quiz1_S2005_solns

quiz1_S2005_solns - MATH 202 QUIZ 1 Covers Chapter 1 and...

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MATH 202 - QUIZ # 1 Covers Chapter 1 and Chapter 2 of the text Time: 60 minutes 1. (10 points) Find the intersection of the three planes x + y + z = 3 2 x y + 2 z = 2 4 x + y + 4 z = 4 Interpret your answer geometrically. Form the augmented matrix and reduce it completely: 1 1 1 3 2 1 2 2 4 1 4 4 −→ ... −→ 1 0 1 1 / 3 0 1 0 8 / 3 0 0 0 0 So the three planes intersect in a line, given parametrically by x y z = 1 / 3 8 / 3 0 + t 1 0 1 2. (10 points) For what values of b does the matrix equation 1 1 1 2 7 7 4 9 b 2 x y z = 1 2 b + 1 have (a) infinitely many solutions? (b) no solutions? (c) exactly one solution? (Justify your answers.) Row reduce the augmented as much as possible: 1 1 1 1 2 7 7 2 4 9 b 2 b + 1 −→ ... −→ 1 0 0 1 0 1 1 0 0 0 b 2 9 b 3 We can divide by b 2 9 provided that this quantity is not zero. So for all b except ± 3 we get 1 0 0 1 0 1 0 1 / ( b + 3) 0 0 1 1 / ( b + 3)
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Thus for b negationslash = ± 3 we get a unique solution.
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