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Unformatted text preview: Eagineering Mathematics (E35 31?) Exam 2 October i0, 200?
This exam ceramics seven multiplenchcﬁce emblems worth two points each, eight {me—fake
preblems worth cne paint each, and four free—response problems worth 18 points altogether, for
an exam iota! of 40 points. Part I. Multipie~€hoice Cleariy circle the only correct response. Each is worth two points. 3 4 5 6
0 8 0 0 .
6 8 New
#0M 3‘ Find the nuility of the matrix
16 12 (A) G of}; I 4: $)1 ((3)2 Wf [
(mg é”f:§ (E) 4 NF) 5 (G) 6 2. Find the eigenvalues of the matrix [—1 1] . 5 3
What is the {arger eigenvalue? I” “’3‘ g _  
1:“ v d— 3”) “.5” é, a”) {’3 i y 3
(A) m2 1
(3)“; 2 wzawwst’w
(C) 0 a a
’2 “Q:&
0:» 1 9‘
(E) 2 {2w Lf)é/2;+2c>:€f§ 3
(£3? 1% 3‘ 3“ a “"2"
(H) 5
(I) The eigenvaiues are repeated, so there is ac “Eager” eigenvaiue. (I) There are me reai eigenvalues. 1
0
3. The water 3 is an eigenvector fer the matrix heiuw. To What eigenvalue does it correspond?
D
1 5 9 M13 3 4 s 14 z
—6 1a '5’ 25 m; a o <3
8 2 W1 1: ~11 % J ‘1 vi? :: wéw‘ S
1 5 W10 wé 9 if? C} a
15 —8 W12 2 —7 e “Hi i; ( (A) m4 E (B) “*3
(C) W2
(D) “1
(E) 0
(F) 1
(G) 2
{H} 3
(I) 4 4. If the vector “if has magnitude 2 and the vector "x? has magnitude 3, anti if 1? and V point in
Oppesite directions, than what are the values of ‘3’? and £33 x V] respectively? (A) “1:1 , . B) 1,—1 $531: 33””?éw5‘f:(z}c/3)é’wﬁ)rwé
(C)m1,0 (mm—1 gfxﬁj:{ﬁffﬁfgmﬁ’xKﬂﬁﬂﬂﬂﬂ
(5)1,0 (F) (Li ((3) 0,0 (H) was {I} sﬁa (K) {3,43 {L} 8,0 {M}§3§ 5. Find the work done by the farce 3 2 £521,513 stating on an objecﬁg mew it fmm the point
A m (£3, 1 2) to the poiat B = {5, 3, G) 31mg the straight Hm: segment AB. x [g 3) “2]
(B) 49
(C)
(D)
(E) (F)
(G)
(H) w “33% “31% gm
H
J;
C:
of“
‘5
90
H
w
DJ M uh
cameo
CD .a.
WK 6. Find the area ofthe triangle determined by the points (m2, 1, 4), (—1, 3, 4), and (1, —1,0)‘ M3) '“ 7 23 *1 3;”2JLU (
f ,3; “g? 3 E J
Eﬁﬁxéi '3’ Z. {3 7. Let M33) 2 [coshi, 2 sinht, 8‘}. Find the tangentiai component 0f acceleratiea at the point
coaesponding totmﬁ. Hint: ceshO: 1 and amaze.
(A) {550$}
w“?  ‘ f ‘  . ~é
(B) §—§,0,%g «6%) '11 L§EVJ2%} Zw$k£)& 5} 5311:} w QSSAZIS} 25;;«;}1%} if]
(D) «A; may «.39 gm 6 'V’ (F) [mwzlieﬂél “£7 w a (G) [Mk] 7 .. ,  r 222; ~
(H) [g 121‘ 605575 35%}2‘5 + #5;‘Vik£m5!zf +6 “2?
1"” :EJ ;: W
"2 _ 2%
(1) {mg} 5msz +4m5£ *2 +63
“,3? a 4,. (:3 4—» § ~01“? ‘
a M
52% {£33 “" U V g} +— 14 “‘L i
i an n
i “E 0) 2) f
' 2 ,L
’— [£2 5”} 5” Part II. True—False
Write Gut the werd “true” or “faise” for each ef the foﬂowing. Each is worth we point. 8. If a matrix has mere colunms than rows, then the dimension of its coiumn space is greater
than the dimensien of 518 row space. a} cj “‘J'LL fifkim (Lg/2.12;ij 9. Suppese the 4X5 masses A is the coefﬁcient matrix of a homogeneous system of equations
Ax = 0. If she rank of A is 4, then the system has Only the tzivial solution. mm) M Me, 10. Suppose the 4x5 matrix A is the coefﬁcient matrix of a nonhomegeneeus system of
equations Ax = b. if the rank of A is 4, then the system must be censistent. _ N i
air/4 11, For ail vectors 3‘, b , and be}, (33}? = 3(EE'). ’ W J
I111 1 a.» M We”; as. yea/71m; 12. Foraiivectorsgand bfgx b = b X 3 E, ’ “711. :2me :9); x Z? s: “" (f w ’2? )
13. Let f be a scaiar function. The notation V2]? means grad(grad(f));
A; es; s, m, ‘71 4:
P I mm) 34. The gravitaticnai vector ﬁeid is conservative. afﬁx mWiafrg fef%‘émf % w s . W} : R
a ,m as” 55%? jimmf'ﬁs a
4% WE 15s The secter ﬁxectien 72%;", y: z} 2 Eye, 32, my; is a petential 0f the seaﬁar inaction ﬁres 2;, z) x mm. = W}?
{E éyw a“; 1/ 6 whee gem ‘ J J
r mtesemi Part III. ﬂee Response In each problem in this section, follow éirections careﬁﬁly. The paint vaiue for each problem is
Shawn is its left, (1) 16. Fiﬁ in the missing entries 36 that the feliewing is a Siochastic matrix. .4 .1 ‘3 t E
.2 a}: ,5 Lam wads} 55L We???
A .6 $2. (6) 17. Consider the following parametric equations.
x23+c036t ytwiwi—sinﬁt .222 (a) Eiiminate the parameter t and identify the type cf curve represented by these
equations. 3: 35.,331éﬁ 22.32 (25MB) +K§L+i3 It! 232.
w£{’£e Umﬁ [3) MIA}
M23154; 2
a“
(b) Carefuily and clearly graph this curve in three~dimensionai space. You may use the axes cirawn. beiow or you may draw your own on the back. Either way, mark
a scale on our axes. .,_,
y 19* "t I (6) 18. Find the iength afﬁne are traced out by the meter ﬁmctien ?(i) m [$12, gig52, 8%] from the point g, 8} to the point (8, ﬁg, 32). Show a}? the steps aeeded to arrive at the earnest answer. if‘é‘e); =3 {Ram +M r: Hwy)1 1‘: 29+?
3 '4 :‘£%[~£¥»§‘)5L¥ :(iézwt 3(9+32>“/'§i +83}
$3 :: 3L§ == m—n
2. (5) 19. Find an equation of the tangent piane to the surface z = grain as + 5 at the point
(1, 2? 5). Show alt your werk. ...
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