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Unformatted text preview: MAT 1302 A Mathematical Methods II Final Exam: Solutions and Marking Scheme April 23, 2003 Q.1. [8 points] (a) [1 pt] Give an example of a matrix in RREF (reduced row echelon form). Marking: 1 point for any correct answer (1s as pivots, echelon, zeros above, below and to the left of any pivot.) (b) [1 pt] Compute 1 1 + 3 4 . Answer: 4 3 Marking: 1 point correct answer (c) [3 pts] Give 3 elements in span 1 1 , 1 1 Marking: 1 point each correct vector; if you gave more than 3, only the first three are checked. (d) [1 pt] Given z = 3 + 4 i compute  z  . Answer: 3 2 + 4 2 = 25 = 5. Marking: 1 point correct answer, deduct .5 for answer 5. (e) [2 pts] Give the roots of the polynomial x 2 4 x + 8. Answer: The quadratic formula gives x = 4 p 16 4(8) 2 = 4 4 i 2 = 2 2 i as the two roots. Marking: 1 point for correct quadratic formula, 1 point for correct simplification; or 2 point for correct answer by any means. Q.2. [6 points] This question has no partial credit. 1 2 The following is the augmented matrix of a system of linear equations, where k a parameter: 1 2 4  2 k + 1 4  2 k 1   1 (a) [2 pts] For which of the following values of k is the system inconsistent: 1 , , 1 , no values Answer: k = 1 (because last row is 0=1) (b) [2 pts] For which of the following values of k does the system have infinitely many solutions: 1 , , 1 , no values Answer: k = 1 (because then 2nd and 3rd rows are multiples, the system is consis tent, and there is a free variable) (c) [2 pts] For which of the following values of k does the system have a unique solution: 1 , , 1 , no values Answer: k = 0 (lots of other values are ok, too; it gives 3 pivots in the coefficient matrix) Marking: No partial credit; 2 points each for correct answer. (Circling more than one answer is incorrect = 0 points.) Q.3. [6 points] (a) [1 pt] Define span { ~v 1 ,~v 2 } . Answer: span { ~v 1 ,~v 2 } = { c 1 ~v 1 + c 2 ~v 2  c 1 , c 2 are scalars } Marking: 1 point for correct definition; .5 for something mostly correct. (b) [5 pts] For which value(s) of the parameter k is the vector 1 k k in span 1 2 3 , 3 2 1 ? Answer: We have to row reduce the following augmented matrix and determine the values of k for which the system is consistent: 1 3  1 2 2  k 3 1  k ' 2 R 1 + R 2 3 R 1 + R 3 1 3  1 4  k 2 8  k 3 ' 2 R 2 + R 3 1 3  1 4  k 2   2(...
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This note was uploaded on 04/12/2010 for the course MATH MAT 1302 taught by Professor Aziz during the Spring '03 term at University of Ottawa.
 Spring '03
 AZIZ
 Math

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