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Unformatted text preview: €OLU €01 Department of Mathematics
University of Toronto WEDNESDAY, NOVEMBER 1, 2000, 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 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: m
STUDENT NO: ___________ SIGNATURE:
TUTORIAL TIME: ___________ TUTORIAL ROOM: T.A.’S NAME: m
M9A T0101A BF323 T0501C
T0101B $31070 T0501D
T0101C BF215 T0601A
T0201A MP118 T0601B
T0201B BF323 T0601C
T0201C RW142 TO701A
T0201D UCS28 TO701B
T0301A RW142 T0701C
T0301B LM157 T0801A
T03OIC PGlOl TO801B
P0301D w1523 T5101A
T0401A LM155 T5101B
T0401B W1524 T5101C
To4o1c BF215 T5201A
T0501A ss2127 T5201B
T0501B UCl44 W3C W3D M9B
MQC
M3A
M3B
M3C
M3D
T3A
T38
T30
T3D
W9A
W9B
WQC
W3A
WSB FOR MARKER ONLY
Multiple Choice B1 Page 1 of 10 PART A. Multiple Choice 1. [4 marks] How many months, to the nearest month, does it take for $1,000 to grow to $5,000 at a rate
of 9% compounded monthly? “ ﬁ
.04
A. 210 2(000: looo<l+ﬁ
B. 200
at 5’ 1 n gt 0* ‘z—‘ﬂ:
@ 215 I
D 225 n ’ % f a: 2 If ‘I
/ '
E 230 , 04 )
% (l 4‘ '71
2. [4 marks]
Which of the following interest rates corresponds to the greatest effective annual rate?
A. 10.4% compounded annually . l0” 2
3 .a
B. 10.3% compounded semiannually ( l ‘l' Luz; ) / I ’ . l0 . ~
@ 10.2% compounded quarterly C l 4. , 02 9" i ' I . . .
L,
D. 10.1% compounded monthly :1,
(1+ .1004 : I’M3I.,
E. 10.0% compounded daily l2. (
36
3:2,? ' J0 Page 2 of 10 3. [4 marks] A person purchases a car by making a down payment of $1,500 followed by 60 monthly pay
ments of $300 each beginning at the end of the ﬁrst month. If money costs 3% compounded
monthly then the price of the car, to the nearest dollar, is: A. 20,894
B. 19,500 lfoo+3ooa 307%
C. 9,803 03 '60
18,196 ‘’ 1920* 3001:!“ U‘PLT; ]
E. 14,987 ﬂ
1 2.
2n§m<41
4. [4 marks] A debt of $2,000 due today and a debt of $2,000 due three years from today are to be paid off
by four annual payments: $1,000 a year from now, $1,000 two years from now, $1,000 three
years from now, and a ﬁnal payment of $ X four years from now. Interest is 7% annually. The value of X to the nearest dollar is compounded
A. $1,000 B. $1,547 0 a 7 3 ‘f C. $1,426 I“ 2000 X
é $1,322 Moo I000 IOOO E $1,101 ( X<m32Lef Page 3 of 10 5. [4 marks]
A $150,000 mortgage is amortized over 20 years at 7% compounded semiannually with
payments at the end of each month. Just after the 60th payment, the principal outstanding
(to the nearest $5) is l n ’1
B. $111,685 r ‘ . ‘
P‘O‘ ' Ra 1804 : [yo/aooalﬂjd
C. $112,500 F
D. $128,775 W1
fl 9'0
D P 0”) : {o)000[l— (LOBgfgoj
[1,0,0'33' AM] Going/13?, 43 6. [4 marks]
On September 11, 2000, a bond issued by the Government of Ontario, maturing on March
11, 2003, with semiannual coupons at an annual coupon rate of 8%, had an annual yield to
maturity of 6%. The price the bond traded at was (to the nearest $0.10): B. $108.40 P, mo (Lﬁyﬁk 0] a
C. $105.40 D. $103.70 96
E. $106.10 ’ 6,03 Page 4 of 10 7. [4 marks]
Find all pairs (2:, y) so that the matrix is in rowechelon form.
A" (227]) = (01 0) and (may) = (120)
B' (37y) = (1’ 1) only E (03,21) = (0,1) only 8. [4 marks]
3 l 4 7 T
If A = _5 6 , B = __2 6 , and B denotes the transpose of B , then I
T_ _ 3 I ‘l '2’ _ ‘1 2)
AB 33— (4 LXI; a) (.4, 18 —2 6
A' [—26 —17 7 ,zu)
—35 —57 2
B' [52 16] (77 73
a 7 —21
28 28
7 6
D‘ [1 28]
10 —30
E [_2 25] Page 5 of 10 9. [4 marks]
The solution to the matrix equation AX = C Where —1 2 2 a:
A1= 1—3—2 ,X=y,
1 ——1 —1 z
isgivenby
A. x=1,y=——3,z=—2
B. $=—1,y=——3,z:3 X’ C. $=3,y=—3,z=0
. ’l 2
D. :z:=0,y=—3,z=—1
I «3
@m=1,y=—3,z=0
: —l
a I
'3
0
10. [4marks]
Consider the linear system Ax = b Whose augmented matrix is Then the system has inﬁnitely many solutions with one parameter
B.
C.
D. no solutions
inﬁnitely many solutions with two parameters
a unique solution E. inﬁnitely many solutions with three parameters Page 6 of 10 HHOH [\Dl—ll—IH N HODJO l—‘NOH l 1  1  1  2 i l o   O
0 l I I
0 l
l O
0 0 / PART B. WrittenAnswer Questions 1. (a) [8 marks]
Sixteen monthly deposits of $50 each, followed by eight monthly deposits of $75 each, are made into an account. What is the accumulated value in the account right after the last
deposit of $75, if the account earned 12% compounded monthly during the ﬁrst year, but
only 9% compounded monthly during the second year? (All monthly deposits are at the end of the month.) 0! ~17. I3..b 17.”
l;
{o {0 5'0 50 7‘ 7f
3
:7. . q {
+ﬂ) 50$ l+.2_)+4 5
{OSmﬂ'U ‘2 + [z baa % (b) [’7 marks]
A person wishes to borrow $5,000 now and another $5,000 at the end of 5 years. He wishes to pay both loans off by 10 equal semiannual payments, the ﬁrst one due at the end of 3
years. If the interest rate is 8% compounded semiannually, then ﬁnd the “hr—payment to the nearest cent. few ’anﬂ ml 0 3’ b 4 7 qu u (L (3 m :(
W
{000 ' 5000 ' 'Rri'RRR Ran R K” Ra ﬂ 0“, Cqu)'(c {000 4“ {OOOCLWr'O Page 7 of 10 2. A loan of $100,000 is paid off with monthly payments (at the end of each month) over 20
years. Interest is 6% compounded monthly. Find [5] (a) the monthly payment;
[7] (b) the interest and principal paid in the 77th payment;
[3] (c) the total interest paid. a) \oo,ooo : R a $37.06
‘2. R = I00,000x.00{ A,
N I \— (Loos/Two Page 8 of 10 3. Ramus has a debt of $10,000 which is accruing interest at 8% compounded annually. He
wishes to pay off the debt in equal annual payments of $1,000 each, starting one year from
now, but he realizes that the very last payment will be less than $1,000. [5] (a) How many annual payments are there?
[10/ (b) How much is the very last payment? a) 10,000 = IOOO a?” Page 9 of 10 4. Consider the system of linear equations $1  2232 + $3 — $4 = 1
—2m1 + 42:2 — 933 + $4 = 3 ' ‘
— $1 + 2232 — (123 + [621134 2 k [6'] (a) Solve the system by using row reduction in case k = 1 . [.9] (b) Is there any value of k such that the system has no solution? Justify your answer. \ — 2 l ’1 "
2. ‘l «I I 3
.4 2 rl km k 'L , of Clo/l
19 l“ W" \ w xﬁa bwo ram M Ill \chl) 19L; [6196’ MW (c 0 ,
My 0/6. wo §a\n§. Awwals 1 , Page 10 of 10 ...
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