School

York UniversityDepartment

Mathematics and StatisticsCourse Code

MATH 1025Professor

AllStudy Guide

MidtermThis

**preview**shows pages 1-2. to view the full**6 pages of the document.**York University

Faculty of Arts, Faculty of Science

Math 1025

Class Test 1

SOLUTIONS

Instructions:

1. Time allowed: 50 minutes

2. There are 5 questions on 5 pages.

3. Answer all questions.

4. Your work must justify the answer you give.

5. No calculators or other aids permitted.

Question Points Marks

1 13

2 7

3 7

4 7

5 6

Total 40

Only pages 1-2 are available for preview. Some parts have been intentionally blurred.

MATH 1025 Test 1

Page 1

January 24, 2005

1. (a) (8 points) Find all solutions to

x1+ 3x2−2x3+ 5x4= 4

2x1+ 6x2−3x3+ 12x4= 9

3x1+ 9x2−7x3+ 13x4= 11 .

Answer: Reduce the augmented coeﬃcient matrix.

1 3 −2 5 4

2 6 −3 12 9

3 9 −7 13 11

1 3 −2 5 4

0 0 1 2 1

0 0 −1−2−1

1 3 −2 5 4

0 0 1 2 1

0 0 0 0 0

13096

00121

00000

From which

x1= 6 −9t−3s

x2=s

x3= 1 −2t

x4=t .

(b) (2 points) Is there a solution with x2= 1 and x4= 0? Find one or explain why not.

Answer: Yes. Set s= 1 and t= 0 to get x1= 3, x2= 1, x3= 1, x4= 0.

(c) (3 points) Is there a solution with x3= 0 and x4= 1? Find one or explain why not.

Answer: No. To obtain x4= 1 we need t= 1. To obtain x3= 0 we need t=1

2.

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