Please answer question (b)

9. (a) Show that the PDE Ux = 0 has no solution which is C1 everywhere and satisfies the side condition u(x,x^3) = x , even though the side condition curve y = x^3 intersects each

characteristic line (y = d) only once.
(b) Part (a) demonstrates the necessity of the transversality condition on the intersections of the side condition curve with the characteristic lines. Explain why.

Hint. At what angle does the curve y = x^3 meet the x-axis?

Find the unique solution u of the parabolic boundary value problem

Ut −Uxx =e^(−t)*sin(3x), 0<x<π, t>0,

U(0,t) = U(π,t) = 0, t > 0,

U(x, 0) = e^(π), 0 ≤ x ≤ π.

can you make me a sample problem of 2 geometric sequence, harmonic sequence, arithmetic sequence so that's all 5 this is for Grade 10 Mathematics all with answers and solutions thankyou

1. Regarding TVM:
   1. What are three solution techniques for solving lump sum compounding problems?
   2. How does the future value of a lump sum change as the time is extended and as the interest rate changes?
2. Why does an investment have an opportunity cost rate even when the funds employed have not explicit cost? How are opportunity costs established?

Consider the ring homomorphism ϕ : Z[x] →R defined by ϕ(x) = √5.

Let I = {f ∈Z[x]|ϕ(f) = 0}.

First prove that I is an ideal in Z[x]. Then find g ∈ Z[x] such that I = (g). [You do not need to prove the last equality.]

For each of the following matrices, find a minimal spanning set for its Column space, Row space,and Nullspace. Use Octave Online to get matrix A into RREF.

A = [4 6 10 7 2; 11 4 15 6 1; 3 −9 −6 5 10]

Prove Longest common subsequence algorithm class finds the optimal solution

Prove that 3 divides n^3 −n for all n ≥ 1.

Estimate the area under the graph of f ( x ) = 1(x + 1) over the interval [ 3 , 5 ] using two hundred approximating rectangles and right endpoints

R n =

Repeat the approximation using left endpoints

L n =

Boise Lumber has decided to enter the lucrative prefabricated housing business. Initially, it plans to offer three models: standard, deluxe, and luxury. Each house is prefabricated and partially assembled in the factory, and the final assembly is completed on site. The dollar amount of building material required, the amount of labor required in the factory for prefabrication and partial assembly, the amount of on-site labor required, and the profit per unit are as follows.

For the first year's production, a sum of $8,200,000 is budgeted for the building material; the number of labor-hours available for work in the factory is not to exceed 215,000 hr; and the amount of labor for on-site work is to be less than or equal to 240,000 labor-hours. Determine how many houses of each type Boise should produce to maximize its profit from this new venture.

Prove that gcd(a,b) = gcd(a+b,lcm(a,b))

Problem 17.5. Consider the function χ(0,1) : R → R (this is the characteristic of Definition 14.4). Find: (a) χ(0,1)((0,1)); (b) χ(0,1)((−1,3)); (c) and (in general) χ(0,1)((a,b)), where a,b ∈ R and a < b; prove that the set you found is correct; (d) χ−1 (0,1) ((−2,−1)); (e) χ−1 (0,1) ((0,2)); (f) and (in general) χ−1 (0,1) ((a,b)), where a,b ∈ R and a < b; prove that the set you found is correct,

Definition 14.4 (for Problems 14.6 through 14.9). Let X be a nonempty set and let
A be a subset of X. The characteristic function or indicator function of the set A
in X is

χA : X → {0,1} defined by χA(x) = 1 if x ∈ A
0 if x ∈ X\A

Consider the following linear optimization model.

Z = 3x1+ 6x2+ 2x3

st       3x1 +4x2 + x3 ≤2

           x1+ 3x2+ 2x3 ≤ 1

      X1, x2, x3 ≥0

               (10) Write the optimization problem in standard form with the consideration of slack variables.

               (30) Solve the problem using simplex tableau method.

               (10) State the optimal solution for all variables.

1) If x, y, z are consecutive integers in order then 9 | (x+y+z) ⟺ 3 | y. (Do proof)

2) Let x, y be consecutive even integers then (x+y) is not divisible by 4. (Show proof and state why it was used)

Let A and C be a pair of matrix where the product AC exists.

1. Show that rank(AC) ≤ rank(A)

2. Give an example such that rank(AC) < rank(A)

3. is rank(AC) ≤ rank(C) always true? if not give a counterexample.