Please answer the following multiple choice questions. Place your letter answer in the indicated space just before the problem number. If more than one answer could be correct, choose the best one.

__ 1. What does the graph of an equation represent?

a) a straight line
b) a relationship between the x-coordinate and the y-coordinate
c) plotting the intercepts and drawing a line between them
d) the Cartesian coordinate system
e) a depiction on a coordinate system of all points that satisfy the equation
f) a curve that you draw by plotting some points and connecting them
g) none of the above

__ 2. Consider the following graph of a polynomial function f(x):

What is the degree of f?

a) 0     b) 1     c) 2    d) >2     e) cannot be determined

__ 3. Consider a polynomial function defined by the equation:

What is the name of this function?

a) z     b) Z     c) y     d) Y     e) f     f) g     g) cannot be determined

__ 4. Concerning question 3, what is the independent variable of the function described?

a) z     b) Z     c) y     d) Y     e) f     f) g     g) cannot be determined

__ 5. Concerning question 3, what is the dependent variable of the function described?

a) z     b) Z     c) y     d) Y     e) f     f) g     g) cannot be determined

__ 6. Consider the following graph of an equation y = f(x):

What are the zeros of f?

a) (0, 4)     b) (-5, 0), (0,4), (5, 0), (10, 0)     c) -5, 4, 5, 10     d) -5, 5, 10
e) (0, -5), (5, 0), (10, 0)                                 f) (4, 0)
g) not enough information given                      h) none of the above

__ 7. Concerning question 6, which of the following list the solutions of the equation f(x) = 0?

a) (0, 4)     (b) (-5, 0), (0,4), (5, 0), (10, 0)     c) -5, 4, 5, 10     d) -5, 5, 10
e) (0, -5), (5, 0), (10, 0)                                  f) (4, 0)
g) not enough information given                        h) none of the above

__ 8. Concerning question 6, which of the following are y-intercepts of the graph?

a) (0, 4)     (b) (-5, 0), (0,4), (5, 0), (10, 0)     c) -5, 4, 5, 10     d) -5, 5, 10
e) (0, -5), (5, 0), (10, 0)                                  f) (4, 0)
g) not enough information given                        h) none of the above

__ 9. Concerning question 6, what is f(0)?

a) 0     b) -5     c) 4     d) 5
e) not enough information to tell     f) none of the above

__ 10. Consider a system of two linear equations whose graphs are depicted as follows:

What is (are) the solution(s) to this system?

a) 10     b) 1      c) no solution     d) (0,10)     e) (5,2)     f) (2, 5)
g) 10 and 1        h) 2 and -5         i) (1, 0)
j) not enough information given     k) none of the above

__11. Consider a system of two linear equations whose graphs are depicted as follows:

What is (are) the solution(s) to this system?

a) 10     b) 1     c) no solution     d) (0,10)     e) (5,2)     f) (2, 5)
g) 10 and 1        h) 2 and 1          i) (1, 0)
j) not enough information given     k) none of the above

__12. What law of real numbers is illustrated by the identity:     x(ab) = (ab)x ?

a) reordering         b) distributive     c) associative     d) multiplication
e) simplification     f) none of the above

__13. Why do we define c⁰ = 1 for all real numbers c not equal to 0?

a) because it is a law of exponents
b) because if we multiply c by itself 0 times, we always get 1
c) because we want the law of exponents = c^m-n (for m>n), to hold for m=n also
d) none of the above

__ 14. What is  ?

a) an imaginary number, if x < 0              b) undefined for some real numbers
c) | x |                         d) x                      e) none of the above

__ 15. Consider a polynomial f(x) having a zero x = 12.  Consider the division . What is the remainder resulting from this division?

a) 12     b) -12     c) 0         d) f(-12)        e) f(0)
f) not enough information to tell                 g) none of the above

__ 16. Consider a polynomial f(x) having a zero x = -1. Name a factor of f(x).

a) x + 1             b) x - 1                   c) 0
d) not enough information to tell        e) none of the above

__ 17. Consider a polynomial f(x) such that the division  results in a remainder of -5. What is f(6)?

a) 6     b) -6    c) 5     d) -5
e) not enough information to tell     f) none of the above

__ 18. Consider a polynomial f(x) such that the division  results in a remainder of -5. What is f(-6)?

a) 6             b) -6             c) 5                 d) -5
e) not enough information to tell                f) none of the above

__ 19. What is the Fundamental Theorem of Algebra?

a) Every equation has at least one rational solution
b) Every polynomial has at least one real solution
c) All polynomials can be factored into linear factors with complex coefficients
d) Every polynomial of degree > 0 has at least one complex zero
e) Every polynomial of degree > 0 has at least one real zero
f)  Complex zeros occur in complex conjugate pairs
g) none of the above

__ 20. Consider the following (only partially drawn) graph:

Which of the following equations best describes this graph?

a) y = x +                     b) y = -x + 2      c) y = e^x
d) y = x² +                e) y = log x         f) y = x² + 3

__ 21. {1, 2, 3} {3, 4, 5} = ?

a) {1, 2, 4, 5}     b) 3         c) {1, 2, 3, 4, 5}         d) f
e) none of the above

__ 22. Consider the following 6 statements:

(1) a  {a}         (2) a  {a}                 (3) {a}  {a}
(4) {a}  {a}     (5) {a}   a                (6) {a}  a

Which are true?

a) all of them     b) 1, 2, 3     c) 2, 3, 5
d) 1, 3               e) 1, 3, 6     f) none of them

__ 23. Consider the following 4 statements:

(1) f {a}         (2) f {a}         (3) f   f       (4) f  f

Which are true?

a) 1         b) 2         c) 3             d) 4     e) 1, 2     f) 1, 3
g) 1, 4     h) 2, 4     i) 1, 2, 3, 4   j) none of the above

__ 24. What is the additive inverse of -2?

a) -2     b) 2     c) 0         d) 1/2     e) -1/2         f) none of the above

__ 25. What is the multiplicative inverse of 2?

a) -2         b) 2         c) 0         d) 1/2         e) -1/2         f) none of the above

__ 26. What is the coefficient of the quadratic term of the polynomial 2x - 3x² - 5 ?

a) 2         b) 3         c) -5         d) -2         e) none of the above

__ 27. Suppose we place our savings of $1000 in two accounts, one earning 12% and the other earning 10% simple interest for two years. If we place x dollars in the account earning 12%, how much interest do we earn on the 10% account?

a) x - 1000         b) 1000 - x         c) .20(1000 - x)         d) .10x
e) not enough information given      f) none of the above

__ 28. Consider the statement: "(a + b)² = a² + b² for all real numbers a and b".

a) it is true, and taking a=1, b=0 demonstrates its truth
b) it is true, and taking a=1, b=1 demonstrates its truth
c) it is false, and taking a=1, b=0 demonstrates its falsity
d) it is false, and taking a = 1, b = 1 demonstrates its falsity
e) none of the above

__ 29. Why is  not a rational number?

a) because it cannot be expressed as the ratio of two integers
b) because it cannot be expressed as the ratio of two real numbers
c) because it is not reasonable
d) because it is not real
e) because it is a repeating, non-terminating decimal number
f) none of the above

__ 30. What does it mean to solve the equation 2x² + 3x - 2 = 0 ?

a) find the value of x         b) solve for x         c) get x by itself
d) find all values of x that satisfy the equation
e) make the equation a true statement             f) none of the above

__ 31. What property of "<" is described by the following:

for all real a, b, and c, if a<b and b<c, then a<c

a) addition property                 b) multiplication property         c) transitive property
d) law of the excluded middle   e) associative property             f) none of the above

__ 32. Consider the statement | x | = -a

a) it cannot be true     b) it can be true, if a > 0         c) it can be true, if a < 0
d) it can be true, if x is imaginary         e) not enough information given

__ 33. Consider the inequality | 5 - x | < -1. How many real solutions does it have?

a) 0         b) 1         c) 2                 d) infinitely many
e) not enough information given       f) none of the above

__ 34. Consider the statement i =  ( i is the imaginary unit )

a) this can be proved         b) it doesn't need to be proved; it is just common sense
c) the Fundamental Theorem of Algebra says it is so
d) it is not true, because then i² = -1, and the square of a number cannot be negative
e) none of the above

__ 35. Consider the following 5 statements:

(1) 2/3 is an integer                     (2) 2/3 is a real number
(3) 2/3 is an imaginary number     (4) 2/3 is an irrational number
(5) 2/3 is a complex number

Which of them is true?

a) 2, 4         b) 1, 2         c) 1, 2, 4         d) 2, 5
e) 4             f) all of the above                g) none of the above

__ 36. The conjugate of the imaginary unit i is given by:

a) doesn't have a conjugate         b) 1         c) -1
d) i                                             e) -i         f) none of the above

__ 37. What is the maximum number of solutions for ax⁴ + bx² + c = 0, for real coefficients a, b, c?

a) none         b) 1         c) 2         d) 3
e) 4              f) 5          g) not enough information to tell

__38. If a boat's speed in still water is x kph, and it is traveling in a river whose current flows at y kph, what is the net speed of the boat if it is traveling upstream?

a) x kph         b) y kph         c) x + y kph           d) x - y kph
e) xy kph       f) not enough information to say    g) none of the above

__ 39. If a rectangle has length 5 ft. and width 3 ft., what are its dimensions?

a) 8 ft.               b) 16 ft.         c) 3 ft. x 5 ft.
d) 8 ft. x 5 ft.     e) 15 ft.          f) none of the above

__ 40. Suppose a polynomial f has exactly two real zeros: 0 and 5. If f(2) = -3, what is the sign of f(3)?

a) +         b) -         c) 2         d) no sign, f(3) = 0         e) not enough information to tell

__41. The graph of a circle whose center is at the origin can best be described as follows:

a) symmetric with respect to the x axis         b) symmetric with respect to the y axis
c) symmetric with respect to the origin         d) a and b
e) a and c                 f) b and c                    g) a, b, and c
h) not enough information to tell                    i) none of the above

__42. Which of the following sets describes all possible slopes of lines?

a) [0, )         b) (0, )          c) [1, )                d) (-, )
e) a plus "undefined"                 f) b plus "undefined"
g) c plus "undefined"                 h) d plus "undefined"
i) none of the above

__43. Consider a function f(x). What is its domain?

a) x                                                 b) all real numbers
c) not enough information to tell       d) none of the above

__44. Suppose the graph of an equation includes (but is not necessarily limited to) the two points (-3,) and (3, ).

a) the equation defines a function             b) the equation does not define a function
c) not enough information to tell

__ 45. Suppose you are given the graph of a function f(x). Consider the function
g(x) = f(x) - 2.

a) the graph of g will be a straight line shifted down two units
b) the graph of g will be impossible to construct
c) neither of the above

__ 46. Evaluate:

 

a) 1         b) 2         c) 3         d) 4         e) 0
f) not enough information        g) none of the above

__ 47. What law of real numbers is illustrated by the identity 3(x + 1) = 3x + 3 ?

a) commutative     b) distributive        c) associative         d) multiplication
e) simplification     f) none of the above

__ 48. How should one describe the Division Algorithm?

a) it says that a polynomial can be represented as a quotient and a remainder
b) it describes a method for doing polynomial division
c) it says that a polynomial can be represented as a specific mathematical combination of an arbitrary linear factor, a quotient, and a remainder
d) it is a computer program used by scientists to do complex division
e) none of the above

__ 49. If f(x) = (x - 5)Q(x) + 4, what is f(5)?

a) 0         b) 5         c) -5         d) 4         e) Q(x)
f) x - 5    g) not enough information         h) none of the above

__ 50. Which of the following 5 laws of real numbers do not hold for matrices:

(1) associative         (2) commutative law of addition         (3) distributive
(4) commutative law of multiplication                                 (5) additive inverse

a) 1 and 2         b) 1, 2, 3         c) 1         d) 2         e) 3
f) 4                   g) 5                  h) 2, 5     i) none of the above

__ 51. Suppose the graph of y = f(x) approaches a horizontal line (call it line "A") asymptotically
as x approaches  +.

a) at the extreme left of the coordinate system, the graph will come and stay close to A
b) at the extreme right of the coordinate system, the graph will come and stay close to A, approaching it from above
c) at the extreme right of the coordinate system, the graph will come and stay close to A
d) A is a vertical asymptote
e) not enough information to say any of the above

__ 52. If f(x) is defined for all real numbers, and g(x) is defined for all real numbers except x = 5, what is the domain of   f o g ?

a) all real numbers                                 b) all real numbers except 5
c) all real numbers in the domain of f
d) not enough information to tell             e) none of the above

__ 53. Suppose the graph of a one-to-one function f is confined entirely to quadrant II. In what quadrant will you find the graph of f ^-1 ?

a) I             b) II         c) III             d) IV
e) II and IV                                      f) I, II, III, and IV
g) not enough information to tell        h) none of the above

__ 54. Suppose the range of a function g is [1, ). If f = g-1, what is the range of f?

a) [1, )             b) (-, -1]            c) all real numbers
d) not enough information to tell         e) none of the above

__ 55. Suppose (2, -3) is a point on the graph of y = f(x), where f is one-to-one. Which of the following points can we say for certain are on the graph of f ^-1 ?

a) (2, -3)         b) (-2, 3)                  c) (-3, 2)         d) (3, -2)         e) (0, 0)
f) not enough information given         g) none of the above

__ 56. Suppose y = f(x) defines a function that increases exponentially as x increases. If f(0) = 0 and f(2) = 4, what can we say about f(4)?

a) f(4) < 8         b) f(4) = 8         c) f(4) > 8         d) not enough information given

__ 57. In what quadrant is the point (-2, -3) ?

a) I         b) II         c) III         d) IV         e) 7
f) not enough information to tell                 g) none of the above

__ 58. Suppose you place $1.00 into a bank account that earns 100% interest per year compounded continuously. How much will your account hold at the end of a year?

a) $2.00         b) an infinite amount         c) neither a nor b
d) not enough information given

__ 59. Suppose I tell you that 2 is a logarithm. Then the number that it is the logarithm of is:

a) 100         b) e²         c)   square root of 2      d) 4         e) 210         f) e¹⁰
g) not enough information given                         h) none of the above

__ 60. Suppose you are asked to compute log₂₃ 17.

a) you can compute it without a scientific calculator
b) you can compute it with a scientific calculator
c) you cannot compute it with a scientific calculator
d) it can't be computed

__ 61. What principle permits the use of row operations when solving a system using Gauss-Jordan elimination?

a) they are legitimate operations
b) they can be proven using the Gauss-Jordan Central Limit Theorem
c) doing row operations on a system does not change the solution set of the system
d) row operations are defined by the way matrices work
e) none of the above

__ 62. What can we say about a system whose coefficient matrix has a determinant of 0?

a) it has a unique solution               b) it has no solution
c) it has infinitely many solutions     d) none of the above

__ 63. Consider two matrices A and B, where AB = C.

Which of the following is true about BA?

a) it is undefined                                            b) it is defined, and BA = AB
c) it is defined, and BA  AB     d) not enough information to state any of the above

__ 64. Consider a system of two quadratic equations in two variables, whose graphs are two distinct circles. How many real solutions does this system have?

a) none         b) 1         c) 2          d) 3         e) > 3
f) 1 or 2        g) 0, 1, or 2             h) 0 or 1
i) not enough information to tell     j) none of the above

__ 65. We compute the dot product of two real matrices. The result is:

a) an n x m matrix         b) a 1 x 1 matrix                               c) a real number
d) a square matrix         e) not enough information given         f) none of the above

__ 66. Suppose A is an n x n matrix. | A | is

a) an n x n matrix, all of whose elements are >= 0
b) a positive number                     c) a real number
d) not enough information to tell    e) none of the above

__ 67. A square matrix A has a column of 0's. The determinant of A is:

a) a real number             b) a positive number             c) 0
d) not enough information given                                  e) none of the above

__ 68. For matrix A, the matrix entry a₇₅ = 0. The cofactor of a₇₅ is:

a) its minor              b) 0                  c) a matrix smaller than A
d) not enough information to tell       e) none of the above

__ 69. Cramer's Rule can be used to:

a) find the zeros of a polynomial                 b) solve systems of linear equations
c) solve systems of quadratic equations      d) compute determinants
e) none of the above                                  f) all of a - d

__ 70. What is the 100th term of the sequence defined by a_i = 3, i = 1, 2, ... ?

a) 100         b) 600         c) not enough information         d) none of the above