Math Answers

5. You have 5 different-colored bottles, each with a distinct cap. In how many ways can these caps be put on the bottles such that none of the caps are on the correct bottles? (Assume that all the caps must be on the bottles.) 

Given an=an−1−6an−2 where a0=1 a2=5

a.) list the first 10 terms of the sequence

b.) find a closed form(solve the recurrence relations) 

 Consider the recurrence relation an=4 an−1−4 an−2

Find the general solution to the recurrence relation and the solution when a0=−2 and a1=3.

There ate 10 students in a cafeteria. If X denotes the number of males, what are the possible values of X?

If F is Lipschitz function and g(x) is monotonically increasing on [a,b] then fog is bounded variation

Consider the function

F(x)= xsin(π/x) 0<x≤1

0 x=0

Show that f(x) is continuous but not of bounded variation

Find the real root of the equation 𝑥𝑒𝑥=2 correct upto two decimal places using the method of false position.

2.8. Let P be the property “is a prime number” and O be the property “is an odd

integer.” Consider the sets A = {x ∈ N : P(x)} and B = {x ∈ N : O(x)}.

1. Examine A and B with respect to the subset relation. What can you conclude? Justify your

answer.

2. Are A and B equal? Justify your answer.

Find out if the inverse exists for the following, give reasoning behind your answer. If you conclude that the inverse exists then find the B ́ezout coefficients and the inverse of the modulo. [Hint: Example 2 of section 4.4 in the book]

(a) - (3 points) 678 modulo 2970

(b) - (3 points) 137 modulo 2350

Find out if the following numbers are prime numbers, show your work using prime factorization. You may use code to verify your answer but do not put it up as your solution.:

(a) - (3 points) 773

(b) - (3 points) 733

(c) - (3 points) 377