math152_april2003

math152_april2003 - Be sure that this examination has 11...

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Unformatted text preview: Be sure that this examination has 11 pages including this cover A The University of British Columbia Final Examinations - April, 2003 Mathematics 152 All Sections Closed book examination - I Time; 2_5 hours Name Signature Student Number____._____ Instructor’s Name " Section Number Special Instructions: No aids allowed. Rules governing examinations 1- Each candidate should be prepared to produce his library/AMS card upon request. 3. Read and observe the following rules: No candidate shall be permitted to enter the examination room after the expi- ration of one half hour, or to leave during the first half hour of the examination. Candidates are not permitted to ask questions of the invigilators, except in cases of supposed errors or ambiguities in examination questions. CAUTION — Candidates guilty of any of the following or similar practices Shall be immediately dismissed from the examination and shall be liable to diBCiPIinary action. (3-) Making use of any books, papers or memoranda, other than those au- th°rized by the examiners. 0’) Speaking or communicating with other candidates. (c) Purposely exposing written papers to the view of other candidates. The plea of accident or forgetfulness shall not be received. 3. Smoking is not permitted during examinations. ‘ Marks l '11 i [20] April, 2003 1. MATH 152 Name Page 2 of .11 pages Let 3’ = [1, 2, -3]_and ‘17 = [3, 2, —1]. (a) Is the smallest positive angle between W and "15’ larger than a right angle or smaller than a right angle? How do you know this? (b) Find a nonzero vector, 3:", that is perpendicular to both 72" and 717. (e) If E) is also perpendicular to both V and 717, how Would E’ be related to ’17 found in part (b) above? Continued on page 3 V April, 2003 MATH 152 Name M Page 3 of 11 pages (d) Find a plane through the point (—1, 0, 1) and containing a vector parallel to '17 and a vector parallel to ‘13. ' V .THE UNIVERSITY OF BRITISH COLUMBlA Counselling Services Student Development & Services 12-00-1874 East Mall Vancouver, B.C. V6T 121 (é) Find thepoint P on the line 5' = 7&7 + t? nearest the point (—1,0, 1). Continued‘ on page 4 MATH 152 Name .____.______..____._____ Page 4 of _11 pages .prfl,‘ 2003 [10]. ,2]. Consider the system of equations 2m1—5m2~a:3=6 -.'1:1 +3$2+4$3 =7 3:111 —- 7m + 22:3 = 19. Write down the augmented matrix of this system. Convert it to a row-echelon form and mark the pivot columns clearly. Then write down the general solution of the System. Continued on page 5 April, 2003 MATH 152 Name ____________ Page 5 of .11 pages Let 1 2 2 B = 2 3 O ——1 1 k where k is a real constant. (a) For what values of k is B invertible? ‘ (b) Find 3-1 "if k = 11. Continued on page 6 Apfil, 2003 MATH 152 Name ______+__ Page 6 of .11 pages , Let A be a 3 X 3 matrix and 71, .172 vectors in R3. It is given that the equation [10] 4 _ _ —_, I u -+ a n n A? = b 1 IS CODSISteIlt and A? = b 2 IS 1ncons1stent. (a) What is the largest number of pivots A can have? (b) Show that the equation A? = 2.53 — 71 is inconsistent. .4 Continued on page 7 April, 2003 MATH 152 Name _____—_—_——_ Page 7 of .11 pages [15] 5. Let C be the standard matrix of the two-step linear transformation of R3 that first projects all vectors into the plane with equation :1: = 0 and then rotates all vectors by 60° counter clockwise about the z-axis when viewed from above. (a) Find 0. (b) Find C‘f if it exists. If it does not exist, explain why. Continued on page 8 April, 2003 MATH 152 Name ________.____ Page 8 of _11 pages (c) Find one eigenvector and corresponding eigenvalue for C. Continued on page 9 April, 2003’ MATH 152 N ame __________——___ Page 9 of .11 pages [15] 6. Find the variables m(t) and y(t) if they satisfy the conditions that $’(t) = $(t) - W) 3/0?) = flit) + W) with$=4andy=0whent=0. Continued on page 10 mil, 2003 MATH 152 Name .._._________ Page 10 of _11 pages Which of the following assertions is TRUE or FALSE. Justify your answers with a. clear reason or a counterexample. ‘ (a) The value of det A is not changed by elementary row operation on A. i (b) An n x n matrix with real entries has n real eigenvalues (counting multiplicities). (c) A 3 x 3 matrix with eigenvalues 1, 5, 2 is diagonalizable. Continued on page 11 April, 2003 MATH 152 Name .______._._._._________ Page 11Lof11 pages (d) A 3 x 3 matrix with eigenvalues 1, 0, 1 is invertible. (e) If the 3 x 3 matrix A has eigenvalues —2, —1, 0, then all solutions of . I 7 71? (t) = approach 0 as t —> oo. ...
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