In lectures, we have discussed first order quasilinear PDEs. That is, PDEs of the general form A(x, y, u) ∂u(x, y) ∂x + B(x, y, u) ∂u(x, y) ∂y = C(x, y, u), (1) for some A, B and C. To solve such PDEs we first find characteristics, curves in the solution space (x, y, u) parametrically given by (x(τ ), y(τ ), u(τ )), which satisfy dx dτ = A(x, y, u), dy dτ = b(x, y, u), du dτ = C(x, y, u). (2) We find solutions to these equations in the form f(x, y, u) = C1 and g(x, y, u) = C2 where C1 and C2 are arbitrary constants. The independent functions f and g are then used to write the general solution to Equation (1) f(x, y, u(x, y)) = F [g(x, y, u(x, y))] , (3) where F is a sufficiently smooth function (that is, you can expect in this question that its derivative exists everywhere). 1. [12 marks] Verify that (3) for any suitable F and for any f and g as described above is actually a solution to the PDE (1). That is, you should show that given (2) which describe the functions f and g and the solution (3), then Equation (1) is always satisfied. HINT: This is not as simple as it sounds. You should first attempt to differentiate f(x, y, u) = C1 and g(x, y, u) = C2 by τ and differentiate the solution (3) first with respect to x and then with respect to y and use the resultant simultaneous equations to deduce (1).

Let K be a cone with a circular bottom, that has a radius r, and the apex is directly above the center of the bottom. Let h represent the height of the cone. Show that the surface area of the cone K without the bottom is equal to

pi * r * sqrt(r^2 + h^2) .

(Use that a sector that is given with angle θ in a circle with radius R has the area  (θ * R^2)/2

Suppose that a large mixing tank initially holds 400 gallons of water in which 65 pounds of salt have been dissolved. Another brine solution is pumped into the tank at a rate of 6 gal/min, and when the solution is well stirred, it is pumped out at a slower rate of 5 gal/min. If the concentration of the solution entering is 3 lb/gal, find the amount of salt in the tank after 10 minutes.

PLEASE SHOW ALL WORK AND USE DIFF. EQs. THANKS!

3. Let G1 and G2 be groups with identity elements e1 and e2, respectively. a. Prove that G1×{e2} is a normal subgroup of G1×G2. (You do not need to prove that G1×{e2} is a subgroup, since this follows from a previous homework problem, just that it is normal in G1 ×G2.) b. Prove that (G1 ×G2)/(G1 ×{e2}) ∼= G2

A tank contains 2800 L of pure water. A solution that contains 001 kg of sugar per liter enters a tank at the rate 9 L/min The solution is mixed and drains from the tank at the same rate. a) How much sugar is in the tank initially? b) Find the amount of sugar in the tank after t minutes. amount = (kg) (your answer should be a function of t) c) Find the concentration of sugar in the solution in the tank after 45 minutes. concentration = (kg/L)

Prove or disprove if B is a proper subset of A and there is a bijection from A to B then A is infinite

How many positive integers less than 1,000,000 have exactly one digit

that is 7 and the product of this digit (7) with the sum of other digits is

between 50 and 65?

A professor wants to determine if there is a difference between students' pre and post test after a semester of learning.

1. What could be the research question for this problem?

2. What could be the null hypothesis and alternative hypothesis for this problem?

3. Looking at the mean of both pro and post test, on the end-results below, does it REJECT or FAIL TO REJECT the null hypothesis and why?

The results of the test showed this following:

B= {1,x,x^2}

B’={1-x^2, x-x^2, 1+2x-x^2}

T(a+bx+cx^2) = 3a+b+c+(2a+4b+2c)x + (-a-b+c)x^2

a) by direct calculation , compute [P]_B’ , p=7-x+2x^2

b) using basis B={1,x,x^2} , compute [T]_B

c_ compute [T]_B’

T(1+2x)=1+x-x^2

T(1-x^2)=2-x

T(1-2x+x^2)=3x-2x^2

a)compute T(-6x+3x^2)

b) find basis for N(T), null space of T

c) compute rank of T and find basis of R(T)

Find a basis of U = span {(1,1,2,3), (2,4,1,0), (1,5,-4,-9)}

Find an injective morphism Z2 × Z2 → S4. Is it possible to find an inclusion into S3?

Given the vectors u1 = (2, −1, 3) and u2 = (1, 2, 2) find a third vector u3 in R3 such that

(a) {u1, u2, u3} spans R3
(b) {u1, u2, u3} does not span R3