MA 262, Spring 2018, Final exam Version 01 (Green)

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1 MA 262, Spring 218, Final xam Vrsion 1 (Grn) INSTRUCTIONS 1. Switch off your phon upon ntring th xam room. 2. Do not opn th xam booklt until you ar instructd to do so. 3. Bfor you opn th booklt, fill in th information blow and us a # 2 pncil to fill in th rquird information on th scantron. 4. MARK YOUR TEST NUMBER ON THE SCANTRON 5. Onc you ar allowd to opn th xam, mak sur you hav a complt tst. Thr ar 13 diffrnt tst pags with a total of 25 problms, plus this covr pag. 6. Do any ncssary work for ach problm on th spac providd, or on th back of th pags of this booklt. Circl your answrs in th booklt. 7. Aftr you hav finishd th xam, hand in your scantron and your tst booklt to your rcitation instructor. RULES REGARDING ACADEMIC DISHONESTY 1. Do not lav th xam during th first 2 minuts of th xam. 2. No talking. Do not sk or obtain any kind of hlp from anyon to answr th problms on th xam. If you nd assistanc, consult an instructor. 3. Do not look at th xam of anothr studnt. You may not compar answrs with othr studnts until your xam is finishd and turnd in, and thn only aftr you hav lft th room. 4. Your bags must b closd throughout th xam priod. 5. Nots, books, calculators and phons must b in your bags and cannot b usd. 6. Do not handl phons or camras or any othr lctronic dvic until you hav finishd and turnd in your xam, and thn only if you hav lft th room. 7. Whn tim is calld, all studnts must put down thir writing instrumnts immdiatly. 8. Anyon who violats ths instructions will hav committd an act of acadmic dishonsty. Pnaltis for such bhavior can b svr and may includ an automatic F on th cours. All cass of acadmic dishonsty will b rportd to th Offic of th Dan of Studnts. I hav rad and undrstand th abov statmnts rgarding acadmic dishonsty: STUDENT NAME STUDENT SIGNATURE STUDENT PUID SECTION NUMBER RECITATION INSTRUCTOR 1

2 1. Solutions of y 2 + cos x + (2xy + sin y)y = satisfy: A. xy 2 + sin x = C B. xy 2 + sin x cos y = C C. D. y sin x + yx2 cos y = C y 3 3 sin x cos y + xy2 = C E. x 2 y 2 + sin x + cos y = C 2. Th gnral solution of y + 3x 2 y = x3 is A. y = (x 2 + c) x3 B. y = (x + c) x3 C. y = (x 2 + c) x3 D. y = (x + c) x3 E. y = (x 2 + cx) x3 2

3 3. Th solution of 3y = sin x and y() = 1 is (1 + y) 2 A. y = (9 cos x) B. y = (5 + 3 cos x) C. y = 1 (3 3 sin x) 1 3 D. y = 1 (1 cos x) 1 3 E. y = 1 (3 3 cos x) Using th substitution u = 1 y, th diffrntial quation y + 3 x y = x2 y 2 bcoms A. u + (3 ln x)u = x 2 B. u (3 ln x)u = x 2 C. u 3 x u = x2 D. u + 3 x u = x2 E. u + 3 x 2 u = 1 3

4 5. A tank initially contains 1 L of a solution in which is dissolvd 5 g of chmical. Watr flows into th tank at th rat of 4 L/min and th wll -mixd solution flows out at th sam rat. What is th amount of chmical (in grams) in th tank aftr 5 minuts? A B. 5 4 C D. 5 2 E Using th substitution v = y x, th diffrntial quation y = y x 2 + y 2 x bcoms A. xv = 1 + v 2 B. xv + xv = v 2 C. xv + v = v 2 D. xv = 1 v 1 + v 2 E. xv = v 1 + v 2 4

5 7. Th gnral solution of y + 1 x y = x 2 is A. y = x4 3 + c 1x + c 2 B. y = x4 3 + c 1 ln x + c 2 C. y = x c 1 ln x + c 2 D. y = x3 4 + c 1 x + c 2x E. y = x c 1 ln x + c 2 x 8. For a ral numbr a, considr th systm of quations Which of th following statmnts is tru? x + 2y + 3z = 2 4x + 5y + z = 3 4x + 5y + (a 2 3)z = a + 1 A. If a = 2 thn th systm is inconsistnt. B. If a = 3 thn th systm is inconsistnt. C. If a = 1 thn th systm has infinitly many solutions. D. If a = 1 thn th systm has at last two distinct solutions. E. If a = 2 thn th systm has a uniqu solution. 5

6 9. Lt A and B b two 3 3 matrics with dt(a) = 3 and dt(b) = 8. Thn dt(2a B 1 ) = A. 3 B. 6 C. 3 4 D. 3 E Lt A b th matrix dfind by A = Th valu of th (2, 1)-lmnt of A 1 is A. 1 4 B. 1 2 C. 1 D. E

7 11. Considr th following matrix: A = Which of th following statmnts is tru? A. Th columns of A ar linarly dpndnt B. Th matrix has dtrminant 1 C. Th matrix is not invrtibl D. colspac(a) = R 3 E. Th nullspac of A has dimnsion Considr th following statmnts about vctor spacs and matrics. (i) If thr vctors v 1, v 2, v 3 in R 4 ar linarly indpndnt, thn any two of thm ar also linarly indpndnt. (ii) If four vctors v 1, v 2, v 3, v 4 span R 3, thn any thr of thm also span R 3. (iii) Th dimnsion of a subspac of R n is at most n. (iv) Evry subspac of R n contains at most n vctors. Th tru statmnts ar A. (i) and (ii) B. (iii) and (iv) C. (i) and (iii) D. (ii) and (iv) E. (i), (iii) and (iv) 7

8 13. Considr th subspac U of R 5 spannd by th st of vctors blow , 2 Thn th dimnsion of U is, 1 1, 1, 5 5 A. 2 B. 3 C. 4 D. 5 E Lt p 1, p 2, p 3 b th polynomials in P 2 dfind by p 1 (x) = 2 2x + 2x 2, p 2 (x) = 3 + 7x 17x 2, p 3 (x) = 3 3x + (6 + k)x 2. Thn p 1, p 2, p 3 ar linarly indpndnt if and only if k is diffrnt from A. 3 B. 3 C. 2 D. 2 E. 8

9 15. Considr th following sts: (i) Polynomials of th form ax in P 2, whr a varis in R. (ii) Points of th form (x, y) with y in R 2. (iii) Solutions of y (3) xy + 5y = in C 3 ([1, )). (iv) Vctors (x, y, z) in R 3 such that 5x = 2y z. Among thos sts, which ons ar subspacs A. (i) and (iii) B. (ii), (iii) and (iv) C. (i), (iii) and (iv) D. (i) and (iv) E. (iii) and (iv) 16. If a linar transformation T : V V satisfis thn T ( 2v 1 + v 2 ) is T (2v 1 + 3v 2 ) = v 1 + v 2, T (2v 1 + v 2 ) = 3v 1 v 2, A. 3v 1 + v 2 B. 5v 1 + 3v 2 C. 2v 1 + 2v 2 D. v 1 3v 2 E. 2v 1 + v 2 9

10 17. Lt A = Thn, A. A has ignvctor (2, 5, 1) T B. A has thr linarly indpndnt ignvctors C. λ = 3 has algbraic multiplicity two D. λ = 3 is an ignvalu of A E. A is dfctiv. 18. Lt A = and lt T : R 3 R 3 b th linar transformation dfind by T (v) = Av. Thn th diffrnc dim(rng(t )) dim(kr(t )) is A. 3 B. 1 C. D. 1 E. 3 1

11 19. Considr th spring-mass systm whos motion is govrnd by th initial valu problm y + 16y = y() = 1, y () = 4 3. Dtrmin th position of th mass at tim t = π 8. A. y = B. y = 3 C. y = 3 D. y = 1 E. y = 1 2. Th function y 1 = t is a solution of th diffrntial quation t 2 y + 5ty 5y =, t >. Choos a function y 2 from th list blow so that th pair {y 1, y 2 } forms a fundamntal st of solutions to th diffrntial quation. A. y 2 = t 5 B. y 2 = t 4 C. y 2 = t 5 D. y 2 = t 4 E. y 2 = t 3 11

12 21. Using th mthod of variation of paramtr, a particular solution to y + 16y = 4 sc(4t) is y p (t) = u 1 (t) cos(4t) + u 2 (t) sin(4t). Thn u 2 (t) = A. 1 B. t C. ln sin 4t D. ln cos 4t E. sc(4t) 22. If th mthod of undtrmind cofficints is to b usd, th suitabl form for a particular solution y p (t) of th diffrntial quation is A. y p (t) = At t + B cos(t) + C sin(t) B. y p (t) = At 2 t + B cos(t) + C sin(t) C. y p (t) = At t + Bt cos(t) + Ct sin(t) D. y p (t) = At 2 t + Bt cos(t) + Ct sin(t) E. y p (t) = At t + Bt sin(t) y (4) y = t + 3 sin(t) 12

13 23. Lt x(t) and y(t) b th solutions of th following initial valu problm: Thn y(1) is qual to: [ x y ] = [ ] [ 1 1 x, 2 4 y] [ ] [ x() 3 = y() 2] A B C D E Th ral [ 2 ] 2 matrix A has an ignvalu λ 1 = 2 + i with corrsponding ignvctor 1 v 1 =. Thn th REAL gnral solution to th systm x 2 + i = Ax is [ ] [ ] A. 2t cos t sin t (c 1 + c 2 cos t sin t 2 ) cos t 2 sin t [ ] [ ] B. 2t cos t sin t (c 1 + c cos t 2 sin t 2 ) 2 cos t + sin t [ ] [ ] C. 2t cos t sin t (c 1 + c cos t sin t 2 ) 5 cos t sin t [ ] [ ] D. 2t 3 cos t 2 sin t (c 1 + c 2 cos t sin t 2 ) cos t + sin t [ ] [ ] 2 cos t E. 2t 3 sin t (c 1 + c cos t sin t 2 ) cos t + sin t 13

14 [ ] 25. If a fundamntal matrix for x 2t = Ax is X(t) = 2t 2t, thn th gnral solution to [ ] 3 th systm of ODEs x t = Ax + is A. [ ] {[ ] [ ]} 2t c1 3 3t 2t 2t + c 2 3 t ; B. [ ] {[ ] [ ]} 2t c1 3 3t 2t 2t + c 2 3 t ; C. [ ] {[ ] [ ]} 2t c1 t 2t 2t + c 2 3 3t ; D. E. [ 2t 2t [ 2t 2t 2t ] {[ c1 c 2 ] + 2t ] {[ c1 c 2 ] + [ ]} 3t t ; [ ]} 3t 3 t. 14