Question 1
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 2
Give the order of the following matrix; if A = [aij], identify a32 and a23.
1
0
-2 -5
7
1/2 ∏
-6
11 e
-∏
-1/5
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 3
Use Gaussian elimination to find the complete solution to the following system of equations, or show that none exists.
3x + 4y + 2z = 3
4x - 2y - 8z = -4
x + y - z = 3
A. {(-2, 1, 2)}
B. {(-3, 4, -2)}
C. {(5, -4, -2)}
D. {(-2, 0, -1)}
Question 4
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)}
D. {(-27t + 2, 17t, -7t + 1, t)}
Question 5
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 6
Use Gaussian elimination to find the complete solution to the following system of equations, or show that none exists.
w - 2x - y - 3z = -9
w + x - y = 0
3w + 4x + z = 6
2x - 2y + z = 3
A. {(-1, 2, 1, 1)}
B. {(-2, 2, 0, 1)}
C. {(0, 1, 1, 3)}
D. {(-1, 2, 1, 1)}
Question 7
Use Gaussian elimination to find the complete solution to the following system of equations, or show that none exists.
8x + 5y + 11z = 30
-x - 4y + 2z = 3
2x - y + 5z = 12
• A. {(3 - 3t, 2 + t, t)}
B. {(6 - 3t, 2 + t, t)}
C. {(5 - 2t, -2 + t, t)}
D. {(2 - 1t, -4 + t, t)}
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.
x + 2y + 2z = 5
2x + 4y + 7z = 19
-2x - 5y - 2z = 8
A. {(33, -11, 4)}
B. {(13, 12, -3)}
C. {(23, -12, 3)}
D. {(13, -14, 3)}
Question 10
If AB = -BA, then A and B are said to be anticommutative.
Are A = 0
1 -1
0 and B = 1
0 0
-1 anticommutative?
A. AB = -AB so they are not anticommutative.
B. AB = BA so they are anticommutative.
C. BA = -BA so they are not anticommutative.
D. AB = -BA so they are anticommutative.
Question 11
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 12
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 13
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 14
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 15
Use Cramer’s Rule to solve the following system.
x + y = 7
x - y = 3
A. {(7, 2)}
B. {(8, -2)}
C. {(5, 2)}
D. {(9, 3)}
Question 16
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 17
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
Question 18
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
Question 19
Find values for x, y, and z so that the following matrices are equal.
2x
z y + 7
4 = -10
6 13
4
Question 20
Use Gauss-Jordan elimination to solve the system.
-x - y - z = 1
4x + 5y = 0
y - 3z = 0
Question 21
Locate the foci of the ellipse of the following equation.
7x2 = 35 - 5y2
Question 22
Locate the foci and find the equations of the asymptotes.
x2/9 - y2/25 = 1
Question 23
Find the vertex, focus, and directrix of each parabola with the given equation.
(x - 2)2 = 8(y - 1)
Question 24
Find the vertex, focus, and directrix of each parabola with the given equation.
(y + 1)2 = -8x
Question 25
Locate the foci and find the equations of the asymptotes.
x2/100 - y2/64 = 1
Question 26
Find the focus and directrix of each parabola with the given equation.
x2 = -4y
Question 27
Find the standard form of the equation of each hyperbola satisfying the given conditions.
Center: (4, -2)
Focus: (7, -2)
Vertex: (6, -2)
Question 28
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)
Question 29
Find the standard form of the equation of each hyperbola satisfying the given conditions.
Foci: (-4, 0), (4, 0)
Vertices: (-3, 0), (3, 0)
Question 30
Locate the foci of the ellipse of the following equation.
25x2 + 4y2 = 100
Question 31
Find the vertex, focus, and directrix of each parabola with the given equation.
(y + 3)2 = 12(x + 1)
Question 32
Find the solution set for each system by finding points of intersection.
x2 + y2 = 1
x2 + 9y = 9
Question 33
Find the standard form of the equation of each hyperbola satisfying the given conditions.
Foci: (0, -3), (0, 3)
Vertices: (0, -1), (0, 1)
Question 34
Find the vertices and locate the foci of each hyperbola with the given equation.
x2/4 - y2/1 =1
Question 35
Convert each equation to standard form by completing the square on x and y.
9x2 + 25y2 - 36x + 50y - 164 = 0
Question 36
Locate the foci and find the equations of the asymptotes.
4y2 – x2 = 1
Question 37
Find the vertices and locate the foci of each hyperbola with the given equation.
y2/4 - x2/1 = 1
Question 38
Convert each equation to standard form by completing the square on x and y.
9x2 + 16y2 - 18x + 64y - 71 = 0
Question 39
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
Question 40
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)
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