if {v1,v2,v3} is a linearly independent set of vectors, then {v1,v2,v3,v4} is too.

Solve each of these congruences after finding their modular inverses

a) 19x ≡ 4 (mod 141)
b) 55x ≡ 34 (mod 89)
c) 89x ≡ 2 (mod 232)

Given the surface x= (3 (y-1)^2)+ (2 (z+ 3)^2)+ 7. Find an equation of the tangent plane at the point(12,2,2) in four ways:

a) By using the surface as given,x=h(y,z)

b) By writing the surface as z=f(x,y)(only keep the square root whose sign corresponds to the point(12,2,2)).

c) By writing the surface as y=g(x,z)(only keep the square root whose sign corresponds to the point(12,2,2)).

The merry-go-round rotates counterclockwise with a constant angular speed u. The distance between the horse on the merry-go-round and the rotational center is r.

(a) Find the position of the horse x and its velocity v, v(t) = d/dt x(t), as vector-functions of time.

(b) Find the acceleration of the horse, a(t) = d^2/dt^2 x(t), as a vector-function of time. What is its direction (in comparison with the direction of x)?

Now the same horse has a non-constant angular speed u(t) (the merry-go- round still rotates counterclockwise).

(c) Find the position of the horse x and its velocity v, v(t) = d/dt x(t), as functions of time.

(d) Find the acceleration of the horse, a(t) = d^2/dt^2 x(t), as a function of time.

(e) What is the direction of a(t) at the moment when the merry-go-round starts to rotate?

find a power series solution

(x-1)y''+y'=0

The results of a blinded study assessing the use of plasma D-dimer levels for diagnosing deep venous thrombosis (DVT) in patients hospitalized for stroke rehabilitation reported increased accuracy over the standard procedure involving ultrasound. To test the claims of this research team, an independent researcher conducted a study to determine if there was a difference in the accuracy of diagnosis for DVT between the two methods. The data in the table below summarize the findings of this study.

Construct a 95% CI for p1 – p2, where group1 is the plasma D-dimer and group2 is the ultrasound group.

Consider the equation t^2 -y"-t(t+2)y'+(t+2)y=2t^3, (t>0). Given that y1(t)=t3, y2(t)=te^t are the two fundamental solutions of the corresponding homogeneous equation, find the general solution of the nonhomogeneous equation.

Describe Menaechmus’s method of duplication of the cube, using parabolas

A researcher wishes to examine the relationship between wage earned and educational level of workers. For a sample of 4000 workers she has data on hourly earnings (measured in Dollar), age of the worker (in years), worker’s gender, years of experience, number of years with the present employer, size of the firm in which the worker is employed, and highest educational qualification (with 4 classifications: no qualification, secondary school certificate, bachelor degree or PhD)

1. Explain how you would set up a model to address the researcher’s key interest.
2. Based on your model, how would you estimate the predicted impact on wages of holding a PhD compared to having a bachelor’s degree.
3. If the researcher uses ln(wage) as the dependent variable, how would the estimated coefficients be interpreted?
4. What general advice would you give the researcher in relation to her work – eg. what econometric issues should she be aware of in conducting this research and interpreting the results?

1. Find the Legendre polynomial P_L(x) for L = 3,4,5,6 where the polynomian is the series solution for Legendre equation

2. Find the other solution Q_L(x) for the Legendre equation for L = 0,1,2

Please explain in full.

Let u and v be two integers and let us assume u^2 + uv +v^2 is divisible by 9. Show that then u and v are divisible by 3. (please do this by contrapositive).

1. The Eleatics say that there is no such thing as the void, but the Atomists say that there must be a void. Explain the reasoning behind both sides of this disagreement.

Solve y'=1-xy, y(0)=1

We are working on functions of complex variables in calculus (Chpt. 17.4 in Advanced Engineering Mathematics), and our prof posed us the following question:

"Find the function w = u + iv = f(z) that maps the region S := {z : 0 ≤ arg z ≤ π 4 } to the upper half plane {w = u + iv : v ≥ 0}."

the answer he gave to this problem was as follows: "such a mapping is given by f(z) = z^4 = r^4*e^4iθ where z = re^iθ ."

I am unsure how he came to this conclusion, could someone please help? Thanks!

Determine whether each statement is true or false, and prove or disprove, as appropriate.

(a) (∀x∈R)(∃y∈R)[xy=1].(∀x∈R)(∃y∈R)[xy=1].

(b) (∃x∈R)(∀y∈R)[xy=1].(∃x∈R)(∀y∈R)[xy=1].

(c) (∃x∈R)(∀y∈R)[xy>0].(∃x∈R)(∀y∈R)[xy>0].

(d) (∀x∈R)(∃y∈R)[xy>0].(∀x∈R)(∃y∈R)[xy>0].

(e) (∀x∈R)(∃y∈R)(∀z∈R)[xy=xz].(∀x∈R)(∃y∈R)(∀z∈R)[xy=xz].

(f) (∃y∈R)(∀x∈R)(∃z∈R)[xy=xz].(∃y∈R)(∀x∈R)(∃z∈R)[xy=xz].

(g) (∀x∈Q)(∃y∈Z)[xy∈Z].(∀x∈Q)(∃y∈Z)[xy∈Z].

(h) (∃x∈Z+)(∀y∈Z+)[y≤x].(∃x∈Z+)(∀y∈Z+)[y≤x].

(i) (∀y∈Z+)(∃x∈Z+)[y≤x].(∀y∈Z+)(∃x∈Z+)[y≤x].

(j) (∀x,y∈Z)[x<y⇒(∃z∈Z)[x<z<y]].(∀x,y∈Z)[x<y⇒(∃z∈Z)[x<z<y]].

(k) (∀x,y∈Q)[x<y⇒(∃z∈Q)[x<z<y]].(∀x,y∈Q)[x<y⇒(∃z∈Q)[x<z<y]].