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 Number:
This test paper is printed on both sides of the page
Calculus for Business, Humanities, and the Social Sciences
MATH 103

Winter 2016
Trots
Lineartlgebry
metals
Recycling
Examples
bicycle
q
*
I
340
f
'
the
wheels
:
Let
b
be
U
t
Zt
pedals
number of
the

.
.
.


.
seats
Zbt
:
b
:
pedals
at
Zb
:
handlebars
340
440
b
( l)
220
=
+
:
We only need
u+2t
+
=
,
(2)
+2T
,
(3)
,
=
220
don't need
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
+
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ow
t
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9
direction
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;]
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we
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5)
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(5
,
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/
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=
+
yo
be
t
coo
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
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m
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Q
Pwjiw (
)
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mm
H
L
A
(
P
T
Calculus for Business, Humanities, and the Social Sciences
MATH 103

Winter 2016
Vectorspacy
( Rough Idea ) A
of objects
collection
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vectors
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with
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1
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space
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t
Calculus for Business, Humanities, and the Social Sciences
MATH 103

Winter 2016
Secltzsubspay
then
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4.8
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,
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R
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"
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where
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,
,
Calculus for Business, Humanities, and the Social Sciences
MATH 103

Winter 2016
numbers
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.EE
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.
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linear
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Calculus for Business, Humanities, and the Social Sciences
MATH 103

Winter 2016
Dene
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sometimes
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we
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e
v
Calculus for Business, Humanities, and the Social Sciences
MATH 103

Winter 2016
#
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Calculus for Business, Humanities, and the Social Sciences
MATH 103

Winter 2016
Equations of
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,
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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 is diagonalizable.
(ii) A is invertible, but is not di
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 both sides of the page.
The questions are on pages 2 th
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 matrix A, where
4 2 6
A= 2 1 0
1 0 1
one of the eigenvalues
Calculus for Business, Humanities, and the Social Sciences
MATH 103

Winter 2016
Secltkdynamicalsysay
(
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Calculus for Business, Humanities, and the Social Sciences
MATH 103

Winter 2016
Now we consider the reverse process of taking power. Given a xed complex number
z = r(cos + i sin ), solve the following equation for w,
wn = z.
I explained in class that there are always n solutions for this equation, which are the nth
roots of z . They
Calculus for Business, Humanities, and the Social Sciences
MATH 103

Winter 2016
Oa
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matrix
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.
of
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det B
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Calculus for Business, Humanities, and the Social Sciences
MATH 103

Winter 2016
Remember
(
l
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@
(A)
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det (A)
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of
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Calculus for Business, Humanities, and the Social Sciences
MATH 103

Winter 2016
NEW
iplicatoy
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Calculus for Business, Humanities, and the Social Sciences
MATH 103

Winter 2016
Invetybematricey
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Calculus for Business, Humanities, and the Social Sciences
MATH 103

Winter 2016
statements
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Calculus for Business, Humanities, and the Social Sciences
MATH 103

Winter 2016
Ex
Gussian/Ganss.JandanEliminaio
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Calculus for Business, Humanities, and the Social Sciences
MATH 103

Winter 2016
Invetybematricey
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9f
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matrices
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.
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.
inverse
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of A
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Calculus for Business, Humanities, and the Social Sciences
MATH 103

Winter 2016
#
Matrix
Operations
when
1st
now
nth
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Calculus for Business, Humanities, and the Social Sciences
MATH 103

Winter 2016
\
Two
motivating
Suppose
by
equations
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ed
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Suppose
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