Download Math 100: Elements of Finite Mathematics and Calculus Σ 100

Math 100: Elements of Finite Mathematics and Calculus
Midterm 2 - Fall 2011
Duration : 90 minutes
Name
Student ID
Signature
#1
20
#2
15
#3
18
#4
20
#5
15
#6
12
Σ
100
• Put your name, student ID and signature in the space provided above.
• Calculators are allowed, but no other electronic device is allowed.
• This is a closed-book and closed-notes exam.
• Show all of your work; full credit will not be given for unsupported answers.
• Write your solutions clearly; no credit will be given for unreadable solutions.
• Mark your section below.
Section 1 (Emre Mengı, MWF 9:30-10:20)
Section 2 (Azadeh Neman, MWF 14:30-15:20)
Midterm 2
2
Question 1. A farm will hire 30 workers to pick cherries from trees. There are
three kinds of workers, type A, type B, and type C. Each worker of type A can pick
5kgs, each worker of type B can pick 15kgs, and each worker of type C can pick 10
kgs of cherries per hour. There are 1680 kgs of cherries to be picked up in 4 hours.
How many workers from each kind must be hired?
Midterm 2
3
Question 2. Indicate whether each of the following is true or false. You don’t need
to justify your answer.
(i) Every n × n matrix has an inverse.
(ii) It is possible that a system of linear equations has two and only two solutions.
(iii) Let A and B be n × n matrices. Then A · B 6= B · A in general.
(iv) The sum 4 + 2 + 1 + 1/2 + 1/4 + 1/8 + 1/16 is a geometric series.
(v) Two friends invest the same amount of money on a bank. The first invests on an
account compounded quarterly. The second invests on an account compounded
semi-annually. Both account have the same annual interest rate. At the end of
the first year the first would have more money in the account than the second.
Question 3.
(a) Find the matrix S such that
2 1
−1 3
(b) Find the matrix C such that

+S =
1 0
0 1

1 1
2
−1
3
C = 1 2 ·
1
5 −2
2 1
Midterm 2
(c) Find the matrix X such that




1 2 −1
1 0 0
 2 1
0 X =  0 1 0 
1 1
3
0 0 1
4
Midterm 2
5
Question 4. Özlem deposits 20,000TL in a saving account that is compounded
monthly at an annual rate of %6.
(a) What is the amount in Özlem’s saving account when she retires 30 years after
the initial deposit?
(b) When Özlem retires after 30 years, she does not withdraw all at once. Instead
she withdraws 5000TL at the end of every month from her saving account
(which is compounded monthly at a rate of %6) until she runs out of money
in her account. For how many months can she continue getting the monthly
payments from her account after her retirement?
Midterm 2
6
Question 5. You talk to the representatives of bank A and bank B to deposit
certain amount of money in a saving account. Bank A is offering %10 compounded
semi-annually. Bank B is offering %8 compounded continuously. Calculate the
annual percentage yield (APY) for each bank. Which bank would you choose based
on APY?
Midterm 2
7
Question 6.
(a) Find x that satisfies the equation below.
log3 5 + log3
81
= log3 x3 − log3 x
5
(b) Simplify
2
log(2−1 ) e + log2 e
(c) Solve the equation below for x.
ln (ln (x)) = 1
.
Midterm 2
Financial Mathematics - Formulas
Simple Interest
F V = P (1 + rt)
P
r
t
FV
:
:
:
:
Present value
Annual interest rate
Time in years
Future value
Compounded Interest
F V = P (1 + i)n
P
i
n
FV
:
:
:
:
Present value
Interest rate per compound period
Total number of compound periods
Future value
Future Value of an Ordinary Annuity
(1 + i)n − 1
FV = PMT ·
i
PMT
i
n
FV
:
:
:
:
Payment per compound period
Interest rate per compound period
Total number of compound periods
Future value
Present Value of an Ordinary Annuity
P = PMT ·
P
PMT
i
n
:
:
:
:
1 − (1 + i)−n
i
Present value
Payment per compound period
Interest rate per compound period
Total number of compound periods
8