Part 1
Question 1
Solve the following system of equations using matrices. Use Gaussian elimination with back substitution or Gauss-Jordan elimination.
x - 2y + z = 0
y - 3z = -1
2y + 5z = -2
A. {(-1, -2, 0)}
B. {(-2, -1, 0)}
C. {(-5, -3, 0)}
D. {(-3, 0, 0)}
Question 2
Use Gaussian elimination to find the complete solution to each system.
2x + 3y - 5z = 15
x + 2y - z = 4
A. {(6t + 28, -7t - 6, t)}
B. {(7t + 18, -3t - 7, t)}
C. {(7t + 19, -1t - 9, t)}
D. {(4t + 29, -3t - 2, t)}
Question 3
Use Cramer's Rule to solve the following system.
x + y + z = 0
2x - y + z = -1
-x + 3y - z = -8
A. {(-1, -3, 7)}
B. {(-6, -2, 4)}
C. {(-5, -2, 7)}
D. {(-4, -1, 7)}
Question 4
Use Cramer's Rule to solve the following system.
12x + 3y = 15
2x - 3y = 13
A. {(2, -3)}
B. {(1, 3)}
C. {(3, -5)}
D. {(1, -7)}
Question 5
Give the order of the following matrix; if A = [aij], identify a32 and a23.
A. 3 * 4; a32 = 1/45; a23 = 6
B. 3 * 4; a32 = 1/2; a23 = -6
C. 3 * 2; a32 = 1/3; a23 = -5
D. 2 * 3; a32 = 1/4; a23 = 4
Question 6
Solve the system using the inverse that is given for the coefficient matrix.
2x + 6y + 6z = 8
2x + 7y + 6z =10
2x + 7y + 7z = 9
A. {(1, 2, -1)}
B. {(2, 1, -1)}
C. {(1, 2, 0)}
D. {(1, 3, -1)}
Question 7
Solve the following system of equations using matrices. Use Gaussian elimination with back substitution or Gauss-Jordan elimination.
3x1 + 5x2 - 8x3 + 5x4 = -8
x1 + 2x2 - 3x3 + x4 = -7
2x1 + 3x2 - 7x3 + 3x4 = -11
4x1 + 8x2 - 10x3+ 7x4 = -10
A. {(1, -5, 3, 4)}
B. {(2, -1, 3, 5)}
C. {(1, 2, 3, 3)}
Question 8
Use Cramer's Rule to solve the following system.
2x = 3y + 2
5x = 51 - 4y
A. {(8, 2)}
B. {(3, -4)}
C. {(2, 5)}
D. {(7, 4)}
Question 9
Use Cramer's Rule to solve the following system.
4x - 5y - 6z = -1
x - 2y - 5z = -12
2x - y = 7
A. {(2, -3, 4)}
B. {(5, -7, 4)}
C. {(3, -3, 3)}
D. {(1, -3, 5)}
Question 10
Solve the following system of equations using matrices. Use Gaussian elimination with back substitution or Gauss-Jordan elimination.
x + 3y = 0
x + y + z = 1
3x - y - z = 11
A. {(3, -1, -1)}
B. {(2, -3, -1)}
C. {(2, -2, -4)}
D. {(2, 0, -1)}
Question 11
Find the products AB and BA to determine whether B is the multiplicative inverse of A.
A = 0
0
1 1
0
0 0
1
0
B = 0
1
0 0
0
1 1
0
0
A. AB = I; BA = I3; B = A
B. AB = I3; BA = I3; B = A-1
C. AB = I; AB = I3; B = A-1
D. AB = I3; BA = I3; A = B-1
Question 12
Solve the following system of equations using matrices. Use Gaussian elimination with back substitution or Gauss-Jordan elimination.
x + 2y = z - 1
x = 4 + y - z
x + y - 3z = -2
A. {(3, -1, 0)}
B. {(2, -1, 0)}
C. {(3, -2, 1)}
D. {(2, -1, 1)}
Question 13
Use Cramer's Rule to solve the following system.
3x - 4y = 4
2x + 2y = 12
A. {(3, 1)}
B. {(4, 2)}
C. {(5, 1)}
D. {(2, 1)}
Question 14
Use Gaussian elimination to find the complete solution to each system.
x1 + 4x2 + 3x3 - 6x4 = 5
x1 + 3x2 + x3 - 4x4 = 3
2x1 + 8x2 + 7x3 - 5x4 = 11
2x1 + 5x2 - 6x4 = 4
A. {(-47t + 4, 12t, 7t + 1, t)}
B. {(-37t + 2, 16t, -7t + 1, t)}
C. {(-35t + 3, 16t, -6t + 1, t)}
Question 15
Find values for x, y, and z so that the following matrices are equal.
2x
z y + 7
4 = -10
6 13
4
A. x = -7; y = 6; z = 2
B. x = 5; y = -6; z = 2
C. x = -3; y = 4; z = 6
Question 16
Solve the following system of equations using matrices. Use Gaussian elimination with back substitution or Gauss-Jordan elimination.
2x - y - z = 4
x + y - 5z = -4
x - 2y = 4
A. {(2, -1, 1)}
B. {(-2, -3, 0)}
C. {(3, -1, 2)}
D. {(3, -1, 0)}
Question 17
Solve the following system of equations using matrices. Use Gaussian elimination with back substitution or Gauss-Jordan elimination.
x + y + z = 4
x - y - z = 0
x - y + z = 2
A. {(3, 1, 0)}
B. {(2, 1, 1)}
C. {(4, 2, 1)}
D. {(2, 1, 0)}
Question 18
Solve the following system of equations using matrices. Use Gaussian elimination with back substitution or Gauss-Jordan elimination.
x + y - z = -2
2x - y + z = 5
-x + 2y + 2z = 1
A. {(0, -1, -2)}
B. {(2, 0, 2)}
C. {(1, -1, 2)}
D. {(4, -1, 3)}
Question 19
Use Cramer's Rule to solve the following system.
x + 2y = 3
3x - 4y = 4
A. {(3, 1/5)}
B. {(5, 1/3)}
C. {(1, 1/2)}
D. {(2, 1/2)}
Question 20
Use Gaussian elimination to find the complete solution to each system.
x - 3y + z = 1
-2x + y + 3z = -7
x - 4y + 2z = 0
A. {(2t + 4, t + 1, t)}
B. {(2t + 5, t + 2, t)}
C. {(1t + 3, t + 2, t)}
D. {(3t + 3, t + 1, t)}
Question 21
Convert each equation to standard form by completing the square on x and y.
4x2 + y2 + 16x - 6y - 39 = 0
A. (x + 2)2/4 + (y - 3)2/39 = 1
B. (x + 2)2/39 + (y - 4)2/64 = 1
C. (x + 2)2/16 + (y - 3)2/64 = 1
D. (x + 2)2/6 + (y - 3)2/4 = 1
Question 22
Find the standard form of the equation of each hyperbola satisfying the given conditions.
Center: (4, -2)
Focus: (7, -2)
Vertex: (6, -2)
A. (x - 4)2/4 - (y + 2)2/5 = 1
B. (x - 4)2/7 - (y + 2)2/6 = 1
C. (x - 4)2/2 - (y + 2)2/6 = 1
D. (x - 4)2/3 - (y + 2)2/4 = 1
Question 23
Find the solution set for each system by finding points of intersection.
x2 + y2 = 1
x2 + 9y = 9
A. {(0, -2), (0, 4)}
B. {(0, -2), (0, 1)}
C. {(0, -3), (0, 1)}
D. {(0, -1), (0, 1)}
Question 24
Find the standard form of the equation of the following ellipse satisfying the given conditions.
Foci: (-2, 0), (2, 0)
Y-intercepts: -3 and 3
A. x2/23 + y2/6 = 1
B. x2/24 + y2/2 = 1
C. x2/13 + y2/9 = 1
D. x2/28 + y2/19 = 1
Question 25
Locate the foci of the ellipse of the following equation.
x2/16 + y2/4 = 1
A. Foci at (-2√3, 0) and (2√3, 0)
B. Foci at (5√3, 0) and (2√3, 0)
C. Foci at (-2√3, 0) and (5√3, 0)
D. Foci at (-7√2, 0) and (5√2, 0)
Question 26
Locate the foci and find the equations of the asymptotes.
x2/100 - y2/64 = 1
A. Foci: ({= ±2√21, 0); asymptotes: y = ±2/5x
B. Foci: ({= ±2√31, 0); asymptotes: y = ±4/7x
C. Foci: ({= ±2√41, 0); asymptotes: y = ±4/7x
D. Foci: ({= ±2√41, 0); asymptotes: y = ±4/5x
Question 27
Locate the foci and find the equations of the asymptotes.
x2/9 - y2/25 = 1
A. Foci: ({±√36, 0) ;asymptotes: y = ±5/3x
B. Foci: ({±√38, 0) ;asymptotes: y = ±5/3x
C. Foci: ({±√34, 0) ;asymptotes: y = ±5/3x
D. Foci: ({±√54, 0) ;asymptotes: y = ±6/3x
Question 28
Convert each equation to standard form by completing the square on x and y.
9x2 + 16y2 - 18x + 64y - 71 = 0
A. (x - 1)2/9 + (y + 2)2/18 = 1
B. (x - 1)2/18 + (y + 2)2/71 = 1
C. (x - 1)2/16 + (y + 2)2/9 = 1
D. (x - 1)2/64 + (y + 2)2/9 = 1
Question 29
Find the standard form of the equation of the following ellipse satisfying the given conditions.
Foci: (-5, 0), (5, 0)
Vertices: (-8, 0), (8, 0)
A. x2/49 + y2/ 25 = 1
B. x2/64 + y2/39 = 1
C. x2/56 + y2/29 = 1
D. x2/36 + y2/27 = 1
Question 30
Find the standard form of the equation of the ellipse satisfying the given conditions.
Endpoints of major axis: (7, 9) and (7, 3)
Endpoints of minor axis: (5, 6) and (9, 6)
A. (x - 7)2/6 + (y - 6)2/7 = 1
B. (x - 7)2/5 + (y - 6)2/6 = 1
C. (x - 7)2/4 + (y - 6)2/9 = 1
D. (x - 5)2/4 + (y - 4)2/9 = 1
Question 31
Find the vertices and locate the foci of each hyperbola with the given equation.
x2/4 - y2/1 =1
A.
Vertices at (2, 0) and (-2, 0); foci at (√5, 0) and (-√5, 0)
B.
Vertices at (3, 0) and (-3 0); foci at (12, 0) and (-12, 0)
C. Vertices at (4, 0) and (-4, 0); foci at (16, 0) and (-16, 0)
D. Vertices at (5, 0) and (-5, 0); foci at (11, 0) and (-11, 0)
Question 32
Find the focus and directrix of each parabola with the given equation.
x2 = -4y
A. Focus: (0, -1), directrix: y = 1
B. Focus: (0, -2), directrix: y = 1
C. Focus: (0, -4), directrix: y = 1
D. Focus: (0, -1), directrix: y = 2
Question 33
Find the standard form of the equation of each hyperbola satisfying the given conditions.
Foci: (-4, 0), (4, 0)
Vertices: (-3, 0), (3, 0)
A. x2/4 - y2/6 = 1
B. x2/6 - y2/7 = 1
C. x2/6 - y2/7 = 1
D. x2/9 - y2/7 = 1
Question 35
Find the standard form of the equation of each hyperbola satisfying the given conditions.
Endpoints of transverse axis: (0, -6), (0, 6)
Asymptote: y = 2x
A. y2/6 - x2/9 = 1
B. y2/36 - x2/9 = 1
C. y2/37 - x2/27 = 1
D. y2/9 - x2/6 = 1
Question 36
Convert each equation to standard form by completing the square on x or y. Then ?nd the vertex, focus, and directrix of the parabola.
y2 - 2y + 12x - 35 = 0
A. (y - 2)2 = -10(x - 3); vertex: (3, 1); focus: (0, 1); directrix: x = 9
B. (y - 1)2 = -12(x - 3); vertex: (3, 1); focus: (0, 1); directrix: x = 6
C. (y - 5)2 = -14(x - 3); vertex: (2, 1); focus: (0, 1); directrix: x = 6
D. (y - 2)2 = -12(x - 3); vertex: (3, 1); focus: (0, 1); directrix: x = 8
Question 37
Locate the foci of the ellipse of the following equation.
25x2 + 4y2 = 100
A. Foci at (1, -√11) and (1, √11)
B. Foci at (0, -√25) and (0, √25)
C. Foci at (0, -√22) and (0, √22)
D. Foci at (0, -√21) and (0, √21)
Question 38
Find the focus and directrix of the parabola with the given equation.
8x2 + 4y = 0
A. Focus: (0, -1/4); directrix: y = 1/4
B. Focus: (0, -1/6); directrix: y = 1/6
C. Focus: (0, -1/8); directrix: y = 1/8
D. Focus: (0, -1/2); directrix: y = 1/2
Question 39
Find the vertex, focus, and directrix of each parabola with the given equation.
(x + 1)2 = -8(y + 1)
A. Vertex: (-1, -2); focus: (-1, -2); directrix: y = 1
B. Vertex: (-1, -1); focus: (-1, -3); directrix: y = 1
C. Vertex: (-3, -1); focus: (-2, -3); directrix: y = 1
D. Vertex: (-4, -1); focus: (-2, -3); directrix: y = 1
Question 40
Find the standard form of the equation of the following ellipse satisfying the given conditions.
Foci: (0, -4), (0, 4)
Vertices: (0, -7), (0, 7)
A. x2/43 + y2/28 = 1
B. x2/33 + y2/49 = 1
C. x2/53 + y2/21 = 1
D. x2/13 + y2/39 = 1
Part 2
Question 1
How large a group is needed to give a 0.5 chance of at least two people having the same birthday?
A. 13 people
B. 23 people
C. 47 people
D. 28 people
Question 2
Write the first six terms of the following arithmetic sequence.
an = an-1 - 0.4, a1 = 1.6
A. 1.6, 1.2, 0.8, 0.4, 0, -0.4
B. 1.6, 1.4, 0.9, 0.3, 0, -0.3
C. 1.6, 2.2, 1.8, 1.4, 0, -1.4
D. 1.3, 1.5, 0.8, 0.6, 0, -0.6
Question 3
A club with ten members is to choose three officers-president, vice president, and secretary-treasurer. If each office is to be held by one person and no person can hold more than one office, in how many ways can those offices be filled?
A. 650 ways
B. 720 ways
C. 830 ways
D. 675 ways
Question 4
k2 + 3k + 2 = (k2 + k) + 2 ( __________ )
A. k + 5
B. k + 1
C. k + 3
D. k + 2
Question 5
Write the first six terms of the following arithmetic sequence.
an = an-1 + 6, a1 = -9
A. -9, -3, 3, 9, 15, 21
B. -11, -4, 3, 9, 17, 21
C. -8, -3, 3, 9, 16, 22
D. -9, -5, 3, 11, 15, 27
Question 6
Use the Binomial Theorem to find a polynomial expansion for the following function.
f1(x) = (x - 2)4
A. f1(x) = x4 - 5x3 + 14x2 - 42x + 26
B. f1(x) = x4 - 16x3 + 18x2 - 22x + 18
C. f1(x) = x4 - 18x3 + 24x2 - 28x + 16
D. f1(x) = x4 - 8x3 + 24x2 - 32x + 16
Question 7
The following are defined using recursion formulas. Write the first four terms of each sequence.
a1 = 4 and an = 2an-1 + 3 for n ≥ 2
A. 4, 15, 35, 453
B. 4, 11, 15, 13
C. 4, 11, 25, 53
D. 3, 19, 22, 53
Question 8
The following are defined using recursion formulas. Write the first four terms of each sequence.
a1 = 3 and an = 4an-1 for n ≥ 2
A. 3, 12, 48, 192
B. 4, 11, 58, 92
C. 3, 14, 79, 123
D. 5, 14, 47, 177
Question 9
Write the first six terms of the following arithmetic sequence.
an = an-1 - 10, a1 = 30
A. 40, 30, 20, 0, -20, -10
B. 60, 40, 30, 0, -15, -10
C. 20, 10, 0, 0, -15, -20
D. 30, 20, 10, 0, -10, -20
Question 10
Find the indicated term of the arithmetic sequence with first term, a1, and common difference, d.
Find a6 when a1 = 13, d = 4
A. 36
B. 63
C. 43
D. 33
Question 11
If 20 people are selected at random, ?nd the probability that at least 2 of them have the same birthday.
A. ≈ 0.31
B. ≈ 0.42
C. ≈ 0.45
D. ≈ 0.41
Question 12
If three people are selected at random, find the probability that they all have different birthdays.
A. 365/365 * 365/364 * 363/365 ≈ 0.98
B. 365/364 * 364/365 * 363/364 ≈ 0.99
C. 365/365 * 365/363 * 363/365 ≈ 0.99
D. 365/365 * 364/365 * 363/365 ≈ 0.99
Question 13
An election ballot asks voters to select three city commissioners from a group of six candidates. In how many ways can this be done?
A. 20 ways
B. 30 ways
C. 10 ways
D. 15 ways
Question 14
Write the first four terms of the following sequence whose general term is given.
an = (-3)n
A. -4, 9, -25, 31
B. -5, 9, -27, 41
C. -2, 8, -17, 81
D. -3, 9, -27, 81
Question 15
If two people are selected at random, the probability that they do not have the same birthday (day and month) is 365/365 * 364/365. (Ignore leap years and assume 365 days in a year.)
A. The first person can have any birthday in the year. The second person can have all but one birthday.
B. The second person can have any birthday in the year. The first person can have all but one birthday.
C. The first person cannot a birthday in the year. The second person can have all but one birthday.
D. The first person can have any birthday in the year. The second cannot have all but one birthday.
Question 16
Find the indicated term of the arithmetic sequence with first term, a1, and common difference, d.
Find a200 when a1 = -40, d = 5
A. 865
B. 955
C. 678
D. 895
Question 17
To win at LOTTO in the state of Florida, one must correctly select 6 numbers from a collection of 53 numbers (1 through 53). The order in which the selection is made does not matter. How many different selections are possible?
A. 32,957,326 selections
B. 22,957,480 selections
C. 28,957,680 selections
D. 225,857,480 selections
Question 18
Write the first four terms of the following sequence whose general term is given.
an = 3n
A. 3, 9, 27, 81
B. 4, 10, 23, 91
C. 5, 9, 17, 31
D. 4, 10, 22, 41
Question 19
If three people are selected at random, ?nd the probability that at least two of them have the same birthday.
A. ≈ 0.07
B. ≈ 0.02
C. ≈ 0.01
D. ≈ 0.001
Question 20
Use the Binomial Theorem to expand the following binomial and express the result in simpli?ed form.
(x2 + 2y)4
A. x8 + 8x6 y + 24x4 y2 + 32x2 y3 + 16y4
B. x8 + 8x6 y + 20x4 y2 + 30x2 y3 + 15y4
C. x8 + 18x6 y + 34x4 y2 + 42x2 y3 + 16y4
D. x8 + 8x6 y + 14x4 y2 + 22x2 y3 + 26y4