Math 152 - Linear Systems Test #1, Version B (MWF sections) Spring, 2010 University Of British Columbia Name:
s~ .
----------------------------------------------------
lD Number:
-------------------------------------------------
Instructions
• You should have six pages including this cover.
• There are 2 parts to the test:
•
•
•
•
•
•
•
•
o Part A has 10 short questions worth 1 mark each
o Part B has 3 long questions worth 5 marks each
Although all questions in each part are worth the same, some may be much
more difficult - do the easy questions first!
Use this booklet to answer questions.
Return this exam with your answers.
Please show your work. Correct intermediate steps may earn credit.
No calculators are permitted on the test.
No notes are permitted on the test.
Maximum score= 25 Marks (attempt all questions)
Maximum Time= 50 minutes.
GOOD LUCK!
Part A
Total
B1
B2
B3
Total
10
5
5
5
25
Math 152 Test #1 version B
Part A - Short Answer Questions, 1 mark each For Questions A1-A4 below, let . AI: Compute 3a - b
(2, 2 , 7)
b
(1, -2, -3)
(CD , ~?-, '2')
A2: Compute
Ilbll 1
a
- l \ I -1-J - ~ )
t
2.4-)
l r; I ~)
~ ~(_~)Z ~(-32)' =~
:::
A3: Compute a· b
'L - \ - '1 \
A4: Compute a x b
j ~ I~
l
- 2
t
-3
:::
- '1-0 =
13 _ l>t)
(':5
l
?
J
J
:1­
A5: Winry and Edwar work at the same restaurant and just have received their new work schedules. Neither will tell you the number of hours they will work each week. Instead, Winry tells you that she will work the same number of hours this week as next week, but this week she works 3 hours fewer than Edward. Then Edward explains to you that he works the same number of hours both weeks, but over the same two-week period , he will work a total of 6 hours more than Winry. Based on these statements from Winry and Edward, would you be able to determine how many hours each of them works each of those two weeks, and why? Circle the correct reasoning below: (a) Yes , because the statements represent a system of two eq uations
and two unknowns and th erefore has a solution.
(b) Yes, because the statements represent a system of equations that
may be red uced to only one solution.
~
No, because the statements represent contradictory equations.
~
No , because the statements represent a system that contains too
many variables and not enough useful equations .
2
Math 152 Test #1 version B
A6: Consider the following lines of MATLAB code:
A
[1 2; 3 4; 5 6; 7 8];
=
Circle the correct description of the kind of object assigned to A.
~
A matrix with two rows and four columns. b A matrix with 2 columns and four rows. e A column vector with 8 components. (d) None of the above.
A 7: Consider the following lines of .MATLAB code that modify a previously
defined 4 x 4 matrix A
for i=1:2 A(i,l)
A(i,1)+A(4,4); end Circle the case below that correctly describes the action of this code: m Theis unchanged.
lower right entry of
®
A
A is added to the entries of the first column of rows 1 and 2. (c) The lower right entry of A is added to the entries of the first row of columns 1 and 2. (d) An error results.
A8: Find the value or values of 'ill (if any) such that the vectors (1, -2 ,3)
and (1, 'ill, -2) are orthogonal.
I
- l vJ - l ~ 0
-":) W = - filL­
A9: The solutions of a system of four linear equations with four unknowns
~ ie on a line. What are the possible valu s of the ra~k of the a\l gr~e nt~d matrix of the system? Justify briefly· {.. ~~
CJ\M..I:\.crv...
~3
06
~~S
L+ - 3
~.t
~~~.
AID: A system of three linear equations in three unk~wns is wnt~~'ii~
augmented matrix [Ale] where A is the 3 x 3 coefficient matrix and cis
the column of right hand sides. It is known that the row vectors of A
all lie on the same plane going through the origin and that the system
has a solution. Write down a possible reduced row echelon form of the
augmented matrix of the system.
i-J
-t
'(
T (-t
o 0
~
3
~~
.
Math 152 Test #1 version B
Part B - Long Answer Questions, 5 marks each
Bl: Consider the linear system for the unknowns x, y and z:
x - y - 2z
x + 5y
3x - y
+ 2z
+ 2z
-5
=
11
=
-3
(a) [1 mark] Write the system in an augmented matrix.
(b) [3] Do row operations (Gaussian elimination) on the augmented
matrix to change it to upper triangular form.
(c) [1] Find the solution to the problem using backward substitution
from the form above. Hint: this system has a unique solution and
all components of the solution are integers.
- \ - 2
(b)
<'v
'?>
-\
\
-
,
-Li ~
\
4­ I Jb
U)-{!)
~ 1 12
2
0
(-?>
L
~
0
(q
2
is
I
-~
(~)
tV
0
0
- 3(()
-1 1-S­
- I
"
0
4 \ Ib
20
3.
1
t
~
3
~- ~
g
CL) X3
~i 2
~
1.
+l.f
:=
"X l ­ 2 - 2.
I~
=.
~
-5
'" 2 -=~
2
4
XI
::
-t .
(3)-~a)
l~- ~
3
Math 152 Test #1 version B
B2: Consider a plane P given in parametric form below:
x = (2,6,8)
+ 8(1,0, -4) + t(3, -2 ,2)
(a) [2 marks] What is the normal (perpendicular) direction to P?
(b) [2] Write an equation form of the plane.
(c) [1] Write a parametric form of a line perpendicular to this plane
that passes through the origin.
lfA.)
1lN. ~ olv1'rt-J\~ V'
Ip () t1A <>tAW c.h 'uvJ '\/1 ~
1\
J
n ­
=
o
-1
(b) Mvs~ ~vv
- ~)(I
r­
\4-,,1.
- Ira - l t.t t ~ )
lLI
~f~) ~o
-2'1.-; = - llto·
_ ~J
\b
(c)
5
Nlath 152 Test
# 1 version
B
B3: Consider the linear systems below, each of which involves a parameter
k. Determine the value (or values , if any) of k so that the systems have the indicated number of solutions. Justify briefly. It is not necessary to find the solutions when they exist . ow
(a) [2 marks] For what k does
x + ky
kx +y
K d-tt
o
C; ~'&-v\. VlAless
(~
I~) ~ 0
2
have exactly one solution?
(b) [2] For what k does
x
x+y+z
4
+ 2y + k z
2
have an infinite number of solutions?
(c) [1] For what k does
x+y
( 1.) x - Y
l~) x + 2y
(I)
0
0
k
have any solutions?
~
S0
(
e,)
3)
1-(1..)
CBvv\..
.:/
ovJcJ
-
~=O
6
'A:::. 0
I
'd ~() .
~~(~dVl w~