# Exam1-A - N Emma—ALHTCIPM MATH 262 I Midterm 1 February,...

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Unformatted text preview: N Emma—ALHTCIPM MATH 262 I Midterm 1 February, Spring 2010.. Section: There are 21 questions on 12 pages. FROM PROBLEMS 17 AND 18, YOU MUST SHOW ALL YOUR WORK FOR FULL CREDITS. NO CREDITS WILL BE GIVEN WITHOUT JUSTIFICATION. You must record your answers of problems 1-16 on the second page and you will get 0 point if you fail to mark. NO CALCULATORS ARE ALLOWED. MATH 262-Midterm 1— 1. (5 points) Find the solution of the initial value problem 2:95”, y<o>=4. A.y=1n(m+1)+4 - : 9.x 519% B. m2+y2=16 3“ xig oy=x2+4 Dw+4=y =5 MN =. Jmcuis)+ c E.y=3x+4 ; 9 «\$113-! ’:C ﬁgs “:9 1 -ll __ 8c; x313 7;) 3—1 :iecmirs) be) 36% D: iec, Hen gig) beams ‘ g I: D (xii—3) :> Swap 9%): +1 : 2322:??- va : i, then the given equation becomes :9 - XL x \/ gm + v+xqu % 0W [+353— ‘cDT: ma) 14 _ * aw ~ H'S‘J >V+Xﬁ— EV 2. L _> X11 J+SV-,__1\’ M 6* : 32v xolV : \lL-Fi d?‘ 1“ 3) 2;, AX: v+s Page 3 of 12 MATH 262—Midterm 1— 3. (5 points) Which of the following is the solution of 7914’ + Zty = sin 75, y(27r) = 0,t > 0? A. y(t) = ‘a CF 9— : 375T? B‘ : 00523—1 g ‘L-z' c. ya) = wt)=€§12;ol: 62a 1:_ 762. D. y(t) = 6—}; 2: 652 sinsds / ~ @110» = > U 14;) = 9M 1‘: Watl'é-Cosi; ta. w): Maj"? C—évszﬁ __ /’ M m o l - God? :2. :2) 3 H; ) = 2. C i 4. (5 points) Which of the following is the solution of (2xy2+2:1:+ 1) dx+(2x2y+2y+1) dy = 0? 11/ M W A. \$2142 —— 4mg + 2 = c, c is any constant @m2y2 —— 2:2 + y2 + a: + y = c, c is any constant gig" ’5 g): '4" 4X3 C. §(xy3 + \$374) + 4323; + a: + y = c, c is any constant ‘ 90 AA: ’60 exact,” D_ gag/3 + \$331) + 2x2 + 2y2 + a; + y = c, c is any constant E. 2m2y2+\$2+y2+x+y=c,cis any constant 2 gr;— 2 mg + u +! 2% = 1x); 113+) 12> Y” :3) 3 X3: X301: Xi"; to Page 4 of 12 MATH 262-Midterm l- 5. (5 points) A body with initial temperature 64°F is placed in a refrigerator whose tem— perature is a constant 0°F. An hour later the temperature of the body is 32°F. What will its temperature be three hours after it is placed in the refrigerator? / A. 16°F THC) = * H1719 *0) , l: jaw 13- 24°F 770/) = 5‘} /> Trt>= 64 6’: tin/2’ 0. 32°F 5 _ V D. 64°F ) THU ‘ l1 Tit) / WW 0 __ I @8F Tia—[Age / -5)“;- \ TC3) : Q4 6 3mm Tfo)=b4’) A: bq' // ‘ é4 M‘f ) Jet: ‘ Wt): Q4 6 / a _3 WW TUB-<32, t [’4 ‘2 _ k 64. G 4‘ :2 3’2: __ ___,_..— 1: 8 ea ( 8 == ’2‘ :2) : “oil/“Q. 6. (5 points) The solution of the initial value proble@ y(1) = g is A. y(ac) — lnx+§ BeYhmw echWl'FOV) B yo) Wu) ‘ _ M; _\ C y(\$)—\$—% M jg '9 ‘3‘ @319“) KT: Then We Vb? E -—% (a: 1) I Q- - 4 31(93) 8 w —-\v “y u __ 1,; a Mk}: 5: yli 321’“: Z Z / gait 1 thﬂX): i f6 {9”}: 21:, wuxl: )i-rC C“ are; i— + i \ 3f XL \ [/2 Page 5 of 12 J,— J_ c, : n/ 1D :16) : X +71 )3”) X+l . / X?» We w" W = 71?: MATH 262—Midterm 1— 7. (5 points) Which of the following is the solution of y” = —2y‘1(y’)2? @éy3=om+D (VWE‘ pkg/elm?) .y+x3+Cx+y=D (ml C.:cy+Cx+Dy+3=0 ; Aw ' + \l I: r; 134? D. xy+y+Cy+D=0 { La a): E. ém2=0y+D // AL 0W \ 2 ’39- Z V M \ dx1 “‘7 .. C \/ "‘ ’3 (\ AV _\ L ‘3 V T ._.. - 219 v a K 7’ ‘21 " C K 7% " "’3 9 ’ A". + g V 2; O :2. .7: C «7le a ‘2) -—- ’ i K‘ s " Zln 2" m; hangar W- 9:35 t / Z 2, :1, l = o r-> v vi :— c 8. (5 points) The number of bacteria in a certain culture grows atllgﬂmW to the number present. If the number is increased from 200 to 1600 in 3 hours, What is the number of bacteria present after 2 hours? A.400 all” __ _&F 19:kou( B. 600 Ht 2 O yﬁe“ r; =1 0 h @800 P [LE—’5 Pﬁe 95’“ D. 1200 kg E' 300 Pit? ‘-'- Pr, 8 = 3.00 g & Slnfe/ P€g>=leoo Iéoo = loDéa — ~ K 61/168 F C 7- :; " )> k: 1m 3 (a): 100 2 ) “Bk ‘: ~l- g [27“; :1 0000 lg: ~ 1% r2. \\ Page 6 of 12 2 t 9.. \\\\\‘ > z 200 6/ Ln \\\\\_, 4: _ ﬁnal 4: V (loo 8 :(200): MATH 262—Midterm 1- 9. (5 points) Let A and B be 4 x 4 matrices such that det(A) = 3 and det(B) = 4. Which of the following is NOT true? A. det(AB‘1) = E B. det(AB) = 12 C. det(AT) = 3 D. det(A’1) = % ® det(2A) = 6 Que/xi: JMW vile-5 ‘4? i go AMLl'XZ-l» 10. (5 points) The solution of the initial value problem 3;” —2:1:“1y’ = 42:2, y(1) = 1, y’ (1) = 015 C? qq5Cm l} : A.y=\$2—2:I:+2 I E B.y=4m3—12x+9 l6:\/ @y=\$4—§\$3+§ "—V’ D.y=x4—-4x+4 \9 _ 3 3 E.y—\$3—§\$2+§ A I “\1 . 2x"\/ =- 4x cw \ Lv e z I 2' l’L i 3 \\ \/ ) 9%) X“ \\ 1/7, a 4X+¢¢ MATH 262—Midterm 1— 11. (5 points) Which of the following is an integrating factor n so that ,u(3my + 3/2) dac + M<£C2 + my) dy = 0 is exact? A. y 99 ObgéYVOL’l/FOV} B. any—1 ‘ _ i w) W E. my '7" (1 Z1210 >95}; :15} w ax mail“ i 12. (5 points) Let ' 1 1 4 —2 A: —3 —3 a ,b= b . 1 8 0 0 For What values of a and I) does the system AX = b have inﬁnitely many solutions? 1. a=—12 andbanynumber I l 4' “Z, @a=—12andb=6 O O IZ+0\ —6+l9 3. a=—1landb=6 O 7' ~41. Z 4.a74—11andb756 5. 0,7512 andbany number l / £+_ _Z a 0 Z} -4 7, Sim AXZL has 0 012m “5+5 Waning mam? gawﬁm 4% We Vial/c, IZ+0\: O I 47+}? Page 8 of 12 MATH 262—Midterm 1- 1 2 3 ’ A 13. (5 points) Which of the following is NOT row equivalent to [ 4 5 6 I? 7 8 9 @J; F7 I_—_——ll—_—Il———'1l__—I .0 [\DU‘OO 0001M CONN .91 xii—uh 00th +4pr ©CDOD RICH?“ Ni: U32 1 lo] 14. (5 points) Let A be an m X n matrix and b be a column n—vector. Denoting by A# the augmented matrix of Ax = b, which of the following is NOT true? @ if mnk(A#) < n, then the system has inﬁnitely many solutions @ the system is always consistent if A is a square matrix C. the number of leading ones in the reduced row echelon form of A cannot exceed m D. if rank(A) < rank(A#), then the system has no solution E. if rank(A) = mnk(A#) = n, then the system has a unique solution Wall "0] Elﬁ] :—> Axel;7 W5 V10 Ho 0 000 } 3 gala/Hon 2W \ M02 “MK (All): 5 44 l ’\ '9 - : : «Bk US) "l; A [,1 I] éageljgofgél] ) {m {We AX: L Mg ha 30 w a WK 01 C2 C3 a1 a2 a3 15. (5 points) Ifdet b1 b2 b3 =3,thendet is A. 1 @3 C. 9 D. ~12 E. —36 16. (5 points) Let A = [GU] = [ OOO'lN) «scum following is NOT true? A. 5021 + 6022 + 7023 = det(A) B. 8C31 + 9032 + 10033 = det(A) C. 4013 + 7023 + 10033 = det(A) D. 2011 + 5021 + 8031 = det(A) @5011 + 6012 + 7013 = det(A) MATH 262—Midterm 1— b1 b2 01 02 Warm) Oﬂrb 1 W Page 10 of 12 and Cij be the cofactor of aij. Which of the 00 £05ch exmwibvx a1+b1—401 a2+b2—402 a3I+b3—403 b3 03 MATH 262—Midterm 1— 17. (5 points) Consider Ax = b, Where 123 4 A: 567,b= 8. 91011 12 Then the number of free variables in the solution set is A. 0 1 Z 3 4 @ 1 S“ {a ‘3' 9“ ] C. 2 if) E} it" D. 3 E. the system is inconsistent #9 ® Z ~ L’L J o I z .3 U 0 O D 1 0 1 18. (5 points) The inverse matrix of [0 1 O ]? 1 1 2 2 0 —1 L? —1 0 1 \7 Wcmmﬁﬂaﬂ/ [1 0 0] 03’ B. 0 1 0 —1 —1 0 -1 i/ “13%) 1 0 1 A “’ MA C. 0 1 0 [1 0 0] 1 0 0 D. 0 1 0 [1011 2 1 —1 @[0 1 —1 ~1 1 Page 11 of 12 \1 (Jo) : 41m, awwwwt gﬁtﬁium (VainJL) r“ we, mm a" t: MATH 262—Midterm 1— 21. (10 points) A huge tank initiale contains 10 gailons(gal) f water with 6 lb of salt in solution. A solution containing 1 1b/ gal of salt is pump d into the tank at a rate of 3 gal/min, and well—stirred mixture ﬂows out at a rate 2 gal/min. Let Q(t) be the amount (g) of the salt in the tank at time t. \/ (a)F1nd adlfferentlal equation for Q(t)? g0 V06 :‘V + (3.2).é dQ — rC " 0 ‘82" 15"r2.‘*2 Q(03=Qo . JO— :MTIA :> 4g; _ 262a) - W/ cit _ 3 Io+t ) QM“ é (b) What is the amount of the salt in the tank after 10 min? at 2 - t + WtQH’)* _ 3:. 0% /L( (t) " 9w.» __ 2 in ( Io+t) L 8 : (lo—Hz) / 22 '3 £Q(t)(ao+t§2”)= 3C’0+t5 :1 3 > QH'JF-Io-l'tf’: (to +1?) + C : _ C > Que) ~ (10+t) + W); ‘ _ , C. Since Q10)- (0 ) 6:|0+% a T55: “4L '0 Q: ~4L'i02 :-> art): lo+t - 4402- CID-Ft); :T. [7 Page 13 of 13 ...
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## This note was uploaded on 08/29/2010 for the course LIN ALG DI 26200 taught by Professor Jinhaepark during the Spring '10 term at Purdue University Calumet.

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Exam1-A - N Emma—ALHTCIPM MATH 262 I Midterm 1 February,...

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