How many subgroups of order 9 and 49 may there be in a Group of order 441

A brine solution of salt flows at a constant rate of 4 ​L/min into a large tank that initially held 100 L of pure water. The solution inside the tank is kept well stirred and flows out of the tank at a rate of 3 ​L/min. If the concentration of salt in the brine entering the tank is 0.6 ​kg/L, determine the mass of salt in the tank after t min. When will the concentration of salt in the tank reach 0.1 ​kg/L?

Solve the Following Equation:

y'' + y' + y = a*sin(ω*t), y(0) = 0 , y'(0) = 0

Thanks

Describe the level surfaces for the 3-variable function: f(x,y,z) = z/(x-y)

Let p and q be any two distinct prime numbers and define the relation a R b on integers a,b by: a R b iff b-a is divisible by both p and q. For this relation R: Prove that R is an equivalence relation.

you may use the following lemma: If p is prime and p|mn, then p|m or p|n. Indicate in your proof the step(s) for which you invoke this lemma.

Show that for any square-free integer n > 1, √ n is an irrational number

Topology question:

Show that a function f : ℝ → ℝ is continuous in the ε − δ definition of continuity if and only if, for every x ∈ ℝ and every open set U containing f(x), there exists a neighborhood V of x such that f(V) ⊂ U.

Use a direct proof to prove that 6 divides (n^3)-n whenever n is a non-negative integer.

Permutations and combinations.

a.)A typesetter has before him 26 trays, one for each letter of the alphabet. Each tray contains

10 copies of the same letter. In how many ways can he form a three letter word that

requires at most two different letter?

b.)Determine the number of ways of forming words which use exactly two different letters.

Solve h(x) =l -2x l

Solve y = 2 (x + 4)2 - 3

Please explain for each of the above provide examples · write down five (5) points on the graph of each equation · For each of your equation, does the graph of each equation have any intercepts? · State the domain for each of your equations. Write them in interval notation. · State the range for each of your equations. Write them in interval notation. · State whether each of the equations is a function or not giving your reasons for the answer. · Select one of your graphs and assume it has been shifted three units upward and four units to the left. · Incorporate the following four math vocabulary words into your discussion. Use bold font to emphasize the words in your writing. Do not write definitions; use sentences describing the thought behind your math.

o Function

o Relation

o Vertical Line test

o Transformation

P.0.2 Show that (a) the diagonal entries of a Hermitian matrix are real; (b) the diagonal entries of a skew-Hermitian matrix purely imaginary; c) the diagonal entries of a skew-symmetric matrix are zero.

P.0.5 Let A ∈ Mn be invertible. Use mathematical induction to prove that (A-1)k = (Ak)-1 for all integers k.

P.0.25 Let A ∈ Mn be idempotent. Show that A is invertible if and only if A = I

P.0.26 Let A,B ∈ Mn be idempotent. Show that tr((A-B)3) = tr(A-B).

Find the area of △ABC. Where:A=(3,2), B=(2,4), C=(0,2)

Suppose that a×b=〈−1,1,−1〉〉 and a⋅b=−4 Assume that θ is the angle between aand b. Find:

tanθ=

θ=

Find a nonzero vector orthogonal to both a=〈5,1,5〉, and b=〈2,−4,−1〉

Find a nonzero vector orthogonal to the plane through the points: A=(0,2,2), B=(−3,−2,2), C=(−3,2,3).

Find a nonzero vector orthogonal to the plane through the points: A=(0,1,−1), B=(0,6,−5), C=(4,−3,−4)..

Integral

Let f:[a,b]→R and g:[a,b]→R be two bounded functions. Suppose f≤g on [a,b]. Use the information to prove thatL(f)≤L(g)andU(f)≤U(g).

Information:

g : [0, 1] —> R be defined by if x=0, g(x)=1; if x=m/n (m and n are positive integer with no common factor), g(x)=1/n; if x doesn't belong to rational number, g(x)=0

g is discontinuous at every rational number in[0,1].

g is Riemann integrable on [0,1] based on the fact that Suppose h:[a,b]→R is continuous everywhere except at a countable number of points in [a,b]. Then h is Riemann integrable on[a,b].

f : [0,1]→R defined by (f(x) =0 if x = 0) and (f(x)=1 if 0 < x≤1)

f is integrable on [0,1]

**Please show all work and explain** Very confused

We can express insertion sort as a recursive procedure as follows. In order to sort A[1..n], we recursively sort A[1..n-1] and then insert A[n] into the sorted array A[1..n-1]. Write a recurrence for the running time of this recursive version of insertion sort.