In 1984, which quotes show that Julia is resourceful?

In 1984, Julia's resourcefulness is one of her most striking characteristics, and this is shown from her very first meeting with Winston. After falling over in front of the telescreen, Julia takes Winston's hand and is able to transfer a slip of paper to him, without anybody noticing, even Winston himself. In fact, it is only after the meeting that he realises what has happened:

"In the two or three seconds while he was helping her up the girl had slipped something into his hand. There was no question that she had done it intentionally."

In addition, Julia is very resourceful when it comes to arranging meetings with Winston. It is her idea, for example, to meet in the woods, and she uses her knowledge of the Party's surveillance techniques to keep their liaisons secret and to avoid detection:

"We can come here once again," said Julia. "It’s generally safe to use any hide-out twice. But not for another month or two, of course."

Finally, Julia is also very adept at overcoming the problem of rationing to secure certain goods for herself and Winston. She procures chocolate from the Black Market, for instance, using her clean-cut image to deflect attention from her activities, as she explains to Winston:

"I always look cheerful and I never shirk anything. Always yell with the crowd, that’s what I say. It’s the only way to be safe."

dy/dx = 2xsqrt(4x^2+1) Use integration to find a general solution to the differential equation

For the given problem:(dy)/(dx) =2xsqrt(4x^2+1) is a first order ordinary differential equation in a form of (dy)/(dx) = f(x,y) .
 To evaluate this, we rearrange it in a form of variable separable differential equation: N(y) dy =M(x) dx .
Cross-multiply dx to the right side:dy=2xsqrt(4x^2+1)dx .
Apply direct integration on both sides: intdy= int 2xsqrt(4x^2+1)dx .
For the left side, we apply basic integration property: int (dy)=y .
For the right side, we may apply u-substitution by letting: u = 4x^2+1 then du=8x dx  or (du)/8=x dx .
The integral becomes:
int 2xsqrt(4x^2+1)dx=int 2sqrt(u)*(du)/8
                               = int (sqrt(u)du)/4
We may apply the basic integration property: int c*f(x)dx= c int f(x) dx .
int (sqrt(u)du)/4= 1/4int sqrt(u)du
Apply Law of Exponent: sqrt(x)= x^(1/2) and Power Rule for integration : int x^n= x^(n+1)/(n+1)+C .
1/4int sqrt(u)du =(1/4) int u^(1/2)du
                 =(1/4)u^(1/2+1)/(1/2+1)+C
                 =(1/4)u^(3/2)/((3/2)) +C
                  =(1/4)u^(3/2)*(2/3) +C
                  =u^(3/2)/6+C
Plug-in u=4x^2+1 on u^(3/2)/6+C , we get:
int 2xsqrt(4x^2+1)dx=(4x^2+1)^(3/2)/6+C
Combining the results from both sides, we get the general solution of the differential equation as:
y=(4x^2+1)^(3/2)/6+C

What did uneven wealth distribution have to do with Social Darwinism?

Social Darwinism, popularized by Herbert Spencer, took ideas Darwin developed about the animal kingdom in his book Origin of the Species and applied them to human society. Specifically, he glommed onto Darwin's theory of natural selection, which argued that the most viable species will survive and thrive, and the least viable die out. Spencer changed the theory somewhat, then used it as a "scientific" explanation for societal wealth.
Spencer came up with the term "survival of the fittest." He argued that the people in society who ended up with the most wealth did so because they were the fittest, or the best of human beings. This ideology said they inherently deserved their great wealth.
We can quickly see the flaws to this argument. It is a circular argument to say you are deserving of your wealth because you happen to have it. This argument ignores such obvious means of getting wealthy as luck or inheritance, which have nothing to do with "fitness." The person who happens to buy the $400 million lottery ticket may be a total loser who has never done anything in life but sit on the sofa eating potato chips: he just got lucky. People who inherit great wealth also are not necessarily people who flourish because of being fit. Some may be extraordinary, but more are likely ordinary people who just happened to pop out of the right womb. Nevertheless, Spenser's theory became popular because it justified the wealthy in having and holding onto their wealth.

Explain why Bud says that "ideas are like seeds." Why does Bud say this?

Bud says that quote in chapter 9:  

It's funny how ideas are, in a lot of ways they're just like seeds. Both of them start real, real small and then ... woop, zoop, sloop ... before you can say Jack Robinson they've gone and grown a lot bigger than you ever thought they could.

He actually carries on with the comparison for the next few pages.  Bud is amazed that something as small as a seed can grow into a huge giant tree that a person could "drive a car into it and kill yourself" with. Bud tells his readers that is how the idea of Herman Calloway being his father started.  He says that the idea was so small that it could have easily been forgotten and lost like a seed being blown away with the "first good puff of wind."  He admits to readers that the idea has slowly grown, and now it is so big that it is a central focus of his.  He plans to make his way across the entire state of Michigan in search of this man that he believes is his father:

The idea first got started when I was looking in my suitcase at one of the flyers showing Herman E. Calloway and his band. That was like the seed falling out of a tree and getting planted.
It started busting its head out of the dirt when me and the other boys at the Home were getting our nightly teasing from the biggest bully there, Billy Burns.
[...]
That little idea had gone and sneaked itself into being a mighty maple, tall enough that if I looked up at the top of it I'd get a crick in my neck, big enough for me to hang a climbing rope in, strong enough that I made up my mind to walk clean across the state of Michigan.

Bud's initially small idea has grown into a hugely motivating force.  It has grown into an impressive "tree," and that initially small idea is why Bud eventually winds up united with his grandfather. 
 

What mark did the second cat have on his chest?

After the narrator hangs his first black cat, Pluto, from a tree in the garden, knowing he is committing a truly vile sin, he discovers a similar cat in the bar where he often drinks. It is as large as Pluto had been, but Pluto was all black, and this cat has "a large, although indefinite splotch of white, covering nearly the whole region of the breast." Initially, then, the white patch has no discernible shape.
As time passes, the narrator begins to dislike this cat as well, and he soon grows disgusted with it. It, too, has only one eye, and it begins to like the narrator more and more. The narrator reminds us that the white mark had, at one time, possessed an "indefinite" shape, but, slowly, "it had, at length, assumed a rigorous distinctness of outline [. . .], the image of a hideous—of a ghastly thing—of the gallows!" Thus, the white "splotch" eventually evolves into a shape that looks like the apparatus by which one is hanged.

At first, the mark on the cat is only a white patch. Later, however, the narrator thinks it has changed shape to appear like the gallows.
The change in the cat's marking likely foreshadows the narrator's fate. At the beginning of the tale, he says it is the last night of his life because he is being put to death the next day for murdering his wife. A gallows is a construct where a person is hanged. It usually includes a platform, two upright posts, and a beam connecting them from which the person is hanged. 
The cat with the mark of the gallows is what ultimately leads the police to find the narrator's murdered wife. When the police are preparing to leave the basement where she is buried, the cat meows in response to the man knocking on the wall. When the police tear down the wall, they find the body of his murdered wife with the cat on top of her. The narrator is taken into custody and ultimately sentenced to death. 

McDougal Littell Algebra 2, Chapter 3, 3.2, Section 3.2, Problem 32

To solve the system of equations 7x + 2y = -3 and -14x - 4y = 6 by linear combination method, the equations have to be modified so that they can be added together to eliminate one of the variables.
7x + 2y = -3
Divide -14x - 4y = 6 by 2, this gives -7x - 2y = 3. This equation is the same as 7x + 2y = -3.
The equations 7x + 2y = -3 and -14x - 4y = 6 represent the same straight line. As a result this system of equations does not have a unique solution. There are infinite combinations of x and y that satisfy both the equations.

Calculus of a Single Variable, Chapter 8, 8.1, Section 8.1, Problem 38

Given to solve
int 1/(cos(theta) - 1) d theta
For convenience, let theta = x
=>
int 1/(cosx - 1) dx
let u = tan(x/2) => dx = (2/(1+u^2)) du
so ,
cos(x) = (1-u^2)/(1+u^2) (See my reply below for an explanation)
so,
int 1/(cos(x) - 1) dx
= int 1/((1-u^2)/(1+u^2) - 1) (2/(1+u^2)) du
= int 1/(((1-u^2)-(1+u^2))/(1+u^2) ) (2/(1+u^2)) du
=int (1+u^2)/(((1-u^2)-(1+u^2)) ) (2/(1+u^2)) du
=int (2)/(((1-u^2)-(1+u^2)) ) du
=int (2)/(((1-u^2)-1-u^2)) ) du
= int (2)/(-2u^2) du
= -int(1/u^2) du
= -[u^(-2+1)/(-2+1)]
= u^-1
= 1/u
= 1/tan(x/2)
= cot(x/2)+c
But x= theta
so,
int 1/(cos(theta) - 1) d theta = cot(theta/2)+c