This preview has intentionally blurred sections. Sign up to view the full version.
View Full DocumentThis preview has intentionally blurred sections. Sign up to view the full version.
View Full DocumentThis preview has intentionally blurred sections. Sign up to view the full version.
View Full DocumentThis preview has intentionally blurred sections. Sign up to view the full version.
View Full DocumentThis preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
Unformatted text preview: NAME MA 262 Spring 2000
FINAL EXAM INSTRUCTIONS INSTRUCTOR INSTRUCTIONS: l. 2. 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 example, 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. . Fill in your name and your instructor’s name above. There are 25 questions, each worth 8 points. Blacken in your choice of the correct an
swer in the spaces provided for questions 1—25. Do all your work on the question sheets. Turn in both the mark—sense sheets and the guestion sheets when you 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 INSTRUCTOR 1. If y(m) is the solution of the initial value problem
3/ = eyas, y(0) = 0, then y(1) = A. 1/4
B. ln(2)
C. —ln3
D. ln(4/3)
E. 3/4 2. Which of the following is the general solution of the differential equation y 1
y, + E = 2;; , .1} > 0 c 1
A. = — — —
y x2 x
B y = 6m + i
1 c
C' y "325 z
1
D. xy — — = c
a:
c 1
E. = — — —
y :1: 23:3
3. The following differential equation is exact. Find the general solution.
(ex —~ y)dz — xdy = 0.
x2 y2
A. m — — — =
e + 2 2 C
B. e“5 — 7111; = C
x2 y2
C. I —  — = C
e + 2 2
D. e1 — xy = C
E. 695 + xy = C' 4. Solve the initial value problem: I x y
=_ _. 1—1,
3/ 2y+xa (Note that the equation is homogeneous of degree zero.) A. y=:1:\/lr1x—ll
B. y=mm+l
C. y=\/a:—lrix—+1
D. y=\/l—n—m
E. y=\/1n—x+l 5. A body with initial temperature 32°F is placed in a refrigerator whose temperature is a constant 0°F. An hour later the temperature of the body is 16°F. What will its
temperature be three hours after it is placed in the refrigerator? A. 1°F
B. 2°F
C. 30F
D. 4°F
E. 8°F
1 2 3 4 5
6 7 8 9 10
6. What is the rank of the matrix 11 12 13 14 15 ?
16 17 18 19 20
21 22 23 24 25
A. 1
B. 2
C. 3
D. 4
E. 5 7. The homogeneous linear system Ax = 0, with coefﬁcient matrix A has only the trivial
'solution. Which of the following statements must be true? A. A is a square matrix and dot A aé 0. B. A is a square matrix and det A = 0. C. The rank of A equals the number of columns of A.
D. The rank of A equals the number of rows of A. E. The reduced row—echelon form of A is 0. 8. Find all the values of k for which the system kx+y +z=1
3x+(k+2)y—z=5
2m+2y +2z=k+1
[It 1 I]
has no solutions. You may use the fact that det 3 [9+2 —1 =(k—1)(k+3).
2 2 2
A. k=0,1,—3
B. k=1,—3
C. MAL—3
D. kaél
E. =—3 9. Let A be a 3 X 3 matrix and let I be the identity 3 X 3 matrix. Suppose that the 3 X 6
'matrix [A I I] can be transformed to
I O 2 —1
I 0 1 _ 0
I 1 0 1 I by elementary row operations. Which statement is correct. COP—1
Ol—‘O
OODN 0 2 —1
A. A is nonsingular and A‘1 = 0 1 O
1 0 1 1 —2 —4
B. A is singular and A = 0 1 3
—1 2 4 —1 2 4 D. A is singular but cannot be determined from the above information. 1 —2 —4
C. A is nonsingular and A = 0 1 3 E. A is nonsingular but cannot be determined from the above information. 10. A and B are 3 x 3 matrices. The equality (AB)2 = A2B2
A. always holds B. only holds when A and B are diagonal matrices
C. holds only if A = 0 or B = 0
D. holds if AB = BA E. never holds 11. The vectors [0,0,1], [0,2,3] and [4,5,6].
' A. are linearly independent and do not span R3
B. are linearly dependent and span R3
C. are linearly independent and span R3
D. are linearly dependent and do not span R3
E. are linearly dependent and form a basis for R3 12. Let T: R4 —> R2 be the linear transformation given by T(x) = Ax, Where 2 —3 4 —5
A:[6 —7 8 —9[‘ Find the dimension of the kernel of T. FPOW?
ﬁwwr—‘o 13. If A and B are 3 x 3 matrices and detA = 2, detB = 3, then det(2A“lB3) = 2—7
2
B. 27 C. 54
D. 108
E. 216 A. 14. Which of the following are vector spaces? ' i) the set of all singular 3 X 3 matrices ii) the set of all polynomials p($) with p(0) = 0 iii) the set of all vectors of the form (r + s, r,r — s), r, s E R
. and (ii)
. and (iii)
. (ii) and (iii)
only (i)
only (ii) mpowi> 15. For how many values of k are the vectors [k, 1, —l], [1, k, —1], [1,—1,16] linearly
dependent ? A. no values
B. one value C. two values D. three values E. all values 16. The sum of the eigenvalues of the matrix [_; 22;] is 14. Which of the following are vector spaces? ' i) the set of all singular 3 X 3 matrices ii) the set of all polynomials p($) with p(0) = 0 iii) the set of all vectors of the form (r + s, r,r — s), r, s E R
. and (ii)
. and (iii)
. (ii) and (iii)
only (i)
only (ii) mpowi> 15. For how many values of k are the vectors [k, 1, —l], [1, k, —1], [1,—1,16] linearly
dependent ? A. no values
B. one value C. two values D. three values E. all values 16. The sum of the eigenvalues of the matrix [_; 22;] is 5 1 —l
17. The eigenvalues of the matrix —3 1 3] are 2 and 4. One of
0 0 4 has dimension one. This eigenspace has a basis consisting of 18. Let L be the differential operator Ly = y’ — xzy. Then Ker(L) = the two eigenspaces A [—2]
B. m
c. [a
D H
E. [a A. span {613/3}
B. span {6—13/3, ems/3} C. span {ta—$2, exz}
D. span {eﬁs}
E. span {6:2} 22. The oscillation of a. spring—mass system is governed by the differential equation d3: with initial conditions 33(0) = 1 and = 3. Then 23(t) = 23. Find the general solution of the system x’ = [ 10 31
13 lx. A. 2cos(t — 7r/2) B. cos(t —— 7r/3) C. 2cos(3t — 7r/2)
D. cos(3t — 7r/6) E. ﬁcos(3t —— 7r/4) 6% +6 e4t
__e2t 2 e4t 24. The 2 X 2 real matrix A has eigenvalues —1 + 2i and —1 — 2i, and corresponding 2 —z
the system x’ = Ax. 'eigenvectors and [ 1.] respectively. Which of the following is a real solution of A e_2t [cost] sint
t sin 2t
B e i — cos 2t]
t cos 2t
C e [— sin 2t]
t cos 2t
D e i sin 2t ]
2t cost
E e i —— sin t]
25. The system
I _ 3 2 et
x (t) — [2 3] x(t) + has fundamental matrix
t 5t _ 1 —t _ —t
X”) = [jet 25:] a and X05) 1 = 5 [:—5t 6551: ] 
Use the method of variation of parameters to ﬁnd a particular solution.
(—2t + 1) et
A' l (t — 1) at
B (—3t — 1) et
' (3t —— 1) eit
C (413 — 3) et
' (t + 1) at
D (3t + 1) et
' (t + 5) et
(3t + 2) et
E' i (t — 1):?t 11 ...
View
Full Document
 Summer '09

Click to edit the document details