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Unformatted text preview: Department of Mathematics
University of Toronto WEDNESDAY, November 1, 2006 6:108:00 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 multiplechoice 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 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 writtenanswer 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 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 semiannually 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 semiannual payments. If the interest rate is 8% per year compounded semiannually, 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 IPRisﬂmszzcwOM Murat/fax )
i ? 19¢ lard /’&W B. $39.41 036/ {/9 (ﬂ (Aﬂfg 0
6w; faker/mi {Mté 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 rowechelon
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 nonzero rows in rowechelon form. E. No two of them have the same number of nonzero rows in rowechelon 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“
[(25 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 1parameter 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 RﬁaRVSR. 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. WrittenAnswer 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 { LLJQ/
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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