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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 multiplechoice 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 writtenanswer 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
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T0501A T0501B Timecode
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LA 341
UC 163
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LA 240
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TF 201
TF 200 Regcode T0501C
T050113
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R6B Page 1 of 10 Room
NF 4
CR 103
UC 52
UC 85
UC 328
UC 87
WI 523
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582130
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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 semiannual payments over 5 years (the ﬁrst payment due in 6 months). If interest is charged at 8% compounded semiannually, 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. WrittenAnswer 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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