The buyer of a piece of real estate is often given the option of buying down the loan. This option gives the buyer a choice of loan terms in which various combinations of interest rates and discount points are offered. The choice of how many points and what rate is optimal is often a matter of how long the buyer intends to keep the property.

Darrell Frye is planning to buy an office building at a cost of $988,000. He must pay 10% down and has a choice of financing terms. He can select from a 7% 30-year loan and pay 4 discount points, a 7.25% 30-year loan and pay 3 discount points, or a 7.5% 30-year loan and pay 2 discount points. Darrell expects to hold the building for four years and then sell it. Except for the three rate and discount point combinations, all other costs of purchasing and selling are fixed and identical.

1. What is the amount being financed?

2. If Darrell chooses the 4-point 7% loan, what will be his total outlay in points and payments after 48 months?

3. If Darrell chooses the 3-point 7.25% loan, what will be his total outlay in points and payments after 48 months?

4. If Darrell chooses the 2-point 7.5% loan, what will be his total outlay in points and payments after 48 months?

5. Of the three choices for a loan, which results in the lowest total outlay for Darrell?

Use the Laplace transform to solve the given system of differential equations.

dx/dt + 2x +dy/dt= 1

dx/dt− x+ dy/dt− y= e^t x(0) = 0, y(0) = 0

Suppose the sets A and B have both n elements.

1. Find the number of one-to-one functions from A to B.

2. Find the number of functions from A onto B.

3. Find the number of one-to-one correspondences from A to B.

Algebraists have proven (using more advanced techniques than ones we’ve discussed) that An is a simple group for n ≥ 5.

Using this fact, prove that for n ≥ 5, An has no subgroup of order n!/4 .

c) Prove that Sn is not simple for n ≥ 3.

d) Is GL(2, R) a simple group? Prove your claim.

1. (5) Taking into account identical letters, how many ways are there to arrange the word CALLITRICHIDAE?
2. (5) Taking into account identical letters, how many ways are there to arrange the word CALLITRICHIDAE that start or end with vowels?
3. (5) Taking into account identical letters, how many ways are there to arrange the word CALLITRICHIDAE that contain the substring LITERAL?
4. (5) Taking into account identical letters, how many ways are there to arrange the letters in the word ANTIDISESTABLISHMENTARIANISM?

8. Definition: A set A is finite if there exists a non-negative integer c such that there exists a bijection from A to {n ∈ N : n ≤ c}. (The integer c is called the cardinality of A.)

(a) Let A be a finite set, and let B be a subset of A. Prove that B is finite. (Hint: induction on |A|. Note that our proof can’t use induction on |B|, or indeed refer to “the number of elements in B” at all, because we don’t yet know that B is finite!)

(b) Prove that the union of two disjoint finite sets is finite.

(c) Prove that the union of any two finite sets is finite. (Hint: A ∪ B = A ∪ (B − A))

7. Let n ∈ N with n > 1 and let P be the set of polynomials with coefficients in R.

(a) We define a relation, T, on P as follows: Let f, g ∈ P. Then we say f T g if f −g = c for some c ∈ R. Show that T is an equivalence relation on P.

(b) Let R be the set of equivalence classes of P and let F : R → P be the derivative operator defined as F([f]) = df/dx. Is F well defined (i.e. is it a function)? Is it surjective? Is it injective?

Let A = {a1, a2, a3, . . . , an} be a nonempty set of n distinct natural numbers. Prove that there exists a nonempty subset of A for which the sum of its elements is divisible by n.

Solve the given initial-value problem. dx/dt = y − 1

dy/dt = −6x + 2y

x(0) = 0, y(0) = 0

1) a) Prove that the union of two countable sets is countable.

b) Prove that the union of a finite collection of countable sets is countable.

Sam got a car financing where he needs to do 48 monthly payments of $5,448.75 each, starting the moment he receives the vehicle. Given an annual interest rate of 18.4% compounded monthly:

(a) Calculate the spot price of the car.

(b) Elaborate the amortization schedule.

This problem is a complex financial problem that requires several skills, perhaps some from previous sections. Clark and Lana take a 30-year home mortgage of $124,000 at 7.4%, compounded monthly. They make their regular monthly payments for 5 years, then decide to pay $1200 per month. (a) Find their regular monthly payment. (Round your answer to the nearest cent.) $ (b) Find the unpaid balance when they begin paying the $1200. (Round your answer to the nearest cent.) $ (c) How many payments of $1200 will it take to pay off the loan? Give the answer correct to two decimal places. monthly payments (d) Use your answer to part (c) to find how much interest they save by paying the loan this way. (Round your answer to the nearest cent.) $

When Maria Acosta bought a car
2
1
2
years ago,
she borrowed $13,000 for 48 months at 7.8% compounded monthly. Her monthly payments are $316.15, but she'd like to pay off the loan early. How much will she owe just after her payment at the
2
1
2
-year
mark? (Round your answer to the nearest cent.)
$