111-2008&amp;2009-2-M20-May2009

# 111-2008&amp;2009-2-M20-May2009 - Kuwait University...

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

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

View Full Document
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

• 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.

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

• 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.

Dana University of Pennsylvania ‘17, Course Hero Intern

• 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.

Jill Tulane University ‘16, Course Hero Intern