Sample Final Exam with
Solution Key - Intermediate
Algebra | MATH 1010
Mathematics
Utah State University (USU)
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MATH1010
SAMPLE FINAL EXAM
FALL 2007
A Number: A_______________________ Name ______________________________________
Instructor__________________________ Section ______________
Instructions: Write your name and student ID number on your Scan-Tron sheet
and on this test booklet. DO NOT USE A CALCULATOR. Mark your answers on your
Scan-Tron sheet AND on this test booklet.
MULTIPLE CHOICE. Circle the correct choice on this test booklet AND mark your
answer on your Scan-Tron sheet.
1. An equation of the line through the point ( 2, 4) perpendicular to 6 x  4 y 5 is
(a) 3x  2 y 2
(d) 3 x  2 y  14
2a 3 15a 5
2. Simplify:
 3
3b 2
4b
45a 8
8b 5
3. Simplify:
(a)
(b)
3
(c)
45a 15
8b 6
(c) 2 x  3 y  8
(d)
8b
45a 2
(e)
5a 2
2b
24a 11 b19
(a) 2a 3 b 6 3 3a 2 b
(d) 3a 3b 6 3 2ab 2
MATH 1010
2a
5b 2
(b) 2 x  3 y  16
(e) 2 x  3 y 8
(b)
(e)
3a 3 b 6
3
2 a 2b
3a 3 b 6
3
6 ab 2
1
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(c) 2a 3 b 6 3 3ab 2
Spring 2007
MATH1010
SAMPLE FINAL EXAM


4. For f ( x)  6 x 2  2 x  5 , find f  
5
9
11

3
(a)
(b)
19
3
FALL 2007
1
.
3
(c)
11
3
(d) 
5
9
(e)
5. Factor r 2  3rs  rt  3st completely. One of the factors is
(a) r  s
(b) r  s
(c) r  3t
(d) r  3s
6. In interval notation, the solution set for the inequality 2 


(a)   ,
7

6


(b)   , 6 


7


(c)  
(e) r  t
2
37
x
is
5
15
7

, 
6



(d)  
7

, 
6

(e)

 7
  6 ,  
7. The domain of the function
MATH 1010
f ( x)  6  3x
is the interval
2
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MATH1010
(a) [2, )
SAMPLE FINAL EXAM
(b) ( ,2]
(c) (2, )
FALL 2007
(d) ( ,2)
(e)
(  , 2]
MATH 1010
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Spring 2007
MATH1010
SAMPLE FINAL EXAM
FALL 2007
8. Solve the quadratic equation 3 x 2  x  2 0 . The solution set is
2
3


(b)   2,3
(a)  1, 
9. Simplify:
75 
(a)
(b)
9 3
48  3 3 .
6 3
 2 
,1
 3 
(c)   3,2
(c)
(e)  1,3
(d) 
The result is
4 3
(d)
3 3
(e)
5 3
10. Identify the center and radius of the circle x 2  8 x  y 2  6 y 0 .
(a) Center: (−8,6), r = 14
(d) Center: (4,−3), r = 25
3
11. Simplify:
(a) 
3
3
MATH 1010
3
(b) Center: (8,−6), r = 14
(e) Center: (4,−3), r = 5
(c) Center: (−4,3), r = 5
 24
.
32
(b) 
3
6
2
(c) 
3
6
8
(d) 
3
3
4
4
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(e) 
3
12
2
Spring 2007
MATH1010
SAMPLE FINAL EXAM
12. Find the quotient:
(a)
1 7
 i
5 5
(b)
5  5i
.
4  3i
1 7
 i
5 5
(c)
1
y x
(b) x  y
14. Solve the equation
2
3
(a)  
MATH 1010
7 1
 i
5 5
(d) 
7 1
 i
5 5
(e) 
1 7
 i
5 5
1 1

x
y
x
y

y
x
13. Write in simplest form.
(a) y  x
FALL 2007
1
1
(c) x  y
(d) x  y
x y
(e) x  y
 t
1

 2 . The solution set is
t 1 t 1
(b)   3
(c)  0
(d)  3
5
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(e) 
Spring 2007
MATH1010
SAMPLE FINAL EXAM
15. Solve for x:
x  3 x  3 .
(a) 1,6
FALL 2007
Check your answer. The solution set is
(b)   1,6
(c)   6,1
(d)  6
(e) 1
16. Solve the quadratic equation (t  5) 2  2(t  5)  24 0 . The solution set is
(a)   4, 6
(b) {−1, 9}
(c) {−9, 1}
(d) {−5, 6}
(e) {−6, 5}
17. Stan invested $12,000 in two accounts, one earning 5% simple interest and the other
earning 7%. T the total interest earned in one year is $750. If X represents the
amount earning 5% interest and Y represents the amount earning 7% annually, write a
system of equations that could be used to determine the values for X and Y.
(a) 0.05 X  0.07Y 12,000 (b) 0.07 X  0.05Y 12,000
0.07 X  0.05Y 750
X  Y 750
(d) 0.05 X  0.07Y 750
X  Y 12,000
MATH 1010
X  Y 750
(c)
X  Y 12,000
(e) 0.05 X  0.07Y 750
Y X  12,000
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Spring 2007
MATH1010
SAMPLE FINAL EXAM
FALL 2007
18. Solve the equation: x 2  4 x  8 0 . The solution set is
(a)   i, i
2 
(b)  1  i,1  i
2i, 2  2i
5 x  2 y 7

2 x  y 5

19. Solve the system:
(a) 11
(c)  1  2 3 ,1  2 3 (d)  1  2i 3 ,1  2i 3 (e)
(b) −3
. The corresponding value of y is
(c) −11
(d) 3
(e) 7
20. Solve the nonlinear inequality x 2  x  8 4 . The solution in interval notation is
(a) [ 3,4] (b)   ,  3   4, 
(e) [3,4]
21. Add:
(a)
(c)   ,  4   3, 
(d) [ 4,3]
2
4

x 1 x  1
6x  2
x2  1
MATH 1010
(b)
4
x 1
2
(c)
6
4
(d)
x 1
x 1
2
(e)
7
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6x  2
x2  1
Spring 2007
MATH1010
SAMPLE FINAL EXAM
FALL 2007
22. Find the distance between the points (5, −4) and (7,3).
(a) 7
(b)
53
(c) 13
(d)
(e)
61
157
23. It takes Kyle twice as long to paint a room than it takes Brendan. Working together,
they can paint the room in 2 hours. How long does it take Brendan to paint the room?
(a) 4.5 hours
(b) 4 hours
(c) 3.5 hours
(d) 3 hours
(e) 2.75 hours
24. Let f ( x) 3x  2 and g ( x) 2 x  5 . Find ( fg )( x ) .
(a) 6 x 2  11x  10
(b) 5 x  2
(c) 6 x 2  10
(d) 6 x 2  11x  10
(e)
6 x 2  11x  10
25. Solve the quadratic equation 2 x 2  7 x  7  x 2  7 x  9 . The solution set is
(a)   4, 4
 i
2, i 2

MATH 1010
(b)   4i, 4i
(c)  
2,
2

(d)   16,16
8
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(e)
Spring 2007
MATH1010
SAMPLE FINAL EXAM
FALL 2007
MATH 1010
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MATH1010
SAMPLE FINAL EXAM
26. Solve the system of inequalities
(a)
(b)
y  3
2 x 

2 x  2 y 2
(c)
FALL 2007
AND
(d)
(e)
27. If the graph of y  x is translated 5 units upward and 3 units to the left, then the
equation of that curve is
(a)
(d)
y 5 
x 3
(b)
(e)
y 5 x  3
28. If
f ( x)  3 x  5
(a) 1
number
(b)
and g ( x) 
2x  1
x 1
2x  1
1
(d) f ( x) 
x 1
x 3
5x  3
, evaluate
2x 1
(c) 0
3
29. Find the inverse of f ( x) 
1
(a) f ( x) 
y 5 
(c)
y  x 3  5
y x 3 5
x 1
,
x 2
f
 g  ( 1) .
(d) 2
(e) Not a real
x 2 .
1  2x
x 1
1  2x
1
(e) f ( x) 
x 1
(b) f  1 ( x) 
(c) f  1 ( x) 
2x 1
x 1
30. The pressure p (in pounds per square inch) exerted by water on a submerged object is
directly proportional to the depth d (in feet) beneath the surface. If the pressure at a
depth of 20 feet is 9 lbs per square inch, at what depth will an egg break if its shell
MATH 1010
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MATH1010
SAMPLE FINAL EXAM
FALL 2007
cannot withstand 36 pounds per square inch?
(a) 90 feet
(b) 80 feet
(c) 60 feet
(d) 50 feet
(e) 40 feet
31. Identify the equation of the exponential function whose graph is shown.
 (1,3)
(a) y 2 x
(b) y 2 x
(c) y 3 x
(d) y 2 x  1 (e) y 3x  1
(c) 3
(d) 9
32. Evaluate log 13 27
(a) −3
(b) −9
33.Rewrite as a single logarithm.
 5 x2  1 


3


(a) ln

(d) ln

5 ( x  1) 


3

MATH 1010
(e) 81
1
ln( x 2  1)  2 ln 3  ln 5
2
 ( x  1) 

 3 5 
(b) ln

(e) ln

 5 x2  1 


9


(c) ln
x 2  1 
45 
11
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MATH1010
SAMPLE FINAL EXAM
FALL 2007
34. It costs a company C dollars per hour to operate its golf ball division. A consultant
group has determined that C is related to the number x of golf balls produced per hour
by the equation
C 0.035 x 2  1.4 x  95
What is the minimum cost per hour (in dollars) of operating the golf ball division?
(a) $77
(b) $81
(c) $129
35. Solve the system of equations.
(a) (2,0) and (0, 2)
(d)  3, 1 and   1,3
(d) $215
x  y 4
y  x  2
2
(e) $305
2
The solutions are
(b) (−2,0) and (0, −2)
(c)
(e)  3, 1 and 1,1

 and (0, 2)
3 ,1
36. The equation of the circle with radius 4 centered at the point (3, −5) is
(a) ( x  3) 2  ( y  5) 2 4
(d) ( x  3) 2  ( y  5) 2 16
(b) ( x  3) 2  ( y  5) 2 4
(e) ( x  3) 2  ( y  5) 2 16
(c) ( x  3) 2  ( y  5) 2 8
37. Simplify (evaluate) the expression 5log 5 ( 55 ) .
(a) −11
MATH 1010
(b)
1
11
(c) 11
(d) 5
12
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(e) 55
Spring 2007
MATH1010
SAMPLE FINAL EXAM
FALL 2007
38. If y varies directly with the square root of x and inversely with the cube of t and
y = 20 when x = 25 and t =5, find t when y = 5 and x = 100.
(a) t = 10
(b) t = 20
(c) t = 30
(d) t = 40
(e) t = 50
39. Which of the following relations has a v-shaped graph opening left (with its vertex
pointing right)?
(a)
y  x
(b)
x  y
(c)
y  x
40. Which of the following is the graph of
(d)
x  y
f ( x) 3 
(a)
(b)
(c)
(d)
2 x
(e)
?
(e) None of the above
EXAM
MATH 1010
x  y
END OF
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MATH1010
SAMPLE FINAL EXAM
FALL 2007
Solution Key Sample Final Exam
1. b
2. d
3. a
4. c
5. d
6. e
7. b
8. a
9. c
10. e
11. b
12. a
13. d
14. e
15. d
16. c
17. d
18. e
19. a
20. c
21. e
22. b
23. d
24. e
25. e
26. b
27. b
28. a
29. c
30. b
31. d
32. a
33. c
34. b
35. a
36. e
37. e
38. a
39. b
40. c
MATH 1010
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