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Unformatted text preview: Engineering Mathematics (ESE 3 E?) Exam i February 10, 2010 This exam contains six multipie«choico problems worth two points each, seven truefalse problems worth
one point each, three short~answor problems worth one point each, and four ﬁeoresponse probiems
worth 13 points aitogether, for an exam Iota} of 40 points. Part I. Muiﬁgle~Choice (two points each) Cieariy ﬁii in the oval on your answer card which corresponds to the oniy correct response. 1. Which ofthe £0110“;ng methods could be used to ﬁnd o1 ((3231):, ) ? a) convoiution V/ (II) differentiation or integration {the “box” method) /
(HI) partiai ﬁactions (EV) tshiﬁing >< (A) (I) and (II) only (B) (I) and (In) oniy
(c) (I) and (IV) only (D) an and (HI) only (E) (11) and (m only (F) (111) and (IV) oniy
((o) (I), (11), and (111) only is (H) (I), (11), and (1V) oniy
(o (I), (III), and (IV) oniy (J) (11), (III), and (IV) onIy 2‘ Lot 16 be a constant. Find £(sin(t + 12)). k A ._
(A) 3244:? ﬁg“ ow Z 5fm£ 5,3311 4 (303%. Sirik
(B) (343)?“ K . o » a 7
(C) W oféiﬁ +1933 : Cﬁ‘ijé jK/ﬁﬁi 59’s,}! Zgéoszir)
(s+k)2+1 i g
03) 32:1 + (92:1. 7: “jig/(I ’ 4»? Vi. 55%;; '53 4— i
(E) 32:; k
(F) 5;“ 32 +1162 1 sink
(G) 32+1+ g (H) 3211+mssk
(I) oosic(8;}:1 “Sink(sg:1)
( 1 ( “ ) (J) cosk 32H +Siflk 32+} 5? q .3. {GD none ofthem E Find 1:0: mm 41:). ~2
(A) 32.46
*4}:
(B) sgwlﬁ
“8
(C) 5216
(B) “'43
32w16 (E) {32318}?
0‘) (G) “I Jigsaw iéYIKZs) (I) I (J) .33 (324“16) . Consider M22, the vectar space of all 2 x 2 matrices, and consider the feiiowing subset S of M22. 19 01 00 11
Sialmee’azzoeﬂ3z1o’a‘w1o Which of the foilowing statements islare true? (I) S spans M22. (11) S is linearly indepencient. {III} S is a basis for M22. _ \1 as”) «m W¢M \}{g) “14%. Wt“; m.
[is 2;} m4 in a; ammmﬁéw ’fiuaaw diasm 53‘: J ‘31)ﬁmi 5&3 '3 (A) (I) only (B) (13081:? (C) (HI) only (D) (I) and (11} only
(E) (I) and (111) only
(F) (II) 336 (HI) oniy 74a" i M9; (G) (3341:), 336611) _; Zia 3W“ {W194 S. The set V of all 2:12 matrices whose determinant is zero is go; a vector space. Which of the
foilowing vector space properties £3}; to be satisfied by V? (1) ciosure under aodition Qua”; “in. E J
(H) ciosure under $03131“ multiplication (III) existence of a zero i (A) (I) one (e) (11) only gigi Loewe:
(C) (HDooly .wé «a; _ E 0 ' :52 i 3
(D) (Dandolmniy g *5 we} ‘Z E aé
(E) (I)and(III)onIy
(F) (II) and (HI) oniy
(G) (I), (II), and (III) A system of iinear equations is. written in matrix form Ax : b, and the augmented matrix K is
then reduced to row echelon form, with the foilowing result: 1 1 2 4
0 1 3 2
O 0 1 5
0 O 0 O
:3
Find the solution of the original system. If it has a unique solution 31 , ﬁnd the value of a:
3 among choices (A) through (H). Otherwise, choose from choices (I) through (K). (113)”? ﬁ§w+ggg (B>~5 “W? W (mu—3 ieoe~2 3w5§32 3;:ng
@237” m (1))”1 X we were x“: f (5)1 (as (3)5 f (H) (I) The originai system has no solutions.
(I) The original system has infinitelyvmany soiutioos. (K) Since row operations do not always preserve soiutions, it is impossibie to determine the
solution of the originai system from the above rowureduced matﬂx. I’art II. Tme~Felse (one point each) Mark “A” on your answer card if the statement is true; mark “B” if it is false. 7. [VI(1:7? (3 )zt4sinst , l 3 ii i ,7 v V A) > : i 55y?— ‘ g J 5f
8. Ever diagonal matrix is nonsingular. ew% 6:6 ﬁve, W‘s 9. If the vector v3 is in the span of the set of vectors {vva}, then the set {V1,V2,V3} is lineme dependent. a  a g
7 V3 (14141. 22.4.. at; joy/mm l0. Any two matrices which are row equivalent have the same column space.
a ?_ gamut" (I?) "y? Wane, s For problems 11—13, you may safer assume that the column vectors 1: and 0 have whatever size is
appropriate for the problem. 11. If A is a 3 x 4 matrix with rank 3, then the system Ax = I) must be consistent.
, K k» , N
'77:}, MM” $12. 3 ié 5 .4
u_.. p' N u . ° '
.23 .3; M“ W; e"; W M¥.¥“&,wxym
12. HA is a 3X 7 matrix, then the system Ax = b cannot have a unique solution.
+0 We so weujux; " féﬂhlms evict fM{Axi' the 13. The set of solutions of a homogeneous linear system Ax m 0 is always a vector space, but the set of
solutions of a nonhomogeoeoos linear system Ax m b is never a vector space. we Part EH. Short Answer (one point each) The answer to each of these is right or wrong: no work is required, and no partiai credit win be given. 14. Given that £(sintsinht) m 33:4, ﬁnd qeaismtsinht). um» ? {’5 g4 a; .4 51;? :, th’gt £173; 5§WE> 4"]? 1 :2 3
15. LetA m What IS the cofactor of the entry (123 = 4‘7
L6 5 E
i Z—«é X x w
5"”f'5} IL; 51" {wf;(/*7} ' 7
~ 4 1 . _E 16 LetA—w[3 2] FandA Part IV. Free Resgonse (point values as shown) Follow directions carefully, and Show 5111 the steps neeiied to arm've at your solution. (4} 17. Find the Lapiaoe transform of the periodic function pictured beiow. You do not need. to simpiify your ﬁnal answer. In other words, do a1} the calculus, but you do not need to do aigebraic
Simpkiﬁcations at the end. 2 Z I! t
: g 63$} V5: ifﬁmnij t1 3 q
:1 ﬁzgz s J!
gen: 3 {7) 18. Soive the foliowing integral} equation for 3105). y(2) ~ 12]: pr) cos 18(1? w» 39)::130 m sin 10: 35923 “ 125] 515') 33'“ cm; ié‘ﬁ—f: =1 §4=m./&é 3’55) « avg» ~ 5 = m—w—i‘im— "~ + ma :5”; 4» ma
2 ' . 12,5 ‘ N m
emf; w J , ‘
:3; +}&& :3 ’9‘» 1635’}
w}; > 5‘2 “32:5 “’“i‘é’cg _ i6”
2 s '2 w ' :1 _
3 +iéa 5 L160:
.  M3
C1355) I  * . as
:5, «125 P5
[‘0
(5%»): ‘F €54:
WW5? ’ 3 (3) (4) 20. emples of polynenﬁals which are in V: exampies (3f poiynomiais which are not in V: 43:2 4:84 19‘ Consider the set V of polynomials 0f degree at most three Whase constant term is zero. 933 + £332 w 2: 9x3+4=x3—$+3 V is a vector space (a subspace 9f P3). (Yam do not need to verify this.) (3:) Find a basis. for V. f (b) What is the dimensian of V? 1 2 m1 0 —2
3 8 m3 4 _5
W A m 0 2 0 a; 1
m1 2 1 8 4
(21) Find a basis for the row space ofA.
' i '2 «4 Ch ’1
:3 2 w 3 "i " g
as; 2 if} ‘4? ‘
7; 3, i ‘8’ ‘f
i ‘2.
32¢} 44» {M13121 C5 '3
(:5 C? 1‘3, [ya/ma: [i 2 *i (b) What is the dimension of the column space QfA‘? (c) What is the dimemien ofthe nun Space ofA? 1" 3 . .3 “ "z
“.31.: Q_x: + «5— (17:: Z ,
) ’3’. , a; '2. '5 G *2
az+Z~ﬁ3>gi C; I G q i
Radfﬁﬁi O 2 a a} I a Li C? E 1 C3 *2
4,} i
(3 O
(3 {:3 3 ( 5m 0EW§W crj gagmﬂ)
.xwﬁﬁ Swwz Z 3 Tabie 0f Laplace Transforms £(f(t)) m F(s) (t) 821%?) ~ 8f(0) — NU)
w m a)
M _ «Wm . 6“““”£(f(t+a))
503%)
N) (with period 2» him gamma f(t)*g(i) F(8} * GT8) 5) : gm) _ S3
V E
a.
I...ﬂ
A
M
W ...
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This note was uploaded on 01/10/2011 for the course ESE 317 taught by Professor Hastings during the Spring '08 term at Washington University in St. Louis.
 Spring '08
 Hastings

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