McMaster University Math 1A03/1ZA3 Fall 2012
Midterm 1
October 4 2012
Duration: 90 minutes
PRACTICE VERSION
Instructors: R. Conlon, A. Contreras, D. Haskell, E. Harper, C.McLean
Name:
Student ID Numbe
Calculus for Business, Humanities, and the Social Sciences
MATH 103

Winter 2016
MATH 1Z03/1 3:
Instructors: Hi 11 ' ,
Date: February 25, 2014  Group A @230
Duration: 90 min. erar’t tin/10
Name: ID #:
Instructions:
This test paper contains 23 multiple choice questions printed on
Calculus for Business, Humanities, and the Social Sciences
MATH 103

Winter 2016
1. Find examples such that
(i) If u v = u w, then v = w.
(ii) If u v = u w, then v = w.
(iii) If A is invertible, then A2 + 2A + I is not invertible.
2. Find examples such that
(i) A is invertible and
Calculus for Business, Humanities, and the Social Sciences
MATH 103

Winter 2016
Equations of
Same
apply
Line~Y

*
line
equation
Ve#m
to
*

equation

y
/
tv
=*ot
Parametric
1123
=
=
*
tat
'
L
yotbt
zotct
z=
f) equation
Ky
90
5ec 3.4 Ex 4
1
_=
Given
=
YT
point
=
Xo
34
(
]
2
equa
Calculus for Business, Humanities, and the Social Sciences
MATH 103

Winter 2016
( Sec
3.4 )
A point
A
and
Xo
direction
W
line
Eqnationsoinesvectorefnatit
t
*o
*
N
+
=
ow
t
,
*
parameter

9
direction
position
Parametriceqnatiof
H
*o=
;]
determine
W=[
=/
a
a
we
then
write
5)
*
(5
Calculus for Business, Humanities, and the Social Sciences
MATH 103

Winter 2016
(
point
OTP
M *
Kot
at
line )
Given
and
line
L
:
*
t N
yo /
Distancing
QIIMM
Disttp
Hmm
HMIP
.
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P
M
=
be
yo
a
a
point
and
a
=
,
Q=*o
direction
orthogonal
P
to
W
/
.4=HPwin#l1/=/0YMm=
The
orthogona
Calculus for Business, Humanities, and the Social Sciences
MATH 103

Winter 2016
Vectorspacy
( Rough Idea ) A
of objects
collection
called
vectors
YEW
with
two
operations
( )
if
(2)
zero
1
(3)
if
( Axiom
1
*
#
Rt
cfw_D
,
4
.
RZ
is
E
number
a
6
.
of
1123
,
,
also
W
be
vector
is
C
s
Calculus for Business, Humanities, and the Social Sciences
MATH 103

Winter 2016
Secltzsubspay
then
18
4.8
Given
of
set
a
span cfw_ W
,
,
.
.
.
,Wk
=
is
ed
a
cfw_
=
cfw_
C
,
N
subspace of Rn
1123
span cfw_
=
x
.
y
.
axis
axis
3. axis
cfw_W
vectors
,
=
=
,
vectors
+
"
it
.
.
.
,W
Calculus for Business, Humanities, and the Social Sciences
MATH 103

Winter 2016
numbers
Aux
.EE
d Sec 3.1
We
say
Def
.
that
C
,
4
Wv
,
.
.
,
.
,
P 139
.
is
CK
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)
of
a
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=
because
@
(cfw_]
W
N
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,
.
.
.
,
Wk
if
we
can
nd
and
K
such
that
N
0
Wi
(]
is
a
a
=
is
a
=
linear
in
C
,
.
i
Calculus for Business, Humanities, and the Social Sciences
MATH 103

Winter 2016
McMaster University Math1ZC3/1B03 Winter 2014
Page 10 of 16
17. For 3x3 matrix A, where
3
0 0
A = 6 6 k
0 3 3
one of the eigenvalues is 0. What is k?
(a) 5
(b) 3
(c) 6
(d) 0
(e) 2
18. For 3x3 matri
Math 1ZA3
lst Sample Test for Suggested Problems #4
Name:
(Last Name) (First Name)
Student Number: Tutorial Number:
This test consists of 26 multiple choice questions worth 1 mark each (no part marks)
Calculus for Business, Humanities, and the Social Sciences
MATH 103

Winter 2016
Dene
for
only R3
IUXW
wr
IYI
Crossres
for
This
a
vectors
dened
is
=
( cfw_)
and
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fdg /
iuxixoatfai
d
=
( bf

e
ce
) ii

( at

cd
)jj
+
(
ae

bd
) 1k
=/ !tdIae
l
Notice that
&
Check
is
P
.
174
.
Calculus for Business, Humanities, and the Social Sciences
MATH 103

Winter 2016
#
Etclideanvectorspacespreviously
.
R
"
numbers
think
we
of
it
vector
a
in
N
as
column
a
like
of
j
Lets
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@
n
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=
book
a
W
=
of
think
(b ]
2
uses
,
them
W
the
because
(
=
as
'
5]
notation
we
geometr
Calculus for Business, Humanities, and the Social Sciences
MATH 103

Winter 2016
Secltkdynamicalsysay
(
A
probability
theory
in
long
the
)
mm
,lity
Mid
vector
called
is
if
,
all
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( a
i
C 1
and
a
,
+
.
"
+
an
=
1
stochasticmatnx/Matkovmatnf
A
matrix
if
all
its
A
is
columns
called
Calculus for Business, Humanities, and the Social Sciences
MATH 103

Winter 2016
Oa
@
Compntationofdeterminiants
!1!
!2!
Cofactor
By
Topics
Sect
for
Nott
last
covered
product of elementary
)
class
matrices
today
Det
if
(
expansion
of elementary
B
EA
=
Properties
Adjoint
matrix
and
Calculus for Business, Humanities, and the Social Sciences
MATH 103

Winter 2016
Remember
(
l
)
(2)
(3)
@
@
(A)
det
A
is
det (A)
A
of
dat
for
=
9
( bjf
ae
=
"
.
=
if
only
and
the
det
A )
(A)
det
,
denominator
then
,
"
ad
it
.
A
then
dhc
+
g
bf
'
Fai

a
hf

.
'
of
form
=/
and A
Calculus for Business, Humanities, and the Social Sciences
MATH 103

Winter 2016
NEW
iplicatoy
Given (
)
mxn
matrix
A
and
(
nxr
)
matrix
B
me
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r
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Calculus for Business, Humanities, and the Social Sciences
MATH 103

Winter 2016
Invetybematricey
5
9f
A
is
El
matrices
then
,
.
.
.
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matrix
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square
a
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elementary
.
,
A
invertible
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say
we
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inve
Calculus for Business, Humanities, and the Social Sciences
MATH 103

Winter 2016
statements
The following
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A
invertible
is
The
(2)
(3)
A
RREF
is
(4) A
*
A
'
=
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trivial
I
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matrices
f
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identity
.
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Calculus for Business, Humanities, and the Social Sciences
MATH 103

Winter 2016
Ex
Gussian/Ganss.JandanEliminaio
a
Sec 1.2
5
li
leading
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!z
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x
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z
ktEtHkdENkecH
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solved
:
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uuiquesohriy
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