2006 - Department of Mathematics University of Toronto...

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Unformatted text preview: Department of Mathematics University of Toronto WEDNESDAY, November 1, 2006 6:10-8:00 PM MAT 133Y 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 10 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 theranswer 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: LM 155 T0501D $51074 T0601A 881084 T0601B T0701A T0701B T0701C T0801A T0801B T5101A T5101B T5201A T0101A TOIOIB T0101C T0201A T0201B T0201C T0201D T0301A T0301B T0401A T0401B T0501A T0501B T0501C M9A NIQE W3D R4A R4B F2A F28 F20 F3A F38 M5A M5B M6A FOR MARKER ONLY BF 323 MP 137 RW 143 MP 137 351087 LM 162 881083 551073 881069 882108 352110 852110 $81083 $81084 SS2127 SS2102 RW 110 MP 203 Page 1 of 10 NAME: 1. [4 marks] STUDENT NO: PART. A. Multiple Choice An interest rate of 8% compounded semi-annually corresponds most nearly to an effective rate of A. 8% B 8.16% C 8.203% D. 8.310% E. 12% 2. [4 marks] 1. \‘fr‘g: I, .0319 r¢= Lezlb If 24 quarterly payments of $200 are deposited into a savings account earning 4% interest per year compounded quarterly starting immediately, then at the end of 6 years, the amount in the account will be closest to: A. $5,648.64 B. $6,094.73 C. $5,448.64 D. $4,848.00 E. $5,394.69 {4 ,0! I 23 2‘! 200 200 Page 2 of 10 NAME: STUDENT N O: 3. [4 marks] A $10,000 loan is amortized over 5 years with equal semi-annual payments. If the interest rate is 8% per year compounded semi-annually, then the principal repaid in the first payment is 10,000 = R 6! 1,0104 A. $762.47 L/ B. $795.38 R '- “9,000 g t0,000r.0 ’IO C. $806.21 C‘ WM 1., ((104) D. $832.91 E. $853.64 Rx12324! J . [S @1er .g -6<¢ ~0th ref“ ; 0 ,or/y [0,.MO 4‘0 3 Zita/400;: 332.9] Prlncifnl refa‘i [7'3 @ 4. [4 marks] A $100,000 loan is amortized over 10 years at 5% per year compounded quarterly with quarterly payments. The interest in the last payment is a Mel/f6; A150 IPRisflmszzcw-OM Murat/fax ) i ? 19¢ lard /’&W B. $39.41 036/ {/9 (fl (Aflfg 0 6w; faker/mi {Mt-é a (y i? C. $107.63 ‘ pr. D. $235.81 RC2,0[95 6% VAL, E $1250 00 a 1 (to, mica/6% M PWMMZZ; R 6mm)" I I gmoa Rx, M Page 3 of 10 NAME: STUDENT NO: 5. [4 marks] What is the market price of a $1,000 bond having 8 years until maturity, semiannual interest payments of $36 each, and an annual yield rate of 8%? ’19 A. $942.60 Pg I000 (L04) +3éa’7 m .04 B. $953.39 C. 935.21 $ at 453,367 @ D. $967.85 E. $929.46 6. [4 marks] ' 9-. 0 7 17 3 . 2 Let A=3 1 ,B=3 18 —3 ,C=l(5 5 5), D=|(1 2).. —1 12 2 7. u Which of the following products of matrices doesn’t exist? ( 3 ) I )Imrofl b . I A. ATC'B A6 [5 1x3; c3 .3 nxz ‘ ‘ 9- B .‘s 3%13‘590653 0K B. ACB AC I: 3‘3) 6‘ . ‘ ' 30K C. BTAC BC .3 2x3; A0534?!” So 9 AC! D. (AD)TB AD is 3:22, E. CAD CA is '1 Page 4 0f 10 NAME: STUDENT NO: 7. [4 marks] 10—2 2—1 3 0—1 3 LetA= 21 0,B=1—1 0,0: 1 3—1. —13 5 1—2 —3 —1 2 0 Which two matrices have the same number of non—zero rows when they are put in row-echelon form (otherwise known as reduced form)? A. A and B B. A and C C. B and C D. They all have the same number of non-zero rows in row-echelon form. E. No two of them have the same number of non-zero rows in row-echelon form. fig; {9 .«Q‘a \0 ~ 1 I o *2 a “W;ch AfiEa 3 saw—efi'q w"? 0‘” rwS King 3 {j 0 '33 O 0"“? I ——I o 1’10 \": 2 “WAC” (aw—7 2,: 3 a o I 3 -—-? 0 '0 NW“ [(2-5 0 «(’3 0 O I «3 «I I s w: I 3 " '3 woW‘éw’O -r W _Cr—7 o ~—l 3 /“)(O l '3 ‘9 D l W3 W .—I Q o 0 5", 0° @ 8. [4 marks] Consider the system 23: —2y +32 +11) = 0 ROW 7' 1: ——9y +72 +21) =0 Rowi 3:r +5y — z +w =0 Raw? :3 +73; —42 = 0 Row ‘f Which of the following statements is true? . R‘ H ‘ S a“ me, A. The system has a unique solution W; H {8M4} all ge'o E . B. The system has no solutions ho v.” WW? ' C. The system has a 1-parameter family of solutions D.' The system has a 2 -parameter family of solutions E. The system has a 3 -parameter family of solutions f . a 2.. . .I _ q 4 I Rx? lavage I ' (f 7 ' Q3”? Ra. 2" fl " w - II 4 -——-—---'> 1 z j I O 12 -2 R "3’ ‘2‘“ '32 Rt 3 5' v! I Rfi-aRVSR. O 39. - v 1 - I! 0 I b - " " l R48 21 " RI 0 ' -: ‘1‘ .. no, a ‘9 Va“ “loll; We: a m , W0 '4’ ‘ I of; Z 3’ W “6‘3"” / Z a o 0 0 W «, q a z I . Q am M9 ‘ 0 0 {7 O h 0. ‘7 Page 5 of 10 NAME: STUDENT NO: 9. [4 marks] For which value of the parameter a does the system {‘15: :2: : 2_ l have no solution? 2 } A. a=2 O llO ) 7<l 2- )a/L) . f B. azé ( I 1 0,11 01 l 0 C. a=0 l ‘2 a/}L > D. (1:1 ““7‘ <0 l/Za Jia’al L E. There is no such a. _ 9 TM 0A7 AtflOMWY 0/665 0 2. mac: l/ZQ?O- l : “h 0) {o WW0 (5.36 RD 0N" L ) 1a a 10. [4 marks] What‘is the value of x in a solution of the following system? 7x—3y=27 11x+5y=23 . 7 —3 5 3 68 0 Hmt‘ [11 5] [—11 7l=[ 0 68] A. 6 fl / 5 3 B. 4 (7 ' ) a 9/3 ,H ? C. 3 H i V D. 5 f 9‘ 2? X .L E_ 30 ‘5 2 ‘ (’3 (ll 7 7'3 Page 6 of 10 NAME: STUDENT NO: PART B. Written-Answer Questions 1. [17 marks] A 10 year, $60,000 mortgage has monthly payments. [5] (a) Find the amount, $R, of each payment if interest is 8% compounded semiannually. . \7. (90,000 1 Ram,- wkua @001: (HO \\ \l [6] ( b ) Immediately after the 5th year of the mortW ‘ . .m,,...........m......_ m.«mw.,mmw rate changes to 6% compounded semiannually. The debtor may change his monthly payment from $R (see part (a)) to a different amount in order to repay the loan in a total of 10 years, as originally agreed. Find the new monthly payment required if the debtor chooses this plan. i a u ‘éM/tflta :. . ' \0 \Cre, Prmc $155 ‘51 R Gib/OTf as Lea IL The MW m‘é’g/cflfi/ “Are, (a r WW6 flail 30’“) New faymw‘éfi are 1’, W“ R Q6“ “(Ta é/Olr T: Ra bog;125.€5[\’6.0‘ll’wj, W1“ fiéql’ob W’ W 0.03) J [6/ (c) Alternatively, when the rate changes as in part (b), the debtor may continue monthly payments of $R 'each (see part (a)) until outstanding principal is less than $R. One month after the last full payment of $R, he would make a smaller final payment. Find the total number of payments required if the debtor chooses this plan. Remember to include the 60 payments made in the first 5 years of the mortgage. - -a -10 v. _. - l.03> RS Ca, eel w {- L-LJQ/ WW am A Fulfillk 4: .QHL‘O’IWOCI / (XI -fl -: %[|”,7/V161600qj h OM65 YeMfith b [3% WWW W 6'? _ , so H”? \h “l Page7of10 NAME: STUDENT NO: 2. [16 marks] X YZ Income Fund issues a bond with $50,000,000 in face value, maturing in 20 years, with semi—annual coupons and an annual coupon rate of 4.8%. Just 5 years after the bond is issued it is trading in the market at $104 per $100 of face value. [11] (a) To within an accuracy of $0.10 in the price for $100 of face value, what is the current annual yield to maturity of the bond? $150,000,000 :5 a red WW] rm ( i .; “w 11.02 P: tor/i9 w“‘1' 6"“ My i 4% l st _ h f, 0 I“ QM” I Halli/W“? between N 4 £017” MSW P tit 11400 low. ' ‘ [L ’ 801 0 l‘f‘lle 1.4306 M 15"” “‘3 J 7‘ and L I P7; “9%: y‘o‘k jld d7 1 , G’odok Wflujlq o ltj AM? 0/0 Pr(C’e a 111 6A! Li'O’UZl ax p.011? ,9 ‘ [5] (b) On the same day X YZ announces that at maturity it will pay $110 per $100 of face value (a premium of $10 per $100). Assuming that this announcement has no effect on the yield to maturity, exactly What happens to the price of the bond? X 65 ffl I . Tl“ f/‘éI/d film lg JZISCWAMM 116/927 6% k/Lal le [Nice wf‘ll {0V Page 8 of 10 NAME: . STUDENT NO: 3. [14 marks] [10] (a) Let 3 -1 0 i {3? 0x Rgém¥€3 (2 —5 1) Q ¥ G 7 1 0 —1 g? {3 i Find A—1 or show that A"1 does not exist. ‘ ' g] g e; .... g Q? 0 § ‘ 5“? § W V fig § W % E 3:; “v z :33: 15m g ” .g, R > A N“ *4 g ‘ fl ‘ § ,V a) “ ' rm? Wat 3 x h‘ ~ ‘ 2 \ [4/ ( b ) Find all solutions (if any) to the system of— equations 33: — y = O 23: —5y +z = 1 (E —z =—1 X I 19/ ’l /l O ‘ O 0 )I : "' 3 '7) /3 I 3 T), O 3 0 fl ' 2 4 a 43 x: n Page 9 of 10 NAME: STUDENT NO: 4. [13 marks] A room contains a collection of ducks, insects, and spiders, but no other living creatures. Each duck has 1 mandible (moving jaw part), 1 pair of eyes, and 1 pair of legs; each insect has 2 mandibles, and 1 pair of eyes and 3 pairs of legs; and each spider has 2 mandibles, 2 pairs of eyes, and 4 pairs of legs. Set up and solve a system of linear equations to determine V the number of ducks, the number of insects, and the number of spiders the room contains, if there are 22 mandibles, 16 pairs of eyes, and 34 pairs of legs in the room. L6€ D ) I) 3 LOW “Why 0‘? 4“ka \mgetfij M 4pm {afrecj‘nuelxl‘ m can/limes: DeQI +25 ‘1 27. D * I + 25 3 Ha Page 10 of 10 ...
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2006 - Department of Mathematics University of Toronto...

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