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ECE220_Exam2_SU09_Sol

ECE220_Exam2_SU09_Sol - ECE 220 Exam#2 Summer 2009 Section...

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Unformatted text preview: ECE 220 Exam #2 Summer 2009 Section 051 Student Name: 12 Q U? L ST NAME (PRINT) FIRST NAME (PRINT) (SIGNATURE) By signing I am stating that I have taken this exam in accordance With the NCSU honor code. Show all work — no credit for answer only Summer 2008 ECE 220 Exam 2 1 HM Problem 1 (10 points) Consider the following matrices and vectors. 2 —1 2 0 1 A: 0 3 B: 4 1 x: 2 y=[3 a 1] 1 0 —1 3 3 a) AB [H] ’X’ (G; 1 AWWMW ”We/:7 77x73? {1767"‘iw’ghbﬁgbté; MEDI YYVV‘“ “LIQVV\ “WI—g) U2 ’9“. {fijiﬁxi Pg“: “.715”; b) B+A ”2, 7'? ”“2 :7 w , ”a K: ,7, \ 7) , H H 1 “‘ K: c) A7 A liq / 7, L75 1 3 2 ”l A [S 7 é: :3 3 ‘ H ‘ 3 Q l 9’ L”; {0 \$0. 23,75 3%.; e) Find the value of a in y that makes y ox orthogonal 1?”; a 1") .ﬂ ,. fa L [email protected] Summer 2008 ECE 220 Exam 2 2 Problem 2 (15 points) For the following matrix A given by Summer 2008 ECE 220 Exam 2 2W1)?” ma iwhcg 3 LIA Problem 3 (15 points) Consider the system of equations represented by the matrix equation Ax = b given below 2 4 4 x1 2 2 3 2 x2=0 —2 —2 1x3 3 a) Find the solution, if it exists, of the system above using the Gaussian Elimination approach. Circle the true statements in the box provided below. Use the following blank page to complete this problem if needed inconsistent C one solution> inﬁnite solutions \ ., we» 0% a a E ZQg+Q.3"%Q/\$L€;5 vZS l J R\”D~{21~3Qi \ 915 “a; v.3 \ % (ll—mam” Lb 5b fig u?) ‘3 t g i b) From the above system of equations, remove the last row of A and the last row (i.e. the last element) of b to get a new system of equations A'x = b'. Is this new system over-determined, under-determined, or" neither? \ x; / ‘ 2 i% le/Jjﬁ/ﬁj/D 1/} ﬁ‘vgﬁath/t‘ (“irﬁﬂii’tf Witt/’34 fax 1:» {(3 9 Would the solution obtained in part a) still be a valid solution to this new system? ;gg K ‘ *’ 3 '5 4 L436 9‘ 7 REV/V01 W9 of {2}” 1/3) Oar/«M4? 35 ff WA 3pqu 3 pal/El “ﬂat-l. witty-4,»: \ J a R Summer 2008 ECE 220 Exam 2 4 45/ Problem 4 (15 points) Consider the following system of equations given below: a 0 1 x1 —-2 0 a 0 x2 = bZ—l . 4 O l 3 1 3) Clearly state the values or conditions of a and b that give one solution‘7 c705 7/77: 07 7/7 a7 a: a 776 97%;; “:7 777 0/77 Valium #79 7172777777 77M- 7 _ r ”7 3 0\ <75 7 ’ 70L ’7 7 7 “ 7'57 0‘7 73 701,7 7 72’0va 707/3”? mom“ 5 9M 3—1577: “LO/W 77;” ,77 «7 7 7 7 77a 7 W 7» £7,» <7 77 7‘77 :m 71 777 ' “7 -« <17“ ’7 J 7’77 i 7 7 Q L7: 3 b 7 5 ()0 Q 4 1’7 ‘3 7/710 £1197 bJMMQW‘x g1!) :37 “55 if?” 7, Gar}. 3(23 7/9757 C’é C57 7 07. ‘7 7 '\ 7 imam WQXa 777727 f 3 :71;ng “77/7877: 7)7«€/A 73:7 ”\ 0‘7 7? P7 7«« <4, [7 7 7 1 4 7 7 07 ’2 7 v «35 J 7 g5 (27 (ﬂ Bk} 5 6L [Q3 U5 7 7 79 7 «g “b ‘P Q“ saw-7 7 , 74" bk , (5,1 7 g0 Q, // (7D ) 76) Bi: :7: 7 7-.‘7 71‘7”? (W572Uwid‘jataymfk 50 07¢ W 7:77 77 7 57671743 {EM/”m 3433\er 7 7 b) Clearly state the values or conditions of a and b that give no solution‘7 7 KW/\ Q7473 viva “mm”, . ‘ 7 07’; 7/7 Own}? \VQ 5 ”@772 7» 7537777,“. {£777. 3 727 .2:- <3 m >4 ‘3 5F 1717 WNW“; v70 73/077777; c) Clearly statet the values or conditions of a and b that give an inﬁnite number of solutions? I \. 7. '"‘ \. 7’ . / f 7. r «3 7 g A. :7 / L? > O M 3:77 Cz‘fyxw/s mi} 777 77 {w V477“ Summer 2008 ECE 220 Exam 2 Problem 5 (15 points) Given the following differential equation, 15(t) +6v(t) = 4cos(3l)u(t) + 3 sin(3z‘)u(z‘) where 12(0) 2 —.5 . Find the particular solution vp (t) by hand. \/p f A w: (@573 M5») + CS 3% 53% mil-M 0 VP ﬁq-ZAMM (39%”) mg newly} "; Mi») Vi” WP ; imwi ) ~ZA\$M31+QDWO +~g<m6ﬂ¢ H0073 )‘i‘ ?g{m€’) (DA-v 2%)mf) :Liossiif) (“EA + {19(3) \$243 :‘ g4m{ j) 0A*\$®ﬂ"l A Q/ZA'i‘LQJﬂ“ 25X mm ﬂ v/\~iim/zl)»r3> a Mm 1‘19.” :3; 9+ “#3 33V“’*f)\ifx§j? ‘0'; ’1 Summer 2008 ECE 220 Exam 2 6 Problem 6 (15 points) Find the total solution by hand for the differential equation given below v'(t)+5v(t)+4v(t)= 6e 3’u(z‘)+36u(t) Where v(0)= 2 and v(0)= l. /’})f a VP: 256 UFO szlﬂ’: ANN I X’vSXa—lﬁ 455 5:2 ”1% 9 A“ » (f) 44 ‘\ A! ”\3 5-! V9: #de Mm V92 “gs, ~ ‘z “*5“ 2. #7 H __ , °' ”3* w W /\ Li W; 5 £6 um VPx > 1 ) X 3‘ VC, " i” 9 nl% I - "a (i All: (1,); u/ WW §<ﬁlalfﬁ {3X 6 HEW C (yﬂ L4H) ““9032 (w A561) (1 a ”‘3 \//M :: VCHX i‘ anéﬂ Arvple’ﬂ A w. )3 f 3 /j( "Lgx ”g/k Al Cl L1“ ‘” 1" 5‘3 *Cle ’ ’16 Ar Clix/\HX/ - C\ -—L\CL;: “(2 Wm: Cw-c-wa :; «305% 9 ”Jr " 31¢ ”a!” 1/ C it." vm ﬂ—Cie ”was: we, hm 1 V \“ L\_)‘.\.,J\ ﬂ” “1'" Summer 2008 ECE 220 Exam 2 7 Problem 7 (15 points) For the differential equation given below, 17(2‘) + 413(1) +20v(t) = te‘g’ cos(3t — 37z/4)u(t) . a) Find the complementary solution, simplify and express it as a real function (i.e. as a function that contains no complex terms). Do not evaluate any constants in vc(t) . L & +H%+AO 4955 W mloi iii/Ewan «1/! +~ j 5’ who W; : ﬁe />\ - w~~mw~€mmwmﬂ .5 iﬁf / i Z. I 5 , 3 j; 3% ' _ (4+ng Magenta «» 39 Se VCHleLClé utﬂ age MHJ. Cl 561 CUM ” 9 ”Di x “\JE , E) “1% wit/{t “i“? ¢ 6 Meg e a‘ MM 5 "UV (aha ,. at up) a C8, E?) + Q \i «ii/kid") ‘ 7 5 163C ‘ a ‘34 “- faﬁv 9C6 057(L’lia-EBEUUO évyvag m(x) c 7%{63 WWW/:3 .3 _' b) Is this system of equations over-damped, under-damped, or critically-damped? ' it" I ‘\ {it \«31' \L fgxxéswvm f“: W‘xd MMVJ‘AX W ﬂ m & \ > H L trio c) For the differential equation below, determine the value of B that makes the corresponding system critically damped. W) + lat/(t) +9v(t) = v5 (t) 1 7, x Aves-ea c mm s x «mm *qu W a} N 3“ \\:”3 T x” <7 Qk‘x ’— 80 \$1: 2/113”(.4% Summer 2008 ECE 220 Exam 2 8 Test Facts Trigonometric Identities cos(—a) = cos(a) sin(—a) = —sin(a) i sin(0!) = c0s(0(n_L g) cos(a i 72') = — cos(a) sin(a i 7:) = — sin(a) i cos(0() = sin(0ti 15—) cos2 (a) + sin2(a) =1 cos2 (a) — sin2 (a) = cos(2a) 0032(0!) 2 %[1+ cos(2a)] sin2 (a) =%[1— cos(2a’)] cos(a i ,6) = cos(a) cos(,6) i sin(a) sin(,6) sin(a i ,8) = sin(05) cos(,B) i cos(a) sin(ﬂ) cos(a) cos(,6) : %[cos(0(— ,6) + cos(a+ ,6)] sin(a) sin(,6) = é—[coswt— ﬂ) — cos(0(+ ,8)] sin(0() cos(,6) = %[sin(a~ ,8) + Sin(a’+ ,8)] a cos(a) + b sin(a) = V a2 + b2 c0s[a— Tarf1 (b / a)] 1N5 Euler’s Identities eij“ = cos(a')1L jsin(0() cos(a) = ——(e’“ + 6"”) sin(a) = 0—(61'0‘ _ 6—”) Summer 2008 ECE 220 Exam 2 9 Quadratic Equation To solve for xi in the equation 61/12 + [9/1 + c = 0 , use the following —bi\/b2—4ac 11912: 262 Second-order Differential Equations Given the above with a = l , system is deﬁned as follows 2 Overdamped b— > 1 4c Underdamped — < 1 . 40 b2 Critically damped Z— = l c Summer 2008 ECE 220 Exam 2 10 ...
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