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Unformatted text preview: Engineering Mathematics (ESE 3 l7) Exam 2 October 7, 2009 This exam contains nine multiplechoice problems worth two points each, ll truefalse problems wetth
one point each, three shortenswer probiems worth one point each, anti two freeresponse problems
worth eight points altogether, for an exam total of 40 points. Part I. MultipleaChoice (two points each) Clearly ﬁll in the oval on your answer card which corresponds to the only correct response. 1. A 3 x 4 matrix A has rank 2. Describe the solution set of the nonhomogeneous system Ax m b. (A) The solution set is a line in R3. (B) The solution set is a line in R4. (C) The solution set is a plane in 33. (D) The solution set is a plane in R4. (E) The solution set might be a line in R3, or it might be the empty set.
(F) The solution set might be a line in R4, or it might be the empty set.
(G) The solution set might be a plane in R3, or it might be the empty set.
(H) The solution set might be a plane in R4, or it might be the empty set.
(I) The solution set is the empty set. 15
42 there is only one eigenvalue with multiplicity two, answer with that eigenvalue.) 2. Find the eigenvaiues of the matrix A = What is the larger of the two eigenvalues? (If (A) «3
(B) ~2
(C) —1
(D) 9
(E) 1
CF) 2
(G) 3
(H) 4
(I) 5
(3) 5 .75 .35
.25 .65 a:
3! 748] [568] [496]  generates the Markov chain [ 20 200] , 272 3. The stochastii: matrix A 2 [ (Y on do net need to verify this.) If [ ] is the limit of this Markov chain, what is the vaiue of a? (A) 301
(B) 328
(C) 320
(D) 323
(E) 326
(F) 442
(G) 445
(H) 448
(1) 450
(J) 467 4. Find the work done by the force 3 m [311:3] acting to move an object from the point
A x (2,2,1) to the point B = (6, 0, ~31) along AB. (A) —s
(B) 4 (C) 16 (D) 20 (E) M573 (F) m (G) m8? — 6}" m 10?
(H) a? + 6}" + 10"}?
(1) W4? + a? w 2"}? A ~+ up
(J) 4i m6} +2k 5. Find an equation of the piane through the points (1, 3, 1), (2, 3,4), and (2, 4, 1). Aﬁer you obtain
an answer, simplify it to the form as: + by + cz m d, where c 2 1. What is the value of d? (A) ——9
(B) —7
(C) "5
(D) “*3
(E) “*1
0:) 1 (G) 3 (H)5
(I) 7 (D9 6. Find the normal acceleration vector 3mm for the curve represented by the vector equation
?(t) m [5,712,2t] at the point (5,4,4). (A) [02 "‘23 “2]
(B)
(C) 5:
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z.__: 5 9 ’7. What type of curve is represented by the vector equation mat) = [16, $30] ‘? (A) line (B) eiﬁpse (including the possibiiity of a circie)
(C) hyperbola (D) helix (E) none ofthe above 8. Find the length of the are traced out by the vector function ?(t) :2 [3 sinh t, t, cosh t} from the point
corresponding to t = O to the point Corresponding to t = 1. (A) 2 cosh 1 (B) 2 sinh 1 (C) 2(coshl ~— 1) (D) 2(sinh1  1) (E) m cosh 1 (F) mainm (G) mwoshl — 1)
(H) ﬁéﬁinhl m 1) (I) 3 sinhl + cosh 1 (J) 3sinh1+cosh1 +1 9. Let Km, 2:) = zinx + 3;, let P = (1, 2, 3), and iet "a" m {2,1,4}. Find (pgfo). (A) 1
(8) g
(C) 3
(D) 7’ (E) 3326 Part II. TrueFake (one point each)
Mark “A” on your answer card ifthe statement is true; mark “8” if it is faise. 5 1 3
10. Consider the matrix A m 0 1 2 . The following set is a basis for the row space of A.
0 G O [1 3 sun 1 2],[o o 0] 11. Let A be 3 5X6 matrix. The dimension of the column space of A pins the dimension of the null
space of A equals 6. 12. Row equivalent matiices have the same eigenvalues. 13. The dot product of two unit vectors equals 1 if and only if the vectors have the same direction. my
14. For any two vectors 3’ and b , a) ~+ —+
15. Foranythreo vectors 3’, b , and ‘3, (Enb)x "3 m ('3 X 3)(b x ‘6’). 16. Suppose z: is a function of :1: and 3;. Then 8:: 2 g—: » ('33:. 17. ’fhe gravitatiooai ﬁeld is conservative. 18. Let f be a scaiar ﬁmction. Then div(grad f) m 0. 19. The function f (x, y, z) = a: + 23; + 52 is a solution ofLaplace's equation. 20. Consider the motion of a ﬂuid in a region around a point P in threespace, and let
ii(m,y,z,t) = p(:c,y,e,t)—J(x,y,z,t), where p is the density ﬁmction and "i? is the velocity
ﬁmetion. If the ﬂuid is compressibie and there are no sources or sinks, then div if m 0 throughout
the region. Part III. Short Auswer (one point each) The answer to each of the following is either right or wrong: no work is required, and no partial credit
Wiii be given. 21. Suppose that the linear system of equations Ax :2 b is inconsistent, where A is a 3x 7" matrix. What
is the maximum possible rank of A? 22. Consider the right triangle pictured below, formed by the vectors ’a"), and '3 having lengths 3, 4, and 5 units respectively. What is the magnitude of "5 x ii? 23. Find a vector in the direction of the line given by the following parametric equations. x=5tm2 ym—t—i zm2t+8 Part IV. Free Resgonse (point values as Shown) Follow directions carefully, and Show all the steps needed to arrive at your solution. cos 8") (4) 24. Let w m , where g: m sin3t and y 2 2528:. Use the chain rule (not substitution) to ﬁnd You (i0 33;: need to simplify your answer. {4) 25. Finci a vector equation mat) = [356), y(t), 265)] for the foliowing curve. x2+8m+22==0 z=2y ...
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 Fall '08
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