤4, x≥0, y≥0, the corners are (0,0),(4,0),(0,4). Evaluate z:
A
z=(0,0)=0; z=(4,0)=12; z=(0,4)=16. Max at (0,4)=16
B
Max at (4,0)
C
Max at origin
D
Max=24
9
The graph of x+y≤5 (with x,y≥0) is:
A
Half-plane including origin
B
Half-plane excluding origin
C
Line only
D
No region
10
Origin test: if ax+by < c is satisfied by (0,0): shade the region:
A
Away from origin
B
Containing the origin
C
Above the line
D
Below the line
11
An infeasible LP problem has:
A
Optimal solution
B
Multiple solutions
C
No feasible region (constraints are contradictory)
D
Unbounded solution
12
An unbounded LP problem:
A
Has a unique optimal solution
B
Has no solution
C
Has objective function → ∞
D
Has only constraints
13
The constraint x≤4, x≥0 in LP is a:
A
Bound constraint
B
Objective function
C
Non-negativity constraint only
D
Resource constraint
14
Non-negativity constraints x≥0, y≥0 restrict the solution to:
A
First quadrant (and axes)
B
Third quadrant
C
All quadrants
D
No restriction
15
If z=2x+3y and feasible region vertices are (0,0),(6,0),(4,2),(0,4), maximum z=
A
z(6,0)=12
B
z(4,2)=8+6=14
C
z(0,4)=12
D
z(0,0)=0
16
Minimum of z=x+y over vertices (2,4),(4,2),(6,0),(0,6):
A
z(2,4)=6
B
z(4,2)=6
C
z(6,0)=6
D
All equal 6 on x+y=6 line
17
A corner point with maximum z is the:
A
Global minimum
B
Local maximum only
C
Optimal solution (for maximization)
D
Infeasible point
18
The solution of 3x−y≥6 includes which side of the line 3x−y=6?
A
The side containing origin
B
The side not containing origin (since 0−0=0<6)
C
The line only
D
Both sides
19
Two-variable LP: number of decision variables is:
A
1
B
2
C
3
D
Unlimited
20
The feasibility of LP is determined by:
A
Objective function value
B
Whether the feasible region is non-empty
C
The number of constraints
D
The number of variables
21
Which point is in region x+y≤4, x≥0, y≥0?
A
(5,0)
B
(0,5)
C
(2,2)
D
(3,2)
22
The boundary line of x+2y≤8 is:
A
x+2y=0
B
x+2y=8
C
x+2y>8
D
x=8
23
Corner of feasible region of x≥1, y≥1, x+y≤5:
A
(1,1),(4,1),(1,4)
B
(0,0),(5,0),(0,5)
C
(1,1),(1,4),(4,1)
D
(0,0),(1,1)
24
Maximize z=x+y for corner (1,1),(1,4),(4,1): max at?
A
(1,1)
B
(1,4)
C
(4,1)
D
All equal
25
Minimize z=2x+y for corners (1,1),(1,4),(4,1): min at?
A
(1,4)
B
(4,1)
C
(1,1)
D
None
26
The word "linear" in LP means:
A
Constraint is curved
B
Objective function and all constraints are linear (degree 1)
C
Only one variable
D
Graphical method only
27
The word "programming" in LP means:
A
Computer programming
B
Planning or optimization
C
Coding
D
Algorithmic
28
Slack variable converts ≤ inequality to:
A
Another inequality
B
Equality (for simplex method)
C
≥ constraint
D
Objective function
29
If a farmer has 100 acres and can plant crop A (profit $200/acre) and B (profit $300/acre) with constraint A+B≤100: maximize profit=
A
Plant all B: z=300×100=30000
B
Plant all A: z=200×100=20000
C
Mix 50 each: z=25000
D
Cannot determine
30
Corner point (0,4) for z=3x+5y: z=
A
12
B
20
C
0
D
24
31
Feasible region bounded (closed): LP has:
A
No solution
B
Always an optimal solution
C
Infinite solutions only
D
Only maximum
32
Unbounded feasible region: LP maximization:
A
Always optimal solution
B
May or may not have optimal (depends on objective)
C
Never optimal
D
Only minimum exists
33
The simplex method is:
A
Graphical only
B
An algebraic method for solving LP
C
Only for 2 variables
D
A non-linear programming method
34
In LP, a solution is basic feasible if:
A
It satisfies only the objective function
B
It satisfies all constraints and non-negativity conditions
C
It is at the origin
D
Only non-negativity holds
35
The dual problem of maximization LP is:
A
Another maximization
B
Minimization problem
C
Equality constraints only
D
Non-linear
36
Region x≥0, y≥0, x+y≥3, 2x+y≥4: vertex at intersection of x+y=3 and 2x+y=4:
A
(1,2)
B
(2,2)
C
(0,3)
D
(2,0)
37
To minimize z=x+2y over region with vertices (1,2),(2,0),(0,4): min at?
A
(0,4)
B
(2,0)
C
(1,2)
D
Boundary
38
Constraint: machine hours ≤ 240, labor hours ≤ 300. This LP has:
A
No solution
B
A bounded feasible region
C
An unbounded region
D
Infinite solutions
39
The profit maximization with two products A and B: if both have equal profit per unit, the optimal point is:
A
Corner with most of A
B
Corner with most of B
C
Any point on the edge between optimal corners
D
Origin
40
Number of corners of feasible region x+y≤6, x≤4, y≤4, x≥0, y≥0:
A
3
B
4
C
5
D
6
41
At (4,2): z=3x+2y=
A
16
B
20
C
12+4=16
D
Both A and C
42
At (2,4): z=3x+2y=
A
6+8=14
B
10
C
14
D
Both A and C
43
At (4,0): z=3(4)+2(0)=12. At (0,4): z=3(0)+2(4)=8. Maximum z occurs at:
A
(0,4)
B
(4,0)
C
(4,2): z=16
D
(2,4): z=14
44
Linear programming was developed by:
A
Newton
B
Gauss
C
George Dantzig
D
Euler
45
Corner Point Theorem: if the feasible region is bounded, the max/min of linear objective function occurs at:
A
Interior points
B
Edge midpoints
C
Vertex (corner point)
D
All points
46
The inequality x+y>5 represents:
A
Region below line x+y=5
B
Region above line x+y=5
C
Line x+y=5 only
D
Empty set
47
The inequality x+y≥5 includes:
A
Region above x+y=5, not including line
B
Region above x+y=5 including the line
C
Below the line
D
Only the line
48
Graphical LP method works best for:
A
Many variables
B
Exactly 2 variables
C
3 variables
D
Non-linear objectives
49
If LP has no optimal solution (maximization), possible reasons:
A
Bounded region
B
Non-empty feasible region
C
Feasible region unbounded in direction of increasing z
D
Feasible region is a point
50
The constraint x+2y=10 with x,y≥0 intersects axes at:
A
(10,0) and (0,5)
B
(5,0) and (0,10)
C
(10,0) and (0,10)
D
(5,0) and (0,5)
51
If constraint is 3x+4y≤24, corner on y-axis:
A
(0,6)
B
(0,8)
C
(0,4)
D
(8,0)
52
Corner on x-axis for 3x+4y≤24:
A
(6,0)
B
(8,0)
C
(4,0)
D
(24,0)
53
For vertices (0,6),(8,0): z=2x+3y: max at?
A
(0,6): z=18
B
(8,0): z=16
C
(0,0): z=0
D
(4,3): z=17
54
LP model must have:
A
Only equality constraints
B
Objective function and constraints (inequalities)
C
Only one constraint
D
Non-linear objective
55
Decision variables in LP are:
A
Constants
B
Variables to be determined (what to produce/allocate)
C
The objective function
D
The constraints
56
The region 2x+y≤8, x≥0, y≥0, x≤3 has corners:
A
(0,0),(3,0),(3,2),(0,8)
B
(0,0),(4,0),(0,8)
C
(0,0),(3,0),(3,2),(0,8): correct
D
(0,0),(3,0),(3,2),(0,8)
57
Maximize z=5x+4y at corners (0,0),(3,0),(3,2),(0,8):
A
z(3,0)=15
B
z(3,2)=15+8=23
C
z(0,8)=32
D
z(3,2)=23 is NOT max
58
Isoprofit line: z=c₁x+c₂y=k is:
A
Always through origin
B
A line for fixed profit k
C
The feasible region boundary
D
The optimal point
59
In LP, the optimal solution is found where:
A
Isoprofit line crosses y-axis
B
Isoprofit line is tangent to or last touches the feasible region
C
Origin
D
Midpoint of feasible region
60
The LP problem with constraints x+y≤1, x+y≥2 (and x,y≥0) has:
A
Unique solution
B
Multiple solutions
C
No feasible region (infeasible)
D
Unbounded solution
61
Transportation problem is a special type of:
A
Non-linear programming
B
Integer programming
C
Linear programming
D
Dynamic programming
62
The LINDO/Excel Solver is used to:
A
Draw graphs
B
Solve LP problems numerically
C
Prove theorems
D
Calculate derivatives
63
For z=0 at (0,0), and feasible region in first quadrant, the minimum of z=2x+3y (not counting (0,0)):
A
Depends on whether (0,0) is feasible
B
(0,0) gives min=0 if it is feasible
C
Minimum is at corner not at origin
D
Cannot determine
64
The two-phase simplex method is used when:
A
Feasible region is bounded
B
An initial basic feasible solution is not obvious
C
Only 2 variables
D
Non-linear constraints exist
65
Shadow price in LP represents:
A
The cost of a resource
B
Change in optimal z per unit increase in constraint RHS
C
The dual variable
D
Both B and C
66
Sensitivity analysis in LP studies:
A
How optimal solution changes with problem parameters
B
Only the optimal value
C
Constraint equations only
D
Non-linear effects
67
The LP relaxation of integer program:
A
Has fewer solutions
B
Allows variables to be non-integer (relaxed)
C
Requires integer solutions
D
Is always infeasible
68
Mixed integer LP (MILP) allows:
A
Only integer variables
B
Some integer, some continuous variables
C
Only continuous variables
D
Non-linear objective
69
If feasible region is a single point:
A
LP is infeasible
B
LP has unique optimal solution
C
LP is unbounded
D
LP has infinite solutions
70
The graph of y≥x+2 shades:
A
Below the line y=x+2
B
Above the line y=x+2
C
The line only
D
Left of y-axis
71
The graph of 2x−y<4: does the boundary belong to the solution region?
A
Yes
B
No (strict inequality: < excludes boundary)
C
Only at x=0
D
Only at y=0
72
The feasible region for x≥0, y≥0, x+y≤5 is:
A
Entire first quadrant
B
Triangle with vertices (0,0),(5,0),(0,5)
C
Square
D
Rectangle
73
For that triangle, maximize z=x+y: max at?
A
(0,0)
B
(5,0)
C
(0,5)
D
(5,0) or (0,5) or any point on x+y=5
74
Minimize z=3x+2y for the triangle (0,0),(5,0),(0,5): min at?
A
(0,0): z=0
B
(5,0): z=15
C
(0,5): z=10
D
(3,2): z=13
0
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