Exam
Name___________________________________
MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.
Write the partial fraction decomposition of the rational expression.
1)
x2– 111
x4–x2– 72
1)
A)
1
x + 3 –1
x – 3 +7
x2+ 8
B)
1
x + 3 +1
x – 3 +7
x2+ 8
C)
1
x + 3 +1
x – 3 –7
x2+ 8
D)
1
x + 3 –1
x – 3 –7
x2+ 8
SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question.
Solve the problem.
2)
Joely’s Tea Shop, a store that specializes in tea blends, has available 45 pounds of A grade
tea and 70 pounds of B grade tea. These will be blended into 1 pound packages as follows:
A breakfast blend that contains one third of a pound of A grade tea and two thirds of a
pound of B grade tea and an afternoon tea that contains one half pound of A grade tea and
one half pound of B grade tea. If Joely makes a profit of $1.50 on each pound of the
breakfast blend and $2.00 profit on each pound of the afternoon blend, how many pounds
of each blend should she make to maximize profits? What is the maximum profit?
2)
3)
A coffee store has available 75 pounds of A grade coffee and 120 pounds of B grade coffee.
These will be blended into 1 pound packages as follows: an economy blend that contains 4
ounces of A grade coffee and 12 ounces of B grade coffee and a superior blend that contains
8 ounces of A grade coffee and 8 ounces of B grade coffee. Using x to denote the number of
packages of the economy blend and y to denote the number of packages of the superior
blend, write a system of linear inequalities that describes the possible number of packages
of each blend. Graph the system of inequalities.
3)
1
4)
Eric’s Carpentry manufactures two types of bookshelves that are 4 feet tall and 3 feet wide,
a basic model and a deluxe model. Each basic bookshelf requires 1.5 hours for assembly
and 1 hour for finishing; each deluxe model requires 2.5 hours for assembly and 1 hour for
finishing. Two assemblers and one finisher are employed by the company, and each works
40 hours per week. Eric can sell more basic models than deluxe models, so he wants the
number of basic models produced to be 50% more than the number of deluxe models
produced. If he makes $50 profit on the basic models and $65 profit on the deluxe models,
how many should he make to maximize the profit? What is the maximum profit?
4)
5)
Your computer supply store sells two types of laser printers. The first type, A, has a cost of
$86 and you make a $45 profit on each one. The second type, B, has a cost of $130 and you
make a $35 profit on each one. You expect to sell at least 100 laser printers this month and
you need to make at least $3850 profit on them. How many of what type of printer should
you order if you want to minimize your cost?
5)
6)
The Fiedler family has up to $130,000 to invest. They decide that they want to have at least
$40,000 invested in stable bonds yielding 5.5% and that no more than $60,000 should be
invested in more volatile bonds yielding 11%. How much should they invest in each type
of bond to maximize income if the amount in the stable bond should not exceed the
amount in the more volatile bond? What is the maximum income?
6)
7)
An artist is creating a mosaic that cannot be larger than the space allotted which is 4 feet
tall and 6 feet wide. The mosaic must be at least 3 feet tall and 5 feet wide. The tiles in the
mosaic have words written on them and the artist wants the words to all be horizontal in
the final mosaic. The word tiles come in two sizes: The smaller tiles are 4 inches tall and 4
inches wide, while the large tiles are 6 inches tall and 12 inches wide. If the small tiles cost
$3.50 each and the larger tiles cost $4.50 each, how many of each should be used to
minimize the cost? What is the minimum cost?
7)
8)
The Jillson‘s have up to $75,000 to invest. They decide that they want to have at least
$40,000 invested in stable bonds yielding 6% and that no more than $20,000 should be
invested in more volatile bonds yielding 12%.
(a) Using x to denote the amount of money invested in the stable bonds and y the amount
invested in the more volatile bonds, write a system of linear inequalities that describe the
possible amounts of each investment.
(b) Graph the system of inequalities.
8)
9)
The Jillson‘s have up to $75,000 to invest. They decide that they want to have at least
$25,000 invested in stable bonds yielding 6% and that no more than $45,000 should be
invested in more volatile bonds yielding 12%. How much should they invest in each type
of bond to maximize income if the amount in the more volatile bond should not exceed the
amount in the more stable bond? What is the maximum income?
9)
MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.
10)
A deli sells three sizes of chicken sandwiches: the small chicken sandwich contains 4 ounces of
meat and sells for $2.50; the regular chicken sandwich contains 9 ounces of meat and sells for $3.00;
and the large chicken sandwich contains 11 ounces of meat and sells for $3.50. A customer requests
a selection of each size for a reception. She and the manager agree on a combination of 52
sandwiches made from 24 pounds 14 ounces of chicken for a total cost of $151. How many of each
size sandwich will be in this combination? (Note: 1 pound = 16 ounces)
10)
A)
18 small sandwiches, 14 medium sandwiches, 20 large sandwiches.
B)
22 small sandwiches, 18 medium sandwiches, 12 large sandwiches.
C)
20 small sandwiches, 24 medium sandwiches, 8 large sandwiches.
D)
24 small sandwiches, 12 medium sandwiches, 16 large sandwiches.
11)
In 1985, in the town of Appleby, 21.2% of Hispanics were overweight, increasing by an average of
0.41% per year. In 1985, in the town of Appleby, 0.16% of whites were overweight, increasing by
an average of 29.0% per year. Write a function that models the percentage, y, of Hispanics who are
overweight x years after 1985. Write a function that models the percentage, y, of whites who are
overweight x years after 1985.
11)
A)
Hispanics: y =21.61x
Whites: y =29.16x
B)
Hispanics: y =0.41x +21.2
Whites: y =29.0x +0.16
C)
Hispanics: y +0.41x =21.2
Whites: y +29.0x =0.16
D)
Hispanics: y =21.2x +0.41
Whites: y =0.16x +29.0
Graph the solution set of the system of inequalities or indicate that the system has no solution.
12)
x2+y2100
5x + 2y 10
12)
A)
B)
5
C)
D)
Solve the system by the addition method.
13)
–9x + y = – 10
–6x – 4y = – 16
13)
A)
4
3, 2
B)
–4, 0
C)
{(1, 14)}
D)
7
3, 11
Graph the inequality.
14)
x – y > – 2
14)
6
A)
B)
C)
D)
Solve the system by the method of your choice. Identify systems with no solution and systems with infinitely many
solutions, using set notation to express their solution sets.
15)
5x + y =4
2y =8– 10x
15)
A)
4
5, 0
B)
{(0, 4)}
C)
{(x, y)|5x + y =4}
D)
7
Solve the system by the addition method.
16)
2x +6y = – 16
–2x –10y=28
16)
A)
{(–1, –3)}
B)
{(–1, 3)}
C)
{(1, –3)}
D)
Write the partial fraction decomposition of the rational expression.
17)
x2+6x – 2
(x2+2)2
17)
A)
–1
x2+ 2
+6x – 4
(x2+ 2)2
B)
1
x2+ 2
+6x – 4
(x2+ 2)2
C)
1
x2+ 2
+
–x – 4
(x2+ 2)2
D)
x + 1
x2+ 2
+6x – 4
(x2+ 2)2
Solve the system of equations by the substitution method.
18)
–2x – 3y= – 21
x= – 5y
18)
A)
{(15, –3)}
B)
{(16, –3)}
C)
{(–3, 15)}
D)
{(15, 3)}
Solve the system by the addition method.
19)
x2+y2– 6x – 6y – 7 = 0
x2–y2– 6x + 6y – 25 = 0
19)
A)
{(8, 3), (2, 3)}
B)
{(3, 8), (3, –2)}
C)
{(8, 3), (–2, 3)}
D)
{(3, –8), (3, 2)}
8
20)
x – 4y = – 15
8x – 5y =42
20)
A)
{(8, 7)}
B)
{(9, 6)}
C)
{(–9, 7)}
D)
Solve the problem.
21)
Steve invests in a circus production. The cost includes an overhead of $117,000, plus production
costs of $6000 per performance. A sold–out performance brings in $15,000. Let x represent the
number of sold–out performances and write the cost function, C and revenue function, R.
21)
A)
C(x) =6000x
R(x) =117,000 +15,000x
B)
C(x) =117,000x +6000
R(x) =15,000x
C)
C(x) =117,000 +15,000x
R(x) =6000x
D)
C(x) =117,000 +6000x
R(x) =15,000x
22)
A doctor has told a sick patient to take vitamin pills. The patient needs at least 25 units of vitamin
A, at least 8 units of vitamin B, and at least 25 units of vitamin C. The red vitamin pills cost 10¢ each
and contain 6 units of A, 1 unit of B, and 2 units of C. The blue vitamin pills cost 20¢ each and
contain 2 units of A, 1 unit of B, and 5 units of C. How many pills should the patient take each day
to minimize costs?
22)
A)
3 red and 5 blue
B)
8 red and 0 blue
C)
6 red and 2 blue
D)
5 red and 3 blue
23)
A person with no more than $1000 to invest plans to place the money in two investments,
telecommunications and pharmaceuticals. The telecommunications investment is to be no more
than 2 times the pharmaceuticals investment. Write a system of inequalities to describe the
situation. Let x = amount to be invested in telecommunications and y = amount to be invested in
pharmaceuticals.
23)
A)
x + y =1000
y 2x
x 0
y 0
B)
x + y 1000
x 2y
x 0
y 0
C)
x + y 1000
2x
y
x 0
y 0
D)
x + y =1000
x 2y
x 0
y 0
Graph the solution set of the system of inequalities or indicate that the system has no solution.
9
24)
y 2x – 4
x + 2y
7
x –2
y 1
24)
A)
B)
C)
D)
Solve by the method of your choice.
25)
x3+ y = 0
10x2– y = 0
25)
A)
{(0, 0), (10, –1000)}
B)
{(–1, 1), (–10, 1000)}
C)
{(0, 0), (–10, 1000)}
D)
{(0, 0), (–10, 100)}
Determine if the given ordered triple is a solution of the system.
26)
(2, 5, 3)
x – y + z =0
x + y + z =10
x + y – z =4
26)
A)
not a solution
B)
solution
Solve the system by the substitution method.
27)
x + y =12
y =x2– 12x + 36
27)
A)
{(3, 9), (8, 4)}
B)
{(3, 15), (8, 4)}
C)
{(–3, 15), (–8, 20)}
D)
{(6, 6)}
Solve the system of equations.
28)
x –y + 3z =5
3x + z =2
x + 3y +z =5
28)
A)
{(2, 0, 1)}
B)
{(1, 0, 2)}
C)
{(2, 1, 0)}
D)
{(0, 1, 2)}
11
Determine if the given ordered triple is a solution of the system.
29)
(1, –1, 0)
x – y + 4z =5
4x + z =1
x + 2y + z = – 1
29)
A)
solution
B)
not a solution
Let x represent one number and let y represent the other number. Use the given conditions to write a system of nonlinear
equations. Solve the system and find the numbers.
30)
The sum of two squares of two numbers is 125, and the difference of their squares is 117. Find the
numbers.
30)
A)
11 and 2
B)
11 and 2; –11 and 2; 11 and –2
C)
11 and 2; –11 and –2
D)
11 and 2; –11 and 2; 11 and –2; –11 and –2
Determine if the given ordered triple is a solution of the system.
31)
(5, 1, 2)
x + 2y + 5z =17
2y + 2z =6
z =2
31)
A)
not a solution
B)
solution
32)
(–1, 5, 1)
x + y + z =5
x – y + 2z = – 4
4x + y + z =2
32)
A)
not a solution
B)
solution
Determine whether the given ordered pair is a solution of the system.
33)
(–4, –2)
4x + y = – 14
2x + 4y =0
33)
A)
not a solution
B)
solution
Solve the problem.
34)
Two kinds of crated cargo, A and B, are to be shipped by truck. The weight and volume of each
type are given in the following table:
A B
Volume 50 cubic feet 10 cubic feet
Weight 200 pounds 360 pounds
The shipping company charges $75 per crate for cargo A and $100 per crate for cargo B. The truck
has a maximum load limit of 7,200 pounds and 1,000 cubic feet. How many of each type of cargo
should be shipped to maximize profit for the shipping company?
34)
A)
18 crates of cargo A and 10 crates of cargo B
B)
0 crates of cargo A and 20 crates of cargo B
C)
10 crates of cargo A and 18 crates of cargo B
D)
20 crates of cargo A and 0 crates of cargo B
Solve the system by the substitution method.
35)
y = – x2+ 3
x2+y2=5
35)
A)
{(2, –1), (1, 2), (–1, 2), (–2, –1)}
B)
{(2, –1), (–2, –1)}
C)
{(1, 4), (4, 19)}
D)
{(1, 2), (–1, 2)}
A
Solve the system by the addition method.
36)
x + y = – 6
x – y =12
36)
A)
{(3, –9)}
B)
{(6, –9)}
C)
{(3, 9)}
D)
{(6, 3)}
A
Solve the problem.
37)
The equation that represents the proper traffic control and emergency vehicle response availability
in a small city is 2P + 3F
23, where P is the number of police cars on active duty and F is the
number of fire trucks that have left the firehouse in response to a call. In order to comply with
staffing limitations, the equation 2P + F
17 is appropriate. The number of police cars on active
duty and the number of fire trucks that have left the firehouse in response to a call cannot be
negative, so P 0 and F
0. Graph the regions satisfying all the availability and staffing
requirements, using the horizontal axis for P and the vertical axis for F. If 6 police cars are on active
duty and 4 fire trucks have left the firehouse in response to a call, are all of the requirements
satisfied?
37)
13
A
A)
No
B)
No
C)
Yes
D)
Yes
Graph the inequality.
14
38)
3x + 5y 15
38)
A)
B)
C)
D)
15
The figure shows the graphs of the cost and revenue functions for a company that manufactures and sells binoculars. Use
the information in the figure to answer the question.
39)
Fewer than how many binoculars must be produced and sold for the company to have a profit loss?
39)
A)
2250 binoculars
B)
2700 binoculars
C)
1500 binoculars
D)
750 binoculars
Solve by the method of your choice.
40)
x2+y2=13
–2x +y2=5
40)
A)
{(2, 3), (2, –3), (–4, 3), (–4, –3)}
B)
{(2, 3), (2, –3)}
C)
{(2, 3)}
D)
{(2, 3), (2, –3), (–2, 3), (–2, –3)}
Graph the solution set of the system of inequalities or indicate that the system has no solution.
41)
x2+ y24
y –x2> 0
41)
16
A)
B)
C)
D)
Write the form of the partial fraction decomposition of the rational expression. It is not necessary to solve for the
constants.
42)
5x – 3
(x – 5)(x2+ x + 5)
42)
A)
A
x – 5 +Bx + C
x2+ x + 5
+D
(x – 5)(x2+ x + 5)
B)
A
x – 5 +Bx + C
x2+ x + 5
C)
A
x – 5 +B
x2+ x + 5
+C
(x – 5)(x2+ x + 5)
D)
A
x – 5 +B
x2+ x + 5
Graph the inequality.
17
43)
2x + y –5
43)
A)
B)
C)
D)
18
An objective function and a system of linear inequalities representing constraints are given. Graph the system of
inequalities representing the constraints. Find the value of the objective function at each corner of the graphed region.
Use these values to determine the maximum value of the objective function and the values of x and y for which the
maximum occurs.
44)
Objective Function z = 3x + 5y
Constraints x 0
y 0
2x + y
15
x – 3y –3
44)
A)
Maximum 38; at (6, 4)
B)
Maximum 33; at (6, 3)
C)
Maximum 22.5; at (7.5, 0)
D)
Maximum 75; at (0, 15)
Solve the problem.
45)
Julie and Eric row their boat (at a constant speed) 45 miles downstream for 5 hours, helped by the
current. Rowing at the same rate, the trip back against the current takes 9 hours. Find the rate of the
current.
45)
A)
2 mph
B)
3 mph
C)
7 mph
D)
1.5 mph
Solve the system of equations.
46)
y– 5z= – 6
5x + y + 3z= – 10
4x + 2z= – 12
46)
A)
{(–2, 6, –2)}
B)
{(–4, 4, 2)}
C)
{(–2, 5, –2)}
D)
{(–4, 9, 3)}
Write the form of the partial fraction decomposition of the rational expression. It is not necessary to solve for the
constants.
47)
3x + 4
(x2+ x + 5)2
47)
A)
Ax + B
x2+ x + 5
+C
(x2+ x + 5)2
B)
A
x2+ x + 5
+B
(x2+ x + 5)2
C)
Ax + B
x2+ x + 5
+Cx + D
(x2+ x + 5)2
D)
A
x2+ x + 5
+Bx + C
(x2+ x + 5)2
Find the maximum or minimum value of the given objective function of a linear programming problem. The figure
illustrates the graph of feasible points.
48)
Objective Function: z = – x – 8y
Find maximum.
48)
A)
maximum: –20
B)
maximum: –34
C)
maximum: –27
D)
maximum: –42
Graph the inequality.
49)
y 6
49)