of Degartment Mechanical En2ineerin2
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MICmGAN
ST ATE UNIVERSITY
ME 391 Exam 1
A dominator's delight!
Fall 2002
Instructions 1. Therearefive problems. Allocateyour time and effort to maximize your points. 2. Layout your work clearly and showyour reasoning. 3. This is a closed-book, openmind, examination.One3*5 card allowed. but
Problem 1. What kind of equation am I?
[5 @ 4= 20 pts]
Classify the following equations with respectto their important attributes.Parameters known are quantities. 2x+7y=14 -x+3xy = 0
Unknowns: X, Y
m*d2xl dt2 +b*dxl dt+k* x=f(t)
Unknowns:x(t) Parameters: h, k m,
a2 *(OT/ax)2
=-7.S*(O2T/c3t2)
Unknowns:T(x,t) Parameters: a
x*dy/ dx =3sin(2x)
Unknowns:y(x)
(e) Give an example of a system of two nonlinear ordinary differential equationsin two unknowns.
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,
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Problem 2. Complex numbers are really not complex. (a)
Let z = a + ib = rei8,
[4@5 each = 20 pts]
where = J:1: Show how (a,b) and (r,B) are related.Which i
form is Cartesian?
(b)
Express 3+i4 )*( 2-i
in the fonn: z= a + ib.
(c)
Express (3+i4)/( 2-i) in the form: z= a + ib.
(d)
Sketch
the
quantity
.
z
=
3 . e
,(-38/2)
. m
the
complex
plane.
Show
. It
. m
(a
+
. Ib)
fonn
. m
the
complex plane, clearlabeling. by
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Page2
September 2002 of 29,
Degartment Mechanical En~ineerin2
Where best the never ... rest
MICmGAN
ST ATE UNIVERSITY
Problem 3. Mashing monster matI"ices(not).
[4@5 each=20 pts]
[ 1 2 3
Find A *B, where A =
2
-1
01,
and B-
~
3J
.
5 8
:1
Find B suchthat A *B = B* A, where A=
1
-2
~J
Find (A *B ), where
't
] A=[~ 3
2
and
B=
-1 ()
3
.2 )
'2
(d)
Find Det A =
IA I, where
A=
-1
0 3
0 3 1 0
0 2 4 2
4 S
-2
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A
N~T
-.J.
A ~.
'T'1j'
A'-'
TOOTV1i'D~rrv -.-,.. . A'-'."-,,a
a
.
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Problem4. Love thoselinear algebraequations.
(a) Find x, where Ax = b, with A ~(~
[6+7+7 =20 pts]
6]
[1 3andb=2 ]
2
(b) Find
X3
by Gaussreduction,where Ax = b, and ;.4=1 1
3 0...