exam 1 spring 2010 solutions

Exam 1 spring 2010 - Engineering Mathematics(ESE 3 E Exam i This exam contains six multipie-«choico problems worth two points each seven

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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 true-false 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) C-ieariy 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) t-shifiing >< (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) 52-16 (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 gig-i 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 dime-mien 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 Rad-ffifii 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.

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Exam 1 spring 2010 - Engineering Mathematics(ESE 3 E Exam i This exam contains six multipie-«choico problems worth two points each seven

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