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Oct 1999 Term #1 - Department of Mathematics University of...

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Unformatted text preview: Department of Mathematics University of Toronto WEDNESDAY, NOVEMBER 3, 1999, 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 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: 1 00 FAMILY NAME: GIVEN NAME: STUDENT N O: SIGNATURE: TUTORIAL TIME: TUTORIAL ROOM: T.A.’S NAME: ————ma T0101A M9A LM157 T05010 W3C T0101B T0501D T01OIC T0601A T0201A T0601B TOZOIB TOGOIC T0201C T0701A T0201D T07OIB T0301A T0701C T0301B T0801A T0301C T080113 P0301D T5101A T0401A TSIOIB T040113 T51OIC T0401C T5201A TOSOIA T5201B T050213 M98 M9C M3A M3B M3C M3D T3A T38 T30 T3D W9A W9B W90 WSA W3B W3D R4A R4B R40 FZA F2B F2C F3A F3B RSA RSB R50 R6A RGB FOR MARKER ONLY Multiple Choice TOTAL Page 1 of 10 mm \”>3 PART A. Multiple Choice [4 marks] A loan, on which interest has been accumulating at the rate of 5% per year compounded I. ‘ _( -~‘,- _ yr 2 .L/ quarterly for a y.ears stands now at $1,955.11. The original loan was A (14".ij : l?{{.“ a; ,A. $1,02519e { '20 B. $1,22596/ A'— ,q§{,u (In? ” C $1,325,063” :5 (€25,003 $1, 525110/ E. $1,625.06’ 2. [4 marks] If the effective annual rate of interest is 8.243216%, and interest is compounded quarterly, then the nominal annual rate is wry/’5 7 '10 A. 16% 1' 1 \ )1 B1 8% 111/; / (We 1 E. 7.8% 1; — (1+ rely" J- D. 4% r 1 L} (Hewl’j E. 2% :L,[[)_082q37'6>:/j 4 0 03 Page20f10 (MA 4 AMC69 @629, 3. [4 marks] 0‘89”” Babar is saving up for the down payment on a condominium. He is making geegykd‘epositgéa in an investment which he expects to pay 6% per year compounded weekly/for the next 5 years. If, at the end of 5 years, he expects to have $100,000. his weekly deposits should be (to the nearest $1) A. $362 $00,000 " R 527,5].% 3' $330 R : loo,ooo "Cob/a) C. $358 W D. $445 52 E. $1,433 :5 330,02 4. [4 marks] 10 semi-annual payments of 8150 are made into an account offering 12% interest compounded semi-annually. The first deposit is made 6 months from now, and the last deposit if/made 5 years from now. The balance in the account 7 years from now is (to the nearest $1) A. $2,003 B $1977 0 ' ” M? w . 1 L___4___;._____..—l C. $2236 {1’0 H 0 ~ K0 0 0(9)” ’ . L___.-’/—’—9 . 15.” $2 496 i *af—fér -/""> 6053104, U ’ 150 57771.0(, E. $1,997 . “ _,‘-- fl- 2400,07 Page 3 of 1.0 ’1‘ k 5. [4 marks] A mortgage of $300,000 is amortized over 25 years at 7.2 % compounded semi—annually with monthly payments. The monthly payments are (to the nearest $1) A. $2,415 B. $2,115 C. $2,800 D. $2,159 ‘1 E. $2,138 J 6. [4 marks] Let (1:,y,z) satisfy Then 2 = A. 2 6 1 C. 0 D. —1 E. —2 fl,03é)2‘ 0" All 300,000 7 R O 73791 t‘ 2 0 * Z“ \ 1 I ’1‘ -1 l1 / '7. U 3 .4 \0 Page4of10 II 1121—.» Q11 / 17— “J l 10 +3 (I, 7. [4 marks] T —1 0 2 4 1 —3 J 1 —3 1 (01) 3—2—55—46 "2—1 0’ A. equals the 1X 1 matrix (——11) , {/0 ”(l 0 ’L ‘f - . B. is not defined 1/ C. equals the 1 X 1 matrix (13) D. equals the l x 1 matrix (21) E. is a 2 x 2 matrix Note: If .M is a matrix, MT denotes the transpose of M. Q x q 6/ rao ’7‘ l -3 _ 4 'IL'01 1-7: ‘2(7~I , 'Lf 7' x 3 -2»? 15"? ,3 (, \ “101.17‘ [ 1.. (’1‘: 3:) (0 ,) (~11 ’1:>(;) (01>('2':>:i‘(2l) 7,! \l 8. [imam/cs] I {B .’L‘ . . 1 0 —3 —-5 If [y] lsasolutxon of the system [3 2 _5] [y] — [_3] then 2 z 3 A. =— —— y 22 2 l 0 '3 '() B y:7z—9 (D ,2 _ 5' —3 C y———Zz+3 r h 2 -5 ’D — 2 6 ' 0 — 3 x/ 1" 2+ 7 O 1 4 ,1 E y=14z412 “#77 Page 5 Of 10 9. [4 marks] The has A. B. C. D": :- E . 10. [4 marks] 2 3 1 A. B. C. D. ,. x‘\ (E, L/ system of equations {111 + 2182 532 no solutions exactly one solution exactly two solutions a one-parameter family of solutions a two-parameter family of solutions —2 1 2 2 —1 9 _20 :1.ZO+?_' 0 5 27 85 + 21133 — $4 + 211:3 + $4 + $3 + 134 fr 0 O ~3 fl Page 6 0f 10 -0 =1 =2 2 7i ‘1 (I) ’2 I ‘ 2 l I ' \ ,1 1 -1 o 0 0"0 I (.2 2’! O \"L' ' (9010 ’l PART B. Written-Answer “Questions 1 [15 marks] Mr. X must pay two debts, the first amounting to $5, 000 in five years time, and the second to $25, 000 in ten years time. He has available, now, $12, 000 in a savings account that earns 6% per year compounded monthly. Six years from now, he will sell certain assets What sum must he obtain from these assets and de ' ' ' ‘ . p051t in his savm account t discharge of these debts? gs‘ 0 complete the M1 1 . ,. , 7 . 0 60 72 120 Le15 X In, {44. WW awoke/“6 ,‘Q agreKS {0w 257000 4 5 . 00{ r” wa—n‘éLI Equa'éuj rafl+Volwej 71 ()vl'o { ( ()v‘io 000, . o 12,000 4' XCI. 00‘) = 50000:” 4— 2 / la wt . (0000.00?) +2{,000(I.007) ’l'2,000 (Loaf) - : #130132 3% 7. (F, We wt, 4m [is {re/A value) X Ltd! l2,000 WILL. accumulakS £0 J1," so “(a A 5 [MI/{000 c |1,ooo(\.00$’lbo aft” 4 734’s (dc/E, ”Is/{Lax yr £0 acoumu lean \2, 000 (f; X 60] Elmo!) CL ooflw 500510 MT)“, He 9"“ “3 Ll Jew-9 '5: ywa l’l/oooo 008') )6” (000-30 ”(3 +>< wlwc: ( W #00 allb:$b?’ aceumqliéfi fa oru-«fla’ L{ yrs '60 wad/x Z (iv-PXECINJY) : 2;,000 ('0 if.g-[gooo w mecca 00$? ](a.ao<7n c /0c125 aw) \«é loo Dre: ‘6‘“ lac/l: pee-elm}? 0r“ .bl/v, 41W la .l‘”+ llve \(fl’ld ‘A 4"“ . ts ~~ '\ A company makes products A, B, and C’. You’ re working for a rival company and have been asked to find out how many of each product they made last year. An informant told you trim Eliey bought 900 pounds of steel, 600 pounds of copper, and 330 pounds of plastic. Your lab'teclimcians have determined that A is made of 2 pounds of steel, 1 pound of copper, and 1 pound of plastic; B is made of 1 pounds of steel and 2 pounds of plastic; and C is made of 3 pounds of steel and 3 pounds of copper. How many of each product was made last year? [gr—n; A; W, w 6'4ch ‘, 1XA+X5+3XC“ qao ‘ , : 00 Car”, _ XA +37% 4’ -: 330 . Vlad"; XA + 2X5 ' 1:55” '3. I (a) [10 'marlcsj Find the inverse of the matrix - H ”H -L/ 1—50 L, l 00 fire?“ I 1 3 _1| 0 . .‘ 7' ( 010 937’) 0" b 30‘ v 0 00 at? "- 7- . Q 3 2 0 l 3—46 ’3 a’ 2.13; , ,1 fl 0 Ia-‘IKI v‘g-YY‘ - (b) [mettle] Y . , Use your solution of part (a) to solve the system a: + 2y —- 42 = '1 1 2x + 3y -— 5z = 2 . —3:I: — y — 47. = 5 Note: No marks W111 be assigned to any other method of: solution. Thosyéé‘“ “ A $)’ i wwe A u #6 ”Ag/(”C 2 { VANS (a) p, mi-annual coupons. Every coupon that Jack receives is immediately deposited into an accgunt ofi'ering interest compounded semi-annually When the bond matures, Jack has ”£11 1‘ plus the accumulated value of the account into which all his coupon payments were deposited The account into which Jack deposits all his coupons offered interest rates as £911?” ' it 3‘ 5”?" 9% compounded semi-annually for the first 3 years, , _11% compounded semi-annually for the next 2 years and 10% compounded semi-annually for the last 2 years. t‘aaafimwi-‘j‘n .. :. {M i” (a) [12 marks] ,. .‘ mIncluding the $1, 000 redemption value of the bond, the coupon payments, and all the 1-: interest earned on the account, how many money does Jack have after 7 years? (b) [3 marks] ' At What effective annual rate would Jack have had to invest his original $1, 000, on the 1 day he purchased the bond, to have the same amount of money after 7 years as the 405'] 0% [405110; (l u’$)+4051u<(]60()*405m0( ‘4— H900 =— b)... 173310 :. \000(l*l)7 CthOfils" L - ,08efe‘f. 396,70 ...
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