(1) Show that the set { 1 m + 1 n : m, n ∈ N} is countable.

(2) Show that the set {a + b √ 2 : a, b ∈ Q} is countable.

(3) Show that the intersection of two countable sets is countable.

(4) Show that the set of all irrational numbers is uncountable.

(5) Let C = {0, 1, 2, . . . , 9}. Show that the set C ×C × · · · is uncountable. [Hint: Imitate the proof we had for E × E × · · ·, where E = {0, 1}.]

Part A

If the voltage across a circuit of constant resistance is doubled, how is the current in the circuit affected? If the voltage across a circuit of constant resistance is doubled, how is the current in the circuit affected?

1-The current is reduced by a factor of 4.

2-The current is doubled.

3- The current is reduced by a factor of 2.

4-The current remains constant.

5-The current is quadrupled.

You roll TWO six-sided die. Find the probability of each of the following scenarios in fractions.

1. P(sum=2)

2. P(sum is less than or equal to 4)

3. P(sum=13)

4. P(product)= 20 (Product means multiply)

5. P(product less than or equal to 12)

Let f(x) and g(x) be two generic functions. Assume

limx→0(f(x)+2g(x))=2 & limx→0(f(x)−g(x))=8.

Compute limx→1(f(lnx)/g(x2−x)).

A. It cannot be computed

B. 2

C. 4

D. -3

E. -4

8) A project costs $1200 immediately. The project yields nominal returns of $200 in years 1 and 2, $250 in years 3 and 4, and $300 in year 5. In addition, the project will have capital worth $400 leftover at the end of year 5. The real discount rate is 4% and the expected inflation rate is 2%. What is the NPV of this project?

You roll TWO six-sided die. Find the probability of each of the following scenarios in fractions.

1. P(sum=2)

2. P(sum is less than or equal to 4)

3. P(sum=13)

4. P(product)= 20 (Product means multiply)

5. P(product less than or equal to 12)

Find the general solution for the following ODEs. If initial condition is given, solve the initial value problem.

3) (D^4 − 13D^2 + 36I)y = 12e^x

4) (D^3 − D^2 − D + I)y = 0, y(0) = 0, Dy(0) = 1, D^2y(0) = 0

Question 40.5 pts

Redox reactions during Qualitative Analysis.

Likewise, there were several instances during the Qual procedure where a redox reaction occurred (which often was also associated with a change in solubility). During the cation procedure, adding a mixture of H₂O₂ and NaOH caused Cr(OH)₃ to be converted to CrO₄^2–. Not accidentally, one of these compounds was quite insoluble while the other one was quite soluble. The balanced reaction equation is:

2 Cr(OH)₃(s) + 3 H₂O₂(aq) + 4 OH^–(aq) → 2 CrO₄^2–(aq) + 8 H₂O(ℓ)

a. For this question, focus on the change in the oxidation state of chromium. What is the oxidation state of chromium in the reactant and product compounds?

Python

1) Show the result of evaluating each expression. Be sure that the value is in the proper form to indicate its type (int or float). If the expression is illegal, explain why.

a) 4.0/10.0 + 3.5 *2

b)10%4 + 6/2

c)abs(4-20//3)**3

d) sqrt(4.5-5.0) + 7*3

e) 3*10//3 + 10%3

f) 3**3

2) Show the sequence of numbers that would be generated by each of the following range experssions.

a) range(5)

b) range(3,10)

c)range (4, 13, 3)

d)range (15, 5, -2)

e)range(5,3)

3) What do you think will happen if you use a negative number as the second parameter in the round function? E.g. what should be the result of round (314. 159265, -1)?

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You have the opportunity of purchasing a farm for $500,000. The owner will allow a $100,000 down payment with the remainder financed in one of the following ways.

1. Equal annual payments with an interest rate of 7% for 20 years.

2. Equal annual payments with an interest rate of 10% for 15 years.

3. Equal quarterly payments with interest rate of 8% for 15 years.

4. Equal principal payments each year with interest rate of 8% for 10 years.

Determine the following: a. The payments for options 1-3.

b. The total interest paid for options 1-3.

c. Principal and interest paid in each of the first 4 years for option 3 and 4. (you may not know how to do option 4).

d. Which option (1-4) would you choose? Why?