Midterm 2019 A PDF

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Question: 1 2 3 4 5 6 7 8 TotalMarks: 8 7 6 7 6 5 3 3 45Score:Name (print):Signature:UWO ID number:Circle the numbers of your section and lab section in the tables below:001 MWF 12:30 Matthias Franz 002 MWF 11:30 Chris Hall003 Thu 1:30 Ashraf+Boudreau 007 Wed 4:30 Herring+Velivasakis 004 Thu 12:30 B...

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Name (print): Signature: UWO ID number: Circle the numbers of your section and lab section in the tables below: 001 MWF 12:30 Matthias Franz 002 MWF 11:30 Chris Hall 003 004 005 006

Thu Thu Thu Thu

1:30 Ashraf+Boudreau 007 Wed 4:30 Herring+Velivasakis 12:30 Boudreau+Velivasakis 008 Wed 1:30 Ashraf+Vergura 2:30 Cizek+Herring 009 Wed 5:30 Cizek+Valluri 10:30 Valluri+Vergura

THE UNIVERSITY OF WESTERN ONTARIO DEPARTMENT OF MATHEMATICS MATHEMATICS 1600B MIDTERM EXAMINATION 7 February 2019 7:00–8:30 PM INSTRUCTIONS: 1. Check that this exam is 4 pages long and report immediately if there are any pages missing. The exam is printed double-sided; there are 8 questions. 2. All questions must be answered in the space provided. Indicate your answer clearly. Should you need extra space, a blank page is provided at the end of the booklet.

Mathematics 1600B

Midterm exam 1

7 February 2019

3. Show all your of your work and explain your answers fully. Unjustified, irrelevant or illegible answers will receive little or no credit. 4. Do not unstaple the exam booklet. 5. No aids are permitted. In particular, calculators, cell phones, ipods etc. are not allowed and may be confiscated. 6. If not stated otherwise, all vectors and equations involve real numbers. 7. In your final answers you must give all numbers in Zn as a number between 0 and n − 1.

Page 2 of 4

Mathematics 1600B

Midterm exam 1

7 February 2019

1. For each of the following statements, circle T if the statement is always true and F if it can be false. Give a one-sentence justification for your answer. (a) (2 marks) If two lines are parallel, then they do not intersect. T

F

(b) (2 marks) If a homogeneous linear system over R has a non-zero solution, then it has infinitely many solutions. T F

(c) (2 marks) The rank of a matrix is an integer > 0. T

F

(d) (2 marks) If u, v ∈ (Z10 )11 represent valid UPC codes, then so does u − v. T

F

Page 3 of 4

Mathematics 1600B

Midterm exam 1

7 February 2019

√ √ 2. Let v = [ 2, 0, −1] and w = [0, 2, 3]. (a) (2 marks) Find a non-zero vector x which is orthogonal to v and w.

(b) (3 marks) Write w as a sum w = y + z of a vector y parallel to v and a vector z perpendicular to v.

(c) (2 marks) Find a unit vector u pointing in the opposite direction as v + w.

Page 4 of 4

Mathematics 1600B

Midterm exam 1

7 February 2019

3. Consider the point P = (1, −1) and vector v = [1, 3] in the Cartesian plane R2 . (a) (2 marks) Give an equation for the line ℓ1 through P and parallel to v.

(b) (2 marks) Give an equation for the line ℓ2 through P and perpendicular to ℓ1 .

(c) (2 marks) Determine all points in the intersection ℓ1 ∩ ℓ2 of the lines. Explain.

Page 5 of 4

Mathematics 1600B

Midterm exam 1

7 February 2019

4. Let P be the plane in R3 given by the parametric equations x = 1 + s y = 2

+ t

z = 3 − s − t (a) (2 marks) Find a normal vector n to the plane P .

(b) (3 marks) Compute the distance from P to the origin.

(c) (2 marks) Find a general equation for the plane Q through the origin and parallel to P .

Page 6 of 4

Mathematics 1600B

Midterm exam 1

7 February 2019

5. Consider the following 3 × 2 system of linear equations over R: 2x + y = 1 −3x + y = 1 4x − y = k

Here k is a real constant. (a) (1 mark) Write down the augmented matrix of this linear system.

(b) (3 marks) Compute the reduced row-echelon form of the augmented matrix and indicate all elementary row operations that you are performing.

(c) (2 marks) For which values of k does the system have a solution? Explain.

Page 7 of 4

Mathematics 1600B

Midterm exam 1

7 February 2019

6. Consider the following system of linear equations over Z5 : 2x +

y + 2z + 3y +

w = 3

z +

w = 3 4w = 2

(Read again the last instruction on the front page!) (a) (2 marks) Show that [x, y, z, w] = [0, 1, 2, 3] is a solution.

(b) (3 marks) Find all solutions.

Page 8 of 4

Mathematics 1600B

Midterm exam 1

7 February 2019

7. (3 marks) Do the following points line in a common plane? Explain. P = (1, 1, 0), Q = (1, 0, 1), R = (0, 1, 1), T = (2, 2, 2).

8. (3 marks) Why are there no valid ISBN-10 codes with exactly one non-zero digit?

Page 9 of 4

Mathematics 1600B

Midterm exam 1

7 February 2019

Use this page if you need extra space for your work.

Page 10 of 4

Mathematics 1600B

Midterm exam 1

7 February 2019

Use this page if you need extra space for your work.

Did you write your name and student ID on the first page? Did you give full explanations and show all of your work? Page 11 of 4...