# N1114 - NAME PID ii iii iv ME 391 FALL 2007 EXAM 2 WEDNESDAY LoLLFHOhLX‘ Note There are 4 questions with two parts each Use of calculators is not

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Unformatted text preview: NAME: PID ii. iii. iv. ME 391: FALL 2007 EXAM 2: WEDNESDAY NOV 14, 2007 LoLLFHOhLX‘ Note: There are 4 questions with two parts each. Use of calculators is not allowed. Only one page (8.5 ” X 11 ”, both sides) of notes is allowed. Please show your work in clear and logical steps in order to get partial credit. 1. (a). (10 points) Find the values of k for which the following matrix is singular k k k A: 3 2 k k 2 k A "b L42 9317va J MLA/r o g k. K K K 7— K C27C2_Cl LL 0 o ‘————> C}->(?—C[ 3 “I 16—} : o k 2~k a g) k [0 — Q40 (w; 1:0 IL (IL—'1) Uta?) >0 '9 LL: 0, 21/? Au 1 (b). (20 points) For k=1, find the inverse of A using any method of your choice. R\-7‘K("(R2 ' O F! ‘2 I o (LL—9 _{LL 0 I 2 3 ¥( 0 2. (a). (13 points) Determine all the eigenvalues of B =[ E—I‘WW/Q‘ASV M9 WW 5] “‘10 CFWMW; o lg”/l1:—l=0 =9 ’Z—A F2 .3 7- lvl Lr :o “l 2 (A ﬂ,~—>€,+K3 v———~—~> ~3”/l o ~3~A “L l—/\ L5. :0 “l 2 «A ég‘fl) | o i 2 lP/I L1 :0 -1 z -,\ C39C3"C‘ l 0 0 Z l”) 30 a1 (-3—rl):o —-l 2 \——,\ ‘9 Qf/UZ’Lr=O 2 /l=-,7 l—V‘Zil 2/13"3 —2 —2 —3 2 1 4 —1 2 2 (b). (12 points) Determine the eigenvector of B corresponding to A=3. To ﬁwWKe egg/cam «f E 28/53] w (New W2 WG-11 kw ) 0F Few (T331) EH? - 2"}? —’Z ——5 o z x»; ‘4- 0 «5 —Z --3 0 K. H (2} I "Z 3 o (ll—aﬂz—QR: l ~—r—> {9'9 (if-SK. o 2 -2 D o \7. 47- 0 ~6Z (124K? 1 ‘ '_'Z ((3 o 0 0 '2 o 0 o o o (Zrﬁﬁﬂﬂz \ o t 0 LL( 1:0 ((2-91;le 0 2) ' 4 ° M249, :0 0 0 o D M FOAM k} 14}: 1—9 k.:—1sz:l 3. (a) (10 points) Given a=xf+yj+zlé,13=f+2j+312, a=3§+41€, 34+} Fonn a system of 3 linear equations in the components of ii i.e. x,y,z, using the following facts: 5.}; =1, compaﬁ = 2, Jam = —13 n r\ A , / ——) n 52Xl4jj+‘ZKJEZL4ZJ+}QJC"3€+Lrﬁjc;-Fltdj gay—J =) X+ Zy+§z=l "‘7 F) A 61;? ~ . - Camp? —J a QC ’2. 5 EX-t-H‘Zzlc) F2) d‘WX'CY): l l o X : l o o 7 Z (ﬁg—ct x ny 3 0 Lf 7 3 ~; Lt 3. (b) (20 points) Solve the system formed in part (a) using Gaussian elimination. Thus, find 21'. “La—lgfl'l _ ___) i 2 3 [ a—\ ((3 *3 {23 O ‘ 5/6 3/5 o I 5% ’3/L. ((3—3 Ry—R? I {2— 3 I o 1 5/6 “7/6 0 o 5/12 5/17. l1 (5—? ER? ‘ 2 3 ‘ O \ 5/6 ‘7/6 0 0 I ' (ZCF EGCLLSt/Erhhvt? 2:! S - a 74'6Z_ Z ay__§_gzﬂz 4. (a) (13 points) The current in a single loop L—R circuit is given by L— + Ki 2 E (t) where L=l h, R=1 S2, E(t) is the periodic Saw-tooth function given by the following figure: If i(0)=0, solve the above differential equation for current i using Laplace Transforms. 4. (b) (2 points) Find the value of the current at t=4 3. Given: I{f’(r)}=sHs)—f(0), 1‘11}=—:-, I{r}=s—12 ﬁe‘s’ﬂt}: F (s—a) 1‘:{f(t—a)u(t—a)}= e‘“ F(s) T For a periodic function f( t) with period T, I{f(t)}= 1 1_ST Ie‘“ f (t)dt For e“, s>0 1 1%. =l+eﬂ+e'2‘”+e‘3“+e"“+ ...... .. —e Lt (a) , Lari: ZEOc} Lu] (x. at / J ” T5 5Q T/Lm fm 0 g, rising—M) + LIU) :iéECﬁjj QM )ICc) : :13 tact/j 1(5) z {CCU} S4\ 1 EW ~_ I ’, f ] beam/QH—t oLt l . — El BQQC :|_I___r[*cg_si) __ (He'jtdtj “ (N775) "Q 0 o —: PM?) l [,eJ’FJ‘ “It/l] .— (re—a5 S 5 PS 0 L ‘ "EL: ~.L(2-S~l) \~€'—; 5 5'2 _r -J‘ _, ' F .1. _ (2‘I 7. y SCI-24} 6 1(5): | _ 0"; §2(HI) STJ—H) (1424/ -———L—’-’ : A 4 4. ___C_‘ Saga) 5 51 5+! \—_ ArCmI) +‘?(J'H)—FC§L 5‘7; A+CZO ->) SI: A+E : D 9) A1”! 5°: 9:! 31(r4l) 5 5 5+1 ' z .L- \_ 50"”) 5 5—” 919‘): ~J~+JE+.L ._ .L__L Q—r 5 S :H 5 J; ~ l+o:’°+Q“ZI+Q—7x+ Q~“9_+ , I—QJ I(J—)Z —% 4— L1 4- ‘_ L _ i Q~; ([+Q‘;+Q_'Zf+ Q'_?I4Q 5 5*! 5 3+1 P __ -S .2; ,2 _L( 5 s y ‘7 5+) "‘5 ~7—J‘ r?r h 4- .8.— + Q A 2 +Q- f+gffwr 5“ \$41 54‘ :4: \$4! ’ r 47 w 2%; w r— » +06” 227% We 3;")? «*6 if?) + W :17] ( —S‘f + I Q-f— p c I I I § 3*. 4 : .4 + k 'EQ—t P u H—I) rum—21 HARP}; Au (kw ,LMJJKW + Q—(’°")u(lc~r) ¢Q-CJ‘”2/u(+~'z) reveal (6—3) + 2'06”) uH;—Lr) + e“ “UFJLWQ‘JJrr H. 00 (Jpn CG): ~‘4E‘FQ‘6 - ifhe’ jquthJ )\':.| CH ka (75 (H h 66+): ~1+L++IL‘L' « i I» o: ’ mph) 71": C 3 “2—” _ m _ (“HUM (w; » ("e“”“Z//u(e~2/ J94) ,me )u(£r3)—(»Pe‘“‘“‘u MH—Lr) , (1r KW”) «40—51 5 3'+Q”” —-(!~ Q“ 2 ﬂ).— (W~ 9"}(b; - O~Qfﬂ)a/ ,0 ~20) 0/ — c9- 10 ...
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## This note was uploaded on 07/25/2008 for the course ME 391 taught by Professor Blazer-adams during the Fall '08 term at Michigan State University.

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N1114 - NAME PID ii iii iv ME 391 FALL 2007 EXAM 2 WEDNESDAY LoLLFHOhLX‘ Note There are 4 questions with two parts each Use of calculators is not

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