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M408M - romasko(qrr58 Assignment 1 luecke(55035 This...

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romasko (qrr58) – Assignment 1 – luecke – (55035) 1 This print-out should have 20 questions. Multiple-choice questions may continue on the next column or page – find all choices before answering. 001 10.0 points Which one of the following conic sections has graph 1 1 2 3 1 2 1 2 1. x 2 2 x + y 1 = 0 2. y 2 + 2 y + x 1 = 0 3. x 2 + 2 x y 1 = 0 correct 4. y 2 2 y + x 1 = 0 5. y 2 + 2 y x 1 = 0 6. x 2 + 2 x + y 1 = 0 7. x 2 2 x y 1 = 0 8. y 2 2 y x 1 = 0 Explanation: The graph is that of a parabola opening upwards and having vertex at ( 1 , 2). Thus it is the graph of y + 2 = A ( x + 1) 2 , for some A > 0. But the graph intersects the y -axis at y = 1 so A = 1. Consequently, the graph is that of the conic section ( x + 1) 2 y + 2 = 0 , which after expansion and simplification can be rewritten as x 2 + 2 x y 1 = 0 . keywords: graph, conic section, parabola, shifted parabola, 002 10.0 points Which one of the following conic sections has graph 1 2 3 4 1 2 1 2 3 4 1 2 1. 9 x 2 + 18 x + 4 y 2 8 y = 23 2. 4 x 2 + 8 x 9 y 2 + 18 y = 31 3. 9 x 2 18 x + 4 y 2 + 8 y = 23 4. 4 x 2 8 x 9 y 2 + 18 y = 31 5. 9 x 2 18 x + 4 y 2 8 y = 23 correct 6. 4 x 2 8 x 9 y 2 18 y = 31 Explanation: The graph is a shifted ellipse centered at (1 , 1) whose major axis has length 6 and mi- nor axis has length 4. Thus one equation for this ellipse is ( x 1) 2 4 + ( y 1) 2 9 = 1 ,
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romasko (qrr58) – Assignment 1 – luecke – (55035) 2 which after expansion and simplification can be rewritten as 9 x 2 18 x + 4 y 2 8 y = 23 . keywords: graph, conic section, ellipse, shifted ellipse 003 10.0 points Which one of the following conic sections has graph 2 4 2 4 6 2 4 2 4 6 1. x 2 2 x + 4 y 2 8 y + 1 = 0 2. x 2 + 2 x 4 y 2 + 8 y = 7 3. x 2 2 x 4 y 2 + 8 y = 7 4. 4 y 2 8 x x 2 2 x = 1 5. x 2 + 2 x + 4 y 2 + 8 y + 1 = 0 6. x 2 + 2 x 4 y 2 8 y = 7 7. 4 y 2 + 8 x x 2 2 y = 1 8. 4 y 2 + 8 y x 2 2 x = 1 correct Explanation: The graph is a shifted hyperbola having asymptotes which intersect at ( 1 , 1) as well as vertices at ( 1 , 0) and ( 1 , 2). Thus one equation for this hyperbola is ( y + 1) 2 1 ( x + 1) 2 4 = 1 , which after expansion and simplification can be rewritten as 4 y 2 + 8 y x 2 2 x = 1 . keywords: graph, conic section, ellipse, shifted ellipse 004 10.0 points Which one of the points P (0 , 2 , 0) , Q ( 3 , 9 , 1) , R (2 , 5 , 3) in 3-space is closest to the xz -plane? 1. P (0 , 2 , 0) correct 2. R (2 , 5 , 3) 3. Q ( 3 , 9 , 1) Explanation: The distance of a point ( a, b, c ) in 3-space from the xz -plane is given by | b | . Conse- quently, of the three points P (0 , 2 , 0) , Q ( 3 , 9 , 1) , R (2 , 5 , 3) the one closest to the xz -plane is P (0 , 2 , 0) . keywords: plane, distance in 3-space, 005 10.0 points Which of the following statements are true: A. in 2 -space the intersection of the graphs of braceleftbigg x = 3 , y = 5 , is a line ,
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romasko (qrr58) – Assignment 1 – luecke – (55035) 3 B. in 3 -space the intersection of the graphs of braceleftbigg x = 4 , z = 5 , is a line .
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