College algebra help please
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college_algebra_2.docx1) Use Gaussian elimination to find the complete solution to the following system of equations, or show that none exists.
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8x + 5y + 11z = 30 -x - 4y + 2z = 3 2x - y + 5z = 12 |
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A. {(3 - 3t, 2 + t, t)} |
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B. {(6 - 3t, 2 + t, t)} |
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C. {(5 - 2t, -2 + t, t)} |
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D. {(2 - 1t, -4 + t, t)} |
2) Use Gaussian elimination to find the complete solution to the following system of equations, or show that none exists.
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5x + 8y - 6z = 14 3x + 4y - 2z = 8 x + 2y - 2z = 3 |
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A. {(-4t + 2, 2t + 1/2, t)} |
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B. {(-3t + 1, 5t + 1/3, t)} |
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C. {(2t + -2, t + 1/2, t)} |
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D. {(-2t + 2, 2t + 1/2, t)} |
3) If AB = -BA, then A and B are said to be anticommutative.
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Are A = |
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0
1 |
-1
0 |
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and B = |
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1
0 |
0
-1 |
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anticommutative? |
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A. AB = -AB so they are not anticommutative. |
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B. AB = BA so they are anticommutative. |
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C. BA = -BA so they are not anticommutative. |
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D. AB = -BA so they are anticommutative. |
4) Give the order of the following matrix; if A = [aij], identify a32 and a23.
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1
0
-2 |
-5
7
1/2 |
∏
-6
11 |
e
-∏
-1/5 |
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A. 3 * 4; a32 = 1/45; a23 = 6 |
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B. 3 * 4; a32 = 1/2; a23 = -6 |
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C. 3 * 2; a32 = 1/3; a23 = -5 |
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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.
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3x1 + 5x2 - 8x3 + 5x4 = -8 x1 + 2x2 - 3x3 + x4 = -7 2x1 + 3x2 - 7x3 + 3x4 = -11 4x1 + 8x2 - 10x3+ 7x4 = -10 |
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A. {(1, -5, 3, 4)} |
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B. {(2, -1, 3, 5)} |
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C. {(1, 2, 3, 3)} |
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D. {(2, -2, 3, 4)} |
6) Use Gaussian elimination to find the complete solution to each system.
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x1 + 4x2 + 3x3 - 6x4 = 5 x1 + 3x2 + x3 - 4x4 = 3 2x1 + 8x2 + 7x3 - 5x4 = 11 2x1 + 5x2 - 6x4 = 4 |
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A. {(-47t + 4, 12t, 7t + 1, t)} |
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B. {(-37t + 2, 16t, -7t + 1, t)} |
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C. {(-35t + 3, 16t, -6t + 1, t)} |
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D. {(-27t + 2, 17t, -7t + 1, t)} |
7) Use Gaussian elimination to find the complete solution to each system.
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2x + 3y - 5z = 15 x + 2y - z = 4 |
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A. {(6t + 28, -7t - 6, t)} |
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B. {(7t + 18, -3t - 7, t)} |
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C. {(7t + 19, -1t - 9, t)} |
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D. {(4t + 29, -3t - 2, t)} |
8) Find the focus and directrix of the parabola with the given equation.
8x2 + 4y = 0
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A. Focus: (0, -1/4); directrix: y = 1/4 |
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B. Focus: (0, -1/6); directrix: y = 1/6 |
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C. Focus: (0, -1/8); directrix: y = 1/8 |
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D. Focus: (0, -1/2); directrix: y = 1/2 |
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9) Find the vertices and locate the foci of each hyperbola with the given equation.
y2/4 - x2/1 = 1
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A. Vertices at (0, 5) and (0, -5); foci at (0, 14) and (0, -14) |
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B. Vertices at (0, 6) and (0, -6); foci at (0, 13) and (0, -13) |
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C. Vertices at (0, 2) and (0, -2); foci at (0, √5) and (0, -√5) |
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D. Vertices at (0, 1) and (0, -1); foci at (0, 12) and (0, -12) |
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10) Convert each equation to standard form by completing the square on x and y.
9x2 + 25y2 - 36x + 50y - 164 = 0
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A. (x - 2)2/25 + (y + 1)2/9 = 1 |
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B. (x - 2)2/24 + (y + 1)2/36 = 1 |
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C. (x - 2)2/35 + (y + 1)2/25 = 1 |
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D. (x - 2)2/22 + (y + 1)2/50 = 1 |
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11) Find the standard form of the equation of the ellipse satisfying the given conditions.
Major axis vertical with length = 10
Length of minor axis = 4
Center: (-2, 3)
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A. (x + 2)2/4 + (y - 3)2/25 = 1 |
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B. (x + 4)2/4 + (y - 2)2/25 = 1 |
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C. (x + 3)2/4 + (y - 2)2/25 = 1 |
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D. (x + 5)2/4 + (y - 2)2/25 = 1 |
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12) 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)
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A. x2/49 + y2/ 25 = 1 |
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B. x2/64 + y2/39 = 1 |
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C. x2/56 + y2/29 = 1 |
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D. x2/36 + y2/27 = 1 |
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13) Find the focus and directrix of each parabola with the given equation.
x2 = -4y
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A. Focus: (0, -1), directrix: y = 1 |
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B. Focus: (0, -2), directrix: y = 1 |
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C. Focus: (0, -4), directrix: y = 1 |
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D. Focus: (0, -1), directrix: y = 2 |
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14) Find the standard form of the equation of each hyperbola satisfying the given conditions.
Foci: (-4, 0), (4, 0)
Vertices: (-3, 0), (3, 0)
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A. x2/4 - y2/6 = 1 |
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B. x2/6 - y2/7 = 1 |
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C. x2/6 - y2/7 = 1 |
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D. x2/9 - y2/7 = 1 |
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15) Locate the foci and find the equations of the asymptotes.
x2/100 - y2/64 = 1
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A. Foci: ({= ±2√21, 0); asymptotes: y = ±2/5x |
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B. Foci: ({= ±2√31, 0); asymptotes: y = ±4/7x |
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C. Foci: ({= ±2√41, 0); asymptotes: y = ±4/7x |
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D. Foci: ({= ±2√41, 0); asymptotes: y = ±4/5x |
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16) Locate the foci and find the equations of the asymptotes.
4y2 – x2 = 1
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A. (0, ±√4/2); asymptotes: y = ±1/3x |
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B. (0, ±√5/2); asymptotes: y = ±1/2x |
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C. (0, ±√5/4); asymptotes: y = ±1/3x |
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D. (0, ±√5/3); asymptotes: y = ±1/2x |
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17) Convert each equation to standard form by completing the square on x and y.
4x2 + y2 + 16x - 6y - 39 = 0
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A. (x + 2)2/4 + (y - 3)2/39 = 1 |
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B. (x + 2)2/39 + (y - 4)2/64 = 1 |
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C. (x + 2)2/16 + (y - 3)2/64 = 1 |
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D. (x + 2)2/6 + (y - 3)2/4 = 1 |
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18) Locate the foci of the ellipse of the following equation.
x2/16 + y2/4 = 1
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A. Foci at (-2√3, 0) and (2√3, 0) |
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B. Foci at (5√3, 0) and (2√3, 0) |
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C. Foci at (-2√3, 0) and (5√3, 0) |
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D. Foci at (-7√2, 0) and (5√2, 0) |
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19) Find the solution set for each system by finding points of intersection.
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x2 + y2 = 1 x2 + 9y = 9 |
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A. {(0, -2), (0, 4)} |
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B. {(0, -2), (0, 1)} |
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C. {(0, -3), (0, 1)} |
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D. {(0, -1), (0, 1)} |
20) Find the vertex, focus, and directrix of each parabola with the given equation.
(x + 1)2 = -8(y + 1)
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A. Vertex: (-1, -2); focus: (-1, -2); directrix: y = 1 |
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B. Vertex: (-1, -1); focus: (-1, -3); directrix: y = 1 |
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C. Vertex: (-3, -1); focus: (-2, -3); directrix: y = 1 |
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D. Vertex: (-4, -1); focus: (-2, -3); directrix: y = 1 |
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21) Locate the foci and find the equations of the asymptotes.
x2/9 - y2/25 = 1
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A. Foci: ({±√36, 0) ;asymptotes: y = ±5/3x |
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B. Foci: ({±√38, 0) ;asymptotes: y = ±5/3x |
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C. Foci: ({±√34, 0) ;asymptotes: y = ±5/3x |
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D. Foci: ({±√54, 0) ;asymptotes: y = ±6/3x |
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22) Locate the foci of the ellipse of the following equation.
7x2 = 35 - 5y2
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A. Foci at (0, -√2) and (0, √2) |
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B. Foci at (0, -√1) and (0, √1) |
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C. Foci at (0, -√7) and (0, √7) |
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D. Foci at (0, -√5) and (0, √5) |
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23) 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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A. (x - 7)2/6 + (y - 6)2/7 = 1 |
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B. (x - 7)2/5 + (y - 6)2/6 = 1 |
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C. (x - 7)2/4 + (y - 6)2/9 = 1 |
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D. (x - 5)2/4 + (y - 4)2/9 = 1 |
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24) 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
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A. y2/6 - x2/9 = 1 |
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B. y2/36 - x2/9 = 1 |
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C. y2/37 - x2/27 = 1 |
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D. y2/9 - x2/6 = 1 |
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25) Find the vertex, focus, and directrix of each parabola with the given equation.
(x - 2)2 = 8(y - 1)
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A. Vertex: (3, 1); focus: (1, 3); directrix: y = -1 |
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B. Vertex: (2, 1); focus: (2, 3); directrix: y = -1 |
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C. Vertex: (1, 1); focus: (2, 4); directrix: y = -1 |
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D. Vertex: (2, 3); focus: (4, 3); directrix: y = -1 |
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26) Find the standard form of the equation of each hyperbola satisfying the given conditions.
Center: (4, -2)
Focus: (7, -2)
Vertex: (6, -2)
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A. (x - 4)2/4 - (y + 2)2/5 = 1 |
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B. (x - 4)2/7 - (y + 2)2/6 = 1 |
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C. (x - 4)2/2 - (y + 2)2/6 = 1 |
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D. (x - 4)2/3 - (y + 2)2/4 = 1 |
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27) Find the focus and directrix of each parabola with the given equation.
y2 = 4x
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A. Focus: (2, 0); directrix: x = -1 |
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B. Focus: (3, 0); directrix: x = -1 |
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C. Focus: (5, 0); directrix: x = -1 |
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D. Focus: (1, 0); directrix: x = -1 |
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