exam 2 fall 2009 - Engineering Mathematics (ESE 3 l7) Exam...

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Unformatted text preview: Engineering Mathematics (ESE 3 l7) Exam 2 October 7, 2009 This exam contains nine multiple-choice problems worth two points each, ll true-false problems wetth one point each, three shortenswer probiems worth one point each, anti two free-response 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: l cam. L—.—l P 100 mm: mm. (JIME- 01m be WWI-mm“: P O W @ A km W r"'—~1 r—-—1 F'—""""l rm"! “0 I-—ll 01 mm: C} W W E} “CD CD «.1 3w E (mifl 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. True-Fake (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, (En-b)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 three-space, 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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exam 2 fall 2009 - Engineering Mathematics (ESE 3 l7) Exam...

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