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test_2001 - Aids Allowed A non—graphing calculator with...

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Unformatted text preview: 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 Department of Mathematics University of Toronto <0iuézfi WEDNESDAY, OCTOBER 31, 2001, 6:10 - 8:00 PM MAT 133Y TERM TEST #1 Calculus and Linear Algebra for Commerce Duration: 1 hour 50 minutes 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: TUTORIAL ROOM: T.A.’S NAME: Regcode TOlOlA T0101B T0101C T0201A TO20lB T0201C TO201D T0301A T0301B T0301C P030113 TO401A T0401B TO401C T0501A T0501B Timecode M9A M9B MQC M3A M313 M3C M3D T3A T313 T3C T3D VVQA WQB W9C W3A SS2111 882128 832130 LA 341 UC 163 382130 LA 240 $82128 881069 852106 VC 206 SS1074 SS2111 LM 123 TF 201 TF 200 Regcode T0501C T050113 TO601A TO6OIB T0601C T0701A T070113 T0701C TO801A T080113 T5101A T510113 T5101C T5201A T520113 Timecode VVBC VVBD R4A R48 R4C F2A F2B F2C FBA F313 R5A R513 RSC R6A R6B Page 1 of 10 Room NF 4 CR 103 UC 52 UC 85 UC 328 UC 87 WI 523 851086 582130 852111 UC 52 UC 85 UC 244 UC 144 UC 244 FOR MARKER ONLY Multiple Choice B1 B3 B2 B4 TOTAL PART A. Multiple Choice 1. [4 marks] If interest is at the nominal rate of 12% compounded every 4 months then the effective quarterly rate is closest to A. 2.874% (1+ ()4: 6+ La); B 4% 3 @2985% 11 (LOWi/vl ”,02235 D 3% E 2.242% 2. [4marks] If $1,200 is invested at the effective annual rate of 5% for 5 years and then at the nominal rate of 5% compounded quarterly for 5 more years, then in 10 years the investment will be worth $1,963.49 { 20 B. $1,629.68 \200 (1.0;) (H '24—?) C. $1,972.34 D. $1,954.67 7—5 [0103.767 E. $9,266.08 Page 2 of 10 3. [4 marks] A $10,000 loan is amortized by equal semi-annual payments over 5 years (the first payment due in 6 months). If interest is charged at 8% compounded semi-annually, then the principal repaid in the first payment is: A. $762.47 101000 1 RC? 707.01, B. $795.38 R t/ C. $806.21 ’ W0 F3 133/2 W '$832.91 I ((1.0%) E $85364 Tflwpfi gm id form} : ’Wr/UMOO :fiq 00,00 4. [4 marks] If a $300,000 mortgage amortized over 25 years at 6% compounded semi—annually has monthly payments, then each payment is clolé‘i‘é ‘60 A. $2,500.00 0 [03) g a J? C) @ $1,919.42 / R Q ~ C. $2,209.71 3 00, €00 300 1 D. $1,932.90 62 00 < E. $2,216.45 ’ 300/ 0 00 f / (I r f ) ’3 L i 7 UO/Wo (1.03)“, J W Page 3 of 10 5. [4 marks] The price of a bond with semi—annual coupons and exactly 18 years remaining to maturity, with a coupon rate of 8.2% and a yield to maturity of 4.6%, is closest to A. $100.00 00( 23), 36 4 l B. $119.10 P’ l 1‘0 4' ‘ £7,023 C. $118.73 $971417? D. $143.43 ' @$143.74 6. [4 marks] Just after the 120th payment, the principal outstanding on a $100,000 mortgage amortized over 20 years at 7.2% compounded semi—annually with monthly payments is closest to A. $ , 50000 \00 000 i R Q - Q 3 lUO/OOU B. $33,019 / zqo t a . «$77 @366981 P 0 : R01 . D. $64,583 | [w t E. $68,762 Z lb0,000 0/101- mi «(”10 =100)0w[1’0*f) j 3 L 0 4034601 20 ' too 000 no : l 03G) #- 041) ( [4—04’47‘7'0 ” W [+4.03é)’10 Page 4 of 10 7. [4 marks] —2 1 0 LetA—[ 0 1 _1],B— r—Awr—J HOr—t J , and suppose C = ABBT . Find the entry in the second row and the first column of C , if C' is defined. A.0 I I 1’1“! #1 l O B. Cisnotdefined C:<C) I4) 10 l 0 l C. —1 -/l .2 I 2 "l E. —2 j (, a, 5 /l 2 8.[4marks/ Find the value of k; , if any, for which the homogeneous system 2331—kCE220 —$1+C132=0 3x1—(k+1)x2 2 O has solutions other than 3:1 : O, 332 = 0, 1'3 2 O. \< A“ brdlk 1 ’ g ,_ l B. there is no such value of k l l > O C. 0 3 ”<(*’ D. 1 l * I / O @ 2 O 1* O < k — 7. O 2’ kl O a ' Page 5 of 10 9. [4 marks] 5 —9 —2 ‘ The determinant of 7 3 1 I5 16 11 3 —23 ® 161 -3 0 1+3 [‘1 ’3 B. —37 ’ .4 3 I :(«0 5’ 2 1 -(’53"§) C. 11 -§ 2 O D. 29 : «23 E. —7 10. [4 marks] Let C be a real number. Cramer’s rule can be used to solve the system 4cx+y=5 C$+C2y=9 ifandonlyif #6 | '5 A. 0750 0%: ’L .1 Lia/C C; C B. c#%andc#—% 1 C’(HCIL'I> Ct Cgé2and07é—2 D. C7£0and07é4 6‘0 Cio CwA Cd; “ii/J1 @ c;£0,c#—;~,and c;£—-:1§ Page 6 of 10 PART B. Written-Answer Questions 1. [13 marks] On January 1, 1985, at the time of his son’s birth, a father deposited $500 into a savings account paying an effective annual rate of 10% and continued to make similar deposits every 6 months thereafter. After January 1, 1995, the account earned 6% compounded semi—annually. How much will there be in the account just before the deposit on January 1, 2003 when his son turns 18? I M f , M94 70“» 1" “W 7W1)” Taxi/q? O I , 2‘ \ g \ 20 1| W 500 $00 500 {2’00 whore (”01:1 IO ”1 ’ "0 :{OO[O+1)ZI/l ————-—-—-——T'—"‘ 1 2 r {00 [WOW/[j ._._7_.—— (#3135422324 (1.10) 1/! / / T 1 ’0'( fix! 1)q( Jaw alga final/a2, fin 1)/03 w ) O i Q ' ’ ’ [5/ lb "0 1.03 I?,é23.2‘1’(1r03> + 5006/fi03( ) (or + 5/00 6Wfl3 " {00) 473?, 3587, 5’? t Page 7 of 10 2. (a) [9 marks] Just after January 1, 2001 the amount of money needed to buy all the outstanding bonds of XYZ Inc. maturing on June 1, 2010 was $53,000,000. The annual coupon rate was 7%; the coupons were semi—annual; the yield to maturity was 5.4%. Assuming that on the date of issue)whenever that was)the yield to maturity rate was 7%, what was the face value of all these bonds? [To the nearest $100] (b) [8 marks] Exactly one year later, just after January 1, 2002, the total value of these bonds falls to $45,000,000. What is the yield to maturity then? [You may stop when the total price is within $1,000,000 of the actual price] a) h=\‘l r=,03( c2027 V: {3, 000/000 : v (1.014) . 73027 ”I Page 8 of 10 cor—4H Nor—l 3 3. Let A = [ :1: J Where 93,3; are arbitrary numbers. y [5/ (a) Find :1; and 3/ such that the matrix A is symmetric (Le. AT 2 A) and invertible. [5] (b) Find the inverse of A in case as = 1, y : 4. [5/( O V Use the result of part (b) to find the matrix X from the matrix equation X A : AT in case 3021, 3/24. 4. Woody Tobias, Jr. has a 1 litre Ehrlenmeyer flask filled with a mixture of three liquids (red, green, and blue) which he has made. The red liquid weighs 1 kg per litre, the green weighs 0.8 kg per litre, and the blue weighs 1.5 kg per litre. Moreoyver, the flask contains exactly twice as much blue liquid as red, by volume. [8] (a) Set up a system of linear equations which will determine the volume of each liquid Woody used to make the mixture, if the total weight of liquid in the flask is 1.3 kg. [7/ (b) Use Cramer’s rule on your system from part (a) to find the volume of red liquid used to make the mixture. Note: & marks will be assigned for a different method of solution. a) L679 R)6’ med 8 L6 W WU. (AD lfiénof 010 red/jraew 0««0{ \olue \xqwlS res/cobweb], o O ”l 4/ - l ‘ ' l .‘3 1K ’1 O " Page 10 of 10 ...
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