1. Use Gaussian elimination to find the complete solution to each system.
|
| x1 + 4x2 + 3x3 - 6x4 = 5 |
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)}
2. Use Cramer's Rule to solve the following system.
|
| x + y + z = 0 |
A. {(-1, -3, 7)}
B. {(-6, -2, 4)}
C. {(-5, -2, 7)}
D. {(-4, -1, 7)}
3. Use Cramer's Rule to solve the following system.
|
| 2x = 3y + 2 |
A. {(8, 2)}
B. {(3, -4)}
C. {(2, 5)}
D. {(7, 4)}
4. Give the order of the following matrix; if A = [aij], identify a32 and a23.
|
| 1 | -5 | ∏ | e |
|
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
5. Solve the following system of equations using matrices. Use Gaussian elimination with back substitution or Gauss-Jordan elimination.
|
| 2x - y - z = 4 |
A. {(2, -1, 1)}
B. {(-2, -3, 0)}
C. {(3, -1, 2)}
D. {(3, -1, 0)}
6. Use Cramer's Rule to solve the following system.
|
| x + 2y + 2z = 5 |
A. {(33, -11, 4)}
B. {(13, 12, -3)}
C. {(23, -12, 3)}
D. {(13, -14, 3)}
7. Use Gaussian elimination to find the complete solution to the following system of equations, or show that none exists.
|
| 2w + x - y = 3 |
A. {(1, 3, 2, 1)}
B. {(1, 4, 3, -1)}
C. {(1, 5, 1, 1)}
D. {(-1, 2, -2, 1)}
8. Solve the following system of equations using matrices. Use Gaussian elimination with back substitution or Gauss-Jordan elimination.
|
| x + y - z = -2 |
A. {(0, -1, -2)}
B. {(2, 0, 2)}
C. {(1, -1, 2)}
D. {(4, -1, 3)}
9. Solve the following system of equations using matrices. Use Gaussian elimination with back substitution or Gauss-Jordan elimination.
|
| x + 2y = z - 1 |
A. {(3, -1, 0)}
B. {(2, -1, 0)}
C. {(3, -2, 1)}
D. {(2, -1, 1)}
10. Use Cramer's Rule to solve the following system.
|
| 4x - 5y - 6z = -1 |
A. {(2, -3, 4)}
B. {(5, -7, 4)}
C. {(3, -3, 3)}
D. {(1, -3, 5)}
11. If AB = -BA, then A and B are said to be anticommutative.
| Are A = |
| 0 | -1 |
| and B = |
| 1 | 0 |
| 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.
12. Use Cramer's Rule to solve the following system.
|
| 4x - 5y = 17 |
A. {(3, -1)}
B. {(2, -1)}
C. {(3, -7)}
D. {(2, 0)}
13. Solve the following system of equations using matrices. Use Gaussian elimination with back substitution or Gauss-Jordan elimination.
|
| 3x1 + 5x2 - 8x3 + 5x4 = -8 |
A. {(1, -5, 3, 4)}
B. {(2, -1, 3, 5)}
C. {(1, 2, 3, 3)}
D. {(2, -2, 3, 4)}
14. Solve the system using the inverse that is given for the coefficient matrix.
|
| 2x + 6y + 6z = 8 |
The inverse of:
|
| 2 | 6 | 6 |
|
is
|
| 7/2 | 0 | -3 |
|
A. {(1, 2, -1)}
B. {(2, 1, -1)}
C. {(1, 2, 0)}
D. {(1, 3, -1)}
15. Solve the following system of equations using matrices. Use Gaussian elimination with back substitution or Gauss-Jordan elimination.
|
| x - 2y + z = 0 |
A. {(-1, -2, 0)}
B. {(-2, -1, 0)}
C. {(-5, -3, 0)}
D. {(-3, 0, 0)}
16. Solve the following system of equations using matrices. Use Gaussian elimination with back substitution or Gauss-Jordan elimination.
|
| x + y + z = 4 |
A. {(3, 1, 0)}
B. {(2, 1, 1)}
C. {(4, 2, 1)}
D. {(2, 1, 0)}
17. Use Gaussian elimination to find the complete solution to the following system of equations, or show that none exists.
|
| w - 2x - y - 3z = -9 |
A. {(-1, 2, 1, 1)}
B. {(-2, 2, 0, 1)}
C. {(0, 1, 1, 3)}
D. {(-1, 2, 1, 1)}
18. Use Gauss-Jordan elimination to solve the system.
|
| -x - y - z = 1 |
A. {(14, -10, -3)}
B. {(10, -2, -6)}
C. {(15, -12, -4)}
D. {(11, -13, -4)}
19. Find values for x, y, and z so that the following matrices are equal.
|
| 2x | y + 7 |
| = |
| -10 | 13 |
|
A. x = -7; y = 6; z = 2
B. x = 5; y = -6; z = 2
C. x = -3; y = 4; z = 6
D. x = -5; y = 6; z = 6
20. Use Cramer's Rule to solve the following system.
|
| x + y = 7 |
A. {(7, 2)}
B. {(8, -2)}
C. {(5, 2)}
D. {(9, 3)}
21. 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
22. 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
23. 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
24. Find the vertex, focus, and directrix of each parabola with the given equation.
(y + 1)2 = -8x
A. Vertex: (0, -1); focus: (-2, -1); directrix: x = 2
B. Vertex: (0, -1); focus: (-3, -1); directrix: x = 3
C. Vertex: (0, -1); focus: (2, -1); directrix: x = 1
D. Vertex: (0, -3); focus: (-2, -1); directrix: x = 5
25. Find the solution set for each system by finding points of intersection.
|
| x2 + y2 = 1 |
A. {(0, -2), (0, 4)}
B. {(0, -2), (0, 1)}
C. {(0, -3), (0, 1)}
D. {(0, -1), (0, 1)}
26. Find the vertex, focus, and directrix of each parabola with the given equation.
(y + 3)2 = 12(x + 1)
A. Vertex: (-1, -3); focus: (1, -3); directrix: x = -3
B. Vertex: (-1, -1); focus: (4, -3); directrix: x = -5
C. Vertex: (-2, -3); focus: (2, -4); directrix: x = -7
D. Vertex: (-1, -3); focus: (2, -3); directrix: x = -4
27. 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
28. Find the vertices and locate the foci of each hyperbola with the given equation.
y2/4 - x2/1 = 1
A. Vertices at (0, 5) and (0, -5); foci at (0, 14) and (0, -14)
B. Vertices at (0, 6) and (0, -6); foci at (0, 13) and (0, -13)
C. Vertices at (0, 2) and (0, -2); foci at (0, √5) and (0, -√5)
D. Vertices at (0, 1) and (0, -1); foci at (0, 12) and (0, -12)
29. 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
30. 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 = ½
31. 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
32. 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
33. 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
34. 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
35. Locate the foci of the ellipse of the following equation.
7x2 = 35 - 5y2
A. Foci at (0, -√2) and (0, √2)
B. Foci at (0, -√1) and (0, √1)
C. Foci at (0, -√7) and (0, √7)
D. Foci at (0, -√5) and (0, √5)
36. Convert each equation to standard form by completing the square on x or y. Then find 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
37. Convert each equation to standard form by completing the square on x or y. Then find the vertex, focus, and directrix of the parabola.
x2 - 2x - 4y + 9 = 0
A. (x - 4)2 = 4(y - 2); vertex: (1, 4); focus: (1, 3) ; directrix: y = 1
B. (x - 2)2 = 4(y - 3); vertex: (1, 2); focus: (1, 3) ; directrix: y = 3
C. (x - 1)2 = 4(y - 2); vertex: (1, 2); focus: (1, 3) ; directrix: y = 1
D. (x - 1)2 = 2(y - 2); vertex: (1, 3); focus: (1, 2) ; directrix: y=5
38. Find the focus and directrix of each parabola with the given equation.
y2 = 4x
A. Focus: (2, 0); directrix: x = -1
B. Focus: (3, 0); directrix: x = -1
C. Focus: (5, 0); directrix: x = -1
D. Focus: (1, 0); directrix: x = -1
39. 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)
40. Find the standard form of the equation of each hyperbola satisfying the given conditions.
Foci: (0, -3), (0, 3)
Vertices: (0, -1), (0, 1)
A. y2 - x2/4 = 0
B. y2 - x2/8 = 1
C. y2 - x2/3 = 1
D. y2 - x2/2 = 0
