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Unformatted text preview: Engineering Mathematics (ESE 317) Exam 1 February 10, 2010
This exam contains six miﬁtiple—choioe problems worth two points each, seven true—faise problems worth one point each, three shortanswer problems worth one point each, and four freeresponse probitems
worth 18 points altogether, for an exam total of 40 points. Part I. MultipleChoice (two points each) Clearly 1:211 in the oval on your answer carei which corresponds to the only correct response.
1. Which of the following methods could be used to ﬁnd £71 (I) convolution (II) differentiation or integration (the “box” method)
(III) partial fractions (IV) t~shifting (A) (I) and (11) only (8) (I) and (III) only (C) (I) and (IV) only (3)) (II) and (III) only (E) (II) and (IV) only (F) (III) and (IV) only (G) (E), (Ii), and (III) only (H) (I), (II), and (IV) only
(I) (I), (III), and (IV) only (J) (H), (HI), and (IV) only 2. Let k be a constant. Find £(sin(t + k)). k
M>Em 03) (3.152% 1
(C) (New: }. 1
m>ﬂﬁ+oo
(B) we 1
(F) 81” 32:;92
(G) 1 + k
m>ﬁq+ﬁ
(I) cosk(32:1)msink(32:1)
(J) cosk(52:1) 3. Find a: sink 4:). (A) 32:26
(B) 323:6
(C) 32116
(D) Effe
(E) (32316)?
(F) (32:26?
<G> (H) (5226? (I) (D 4. Consider M22, the vector space of all 2 x 2 matrices, anti consider the foliewing subset S of M22. 10 01 00 11
S: “WLJ 0]3a2:[0 0]”“3=[1 OJ’MWL 0] Which of the following statements is/are true? (I) S spans M22. (11) S is linearly independent. (III) S is a basis for M22. (A) (I) only (8) (H) Oniy (C) (III) enIy CD) 0) and (I!) one!
(E) (I) and (III) only
(F) (II) and (III) only
(G) (I), (II), and (III)
(H) none of them 5. The set V of all 2x2 matrices Whose determinant is zero is 39,; a vector space. Which of the
following vector space properties gig to be satisﬁed by V? (A)
(B)
(C)
(D)
(E)
(F)
(G) (I) closure under addition
(H) closure under scalar multiplication
(Ill) existence of a zero (I) only ODGMY (111) only (I) and (11) only
(I) and (Ill) oniy
(II) and (111) only
(I), (II), and (III) A system of lineax equations is written in matrix form Ax = b, and the augmented matrix K is
then reduced to row echelon form, with the following result: OWWM 4
2
5
O DOOM
Oat4H 33 Find the solution of the original system. If it has a unique solution 3; , ﬁnd the value of x Z among choices (A) through (H). Otherwise, choose from choices (I) through (K). (A)
(B)
(C)
(D)
(B)
(F)
(G)
(H)
(D U) (K) 1
3
5
7
The original system has no solutions. The original system has inﬁnitelywmeny solutions. Since row operations do not always preserve solutions, it is impossible to determine the
solution of the original system from the above rowreduced matrix. i’art {1. True—False (one point each) Mark “A” on your answer card if the statement is true; mark “B” if it is false. 7. [it . 8219) = t‘gsin3t 8. Every diagonal matrix is nonsingular. 9. If the vector V3 is in the span of the set of vectors {V1,V2}, then the set {V1,V2,v3} is linearly
dependent. E0. Any two matrices which are row equivalent have the same column space. For problems 3—13, you may safely assume that the column vectors b and 0 have whatever size is
appropriate for the problem. 11. If A is a 3 x 4 matrix with rank 3, then the system Ax = b must be consistent. l2. If A is a 3 x 7 matrix, then the system Ax m b cannot have a unique solution. 13. The set of solutions of a homogeneous iinear system Ax 2: 0 is aiways a vector space, but the set of
solutions of a nonhomogeneous linear system Ax = b is never a vector space. Part III. Short Answer (one point each) The answer to each of these is right or wrong: no work is required, and no partial credit will be given. 14. Given that £(sin t sinh t) m 842:4 , ﬁnd £(e3tsin t sinh t). 1 2 3
15. Let A w 5 O 4 . What is the cofactor of the entry egg m 4?
6 5 1 Part IV. Free Response (point values as shown)
Follow directions careﬁlily, and Show all the steps needed to arrive at your solution.
(4) 17. Find the Laplace transform of the periodic ﬁmction f (t) pictured below. You do not need to simplify your ﬁnal answer. In other words, do all the calculus, but you do not need to do algebraic
simpliﬁcations at the end. (7) 18. Solve the following integral equation for ya). Mt) ~— 12fl: Mp) cos 10(t — p) tip :2 sﬁn 10t (3) 19. Consider the set V of poiynomiais of degree at most three whose constant term is zero. exampies of polynomiais which are in V: 91:3 + 4x2 — a:
4x2 exampies of polynomiais which are not in V: 91:3 + 42:2 m 2: + 3
4x2 w 1 V i_s_ a vector space (a subspace of P3). (You do not need to verify this.) (a) Find a basis for V. (b) What is the dimension of V? I 2 ml (3 m2
3 8 w3 4 m5
(4) 20. LetA: 9 2 0 4 1
m1 2 1 8 4 (3) Find a basis for the row space of A. (b) What is the dimension of the column space of A? (c) What is the dimension of the null space of A? Table of Laplace Transforms .—
0.)
c» —' (4)
— :
 0
O
m
g 1 2 52_a'2
3—ﬁ (6) a. a a.
’* a
"h
H a
it n. a ' (8) e coswt (swapwmg at ' {A} 9)
m Sammy
<2) 8%) w» w» « we) 
mww
was s
N) ) mum»
am) '
f(t) (with period 39) hips f9? 6""5‘ f (ﬂdt
ma) . F<s>a<s) (7) 6—623 Q
m
% I
9 !
Pi
“15
ﬂ
H:
V 1%) = 23%) ...
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 Spring '08
 Hastings
 Linear Algebra, Vector Space, Laplace, vector space properties, matrix form AX

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