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Unformatted text preview: Engineering Mathematics (ESE 317) Exam I September 21, 26“) This exam contains four multiple~ehoice problems worth two points each, £2 ﬂuefalse probiems worth
one point each, ﬁve short—answer probiems worih one point each, and three ﬁee~response problems
worth 15 poiots altogether, for an exam Iota} of 40 points. Part1, Multigie—Choice (two points each) Cleariy ﬁll in the oval on your answer card which corresponcis to the oniy correct response.
 M1 i ~63
I . F 1nd I: ( W 8 ) . (A) etcos 2t 6(t —— 6) (B) 85331215 5(15 M 6) (C) egtcostﬁt m 6) (D) egtsintﬂt —— 6) (E) et*ﬁcos(2(t — 6))u(t ~ 6)
(F) eiﬁsin(2(z m 6))u(t — 6)
(G) 82{t“§)cos(t w 6)u(t ~ 6) (H) 636435620? — 6)u(t —— 6) 0 if 0 < t <1 .
2. Let r(t) x {egt if‘t‘> 1 . F1nd£(r(t)). (A) w 5—5;; (B) 938 (C) 8 9+3 5—1—3 0)) 845+?) 11:3, (E) gi—gmeé (F) gig—W :33
(G) :3 8—H 3. Find the determinant of the following manix: (A) m1:
(B) “9
(C) W5
(13} *3
(E) M}
(F) 1
(G) 3
(H) 5
(I) 9
(J) 11 4. Solve the following system of equations. If there is no solution, select (A). If thore are inﬁnitely~
x many solutions, select (B). If there is a unique solution [y , ﬁnd the sum 33 + y + 2 among 7 at x + 23; Him 3:5 2 6
choices (C) through (I). Work oareﬁllly! { 333 + y  42 2 2 25:: + 4y + 3z 2 O
(A) no solution (B) inﬁnitely—many solutions
(C) w6
(D) —4
(E) 2
(F) O
(G) 2
(H) 4
(I) 5 Part ll. Tme~False (one point each) Mark “A” on your answer card if the statement is true; mark “B” if it is false. 5. iff and g are contiguous functions such that £(f(t)) m £(g(t)), then ﬁt} m 9(3). 6. For any two n x 2*; matrices A and B, det(AB) m det(BA). 7. The set of all solutions ofthe differential equation y” + 43; 2 cos 5x is a vector space. For problems 8—«10, cansider the vector space R3 of all ordered triples v m [01,32503] of real numbers. 8. The span ofthe set {[1, 0, 1], [0, l, 0]} is a subspace (£1323. 9. The set {[130, 0] [0, 1,0},{15L0l} spans R3. 3 H). The set {€13,119}, [0, LG], {1, LOE} is linearly independent. ., For problems 1143, consider the set V ofai‘l eiements v 2 {v13 2);, v3} in R3 such that ’02 2 0. 11. V is closed under addition. (in other words, vector space property 1.0 is satisﬁed by V .) 12. There exists a zero element in V. (In other words, vector space property 1.3 is satisﬁed by V.) 13. V is closed under scalar multipiication. (In other words, vector space property 2.0 is satisﬁed by
V.) 14. Suppose a system of equations is represented by an augmenteci matrix, and then this matrix is row—
redueed. If the rowreduced matrix contains a row of zeros, then the system of equations has
inﬁnitelyumany solutions. 15. If the n x n matrices A and B are row equivaient, then det A. 2 det B. 16. If A is a nonsinguiar n x n matrix, then its rank is n. Part HE. Short Answer {one point each) The answer to each of these is right or wrong: no work is required, and no partial credit wiil be given.
Give oniy one answer to each. (If you give more than one answer, the poorer one will count.) 17‘ The Lapiace transform is useful for solving differential equations because it transforms a
differentiai equation into what other type of equation? (Note that this other equation goes by the
name of the “subsidiary equation,” but that is not the question. What two of equation is it?) 18. On the axes below, draw the haiﬁwave rectiﬁcation of the function f (t) = sin t. Make your
picture accurate and neat so you get credit for this point. e e is
g _
,_.Wvéwmwww=mmg§.ic“Wsummons,mécwmcmWe,..§Lw.HWQWW M e“ we
1 2 0 3 G 2 1
For problems I9w21, refer to the matrix A z 0 0 1 O 1 O 0
G 0 O O G 0 0 19“ Find a basis for the coiunin space of A. 20. What is the rank of A? 2 1. Give an exampie of an eiernent of R3 which is not in the column space of A. Part IV. F 1‘86 Response (point values as shown)
Foliow directions carofuliy, and Show oil the steps needed to arrive at your solution.
(5) 22. Work out the following convolution. Simiaiify your answer compiet‘eiy. 1* t cosht (6) 23. Solve the following integral equation. (The equation is given in two equivalent forms. Look at
whichever seems friendiier to you.) w) + 4 1;:th m T)dr m 1 + 3t y(t) + 4f;y(p)(t m pup z 1 + 3: (4) 24. Let V be the get of all 3 x 3 matrices such that the entries in the four Gamers all match the entry in
the cenier, in ether words, such that a” m (213 m (122 w (231 x {133. Examples of such a matrix 2’ 1 2 2 2 2
weuld be ~5 2 G at even 2 2 2 . Then V is a vecter space. (You do not need to
2 0 2 2 2 2 verify this.) (21) Find a basis for V. (b) Find the dimension of V. Table of Laplace Transforms bi..” jg “31mm
F13) 9(3) ms) :2 ﬁﬂs) ...
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 Fall '08
 Hastings
 Laplace, LG, 0 g, 5 2 G

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