Fall 07 Exam 2 Solutions - Eagineering Mathematics (E35...

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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 W-f [ (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 1-4 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}, (3-3}? = 3-(E-E'). ’ 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 sect-er fixectien 72%;", y: z} 2 Eye, 32, my; is a petenti-al 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: 3-5.,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, gig-52, 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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Fall 07 Exam 2 Solutions - Eagineering Mathematics (E35...

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