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111-2007&2008-2-M20-April2008

111-2007&2008-2-M20-April2008 - Kuwait University...

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Unformatted text preview: Kuwait University Math 111 Date: April 29, 2008 Dept: Math 8; Comp Sci 2nd Exam Duration: 90 minutes Calculators and mobile telephones are NOT al]owed. 2 2 l (3) Compute det(A). (b) Does A‘1 exist? ((1) Find ade . 1 0 1 1. (Epts)LetA= 0 —1 1. 2. (4 pts) Let X 2 [1 1 1 l] and Y = [2 *2 2 2]. Find escalar a such that Y —aX is orthogonal to X. 3. (4 pts) Consider the four points P(1,1,D), Q(2, 37 —1.), R(1,1,2) and 8(2, 2,2). (3,) Find the area of the triangle with vertices P, Q and R. (b) 30 the four points P, Q, R and .5' lie in the same plane? Justify your answer. 4. (4 pts) Find parametric equations of the line B which passes through the points B(1,2,1) and C(5, 1, w1). 5. (4 pts). Determine whether V is closed under the operations EB and G if V = {(30, y) : my 2 0} with (miy)®(mfsyl)=(I+x,1y+yl) and C®(m,y)=(cas,cy). 6. (4 pts) Show that for all X,Y E R", ||X+Yl|2 — llX m: = (Mo-Y O. 09 "" vmrjws L~AHN u. LINEAR HLLrL—L‘SKH HV'JHIL-gla’élgg 6 ,; A’ .. +_fl —-—-—w-—a—_. ii: [0,, L 0919i:i(~|--2)+f(u+a):—“?>-+Q:_ 0+”, L is) Hill-7:40 1:) H"ew.g‘i‘g. Q Q a C) RI] 2 - 3 3 HI). I Q) 9173 I J —3 [l 1 9,2,: '2 ) A33 ‘4) 933: Q QC’th): a? l —t 93) Z l J 93:1 _..EJ g33:-| .2 31 —l L X1Ll 1 rlj gig/2C2 —3_—l 2] Fjozx a.” _‘ Y—K‘A2EQ—x —-Q-u Q 54 (WW) x -(3 «MK—.09 Exp/:2 9442 2x): 792%: o ,w,___,,__77 __,_ ,,_, ex _ /_ _ w’P/I/fl) cam/3]-») P(Z),.2)J?5(a? :22) WWW—W 1/ flrflwélé :— (’3‘) KPQD : <II/ ‘Q/—/> //<')) : <0; 0/ 52> 7513/? __J _M 1’ 3’ TC? 2 mm: . :1 «am 3LQ>+W>=<WQO> o O Q magnum”, :VJQZ-2u993“. H _—, ELI/:55 AP72)(_ V57, 2) IS; fifli'p 5? S “7 7/1: 50:316—- Piccoa'fl £62 0F 71?, Pita-13.- pQS-grwc- 'Tdkn-anx P’Q’ER 31—71:). - = <4,-3,0> , 7: 4’ 0/ 4/X—/)«Q[;/_r) Is mam) «HT. 7? ~41 907:11 ifii_’lii/";f’7' (if-“U Ma): 4 7(3)" .I/_ g/s 0/ via/0c. é LU/(r M /9¢i.§5é‘r ’77-‘16an 861(2))5} (SI Hf) EU- BC:<4,-/ ~1> 5MG_ 3%,,an i1 (2 (I . OWQQ MEI/r16. é/f 09/1, K: I‘LL“: a: Q-Jc 2: [~th 00 09 .1 ‘V: {(14) 2 91.3203 Wifl G; J; (D 539?..qu .5ch0! (71.) La) (9 (xx 51’) -: (7H 7% $7”) 3 C(DUIng): (CAGE) (i) V in wok dust} nab—ca. (f) '_ (5—106 1/ SMQ. (6“)(01320 (“l/'2) 6 V 9 ”7c“ (-r){-2) : Q_>/ o. 5641/— (5.» /)@(‘//42):(Z//”/)§/// $1276,— (”(4)40 (ii) \/ L'o CZpsaQ WALQ Jill. M : (by) é\/, (7‘32”) 5 C {R (3906;) ; (age?) 6 a\/ Wmmi”:___€’i_;_f§_ifi:i>/ 5 ”6‘ >0 H x ”/119an y111:(;/,-x).y ”N ““W‘t ”LY“: Wylmfl- MAO-(xv) :(x ><+ X-w w+ w) ”(K‘x—x-ww-firxy) 1013944 + flx-y + Ilia-IQ)- {WK 2 ><-\/+ IMF) : 3x. yaw/:- 4Xy (MANN 09 ...
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