Assume that you are breaking a stick into 3 pieces by uniformly and independently selecting two break points ? and ?. If we denote the event that these pieces form a triangle by T and lengths of the three pieces by ?, ? and ?, calculate: a. ?(? ∩ (? < ? < ?)), b. The probability that the pieces form an equilateral triangle.

Bezout’s Theorem and the Fundamental Theorem of Arithmetic

1. Let a, b, c ∈ Z. Prove that c = ma + nb for some m, n ∈ Z if and only if gcd(a, b)|c.

2. Prove that if c|ab and gcd(a, c) = 1, then c|b.

3. Prove that for all a, b ∈ Z not both zero, gcd(a, b) = 1 if and only if a and b have no prime factors in common.

Let S_k(n) = 1^k + 2^k + ··· + n^k for n, k ≥ 1. Then, S_4(n) is given by

S_4(n)= n(n+1)(2n+1)(3n^2 +3n−1)/ 30

Prove by mathematical induction.

Adam is loaning $5000 to Bert for a period of 2 years. Suppose Bert will repay the loan with a $5000 balloon payment at the end of the 2 years and will pay monthly interest payments each month which will end the month before the $5000 balloon payment. e. If interest is 6% effective interest, how much will Bert’s interest payments be to Adam each month? f. If Adam takes the payments he receives each month from Bert and reinvests them at 3% annual interest compounded monthly, how much will he have in his savings account at the time that Bert pays the balloon payment? g. What is Adam’s rate of return on his initial investment of $5000? That is find the yield rate as an effective interest rate.

Part II: Choose the best answer

7. Which of the following is a fundamental product adjacent to xy'zw
   1. xyzw'
   2. xy'zw'
   3. x'y'z'w
   4. x'yzw
   5. None
8. The quantity of fundamental products when you have 3 Boolean variables is:
   1. 4
   2. 6
   3. 15
   4. 8
   5. None
9. Which of the following is not a fundamental product adjacent to xy'zw
   1. xy'z'w
   2. x'yz'w
   3. x'y'zw
   4. xyzw
   5. None
10. Morgan's 1st law is defined (x + y) '= x'y' How do you simplify (x + y + z + w)' using this law?
    1. x'+y'+z'+w'
    2. (x+y)'(z+w)'
    3. x'y'z'w'
    4. (xy)'+(zw)'
    5. None
11. Morgan's 2nd law is defined (xy)'= x' + y ' How do you simplify (xyz)' using this law?
    1. x'+y'+z'
    2. x'+y'z''
    3. x'y'z'
    4. x'y'+z'
    5. None
12. Use Morgan's 1st and 2nd law, to simplify
    [(w + x) y] '
    1. w'+x'+y'
    2. w'+x'y'
    3. w'x'y'
    4. w'x'+y'
    5. None
13. Use Morgan's 1st and 2nd law to simplify [(x + y)'z']' Remember that (x')'= x
    1. xyz
    2. (x+y)z
    3. x+y+z
    4. (xy)+z
    5. None
14. Use Morgan's 1st and 2nd law, to simplify
    [(w + x + y) z] '
    1. w'x'y'z'
    2. w'+x'+y'+z'
    3. (w'x'y')+z'
    4. (w'+x'+y')z'
    5. None
15. The quantity of fundamental products when there are n Boolean variables is
    1. n+2
    2. 2n
    3. n²
    4. 2ⁿ
    5. None

find Lagrange polynomials that approximate f(x)=x^3,

a) find the linear interpolation p1(x) using the nodes X0=-1 and X1=0

b) find the quadratic interpolation polynomial p2(x) using the nodes x0=-1,x1=0, x2=1

c) find the cubic interpolation polynomials p3(x) using the nodes x0=-1, x1=0 , x2=1 and x3=2.

d) find the linear interpolation polynomial p1(x) using the nodes x0=1 and x1=2

e) find the quadratic interpolation polynomial p2(x) using the nodes x0=0 ,x1=1 and x2=2

Find the product from the multiplication table for the symmetries of an equilateral triangle the new permutation notation to verify each of the following:

A) R(240)R3 B) R3R(240) C) R1R3 D) R3R1

In the new system under development for Benny's Used Cars, seven tables will be implemented in the new relational database. These tables are New Vehicle, Trade-in Vehicle, Sales Invoice, Customer, Salesperson, Installed Option, and Option. The expected average record size in characters for these tables and the initial record count per table are given next.

Perform a volumetrics analysis for the system. Assume that the DBMS that will be used to implement the system requires 35% overhead to be factored into the estimates. Also, assume a growth rate for the company of 10% per year. The systems development team wants to ensure that adequate hardware is obtained for the next 3 years.

Let E = Q(√a), where a is an integer that is not a perfect square. Show that E/Q is normal

A field F is said to be perfect if every polynomial over F is separable. Equivalently,
every algebraic extension of F is separable. Thus fields of characteristic zero and
finite fields are perfect. Show that if F has prime characteristic p, then F is perfect
if and only if every element of F is the pth power of some element of F. For short we
write F = F p.

prove that a translation is an isometry
"i want the prove by using a parallelogram and proving that the two side are congruent please"

Multivariate analysis

Using the data provided, perform the following analysis:

• Determine the explanatory and response variables.
• Run a multivariate regression analysis on all three variables.

Interpret the results by answering the following questions:

• What is the calculated correlation coefficient? Do the sales figures correlate with the marketing expenditure and price?
• Comment on the coefficient of determination. What percentage of the response data can be explained by the explanatory variables?
• Determine the multiple regression line equation in the form:

sales^ = (intercept) + (coefficient)× marketing + (coefficient)× price

• Using the regression equation formulated, what is the amount of expected sales (in pounds), if the price is set at £3.50 and the amount spent on marketing is £300?
• Interpret the variables in the regression equation. What impact does each of the factors (marketing and price) have on the sales figures?

What is the last digit of 123456789012345678×609 in base 6. Don’t try to compute the product.