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Oct 2002 Term #1 - 5 ohm Department of Mathematics...

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Unformatted text preview: 5 ohm! Department of Mathematics University of Toronto WEDNESDAY, OCTOBER 30, 2002, 6:10 — 8:00 PM MAT 133v TERM TEST #1 Calculus and Linear Algebra for Commerce Duration: 1 hour 50 minutes Aids Allowed: A non—graphing calculator, with empty memory, to be supplied by student. Instructions: Fill in the information on this page, and make sure your test booklet contains 11 pages. In addition, you should have a multiple-choice answer sheet, on which you should fill in your name, number, tutorial time, tutorial room, and tutor’s name. This test consists of 10 multiple choice questions, and 4 written—answer questions. For the multiple choice questions you can do your rough work in the test booklet, but you must record your answer by circling the appropriate letter on the answer sheet with your pencil. Each correct answer is worth 4 marks; a question left blank, or an incorrect answer, or two answers for the same question is worth 0. For the written-answer questions , present your solutions in the space provided. The value of each written-answer question is indicated beside it. ENCLOSE YOUR FINAL ANSWER IN A BOX AND WRITE IT IN INK. TOTAL MARKS: 100 FAMILY NAME: GIVEN NAME: STUDENT NO: SIGNATURE: TUTORIAL TIME and ROOM: REGCODE and TIMECODE: T.A.’S NAME: TOlOlA T0501C LA341 T0101B T0501D NF 6 TOlOlC T0601A RW142 T0201A T0601B UC244 T0201B T0601C 882130 TOZOlC T0701A MP118 T0201D T0701B 882 130 T0301A T0701C RW229 T0301B T0801A RW142 T03OIC T0801B 88211 1 T0301D T5101A MP118 TO401A T5101C UC244 T0401B T5201B LM157 T0401C T0501A T0501B FOR MARKER ONLY Page 1 of 11 PART A. Multiple Choice 1. [4 marks] The nominal annual rate of interest at which $2500 Will grow to $7000 in eight years com— pounded quarterly is closest to 32 .13.08% .7000: ZKOOCH'fi) B. 13.74% 7000 ”L7. 1. 3 ., C. 13.15% r /I X4’v’307q'” 2500 D. 13.29% E. 12.94% H.035 0/o 2. [4 marks] If $2000 is deposited at the end of each quarter into a fund paying 8% compounded quarterly, then the fund will first contain at least $50,000 after A. 6 years and 3 months 50, 000 = 2000 Sm 01 B. 5 years and 9 months ‘ © 5 years and 3 months wlneae, h : VIO‘ op quawtéilg D. 4 years and 9 months E. 4 years and 3 months 2.; f' 0 OZ) ’- l n: gt. l 5 I); 9.0.”? qualée/J ’L I wtflf "0 4:. 57:5 4'1 7“ I wan/7w Wd-b‘ gmauL L00 4 [5* 7 {ny if 7 wtl CID Page 2 of 11 3. [4 marks] A $15,000 car sells for $2,000 down with the remainder to be paid by equal payments made at the end of each month for 3 years. If interest is 8.4% compounded monthly, then each payment is A. $423.36 [3 000 7.. R a a $4 ) 3707. 007 . 09.78 C. $417.44 1: [5/ 000% ,007 F, " 09 313 M N’ I D. $395.21 l“ U OO4)/3é E. $429.03 4. [4 marks] The price on September 5, 2002 of the following bond Issuer Coupon Rate Maturity Date Yield to Maturity Manitoba 10.500 March 5, 2031 5.90 is closest to -5? _ A. 133.19 , + P” 1000,0245) 5.2m mom; B. 148.14 C. 155.62 x I (a 3 . O (i q ’1‘ D. 162.75 (V @m ~- Page 3 of 11 5. [4 marks] 0 ——1 3 2 4 2 5 —1 + 0.5 —6 0 1 1 O 2 0 0 —-1 A. [2 5] O 0 —1 3 0 2 1 7" B. 2 5 —1 —3 0 0 1 1 0 1 O ——2 F—fir—fir—‘fi l tor—4o r—Icnr—A ll NH“; L___.._l l .mmwMM,,.,m.~.«...-....~w.~ , . .52.“. , 6. [4 marks] The system 751: —+— 2y — z : 5 x + y + 2 = 2 { -3:E — y = 0 has A. no solution. a unique solution. C. infinitely many solutions depending on one parameter. D. infiniter many solutions depending on two parameters. E. infinitely many solutions depending on three parameters. II I 2 | '1 l 2 77f'/ 55:12") (2-5.: —q ’23:? 015:: 3 '3‘l00%:79|013(¢221100"%é 3*93‘95 R: {4R3 “21 _. e Salm JUN/Hill 7:41 Y13-3i2:2l or Cawwée‘ 4&540 l x< 2-2’7 r? a Wtqwc WV" Page 4 of 11 3 1 0 531 +962 -1- $3 +2233 2$1 - $2 —||I._2 /1 1 4. / 1 _ 2 4 /O/ 00 1 1.... _ 2 / 1 1 1 0 1/2 0 0 1/4 —1/4 1 1/2 —1 0 0 1/2 1 0 0 1 1/2 1 0 0 2 0 —1 1 0 0 1/2 —1/4 —1 ,1 _ 1 0 2 1/2 0 1 1 1 E 8 [4 marks] 1 A B. C In the solution to the following system: Then F 7. [4 marks] Let 3" 2/ a) , 31/4 117, ,3137 1.10 1 , 1.1/1 H. / ([0 0 112’ 5 (100100 Mb 1 \l 6 .m 3 $ fl. 0 e4 2 4. m__74_5__74_7_ V e . . . . m A B can” 9. [4 marks] Let 1 0 1] [1 0 1/2] H: 2 —1 0 0 —1 5 0 —2 0 0 0 7 Then the determinant of H is |/ A. 0 I C? ' l O 7' 28 lHl” 2 “l O 0 I. { C. —5 0 '1 O 0 O 7 D. 7 .. J iii/J {‘01 j “A; QM 1/613 [0/ . g j E —14 @460” ‘” r 04 (If 10 Kl ‘9‘ A the ll» " 7 10. [4 marks] If A is a 3 X 3 matrix such that [A] gé 0, then which of the following statements is false? A‘ l-A|=-IAI l~A|='(-‘I)3lAl¢ «IN 5.!» Me 3 IAlelAy true C. |2Ay=81Al liAl‘ leAlt'Zl/Oll 60 {Sue D. lA“1y:% ewe, @There is a 3 X 1 non—zero matrix X such that AX = 0. El: IAHO} A IS (”W/{Me} Lat: ‘tlem ya A"A>< 7 A’tflco W..ze/o X cam/2'6 V be “VI Page 6 of 11 PART B. Written-Answer Questions 1. A person Wishes to save $50,000 by making equal deposits at the end of each month for 12 years. Interest is 6.6% per year compounded monthly. [5] (a) What should be the amount of each deposit? [10] (b) Immediately after the 84th deposit, the interest rate changes to 5.4% compounded monthly. Accordingly, the remaining deposits (which are to be monthly and equal to each other) must be adjusted to still achieve the goal of $50,000 in a total of 12 years. What should be the amount of each of the remaining deposits? L: 9%:1002’5/ a‘l’ 45‘»; wéjfiméflj Fl WLHJA “1"” aCCumulcfke/ >650 fl 3’) fig 3. (9(0 0.0045 '23 Page 7 of 11 2. An $80,000 mortgage is to be repaid by equal payment made at the end of each month for 10 years. Interest is 7% compounded semi—annually. [5] (a) What is the amount of each payment? [5] (b) How much principal is outstanding at the beginning of the 9th year of the repayment schedule? [5] (c) How much interest is included in the 97th mortgage payment? 0,0301 :QM)” a) $0,000 ‘ R 0.707% fig R: 30 Omit _, 30/000 [0‘03{ {)‘20 W \ 0 ”‘7’ co # 24.75 Page 8 of 11 [13] 3. A rental office has two types of apartments for rent. To prepare an apartment of type A, 2 labor hours of painting and 3 labor hours of cleaning are needed. For an apartment of type B , 4 labor hours of painting and 3 labor hours of cleaning are needed. There are 20 labor hours of painting and 18 labor hours of cleaning available. Use inverse matrices to solve a system of equations that determines how many apartments of each type can be prepared if all the available labor hours are to be u 'lized. L66“ YA: mama 0155 a, 17/6‘: f ZXA’l'Lle 17.0 3XA+ BXBclfi' Pam‘lu/‘f, 0 [50%le Page 9 of 11 1 0 —1 0 0 a 1 2 4. Let A— (a 0 _1 3) 1 —a 0 1 [5] (a) Find the determinant of A in terms of a. [Suggestion: Use some row and/or column reduction] [3] (b) For what value(s) of a will A not have an inverse? [3] (c) If AX = B Where then find :33 when A has an inverse. [Suggestiom Use Cramer’s Rule] [6] (d) For each value of a for Which A does not have an inverse, find all possible values of :33 . Why can’t you use Cramer’s Rule? o »I 0 a ' 2 l O “l O [O a | 7. 1 0 0,4 3 k7 l5 0 a ll ’7; c O 0 a" 3 ~a | l 60‘: a 00x ’67 | O (a l l J ‘ P a/l 3 “bk/5% [‘7 a | 1 b7 ‘96 5V / 0 ad 3 j a 2 3 0 1 3 , BQCQ’S) [3,») A \AaS mo own/668 0 1| 0 (1) far aéFO/(5 \ I 2 W60 0 g ”I 3 6'1/164 WW" 3%“ W 3 fl owl a/Cfr/‘(Jw‘me X3 ’ | 0 "I 5: gig [Mariam Z 39 ,. a I _a 0 ' 0]) TWO ca1€553)0?0 (f) ab/Z” 35/ 6 W144 ) l/ C «49606 (015 ”£ng e04“ [E (C, CUL‘Mg’S Qwe’ a na‘éa/ (A Claude/'5 u~ W56: car/s {fl/w Ohm)»: [s (9‘ Page 10 of 11 (Extra, page if necessary) .«I I l 7. W) 3 | 1| Page 11 of 11 ...
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