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Unformatted text preview: Department of Mathematics 600/61 '
University of Toronto MONDAY, OCTOBER 26, 1998, 6:10  8:00 PM
MAT 133Y TERM TEST #1 Calculus and Linear Algebra for Commerce
Duration: 1 hour 50 minutes Aids Allowed: A nongraphing 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 writtenanswer 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 questi0n is worth 0. For the written—answer questions , present your solutions in the space provided. The value of each writtenanswer question is indicated beside it.
ENCLOSE YOUR FINAL ANSWER IN A BOX AND WRITE IT IN INK. TOTAL MARKS: 100 TAU 1 FAMILY NAME: WW0 Cum": W
GIVEN NAME: (w) are m/ﬁt‘éﬁa M 0’ STUDENT NO: m 9” quwﬁzm g7 )MVL
SIGNATURE: Ltan 1% diam &Q @1490? TUTORIAL TIME: Mi [Mi A360 or E. TUTORIAL ROOM: T.A.’S NAME: P0503 Regcode Timecode Room $51086
$31088 WE52
$31074 P0102 FOR MARKER ONLY 
P0103 $52105 $52106
P0201 LMi57 $52108 
P0202 RW143 UC152
P0204 UC177 $52128 _ P0302 MP202 551085 ' P0303 55620 $32,127 P0304 UCi4o 551085 P0401 531085 531086 ,
P0402 $51086 $52128 
P0403 531088 551086 i P0501 332105 110152 “[1523 Page 1 of 10 PART A. Multiple Choice \/1. [4 marks]
If $1000 is invested for ﬁve years at a nominal rate of 7% per year, compounded semiannually,
then the effective annual rate of interest is closest to A. 7.21% (c 1 ("03§)1 '2 .07 l 22; f 2. [4 marks]
A car is purchased by borrowing $20,000 and paying it back with monthly payments (at the end of each month) over four years, with interest at 8% per year compounded monthly. The
total amount of interest paid during the four years (to the nearest $10) is A. $3,430 9 a
$3,440 20/000 ’ ‘7”—,9§_
0. $3,450 R c Zoioooxﬂg
D. $3,460 W
E. $3,470 I ’ (H 'ﬁ
R : 433729
0 : 3436.1”
439, 20/00
9/3940 Page 2 of 10 / 3. [4 marks]
A $100,000 mortgage is amortized over 15 years at 7.4% per annum, compounded semi
annually, with monthly payments. The monthly payments (to the nearest dollar) are $617 =11xK2YO
7. '7'
$921 (1.037) ‘0”) $938 IOOIOOO :— R 0 W4
$910 A B C D. t
@0915 R: I00,000M
[l , 0.034) J R : 01/{03
x 67/5” /4. [4 marks] A car is advertised for $1,000 down and $500 per month for 36 months. Suppose the interest
rate is 12% per annum, compounded monthly. What is the increase in the down payment that you should offer (to the nearest dollar), in order to reduce the monthly payments to
$300 per month for 36 months? a $6,022 yaw WC M loan/ F017“ ZOO/wmtt. B $6,032 W (“ﬂat Wt!“ J .y K W
C $7,200 . “I “an?
D. $6,027 Una/0‘54 M W56“, fly
E. $6,019 1 OWIOI
’34
’ 200 [1’0'00 1
I //
,0!
, GOZL§01
$0012 Page 3 of 10 / 5. {4 marks] A $100 bond with a. coupon rate of 12.0% is quoted at one time as $148.00 to yield 6.0% and
at another time as $140.00 to yield 8.0%. The price of the bond to yield 12.0% is A. $144.00 (l:r' Q P._—_\/
67405 V7’ @ $100.00 P ,1 (00 D. $124.60
E. $96.50 ‘/6. [4 marks] The price per $100 of a bond (bearing semiannual coupons) is quoted as $62.18. If its nominal annual coupon rate is 8%, and there are exactly 23 coupon periods remaining, then
the nominal annual yield rate is A. 11% TC dd 1 30/"
B. 12% 7 3
—2
:13: P: WOUHY) + 407;],07f
. 140
@1593 ; _, 6L! 5
u 6"
(/loﬁ‘e/ ﬁlm my ﬂaw Page 4 of 10 1 3
Let S: 2 4) andR=<i (IfMisamatrix, W denotesthetranspose
3 5 '
T
ofM). Then (ST+R)T= 5 1’ R 3 , u
3 2 —2 l2 ‘5 (7‘ " ’ : >
A.31—2 §+3L 413
‘5‘!
3 5 0
T 4
3 4 T 1 3
(1 I) So (5' 1K) ‘ I
1 3 I 3
2 5
C. —1 —3
—5 3
5 —10 8
D. 8 —14 —12
11 —18 —16
3 1 1
E' (4 1 3) ~/ 8. [4 marks]
Which of the following is always true, for any 2 x 2 matrices A and B ? (If M is a matrix, MT denotes the transpose of M .) A (AB)T=ATBT CANT" STAK 50 “well {7qu A B. ATB=BTA A13: 51’0" Mk BrA a BA A6
""" “k” v a _ a: 0 (A+B)(A—B)=A2—BZ (NBYAIBL A’mﬁ 7‘6 61—13% 4/ . (A+B)2=A2+AB+BA+BZ E. (A+B)2=A2+2AB+B2 ganw comment as C. Page 5 of 10 1 21 ’1. "*0 1 “'40 000\
A.S g". u—zo 1‘20=0‘:O
BI 11' 12' 5 l/
C.0 000 4 000 0"” 1
000 S= 000 000 C 000 «.g
D. 000
0 0 1 {—L I {’0' §'0l
0 00 1 L 1, (900
o 00 g H H L :g:
@(5 451) 53(6) ‘(6) 000
6'/GI J10. [4 marks] Adri lends Sylvie $1,500 at an interest rate of 8% per annum, compounded annually. Sylvie
repays Adri $200 at the end of each year for as long as it takes to discharge the debt. Adri invests these payments at the same interest rate, namely, 8% per annum, compounded
annually. When the debt is ﬁnally discharged, Adri will have (You may assume that the interest rate in all the answers is 8% per annum, compounded
annually.) A. $1,500
B. the future value of an annuity of $120 per year for 14 years. C. the present value of an annuity of $200 per year for the same period as the loan. D. the future value of an annuity of $200 per year for 9 years.
@ the future value of $1,500 at the end of the same period as the loan. Sﬁurrosa p I500 1 ’1 000m; "I 200 "Of/KW)“: 100§ﬂ5 Ww&.<wwf
Aw. WM We {like W/ QM<WS Aye Wlé/‘j. Page 6 of 10 PART B. WrittenAnswer Questions 1. (3)48 marks]
A debt of $1,000 due in 3 years and $5,000 due in 7 years is to be repaid by a single payment
of $2,000 now and two equal payments that are due 1 year from now and 3 years from now.
If the interest rate is 8% per annum, compounded semiannually, how much are each of the
equal payments? 0 I 3 ‘9' ..___.p————o————"'—_“"‘ ' .000 {000 1000 X X q ... «L
= 2000+XCI0‘f) M00") 40 «'
\0000‘0”) + {000 (UN) 4 50000.07) "—7: Z 000 (1 0 ‘4) W x4 1 +0. w)” xc 978.3185. 
X“, (b) [7 marks]
Find the present value of an annuity due, consisting of 10 yearly payments of $2,000, if the
interest rate is 6% compounded annually. 0 1 q 40
W
W, mu . V x 2000 : 15,403.39w a Page 7 of 10 1 2_ (a) [5 marks]
A loan of $7,500 is amortized over 36 months at a rate of 12% per annum, compounded
monthly, with monthly payments at the end of the month. How much is the principal repaid
in the ﬁrst payment? 7500‘ Ram. R= : 2492107,“
l” U'U')‘ 6 xii/1.44.1! ‘ ’lV‘U l w. ou‘liﬁhlt‘)
Du/(nj $8 IW'IJ ) Men‘f‘v’c { r! S 7 «5
4o Leda/art was 43/00 veal « 7f 12 fmtdﬁ: 244,1/«7f: An9~5l ‘5 (b) /[10 marks]
Semiannual deposits of $100 are made on May 1 and November 1 for each year from 1994 to
1997 inclusive, followed by semiannual deposits of $125 on May 1 and November 1 for the
years 1998 and 1999. Find the balance in the account on November 1, 1999, just after the
last deposit, if interest is 6% per annum compounded semiannually for the entire six years. q M”qu NJ! 44
H“ LI 2 ‘ ’ 3 q '0 H r; r
0W __ _
\00 I00 v « — mo 195/sz m [x Page 8 of 10 3. (air/[7 marks]
Suppose a 20 year bond is issued on October 1, 1995, with a nominal annual coupon rate
of 8% and semiannual coupons. Find the market price per $100 of this bond on October 1,
1998, if the current nominal annual yield to maturity on this bond is 5.8%. an 00% '1}iqqy) ' r e3 1: rv: 4
Up" at
Q 0‘0 {Le 40 aa(  34
P” “900024) ’4' Lida/41.029 P :1!23.5’€ 0%&. /(b) [8 marks]
On October 1, 1998 an exchange for the bond in (a) is offered. The new bond, with the
same face value as the old bond, will be due on October 1, 2018. The new bond will
carry a nominal annual coupon rate of 5.8%. What cash premium per $100 of face value,
payable at maturity of the new bond, should be offered with the new bond to encourage holders of the existing bond to e change for the new bo d?
Snag 6L2, Coufw {4% pg do WW V1,; "g = /l$)
W weal WAX U “Ii/#4 I00 On 006 1) qa ‘ 4,4 M
ail (of/"01 '5 Wﬂ’éi" dilg/yg, You dei/ {ail
the} a.» #40 wry/i634 4.4x nan/(415’ 4100 a! 156% {mow/5
0mm new b’ﬂx} M Invert M TW(A.,:7 (M 45% mew VIA/5. i A5730” PO’ Wip 764’ 66M: xix/HA accmmai aft? ‘60 13,580,024)” Ly Oetjlzoza’ .
: 9, ﬁlm (5 “we MSLx ﬁfW‘MM
necessary 46' who «A;
We We Mk all \mkw qu <56 705” 5wc 04%“ Page 9 of 10 (a) {8 marks]
For how many months can you pay $500 at the end of each month out of a fund of $10,000,
deposited today at 10.5% per annum compounded monthly, if the payments are deferred for
9 years? (This means that the ﬁrst payment of $500 is not made until the end of the ﬁrst
month of the 10th year.) 4.: .125 61 Yrs a to? W“
.l’l '
o :06
'_,__._~_________.——o '
10,000 M)
w
.   , )
M004rx (€56“"“‘lf0u&“
l “W ' [x
 l+1 MI Awe We“ (1an (b) [7 marks] How much money is left in the fund at the moment after you make the last full
payment of $500? 68
'03 a ~+XCI+0
10,0000”) ’ 500 ‘7“
Fit .
(0,0000+t) ‘5 {Oosﬁh +X by
L ' JOY), ]
Muﬂ "I WM
)(: {0/00 17, LL02:
:7. )( :1’1’4339 Page 10 of 10 ...
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 monthly payments, annual coupon rate, nominal annual coupon

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