38-1E-09Fsols - Mathematics 38 Differential Equations...

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Mathematics 38 Differential Equations Examination 1 October 5, 2009 No calculators, notes, or books are allowed. Please make sure all electronic devices are turned off and out of sight. Show all work and cross out work you do not want graded! Remember to sign your blue book. With your signature you are pledging that you have neither given nor received assistance on this exam. Good luck! 1. (5 points, answer only, no partial credit) Can the system below be solved by Cramer’s rule? x 1 + x 2 + x 3 = 0 2 x 1 +3 x 2 + x 3 = 0 x 1 3 x 2 + x 3 = 0 Solution: No, the determinant of the matrix of coefficients is zero. 2. (5 points, answer only, no partial credit) Consider the following system of equations: x 1 + x 2 + x 3 = 0 2 x 1 +3 x 2 + x 3 = 0 x 1 3 x 2 + x 3 = 0 Choose one answer: The system has a. a unique solution, b. no solution, c. more than 1 solution. d. None of the above. Solution: c. : d. is clearly impossible, a. is impossible because the previous problem shows that Cramer’s test gives a zero determinant, and b. is not true because x 1 = x 2 = x 3 = 0 is an obvious solution. 3. (5 points, answer only, no partial credit) Choose one answer. det 5 1 sin t t 2 + 3 1 0 4 e t e t 0 0 0 3 ln t 8 0 0 0 2 t 0 0 0 0 1 = a. 5 , b. 2 t , c. 120 , d. 120 sin t 1 4 e t 1 4 ( t 2 + 3) e t . e. None of the above. Solution: c. ; this is easy because the matrix is triangular.
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