MA 262 Spring 1993 FINAL EXAM INSTRUCTIONS. 1. You must use a #2 pencil on the mark sense sheet (answer sheet).

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1 MA 6 Spring 993 FINAL EXAM INSTRUCTIONS NAME. You must use a # pencil on the mark sense sheet (answer sheet).. On the mark sense sheet, fill in the instructor s name and the course number. 3. Fill in your name and student identification number and blacken in the appropriate spaces. 4. Mark in the section number, the division and section number of your class. For example, for division, section 3, fill in 3 and blacken the corresponding circles, including the circles for the zeros. (If you do not know your division and section number ask your instructor.) 5. Sign the mark sense sheet. 6. Fill in your name and your instructor s name above and on the first page of the question sheets. 7. There are 5 questions, each worth 8 points. Blacken in your choice of the correct answer in the spaces provided for questions 5. Do all your work on the question sheets. Turn in both the mark sense sheets and the question sheets when you are finished. 8. 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. 9. NO CALCULATORS, BOOKS OR PAPERS ARE ALLOWED. Use the back of the test pages for scrap paper.

2 . If y = 4(x )3, y >, y() = 3, then y() = y A. 5 B. 5 C. D. E. 9. If xy + y = 4x and y() =, then y(e) = A. e B. e C. e D. e /4 E. e 3 3. If dy dx = xy, y and y() = then y() = x + y A. B. C. D. + E. 4. The general solution of y (4) 6y = is A. c e x + c xe x + c 3 e x + c 4 xe x B. c e x + c xe x + c 3 cos x + c 4 sin x C. c e x + c xe x + c 3 cos x + c 4 sin x D. c e x + c e x + c 3 cos x + c 4 sin x E. c x sinx + c x cosx

3 5. A clay pot is removed from the kiln at PM and allowed to cool in a large room of temperature 75 degrees F. After minutes the pot has cooled to 475 degrees F. and after 4 minutes, to 75 degrees F. What was the temperature of the pot when it was first removed? 6. An annihilator of x e x + 3e x cos(x) is 7. A particular solution of y 6y + 9y = xe 3x has the form 8. If x y 6xy + y =, y() =, y () = 3, then y() = A. 675 B. 75 C. 85 D. 875 E. 95 A. (D + ) + (D ) 3 B. (D ) 3 (D + ) C. (D ) 3 (D D + ) D. (D 3 )(D D + ) E. (D ) (D i) A. Ae 3x B. Axe 3x + Be 3x C. Ae 3x + Be x D. Ax e 3x + Bxe 3x E. Ax 3 e 3x + Bx e 3x A. B. C. 4 D. 8 E. 6 3

4 9. A body attached to the lower end of a vertical spring with no external forces has acceleration d x dt = 5x ft/sec, where x = x(t) is the distance of the body from equilibrium position at time t. If the spring is stretched so that at t =, x() = 5f t and the initial velocity is v() = ft/sec, then x = A. 5 cos5t B. 5 cos5t + 4 sin5t C. 5 sin 5t D. 5e 5t E. 4 cos 5t + 3 sin5t 3. Find the determinant of A if A = A. 8 B. C. D. -6 E. 6. If A = then the third column vector of A is A. C. E B. 8 D

5 . Which of the following are subspaces of C, (the set of functions from the unit interval to the reals that have at least derivatives)? i. set of all functions f such that f(.5) = ii. set of all functions f such that f + f = e x iii. set of all functions f such that f + f + f = iv. set of all functions such that f() = f() A. i and iii B. ii and iii C. i, iii and iv D. ii E. ii and iv 3. Let T: R 4 R 3 be defined by T(x) = Ax where A = Then a basis for ker(t) is 5 A., B.,,, 4 C. 3 D., 3 E., 4

6 4. Find the dimension of the range of the linear transformation given by the matrix A, R 4 R 4 where A = A. 3 B. C. D. E Two liter mixing tanks are initially full of salt water and connected as follows: ) Tank has an input pipe with a solution containing grams/liter of salt flowing into tank at 6 liters/second. ) Tank has an outlet pipe so that mixed solution drains out of tank at the rate of 6 liters/second. There are two pipes between tank and tank 3) In the first one, mixed solution is pumped from tank to tank at the rate of 8 liters/minute 4) In the second pipe, mixed solution from tank is pumped to tank at the rate of liters/minute. For what choice of matrix A and vector b does the system of diff. equations x = A x + b best describe the change in ammounts of salt (x, x ) with time? 8 A. A = (/) C. A = (/) 8 8 E. A = (/) 8 8 8, b =, b =, b = 8 B. A = (/) D. A = (/) 8 8, b =, b = 6. If T: R 3 R 3 is a linear transformation such that T(,, ) = (, 3, 5) and T(,, ) = (3, 4, 5), then T(3,, ) = 6 A. (8,, 5) B. (7,, 5) C. (5, 7, ) D. (,, ) E. (,, )

7 7. If A is an n n matrix with det(a) = then which of the following statements are true (i) Ax = has only the zero solution (ii) RankA < n (iii) At least one eigenvalue of A is. (iv) A is non singular (v) Ax = b has a solution for every b in R n. A. (i) and (iv) B. (ii) and (iii) C. (ii) D. (iii) and (v) E. (iii) 8. Given the vectors v = (,, ) and v = (,, ) which of the following choices of v 3 will make the set { v, v, v 3 } a basis for R 3 : i) v 3 = (,, ), ii) v 3 = (5, 5, 6) or iii) v 3 = (7, 8, 9) A. (i) and (iii) B. (ii) and (iii) C. (i) D. (i) and (ii) E. (iii) 9. Determine which of the following transformations T: R 3 R 3 are linear transformations. (i) T(x, x, x 3 ) = (x, x + x + x 3, x ) (ii) T(x, x, x 3 ) = (x + x,, ) (iii) T(x, x, x 3 ) = (x + x, x x 3, x ) 7 A. (i), (ii) and (iii) B. (i) and (iii) C. (ii) and (iii) D. (i) and (ii)

8 8 E. (i)

9 . If A =, then the eigenvalues of A are 4. The matrix A = 4 is A. 3 and B. and 4 C. and 3 D. and 6 E. None of the above 7 has eigenvalues 3 and 4. An eigenvector corresponding to 6 A. B. 3 3 C. D. E.. The matrix A = 3 has as an eigenvalue. What is the dimension of the 7 7 eigenspace corresponding to eigenvalue? A. 4 B. C. D. 3 E. 9

10 3. A real matrix A has an eigenvalues and. The corresponding eigenvectors are and. A solution to x = A x such that x() = is 4 A. B. C. D. E. 3e t e t 3e t + e t 3e t e t 3e t + e t 3e t e t 3e t + e t 5e t 3e t 5e t e t e t 4e t 4 e 4. If A = and one solution of the equation x 4 4t = Ax is x = independent solution is A. B. C. D. E. te 4t + e 4t e 4t te 4t e 4t e 4t te 4t + e 4t e 4t te t, then another

11 e 5. The system x t e = Ax has the fundamental matrix X(t) = t e t 3e t e solution to x t = Ax + is x(t) = X(t)u(t) where u (t) is e t A. B. C. D. E. + e t e t + e t + e t e t e t e t e t e t 3e t e t e t + e t. A particular

June 2011 PURDUE UNIVERSITY Study Guide for the Credit Exam in (MA 262) Linear Algebra and Differential Equations

June 2011 PURDUE UNIVERSITY Study Guide for the Credit Exam in (MA 262) Linear Algebra and Differential Equations June 20 PURDUE UNIVERSITY Study Guide for the Credit Exam in (MA 262) Linear Algebra and Differential Equations The topics covered in this exam can be found in An introduction to differential equations