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Winter 2007 - Takeda's Class - Practice Exam 1

# Winter 2007 - Takeda's Class - Practice Exam 1 - Math 20C...

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Math 20C, Practice Midterm 1 Solutions January 27, 2007 Name: Section: This exam consists of 7 pages including this front page. Ground Rules 1. No calculator is allowed. 2. Show your work for every problem. A correct answer without any justiﬁcation will receive no credit. 3. You may use one 4-by-6 index card, both sides. 4. You are not allowed to leave your seat if the remaining time is less than 5 minutes. Score 1 10 2 10 3 10 4 10 5 10 6 10 Total 60 1

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(a) Find the center and radius of the sphere give by the equation x 2 + y 2 + z 2 - 2 y = 0 By completing the square, x 2 + y 2 + z 2 - 2 y = 0 becomes x 2 + ( y - 1) 2 + z 2 = 1 . So the center is (0 , 1 , 0) and the radius is 1. (b) Find the number a so that the two vectors a = h 1 , a, 2 i and b = h- 5 , 3 , a i are orthogonal to each other. For a and b to be orthogonal, we must have a · b = 0. Now a · b = - 5 + 3 a + 2 a = - 5 a + 5. So a = 1. 2
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Winter 2007 - Takeda's Class - Practice Exam 1 - Math 20C...

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