In each case below either prove that the statement is True or disprove it by giving an example showing that it is False.

(i) If B is a fixed 2 × 2 matrix, then the set U = {A|A ∈ M22, AB = 0} is a subspace of M22. T or F

(ii) If u, v, w are vectors in a vector space V , then span {u, v, w} = span {u + v, u + w, v + w} T or F

(iii) The set {x,sin2 x, cos2 x} is independent in F[0, 2π] T or F

(iv) {1 + x, x + x 2 , x2 + x 3 , x3} is a basis of P3. T or F

Classify: homogenoues, inhomogenous or non linear:

1. Ut=X^2Uxx + 2XUx

2. - Uxx - Uyy = sineU

3. Uxx + 2YUyy =3

4. Ut + UUx =3U

5. e^yUx=e^xUy

6.Ut =5Uxxx + X^2U + X

Note: small letters represent subscript

5. Without using the method of mathematical induction, prove that 5^ⁿ − 3^ⁿ + 2n is divisible by 4 for all natural n.

Mortgage payments

Mortgages, loans taken to purchase a property, involve regular payments at fixed intervals and are treated as reverse annuities. Mortgages are the reverse of annuities, because you get a lump-sum amount as a loan in the beginning, and then you make monthly payments to the lender.

You’ve decided to buy a house that is valued at $1 million. You have $100,000 to use as a down payment on the house, and want to take out a mortgage for the remainder of the purchase price. Your bank has approved your $900,000 mortgage, and is offering a standard 30-year mortgage at a 12% fixed nominal interest rate (called the loan’s annual percentage rate or APR). Under this loan proposal, your mortgage payment will be   per month. (Note: Round the final value of any interest rate used to four decimal places.)

Your friends suggest that you take a 15-year mortgage, because a 30-year mortgage is too long and you will pay a lot of money on interest. If your bank approves a 15-year, $900,000 loan at a fixed nominal interest rate of 12% (APR), then the difference in the monthly payment of the 15-year mortgage and 30-year mortgage will be   ?(Note: Round the final value of any interest rate used to four decimal places. )

It is likely that you won’t like the prospect of paying more money each month, but if you do take out a 15-year mortgage, you will make far fewer payments and will pay a lot less in interest. How much more total interest will you pay over the life of the loan if you take out a 30-year mortgage instead of a 15-year mortgage?

$1,777,192.70

$1,916,035.88

$1,388,431.80

$1,638,349.52

Which of the following statements is not true about mortgages?

The payment allocated toward principal in an amortized loan is the residual balance—that is, the difference between total payment and the interest due.

Mortgages always have a fixed nominal interest rate.

The ending balance of an amortized loan contract will be zero.

The revenue and cost functions for a particular product are given below. The cost and revenue are given in dollars, and x represents the number of units .

R(x) = −0.8x² + 704x

C(x) = 264x + 57120

(a) How many items must be sold to maximize the revenue?
________________
(b) What is the maximum revenue?
________________

(c) Find the profit function.
P(x) = ___________

(d) How many items must be sold to maximize the profit?

_______________

(e) What is the maximum profit?

________________

(f) At what production level(s) will the company break even on this product? (Enter your answers as a comma-separated list.)
units

________________

An oil tanker has hit a sand bar and ripped a hole in the hull of the ship. Oil has begun leaking from the tanker. The oil is leaking from the ship forming a circle around it. The radius of the circle is increasing at a rate of 2.3 feet per hour. Please assist the Wild Life Federation with the following calculations. Thank you in advance for your assistance. (Round all answers to the nearest tenths place unless otherwise specified, use the π button on your calculator for calculations.) 1) A) Write the radius of the circle as a function of time. Use t as the symbol to represent time (you will need to use the information found in the news clip above.) B) What is the radius of the circle after 2 hours? C) What is the radius of the circle after 2.5 hours? D) If the oil tanker is 250 yards from shore, when will the oil first reach the shoreline? (Remember to convert to feet) 2) Write the area of the circle as a function of the radius. Use the symbol r to represent the radius. 3) A) Using the functions found in questions 1 and 2, write a function that represents area as a function of time. B) What is the area of the circle after 2 hours? C) What is the area of the circle after 2.5 hours? D) What is the area of the circle after 3 hours? E) What is the area of the circle after 3.5 hours? Math 171 Unit 1 Lab 4) Compute the average rate of change per hour for the area from 2 to 2.5 hours. 5) Compute the average rate of change per hour for the area from 3 to 3.5 hours. 6) Based upon the results found in questions 4 and 5 what is happening to the average rate of change of the area of the circle as time passes? (Increasing, Decreasing, Constant?) 7) If the tanker is 250 yards from the shore, how long will it be until 8 miles of shoreline is contaminated with oil? Round to the nearest number of days. (For the benefit of this problem, the shore is straight and if you were to draw a line perpendicular to the shore 250 yards you would find the boat, the spread will be 4 miles to either side of the ship, refer to the diagram below. Hint 1mi=5280 ft.) A) Use the figure below and find the length of the radius in feet. B) Using your formula from problem 1A, find the time in hours. C) The original question asks for the time measured in days. Convert your answer from 7B to days, rounding to the nearest number of days. Radius Radius 250 yds 4 miles 4 miles

Prove or disprove:
(a) If G is a graph of order n and size m with three cycles, then m ≥ n +
2.
(b) There exist exactly two regular trees.

Problem 3. Consider the unit interval [0, 1], and let ξ be fixed real number with ξ ∈ (0, 1) (note that the case ξ = 1/3 corresponds to the regular Cantor set we learned in our lectures). In stage 1 of the construction, remove the centrally situated open interval in [0, 1] of length ξ. In stage 2 remove the centrally situated open intervals each of relative length ξ (i.e. if the interval has length a you remove an interval of length ξ × a), one in each of the remaining intervals after stage 1, and so on. Let C_ξ denote the set which remains after applying the above procedure indefinitely

(a) Prove that C_ξ is compact.

(b) Prove that C_ξ is totally disconnected and perfect.

(c) Atually, prove that the complement of C_ξ in [0, 1] is the union of open intervals of total length equal to 1.

(d) Show directly that m_∗(C_ξ) = 0.

Increased Efficiency, Inc. is looking for ways to shorten its cash conversion cycle. It has annual sales of $36,500,000, or $100,000 a day on a 365-day basis. The firm's cost of goods sold is 75% of sales. On average, the company has $9,000,000 in inventory and $8,000,000 in accounts receivable. Its CFO has proposed new policies that would result in a 20% reduction in both average inventories and accounts receivable. She also anticipates that these policies would reduce sales by 10%, while the payables deferral period would remain unchanged at 40 days. What effect would these policies have on the company's cash conversion cycle?

Each year, Florida's Best Salad Dressing, Inc. (FBSD) purchases 50,000 gallons of extra virgin

olive oil. Ordering costs are $80.00 per order, and the carrying cost, as a percentage of inventory

value is 80 percent. The purchase price to FBSD is $0.50 per gallon. FBSD’s management currently

orders the EOQ each time an order is placed. No safety stock is carried. The supplier is now offering a

quantity discount of $0.03 per gallon if FBSD orders 10,000 gallons at a time. What is the net benefit

in dollars if FBSD takes the discount?

A single card is chosen at random from a deck of 52 cards, the probability that a club card is selected is 1/4. Does this probability mean that, if you choose a card at random 8 times, a club will appear twice? If not, what does it mean?

Probability?

How much interest (to the nearest dollar) would be saved on the following loan if the condominium were financed for 15 rather than 30 years? A $269,000 condominium bought with a 30% down payment and the balance financed for 30 years at 3.05%

Implement the TF-IDF pivoted normalization method and Okapi retrieval formula above. For the Okapi formula, set k3=1000 and k1=1.2, so that you have only one parameter b to vary. In this way, both algorithms have precisely one parameter to tune.

On January 1, Landan, Inc. issued a 20-year semiannual bond with a face value of $915,000 paying 8%. On the date of issue, the market rate for similar bonds was 8%.   Landan would like to know how much this issuance would raise.

The following table is a SCRATCH table and will NOT be graded. You may use it, if you wish, to assist you. Provided you answer the other questions correctly, you could leave it blank and still receive full credit.

What is the value of the ordinary annuity factor, AO() that is used in this problem?

What is the value of the discount factor, DF(), that is used in this problem?

How much does Landan raise from this issue? . You may ignore transaction costs.

NEED ANSWER IN 20 TIMED QUIZ