Let p be an element in N, and define d _p to be the set of all pairs (l,m) in N×N such that p divides m−l. Show that d_p is an equivalence relation

Good afternoon interns,

As you all know, you are all vying for the 15 openings to be offered at the end of the internship. Another chance has come up for you to show us that you deserve one of those positions here at Firmament Financials Inc. New clients, Mr. and Mrs. Perez are planning on starting a family and would like to start saving for their child’s college education. The want to know what they can afford when their child is ready for college. They would like to utilize our Yearly College Savings Plan which grows a yearly deposit at an APR of 15% with continuous compounding. Your job is to analyze their information and come up with feasible recommendations. Their information can be found in the email they sent. Of course, you will be graded on your analysis. A grading rubric has been provided.

Good luck interns!

Dear Ms. Robles,

As satisfied customers of Firmament Financial, we of course look to you to help us save for our upcoming child’s college education. We feel that we can afford $100 per month for the 18 years until college in your College Savings Plan. We have a list of universities that we have researched with the approximate cost per year for each. Which if any of the university options will we be able to afford for our child? We would prefer to send them to an elite private university if we can afford it.

Anthony & Veronica Perez

Let b be a vector that has the same height as a square matrix A. How many solutions can Ax = b have?

Suppose your instructor asked you to look at the influence of age (X) on automobile insurance premium rates (Y). List three variables you might control for in your regression analysis and why you would control for them (that is, what effect would these control variables have on insurance rates?).

7. Find the solution of the following PDEs:
ut−16uxx =0
u(0,t) = u(2π,t) = 0

u(x, 0) = π/2 − |x − π/2|

Write a funtion and apply multi-var. Newton-Raphson method.

QUESTION 1

Vector Space Axioms

Let V be a set on which two operations, called vector addition and vector scalar multiplication, have been defined. If u and v are in V , the sum of u and v is denoted by u + v , and if k is a scalar, the scalar multiple of u is denoted by ku . If the following axioms satisfied for all u , v and w in V and for all scalars k and l , then V is called a vector space and its elements are called vectors.

1) u + v is in V

2) u + v = v + u

3) (u + v) + w = u + (v + w)

4) 0 + v = v

5) v + (−v) = 0

6) ku is in V

7) k(u + v) = ku + kv

8) (k + l)u = ku + lu

9) k(lu) = (kl)(u)

10) 1v = v

Task: Show that the set V of all 3×3 matrices with distinct entries and also combination of positive and negative numbers is a vector space if vector addition is defined to be matrix addition and vector scalar multiplication is defined to be matrix scalar multiplication.

QUESTION 2

Suppose u, v, and w are all vectors in a vector space V and c is any scalar. An inner product on the vector space V is a function that associates with each pair of vectors in V, say u and v, a real number denoted by u, v that satisfies the following axioms.

(a) < u, v > = < v, u > (Symmetry axiom)

(b) < u + v, w > = < u, w + v, w > (Additive axiom)

(c) < cu, v > = < c u, v > (Homogeneity axiom)

(d) < u, u > ≥ 0 and < u, u > = 0 if and only if u = 0 (Positivity axiom)

A vector space along with an inner product is called an inner product space.

Task: Show that the set V of all 3×3 matrices with distinct entries and also combination of positive and negative numbers is a inner product space if vector addition is defined to be standard matrix addition and vector scalar multiplication is defined to be matrix scalar multiplication.

4.Maximize: Z = 2X1+ X2-3X3

Subject to: 2X1+ X2= 14

X1+ X2+ X3≥6

X1, X2, X3≥0

Solve the problem by using the M-technique.

Suppose your vehicle’s speedometer is geared to accurately give your speed only using a certain tire size: 17” diameter wheels (the metal part) and a 6.5” tires (thickness of rubber part).

a. If your vehicle’s instruments are properly calibrated, how many times should your tire rotate per second if you are travelling at 55 miles per hour?

b. Now find the angular speed from part a.

c. Suppose you bought new 7.1” tires for your 17” diameter wheels. You took the for a test run and drove at a constant speed of 55 miles per hour (according to your vehicle’s instrument). However, a law enforcement officer stopped you and claimed that you were speeding in a 55 miles per hour zone. How fast did the officer’s radar clock you at, in miles per hour?

Solve the following equations for the unknown.

1. G - 24 = 75

2. 3(2c - 5) = 45

3. n/4 - 7 = 8

For the following statements, underline the key words and translate into an expression.

4. 15 less than one-ninth of P

5. 3 times the quantity of H less 233

For the following statement, underline the key words and translate into an equation.

6. A number increased by 11 is 32

7. The sum of 2 more than 6 times a number and 7 times that number is that number decreased by 39

Set up and solve equations for each of the following business situations.

8. At Telepower Plus, long-distance phone calls to China cost $0.59 for the first minute and $0.25 for each additional minute plus an additional roaming charge of $2.50. If the total charge of a call to Beijing was $11.84, how long did the call last?

9. A Cold Stone Creamery ice cream shop sells sundaes for $3.60 and banana splits for $4.25. The shop sells four times as many sundaes as banana splits.

a. If total sales amount to $3,730 last weekend, how many of each were sold?

b. What were the dollar sales of each?

Use ratio and proportion to solve the following business situation.

10. Angeles Hatcher is planting flower bulbs in her garden for this coming summer. She intends to plant 1 bulb for every 5 square inches of flower bed.

a. How many flower bulbs will she need for an area measuring 230 square inches?

b. If the price is $1.77 for every 2 bulbs, how much will she spend on the flower bulbs?

suppose a and b are positive integers. price the following by biconditional statement

---- a +1 divides B and B divides b + 3 if and only if a = 2 and b = 3

a) Define which differential equations are called linear and which are called nonlinear? [10 marks]

b) Define which systems of differential equations are called homogeneous and which are called non-homogeneous? [10 marks]

c) What is the general structure of solution of a linear nonhomogeneous differential equation? [10 marks]

d) What is the principle of superposition in application to the solution of ordinary differential equations (ODEs)? To which kind of ODEs is it applicable? [10 marks]

a) Merchandise is sold at concerts. The manager of a concert claims that the mean value of merchandise sold to premium ticket holders is more than the mean value of merchandise sold to standard ticket holders. The mean value of merchandise sold to a random sample of 60 standard ticket holders at the concert is $15 with a standard deviation of $10. The mean value of merchandise sold to a random sample of 55 premium ticket holders at the concert is $23 with a standard deviation of $8. Test the manager’s claim at the 5% level of significance. State your hypothesis clearly.

b) For the test in part a), state whether or not it is necessary to assume that values of merchandise sold have normal distributions.Give a reason for your answer.

c) A machine fills packets with almonds. The weight, in grams, of almonds in a packet is modelled by ?(?,?2). To check that the machine is working properly, a random sample of 10 packets is selected and unbiased estimates for ? and ?2 are ?̅ = 202 and ?2 = 3.6
Stating your hypothesis clearly, test, at the 1% level of significance, whether or not the mean weight of almonds in a packet is more than 200g.

d) In order to test ?0: ? ≤ 35 versus ?1: ? > 35, a random sample of ? = 15 is obtained from a population that is normally distributed. The sample had a standard deviation of ? = 37.4. Test this hypothesis at the level of significance of 1%.

Explain using complete sentences the advantages of providing an EXACT solution to a quadratic equation rather than a decimal approximation.