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Unformatted text preview: NAME 1 2 MA 262 Fall 2002
FINAL EXAM INSTRUCTIONS INSTRUCTOR . You must use a #2 pencil on the mark—sense sheet (answer sheet). . On the mark—sense sheet, ﬁll in the instructor’s name and the course number. . Fill in your name and student identiﬁcation number and blacken in the appropriate
spaces. . Mark in the section number, the division and section number of your class. For ex
ample, for division 02, section 03, ﬁll in 0203 and blacken the corresponding circles, including the circles for the zeros. (If you do not know your division and section
number ask your instructor.) . Sign the mark~sense sheet. There are 25 questions, each worth 8 points. Blacken in your choice of the correct an—
swer in the spaces provided for questions 1725. Do all your work on the question sheets.
Turn in both the mark~sense sheets and the question sheets when _ygu are ﬁnished. . No partial credit will be given, but if you show your work on the question sheets it may be considered if your grade is on the borderline. . NO CALCULATORS, BOOKS OR PAPERS ARE ALLOWED. Use the back of the test pages for scrap paper. NAME 3(33 +1)2
3/ 1. If y’ : ,y(—1): 2, then y(0) : 2. If my' — 3y 2 $3 and y(1) : 1 then y(e) : macaw? EDQW? NAME 3. The general solution of   dsc 312 — 51:2 is
A. y§—$2y~$3:c B. y—+x2y+$3:c C. 2my+y2+3x2 :c
D. log(;1:2 + y?) = c E. none of the above 4. The population of a certain city is increasing at a rate inversely proportional to the
population. At t : 0 the population is 1000 and at t 2 1 it is 2000. The population
as a function of t is p(t) = A. 1000x/3t + 1
1000 ‘/1—§:t
(3. 1000(1—Ft) 1000
l—it
E. 1000(1—+t2) B. D. NAME 5. A rocket traveling straight up has velocity 110 when it is at a distance of 212 from the
surface of the earth where R is the radius of the earth. The engines are shut off and
the only force on the rocket is due to the gravitational attraction of the earth. The
velocity v of the rocket when it is a distance a: from the surface of the earth satisﬁes a
differential equation where g is a constant. For what values of 110 will the velocity be positive for all 2:? A. 120 > «291%
B. 110 > VgR /2R
C. ’Uo> 9?
(R
D. ’U0> 1,3— R
E. ’U0> 93— 6. Initially a IOU—gallon tank is half full of pure water. A salt solution containing 0.2 lb.
of salt per gallon runs into the tank at a rate of 3 gallons per minute. The well mixed
solutions runs out of the tank at a rate of 2 gallons per minute. Let 17(t) be the amount
of salt in the tank at time t. Then $(t) satisﬁes the differential equation A :‘fzo‘4_503:t
B 3%:0'6_502it
C %:0'6+502:t
d 2
D'd—f: ~50L
E 2—:20'4_502:t NAME 3 4 7
7_IfA: 2 6 1 ,thendet(—2A): 3 14 —1
A. 60
B. —60
C. —240
D. 240
E. 30 8. If A and B are 3 x 3 matrices such that det A .2 3, detB : ~4, then det(~2A_lB) 32
h 3
32
3
C. 96 8 D. __
3 A. B. E.§
3 NAME 813 6—2:
9. If A : [at “2644, then det(A_1):
l
A. —§€t
1
B. ~§€Wt
C. —3et
D. 3e‘
E. —3e“‘
10. Determine all values of k so that the system
1171 + $3 I 0
11:1 +2172 +kLL'3 =0
k$1+ [$132 + 61133 =k+4
has inﬁnitely many solutions.
A. k 75 —4 B. k7é3andk5£~4
c. k23andk2—4
D. k=3
E. k:—4 NAME 11. Which of the following sets is a subspace of the given vector space? A. V = 03(R), S : {ym — (ac2 ~1)y”+ 3x(y’ — 2) — y = 0}
B. V : COOK), S : {ylll _ ($2 _1)y// +3xy/ _ y : 0}
C. V2R3.S:{(m,y,z)ER3:3x—y:z+1} a b D. V=M2X2(R), S: {A2 [C d EM2X2(1R):a+d= 1} E. V = Ola—1,11), S = {f <—: v, f’(—1) = 21m) — 1} 12. Consider the system Ax=b given by then xT : [$1, 5102, 2:3] is NAME 13. The vectors (1, 2, 1), (3, 4, 5), and (2, —2, k) are linearly dependent if k equals A. 4
B. 8
C. —5
D. —1
E. 0 14. Let
1 —1 2
A :
[—3 3 —6]’
and let T : R3 —> R2 be a linear transformation given by T(:r) : Am. Then Ker(T)
and Rng(T) are A. Ker(T) : span{(0, 0, 1), (1, 0, —2)}, Rng(T) : span{(0, 0, —1), (1, 2, 3)}
B. Ker(T) : span{(1, 1, 0), (—2, 0, —1)}, Rng(T) =span{(0, —3), (1, 2)} C. Ker(T) = span{(1, 1, 0), (—2,0,1)}, Rng(T) :span{(1, ~3)} D. Ker(T) : span{(1,0), (0, —1)}, Rng(T) =span{(1, 1,0), (2,0, 2)} E. Ker(T) 2 span{(0, —3,0), (—2,0, 0)}, Rng(T) :span{(0,0)} NAME 15. A basis for the kernel of the linear transformation T : 02(R) ——) C(R) given by T(y) :
y" + 43/ + 8y is
A. {6%, teZ‘}
B. {6"2t cost, 6—2‘sint}
C. {8—4t cos 2t, 6““ sin 2t}
D. {ta—4‘ cos t, e—4tsint} E. {ti—2‘ cos 2t, 6‘” sin 2t} 16. The general solution of y” + ay’ + by = 0 is y : Clem + 62622. To ﬁnd a particular
solution by the method of undetermined coefﬁcients of the equation y”+ay'+by:ez+e3$+1, one should try a solution of the form 616$ + 6263“ + C3
0111263” + 0263” + 03
clrcex + 0263:” + 03$ 611,123: + C263m $.50??? C182 + 0263“: + 0331: + C4$2 NAME 17. A trial solution to use for ﬁnding a particular solution of the differential equation
(D2 —1)(D2 — 4D + 3):; : cosx — area“ is: A. C1 cosa: + Cg sinar + 0316269” + 04x36” B. (3'1 cos :c + Cg sinx + 03:36” + C4$2€$ C. 0115 cos a: + Czar sins: + C3$€z + 04.73%” + 05x36;
D. 01 cos a: + 02 sin a: + 032:6z + C463z E. 01 cos a: + 02 sin :1: + C'3em + C4956” + 05113261; 18. If y : ulyl + uzyz, where yl : 8t and 3/2 = t, is a particular solution of the equation I/ t l
y +my’—~1Tty:2(1—t)e_t, 0<t<1, then by applying the method of variation of parameters one ﬁnds that U2 : A. —Ze_t
B. 264
C. e‘t 1
D. §€_t
E. te—t 10 NAME 19. One solution of the differential equation is yl 2 :17. Another solution is of the form yz : m: where 1} satisﬁes the differential
equation A. v" + :w' z 0 $21)” + 2131/ = 0 . 11:21)" + (:1: +1)U' = 0 . 1:21)” + (2x + 1)v' : O .mcow (2:1: +1)v” + $21), = 0 20. Let y(x) be the solution to the initial value problem
34” * 311’ + 2y = 43:, 31(0) = 4, y'(0) = 3 What is y(1)? 11 NAME . . . . d2 d .
21. The oscillation of a spring—mass system [S determined by i + 3—3: + 2:1: 2 0, w1th dt2 dt
d
initial conditions 33(0) 2 1 and : ~3. Then a. sketch of the motion x(t) is A. 9( B. ’X
C. ,x D. 9‘ 12 NAME 22. It is observed that an eigenvalue of the matrix ~1 0 O
2 1 4
O —1 —3
is /\ = —1. Let m_1 denote the multiplicity of this eigenvalue, and d_1 denote the dimension of the eigenspace corresponding to this eigenvalue, then which one of the
following is true? 0 2 k
23. Determine all values of k so that the matrix 0 2 k is defective.
0 2 k
A. k = O
B. k : 2
C. k 2 —2
D. no k E. all k’s 13 is 14 _e—2t e4t
0 0
8—2: ecu
e—Zt _e4t
0 0
6—21: e4t
“6—2t 64¢
0 0
6—2t 64¢
_e—2t e4:
0 O
e—Zt e4: NAME _ —2t 0
25. Ifa fundamental matrix for x’ : Ax is X(t) : [ e_2t 2t J , then the general solution
6 e . . 0 .
t0 the system of differential equations x’ = Ax + [ J [S A. [Le23‘ :il{[:;l+[£l}
3 [Le—23‘ Gama—{:1}
0 [:3 ef’quzﬂl‘il}
D. [:35 eithﬂle‘il}
E hi?" ail {liil — [Jill 15 ...
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