7DgDJGGGGWssxG '=*gMath 221Fall 1989
Students name: .
Test No. 1
Makeup
October 9, 1989
The test consists of two parts.
Part 1 (questions 1 - 7) is multiple choice. No partial credit will be given for wrong answers. Please mark your answers in the table below.
Part 2 (questions 8-11) consists of questions, for which partial credit will be given. Please write your solutions in such a way that the reader can follow your reasoning and check your computations. If the space under the question is insufficient, use the reverse of the preceding sheet or the blank sheets at the end of the booklet.
Answers
Question 1 a b c d e
Question 2 a b c d e
Question 3 a b c d e
Question 4 a b c d e
Question 5 a b c d e
Question 6 a b c d e
Question 7 a b c d e
Scores
Missed on multiple choice . . . . .............
question 8 . . . . . . . . .............
question 9 . . . . . . . . .............
question 10. . . . . . . . .............
question 11 . . . . . . . .............
Score. . . ............. Total . . . . . . . . . . . . .............
Part 1. Multiple choice (7 points each)
Given the matrix A = b(aco4hs9(1,1,3,2,3,3,1,1,2,2,1,1,1,1,2,1)) , find the following:
1. The determinant det A.
Answers: (a) -3, (b) -1, (c) 0, (d) 2, (e) 5.
2. The echelone form of the matrix A is
(a) b(aco4hs9(1,1,0,0,0,0,1,0,0,0,0,1,0,0,0,0)) , (b) b(aco4hs9(1,1,0,0,0,0,1,0,0,0,0,1,0,0,0,0)) , (c) b(aco4hs9(1,0,0,0,0,1,0,0,0,0,1,0,0,0,0,1)) ,
(d) b(aco4hs9(1,0,0,1,0,1,0,0,0,0,1,0,0,0,0,0)) , (e) b(aco4hs9(1,0,1,1,0,1,0,0,0,0,0,0,0,0,0,0)) .
3. Which of the following matrices is not an elementary matrix?
(a) b(aco3hs7(1,2,0,0,1,0,0,0,1)) , (b) b(aco3hs7(1,0,0,0,3,0,0,0,2)) , (c) b(aco3hs7(1,0,0,0,1,0,1,0,1)) , (d) b(aco3hs7(0,1,0,1,0,0,0,0,1)) .
4. Find the angle between the lines
f(x,2) = f(y,1) = f(z1,2) and x = f(y , 2) = f(z -1,-2) .
Answers: (a) 0, (b) /6, (c) /4, (d) /3, (e) /2.
5. The inverse matrix of the matrix b(aco2hs5(1,3,2,4)) equals
(a) f(1,2) b(aco2hs5(-4,2,3,-1)) , (b) f(1,2) b(aco2hs5(1,-2,-3,4)) , (c) f(1,2) b(aco2hs5(4,3,2,1)) , (d) f(1,2) b(aco2hs5(1,-3,-2,4)) , (e) b(aco2hs5(0,1,1,0)) .
6. The symmetric equation of the line through (1,0,2) perpendicular to the plane
2x y = 5
is
(a) x - 1 = f(y,2) = f(z - 2,-2) , (b) f(x - 1,2) = f(y,-1) , z = 2,
(c) x - 1 = y = z2, (d) f(x - 1,4) = f(y, -1) = z -2, (e) x = 1, z = 2.
7. The distance of the point (1, -3, 3) from the plane
2x - y + 2z = 5
equals
(a) 0, (b) 1, (c) r(3), (d) 2, (e) f(7,r(5)) ,
Part 2. Partial credit given
8. (6 points) Represent the matrix b(aco2hs7(2,1,3,4)) as a sum of a symmetric matrix and a skew-symmetric matrix.
9. (5 points) Write the parametric equations and the symmetric equation of the line which goes through the points (1,2,3) and (2,3,4).
Parametric: Symmetric:
10. (20 points) Determine whether the lines
f(x - 1,2) = f(y + 1,-1) = f(z - 2,3) and x = [1, 2, 1] + t [1,1,1].
intersect. If they do, find the intersection point.
11. ( 20 points) Find the equation of the line through the point
P (1, 1, 1), which intersects the line
x - 1 = y + 6 = z + 1
with a right angle.
Hint: Write the equation of the plane through P perpendicular to the line and find the intersection of the plane with the line.
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