Let E be a projective conic through the quadrilateral ABCD, and let the tangents to E at A and C meet at P on the Line BD. Show that the tangents to E at B and D meet at a Point Q on AC.

Hint: Consider poles and polars.

Solve this system using LU-factorization.

3x_1-2x_2+ x_3 =-5

x_2- x_3= -2

-3x_1+ x_2 + 2x_3 = 13

Theorem (Three Tangents): Let a non-degenerate plane conic touch the sides BC, CA, and AB of a triangle ABC in R² at the points P, Q, and R respectively. Then AP, BQ, and CR are concurrent.

Please provide a proof of the Three Tangents Theorem without reference to Ceva's Theorem.

Hint: Consider the Three Point Theorem

a) Using the method of undetermined coefficients, find the general solution of yʺ + 4yʹ − 5y = e^−4x

b) Solve xy'=(x+1)y^2

c) Solve the initial value problem : (x−1)yʹ+3y= 1/ (x-1)^2 + sinx/(x-1)^2 , y(0)=3

What is the dominant color with these x and y values? x=7/15, y= 7/15 (z= 1/15 of course)

Select one:
a. Reddish green
b. Greenish blue
c. Violet
d. Yellow

Please, give an example of at least 1 case where rational functions can be used in each of the next careers:

-Photography

-Economics

-Medicine

-Music

Which one uses them the most?

Consider the nonhomogenous heat equation with time dependent sources u_t = u_xx + xt, an initial condition u(x, 0) = x² and inhomogenous boundary conditions u_x(0, t) = 2t and u_x(1, t) = 4. Use eigen function expansion to find the solution u(x, t).

Select a 12 ft cantilever beam with p = 0.012, to support a dead load of 2k/f (including self weight) and Live load of 3 k/ft. fy= 60000 psi, fc'= 3500 psi (hint your answer should be Width = XXXX, Effective depth= XXXXX, Area of reinforcement = XXXXX)

Vector Error Correction Models (VECM) provide a useful way to find the co-integration vector between two or more series, please explain it with the help of equations. What is an Error Correction term? and why do we call it so?

Let ?+?+?=7 where ?,?,?∈ℝ. Prove that ?^2+?^2+?^2≥7/3

Building on Group Theory #1(topics of "Operators and Postulates" under Group Theory) for the following 3 postulates:

1) Commutative, 2) Distributive, 3) Associative: Give 2 examples for each not provided on the website (Tutorial point) with your own explanation and only own answer appreciated.

Solve each of the following IVP’s:
y'' - y' - 3y = 0, y(0) = 0, y'(0) = -1
  y'' + 4y' + 2y = 0, y(0) = 1, y'(0) = 0
  y'' + 11y' + 30y = 0, y(0) = -2, y'(0) = 0
y'' + 3y' = 0, y(0) = 1, y'(0) = 0
  y'' - 16y = 0, y(0) = 1, y'(0) = 2

1. The total market value of fish caught by a fleet of fishing boats (per month) is given by:

V = $40,000B - $1000B²

Where B is the number of boats engaged in fishing.   The cost of running one fishing boat for one month is $18,000, giving the total cost of running B fishing boats as $16,000B

1. Write down an expression for total net profit for the entire fishing fleet
2. Find the number of boats that would maximize this total net profit
3. Suppose that when B boats operate, each boat gets V/B (catch is split equally). Each boat captain can decide to operate in a given month or not. How many boats will choose to operate?
4. What is V and V/B with the number of boats you found in 3)c)?

1. Diseases tend to spread according to the exponential growth model. In the early days of AIDS, the growth factor (i.e. common ratio; growth multiplier) was around 2.1. In 1983, about 1800 people in the U.S. died of AIDS. If the trend had continued unchecked, how many people would have died from AIDS in 2005?

   ______________________people

(Note: once diseases become widespread, they start to behave more like logistic growth, but don't worry about that for the purpose of this exercise)

2 . A population grows according to an exponential growth model. The initial population is P0=10, and the growth rate is r=0.45.

Then:

P1 =

P2 =

Find an explicit formula for Pn. Your formula should involve n.

Pn =   

Use your formula to find P12

P12 =

Give all answers accurate to at least one decimal place

3 . Assume there is a certain population of fish in a pond whose growth is described by the logistic equation. It is estimated that the carrying capacity for the pond is 1800 fish. Absent constraints, the population would grow by 240% per year.

If the starting population is given by p0=500, then after one breeding season the population of the pond is given by

p1 =   

After two breeding seasons the population of the pond is given by

p2 =