math2243 - Math 2243 Name (Print) Spring 2003 FINAL EXAM

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Unformatted text preview: Math 2243 Name (Print) Spring 2003 FINAL EXAM Signature_______________.____._——— Recitation Instructor Section__.__. l.D.# READ AND FOLLOW THESE INSTRUCTIONS This booklet contains 13 pages,including this cover page. Check to see if any are missing. PRINT on the upper right—hand corner all the requested information, and sign your name. Put your initials on the top of every page, in case the pages become separated. Textbooks, notes and calculators are not permissible. Do your work in the blank spaces and back of pages of this booklet. Show all your work! There are 11 machine-graded problems, each worth 9 points. There are 6 hand-graded problems of varying worth for 101 points. This gives a total of 200 points which will be rescaled to 300 INSTRUCTIONS FOR MACHINE-GRADED PART (Questions 1-11): You MUST use a soft pencil (No. l or No 2) to answer this part. Do not fold or tear the answer sheet, and carefully enter all the requested information according to he instruc- tions you receive. DO NOT IVIAKE ANY STRAY MARKS ON THE AP-T'SWER SHEET. When you have decided on a correct answer to a given question, circle the answer in this booklet and blacken completely the corresponding circle in the answer sheet. If you erase something, do so completely. Each question has a correct answer. If you give two different answers, the question will be marked wrong. There is no penalty for guessing, but if you don’t answer a question, skip the corresponding line in the answer sheet. Go on to the next question. INSTRUCTIONS FOR THE HAND-GRADED PART (Questions 12-17): SHOW ALL WORK. Unsupported answers will receive little credit. Notice regarding the machine graded sections of this exam: Either the student or the School of Mathematics may for any reason request a regrade of the machine graded part. All regrades will be based on responses in the test booklet, and not on the machine graded response sheet. Any problem for which the answer is not indicated in the test booklet, or which has no relevant accompanying calculations will be marked wrong on the regrade. Therefore work and answers must be clearly shown on the test booklet. AFTER YOU FINISH BOTH PARTS OF THE EXAM; Place the answer sheet between two pages of this booklet (make a sandwich), with the side marked “GENERAL PURPOSE ANSWER SHEET” facing DOWN. Have your ID card in your hand when turning in your exam. Multiple choice part ______ Hand—graded part Total 16 1 Letter Grade _____ Mmm I ,7 , I" l 11 H 2a; L_.zl!:‘.‘1,.U:$U Q}; gugyz H5“; My): fl 1 {fine '3'}, 33,5, {SQQEE _ DUthtal . t 1Le 3 y’ — —y : sin .73. :1; Then an integrating factor for this linear differential equation is given by: A. 11:3 B. g C. m‘3 E :1: Di e— 81:» Be 2. The differential equation, d2y dy _ 2— = dtz + dt +53; 0, describes the motion of a loaded spring from its rest position, y = 0. If y(0) = 0 and %(0) = —1, then the first time the spring returns to y = 0 is: Ala :4 .U .0 w .> Ali? wl‘s’ we 3. P is the family of parabolas y 2 09:2. Then the orthogonal trajectory through (1, l) is: A. y2 + 2x2 = 3 B. y = —-%—:c + g C. 3:2 + y2 = D. 11:2 + 3y2 = 4 E.x2+2y2=3 4. A change of variables that Will turn into a linear differential equation is: A.u=y% "% B Uzi—mm C.u=}i D.u=cc3y E.u=y-% '5. Let T2R2~+R3 be a linear transformation and let T(1, —2) = (1, ——1, 1), T(2,3) = (2,1,1). Then T(1,0) = A “00 H 01 \_/ \lh—I ash- ‘1'“ (2,5, —2) ( . (7,2,5) 6. Let Then the cofaetor of a2,3 is: A. 10 B. 20 C. —20 D. —10 E. —8. 7. Let 2 A: l b—‘lOO 03030 —1 We know that there is an invertible matrix S and a diagonal matrix D, such that S"1AS = D. Then this diagonal matrix D can be chosen to be: A. diag(0,2, 5) B. diag(2, —2, 3) C. diag(—-2,0, 5) D. diag(5,2,2) E. dia.g(5,2, —3) 8. Let M2 (R) denote the vector space of real 2 x 2 matrices. Then a basis for this vector space is given by: At 8% [8 8H: ‘5}- 3. a], [3 3], -3 ‘2’], i8 3]- 9. Let Then, detA 2: A. —20 B. 10 C. ~10 D. 20 E. 0 10. Let Then 3:1 2: A. clet +‘cze—2t B. 01th + age—t C. cle’at + C282t D. C1€3t + c262t E. 016‘ + 02 1 1 2 —1 —l 1 1 2 A —_ 2 -2 1 1 1 ——1 1 —2 (D — 1):c1— 2$2 = 0 and —2:1:1 + (D + 2)a:2 = 0 11. Let P4 be the vector space of polynomials of degree 3 3. Then dimP4 2: A.4 B3 (3.2 D.5 E6 12. (26 points) Find the solution to the initial value problem: gig: ‘flx, 32(0): l3.(15 points) Are the functions 62‘, e‘3t,t linearly independent on the line (—00, +00)? Prove your answer. 14 (15 points) Let Find the rank of A- 15. 15 points) Let A be the matrix of Problem 14. Find all the solutions to the system of algebraic equations: A11: = O. 16.(15 points) Let A be an arbitrary 3 X 5 matrix. Show that the vectors Arr, as :1: varies over R5, form a subspace of R3. 17. (15 points) Find the general solution for the differential equation 2 e: 2/ d3: $24-1- ...
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math2243 - Math 2243 Name (Print) Spring 2003 FINAL EXAM

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