1. (2 pts each) Consider the following algorithm:

procedure polynomial(c, a0,a1,…an: real numbers)
power≔1
y≔a₀
for i=1 to n
  power≔power*c
y≔y+a_i*power
return y
(Note: y=a_ncⁿ + a_n-1 c^n-1 +. . . + a₁C +a₀ so the final value of y is the value of the polynomial at x=c)

a. Use the algorithm to calculate f(3), where f(x)=2x²+3x+1 at x=3. Show the steps of working through the algorithm – don’t just plug 3 in for x in f(x).

b. How many multiplications and additions are needed to evaluate a polynomial of degree n at x=c?

Exercise 9.9.1: Breaking RSA by factoring.

Bob publishes his public key (e, N) = (109, 221)

(a)

Show that if Eve can factor N (N = 13 · 17), then she can determine Bob's private key d. What is Bob's private key?

(b)

Now suppose that Eve intercepts the message 97. Use Bob's private key to decrypt the message.

Imagine an octagon(8 vertices) with all its diagonals. Is it possible to draw this without lifting your pen (and no backtracking). If we attach a triangle for each boundary edges of the Octagon, can you draw this without lifting your pen? Explain

Combinatorics:

6. A mathematician picks and integer i from the set {1,2,3,...,15} and a computer scientist tries to find the number by asking questions of the form: Is i<x, i>x, or i=x? Show that the number can always be found using three questions

Determine the eigenvalues and eigenfunctions of the following regular
Sturm–Liouville systems:

(a) y′′ + λy = 0,
y′ (0) = 0 , y′ (π) = 0.

(b) y′′ + λy = 0,
y (1) = 0 , y(0) + y′ (0) = 0.

(5.1.24) Prove that 1/(2n) ≤ [1 · 3 · 5 · · · · · (2n − 1)]/(2 · 4 · · · · · 2n) whenever n is a positive integer

A student is taking a standardized test consisting of several​ multiple-choice questions. One point is awarded for each correct answer. Questions left blank neither receive nor lose points. If there are six options for each question and the student is penalized 1/3 point for each wrong​ answer, how many options must the student be able to rule out before the expected value of guessing is zero?

1. Sorur Inc. has gone out on bid for an electronic component. Expected demand is 500 units per month. The item can be purchased from either Asha Manufacturing or Brian Manufacturing. Their price schedule is shown below. Ordering cost is $25, and annual holding cost per unit is $15.

Ahsa Mfg.                                                           Brian Mfg.  

Quantity              Price/unit                           Quantity              Price/unit

1- 199                    $20.00                                   1 – 299                 $20.00

200 – 399             $19.80                                   300 – 599             $19.75

400+                      $19.60                                   600+                      $19.50

What is the optimal order quantity? Which supplier should be used?

: Describe the different models used to model the distribution of particles in statistical mechanics, including Maxwell–Boltzmann, Bose–Einstein, and Fermi–Dirac statistics. In each case, describe the counting techniques used in the model.

1 Over the past decade, there has been a strong positive correlation between teacher salaries and prescription drug cost.

(a) Do you think paying teachers more causes prescription drugs to cost more? Explain.
(b) What lurking variable might be causing the increase in one or both of the variables? Explain.

2 When we use a least-squares line to predict y values for x values beyond the range of x values found in the data, are we extrapolating or interpolating? Are there any concerns about such predictions?

WRITE IN YOUR OWN! WORDS and dont copy from another question that have been answer plz thank you

Let G be a group and let C={g∈G|xg=gx for all x∈G} be the center of G. Prove that for any a ∈ G, aC = Ca.

Prove that {??+?:?,?∈?} is dense in ? if and only if  r is an irrational number.

Explain why every permutation in S(n) can be represented by a product of n-1 or fewer cycles of length 2 (transpositions). Represent the permutationσ in problem (1) above as a product of 8 or fewer transpositions. Is σ an even or an odd permuation?