111-2008&2009-2-M20-May2009

111-2008&2009-2-M20-May2009 - Kuwait University...

Info icon This preview shows pages 1–2. Sign up to view the full content.

View Full Document Right Arrow Icon
Image of page 1

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full Document Right Arrow Icon
Image of page 2
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: Kuwait University Math 111 Date: May 19, 2009 Dept of ‘Math and Comp Sci“ Second Midterm Exam Duration: 90 minutes Answer all of the following questions. Calculators, pagers and mobile telephones are NOT allowed. 1. (6 pts.) Let A = Ob—‘O AXQN 01000 a) Show that A is nonsingular. b) Find adj(A). c) Find adj(adj(A)). 2. (6 pts.) a) Let a, b, c,a:,y,z E R. Use the Cauchy—Schwarz inequality to prove that (a2 + b2 + c2)(a:2 + 3/2 + z2) 2 (am + by + cz)2. b) Find the area of the triangle ABC with vertices A : (1,1,3), B = (2, 2, 2), C = (4, 6, 1). c) Find equations for the straight line passing through the point (1,2,3) and which is perpendicular to the lines a: = y 2 z and :5 2 2t, y = 1 + 3t, 2; = —2 + 425. 3. (4 pts.) a) Determine whether W = {(a, b, c)|cL2 ~ b2 = O}Iis a subspace of R3. b) Determine whether R3 is a vector space with the operations EB and, (D, where XEBYzX—l—Yand C®X=X for every X,YE R3 and everyCER. 4. (3 pts.) If X, Y E R3, prove that X x Y is perpendicular to 2X + 31". 5. (6 pts.) Answer True or False and justify your answer. a) If X, Y E R3 are orthogonal vectors and if HXH : 2, HY|| = 3, then [IX — ZYH :2 9. b) If A is a 3 x 3—matrix and if adj(A) = —A, then |A| = 0, or |A| : —1. c) The planes at + 23/ + 32 — 6 = 0 and —31: — 6g + 52 + 4 = 0 are perpendicular. d) The set of all unit vectors in R3 is a subspace of R3. 00 CO ~fr€¢4ims — I) 4) \M=—40=1Fo)swmsik E) u— \F-CJuDL'm {was E: k 5% IN Hat] 4 o — 0 ~ -\ -1 — _ :1fi-‘A=-\OA:¥“1 ‘6 a] *3 0 O . Ll 0—2 2. if]; wL‘QML ”‘41 (Ax) = ‘3: L1 _ a -\ 'L g ' 2-10 C) “‘8' (411/0) =\aa4'm\('-a1'(k)‘= ”$6M ) = W T1,“ A ”“6‘ A : O — 20 ’0 -119 0 ~30 5 —1‘0 -SD 1) ‘9 “= MN); Wham?) ~> \ums M\\vn'>LU-v wheat; in. Requdahl . Q u:?§'= (Am—0) u: 1117 = (3,L,)—7_)) Lu; \r: (2,510) .L “LLJI- v“ :1 fl Ctr-2.9. c c) Aaa‘firf’vgdm NU. ;=(1)\’\)N=L773)L3)) AXreCJ-{ma ”If WGM‘i‘e/L flue. {5. M0} = (5—2“); 53mme'1rt equcflms m: 5%} Egg: :17:35 3) A) \N *3/1le ten”. (J-WL *namreé (l)—2)o)) (2,1,0) 5 w M (150”): L2;2 03.14;): 03¢ W- , ‘ 3°) \ifi q Veda; 5‘3abgdrwfi-QL {4 Waktfi) . $uk—Qrgzq‘fluJ-kw 060,29) = tings) M .213 mm» wL shad/L Am Q3(1)%)3.)= (0,0,0) JD We Know Cu". S’anLLfl “was {1' \L 4K4u.) 741mg} T: Xxfi’ )g 1%A;¢..Qn,h i; M 73% sv— T.(1>!+SY>=QCT‘x)+3(T-Y) 2-04-01 0) W? M $4+Emeui _ 5) 9F "Em \\>(*l‘(\[email protected]=f‘i. QT W ham}: \-A‘\4=.=»m\2=-1Mc=3 \A\C1A—H|}=0¢=3 \AMS {—4) oi. ‘1- '2. 315 M w u ...
View Full Document

{[ snackBarMessage ]}

What students are saying

  • Left Quote Icon

    As a current student on this bumpy collegiate pathway, I stumbled upon Course Hero, where I can find study resources for nearly all my courses, get online help from tutors 24/7, and even share my old projects, papers, and lecture notes with other students.

    Student Picture

    Kiran Temple University Fox School of Business ‘17, Course Hero Intern

  • Left Quote Icon

    I cannot even describe how much Course Hero helped me this summer. It’s truly become something I can always rely on and help me. In the end, I was not only able to survive summer classes, but I was able to thrive thanks to Course Hero.

    Student Picture

    Dana University of Pennsylvania ‘17, Course Hero Intern

  • Left Quote Icon

    The ability to access any university’s resources through Course Hero proved invaluable in my case. I was behind on Tulane coursework and actually used UCLA’s materials to help me move forward and get everything together on time.

    Student Picture

    Jill Tulane University ‘16, Course Hero Intern