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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 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 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 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: 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 semiannual payments of 8150 are made into an account offering 12% interest compounded
semiannually. 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 17: ‘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 oneparameter family of solutions a twoparameter 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. WrittenAnswer “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 Jew9 '5: ywa l’l/oooo 008') )6” (00030 ”(3 +>< wlwc: (
W #00
allb:$b?’ aceumqliéﬁ fa oru«ﬂa’ L{ yrs '60 wad/x Z (ivPXECINJY) : 2;,000 ('0
if.g[gooo w mecca 00$? ](a.ao<7n c /0c125 aw) \«é loo
Dre: ‘6‘“ lac/l: peeelm}? 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‘gYY‘ 
(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, miannual coupons. Every coupon that Jack receives is immediately deposited into
an accgunt oﬁ'ering interest compounded semiannually 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 semiannually for the ﬁrst 3 years,
, _11% compounded semiannually for the next 2 years and 10% compounded semiannually 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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