How did the colonists break ties with Great Britain, win the Revolutionary War, and create a national identity?

The colonists did not entirely sever ties with Great Britain, as Great Britain continued to be the United States's chief trading partner after the American Revolution. The United States was successful in its war with Britain due to its ability to make Britain fight a longer war than it anticipated. The United States was able to capitalize in shorter supply lines than the British.
The United States also had powerful allies in France, Spain, and the Netherlands who wanted a victory over the British more than they wanted the colonies to succeed in gaining self-rule. Britain lost the Revolutionary War due to a lack of will and the possibility of fighting yet another expensive war with other European powers.
The United States was already starting to form a national identity before the American Revolution. American colonists already had different patterns of speech than people from Britain. Americans were also starting to form their own culture. Religious freedom became important in the colonies and later in the United States as religious groups such as the Baptists and Lutherans came to America in order to find freedom. In time, different denominations arose from these smaller religious sects.
Criticizing political leadership also became a key part of the American culture as the American newspaper industry grew exponentially between the end of the Revolutionary War and the Civil War. Compared to Europeans, most Americans were quite politically aware and involved. This focus on freedom and a willingness to criticize officials were key parts of the American identity.

In 1775, fighting broke out in Massachusetts between the British and patriots. There were battles at Lexington and Concord and at Bunker Hill. The colonies faced a stark choice: try to reconcile with Britain or declare independence. Thomas Paine's pamphlet, Common Sense, argued for independence, but many loyalists were opposed to that idea. The Declaration of Independence (July 4, 1776) settled the question and broke the colonies' ties with Britain.
Winning the war against Britain was extremely difficult, however. Britain was the world's greatest power; it had the largest navy and a professional army. George Washington's army lost numerous battles, and American prospects seemed dim. At this point, the British miscalculated, and an entire British army surrendered at Saratoga in 1777. Saratoga showed that the patriots had a chance to win, so France decided to join the war against its old enemy, Britain. French assistance was crucial in Washington's great victory at Yorktown in 1781. Britain had lost the war, and America was independent.
America did not have a strong national identity at the time of its inception as a nation. The thirteen colonies had had a great deal of independence, and they were reluctant to give that up to a national government. The Founding Fathers, however, created the Constitution of the US in 1787 after compromises were made to placate the doubters. Washington became the first president of the new nation.

There was unrest in the Thirteen Colonies for many years before the Revolutionary War.  New taxes, such as the Sugar Act and the Stamp Act, were considered unfair.  The Colonists were not represented in Parliament, and because of this they had no one to speak on their behalf.  The Townshend Revenue Act of 1967 taxed tea and other common commodities.  All of the taxes were eventually repealed except for the one on tea.  Protests in Boston led to British troops occupying the city.  The unrest continued, and led to the Boston Massacre.
The Tea Act was made to benefit the East India Company, which was having financial difficulties.  They had an overstock of tea, which was to be sold cheaply in Boston.  Concern that this would have a negative impact on local merchants combined with the unrest led to the Boston Tea Party.  Tea from ships in the Boston Harbor were thrown into the water in protest.  
Troops were quartered in private homes and buildings, which led to further dissatisfaction with the British government.  Samuel Adams, Patrick Henry, Thomas Paine, Paul Revere, and other people began to talk of independence from Britain.  The Sons of Liberty and other organizations were formed.  Local militias were established.  Shots were fired at Lexington and Concord and the Revolutionary War began in 1775.
In 1776, the Second Continental Congress made the decision to officially declare independence from Great Britain.  On July 4th, 1776, the Declaration was Signed.  The Revolutionary War raged on until 1783 with the signing of the Treaty of Paris.  General George Washington and other generals led troops into battle over a period of eight years.  Though the Patriot troops were initially less organized, less trained, and had less supplies, they fought on with determination.  The fact that the Thirteen Colonies spanned a large area filled with countryside benefitted the Patriot troops.  Also, the Patriots did not have a national capital.  A capital could have been captured and would have been a difficult blow to recover from.  The British wore bright red uniforms, which were easy to spot from a distance.  The Patriot troops wore ordinary clothing, and thus looked similar to Loyalist civilians.   Other countries, such as France, also sent troops to fight alongside the Patriots.
Once the war was over, the independent Thirteen Colonies  had to decide exactly what sort of nation they wanted to be.  Many signers of the Declaration of Independence and other men met in 1787 to write the U.S. Constitution.  There were disagreements, but decisions were made.  It was decided that the United States would have a system of checks and balances with three branches of government.  They did not want the tyranny they had experienced under King George III to occur again in their nation.  The Constitution also emphasized personal liberties, which was an uncommon idea at this time.  General George Washington was elected President.  In another decision to set themselves apart from other nations, the founders of the United States decided against having a king.  Instead, they wanted a leader who was elected by the citizens.

What hints are there about George and Lennie about their relationship?

George and Lennie's friendship doesn't conform to the audience's picture of a traditional friendship because the balance of power is weighted in George's favour. The dialogue and discourse that George shares with Lennie sees George characterised more as a 'critical father' than a friend. Lennie follows George's orders and relies upon him for the most basic of things: food, water, shelter and work. Having Lennie report to George in this way demonstrates a void in status between them and shows the audience that this relationship doesn't follow the normal conventions of companionship.

The relationship between George and Lennie begins when they are both young and Lennie's aunt, Clara, dies. George promises her he will take care of Lennie, a man who is cognitively delayed. By taking care of Lennie, George essentially gives up his personal freedom as well as his job security; Lennie will prove to be problematic everywhere they go due to his uncontrollable fits, his use of extreme strength, and his imposing build. 
Still, their relationship is symbiotic; both men need something from one another the very things that sometimes come as obstacles in their relationship. For instance, from the start of the novel, we learn that George, a relatively small-sized man, is in control of the duo. George calls the shots, decides where and when they will stop for the night, and he scolds Lennie when the latter acts out his strange behaviors. In the first part of the story, we see how George calls out Lennie about the mouse that Lennie is carrying in his pocket, which he accidentally kills from petting it too hard. George consistently accuses Lennie of being slow, of ruining his (George's) life, and for getting them in trouble all the time; all of these accusations, however cruel, are also true. 
Still, they remain together. Why? Because Lennie's strength and size compensates for George's smaller, less impressive frame. On the other hand, Lennie depends almost entirely on George's wit and quick thinking because Lennie's own mental capacity is limited. As such, he is the body while George is the brains. Together, they form one whole person. This is where the symbiotic nature of the relationship is evident. 
Moreover, they also remain together because they have absolutely nobody else to rely on. They have no family, no other friends and, as such, they greatly benefit from being each other's protectors and supporters.

We got a future. We got somebody to talk to that gives a damn about us [one another]. We don't have to sit in no bar room blowin' in our jack jus' because we got no place else to go. If them other guys gets in jail they can rot for all anybody gives a damn. But not us."

On top of it, they do share a dream together: the dream of living "off the fat of the land" and being able to enjoy the benefits of farming for themselves, rather than doing it for others. These variables keep the men together. All of this gives us hints that their relationship is best considered fraternal because they act as if they are brothers that watch out for one another (probably better than brothers do), and they are also interconnected at many other levels: for protection, for support, for friendship, for company, and to help one another keep the dream of the land alive. 

What was the name of the program developed to deal with contact from an alien?

The name of the program is Wildfire.
In the book, the Andromeda Strain is said to be the first extraterrestrial-based biological crisis on earth. Accordingly, the Wildfire Project was created to deal with such a crisis.
In Chapter 4, the text reveals one Arthur Manchek as a senior Scoop officer. Scoop is the name of a series of American satellites. If we go back to Chapter 1, we read that Lieutenant Roger Shawn and Private Lewis Crane are looking for a Scoop satellite. Neither man knows much about the Scoops, except that the satellites are tasked with exploring and analyzing the upper atmosphere.
In the story, the two men look for the Scoop satellite that has just landed and are shocked to discover dead bodies in the surrounding area. This is the sort of biological crisis the Wildfire Project has been created for. In Chapter 4, we learn a little more about Project Wildfire. In the event of a crisis, a 5-man laboratory team is tasked with investigating the presence of extraterrestrial forms on any American spacecraft (like the Scoop) that has returned to earth.

In lines 8–13 of "The Pit and the Pendulum," describe the narrator’s state of consciousness and how it changes.

Allowing six lines for the introductory quatrain and its translation, lines 8–13 of Edgar Allan Poe's "The Pit and the Pendulum" encompass approximately this section of the story:

I felt that my senses were leaving me. The sentence—the dread sentence of death—was the last of distinct accentuation which reached my ears. After that, the sound of the inquisitorial voices seemed merged in one dreamy indeterminate hum. It conveyed to my soul the idea of revolution—perhaps from its association in fancy with the burr of a mill-wheel. This only for a brief period, for presently I heard no more.

From this passage, we can deduce that the narrator is fully conscious at first. Since he feels his senses leaving, he was aware of his mental state and surroundings initially. He hears his death sentence distinctly. However, whether from mental distress at hearing the pronouncement or from some sort of physical trauma or both, he fades into a state of semi-consciousness. He is not completely unconscious, for he has a vague perception of the voices of the inquisitors and associates the "dreamy indeterminate hum" of their speech with a mill wheel. From that point, he loses more consciousness without completely passing out, for he "heard no more," even though he continues to visualize the proceedings without properly interpreting what he sees.
In this passage, the narrator moves from being fully conscious to losing some consciousness (his ability to understand speech) to losing more consciousness (his ability to hear and to correctly interpret visual stimuli). In this section, however, he never fully loses consciousness but remains in a state of semi-consciousness.

Lines 8-13 will differ depending on how the text has been printed in a particular book that contains this Poe story.  I will have to analyze the first paragraph of the story to make sure that I get these lines.  
Physiologically, the change in the man's consciousness is an easy question.  He is conscious when the paragraph begins, and he is unconscious when the paragraph ends. 
However, I believe that the question is probably asking about the man's psychological state of mind. He starts off okay.  He's scared when the paragraph begins, but he's in control of his mind; however, that doesn't last long.  The text tells us that he felt his senses leaving him.  

I was sick—sick unto death with that long agony; and when they at length unbound me, and I was permitted to sit, I felt that my senses were leaving me.

His physical senses start blurring the lines of reality and unreality.  He clearly sees the judges, and he hears their verdict, but after that his mind starts playing tricks on him.  He sees the candles that are on the table, and he envisions them as angels. The angels then transform from beautiful signs of hope to "meaningless spectres" with flaming heads.  It's at this moment that he acknowledges to himself that death might be good and restful. 

And then there stole into my fancy, like a rich musical note, the thought of what sweet rest there must be in the grave.

The narrator's final thoughts convey his knowledge that he's losing consciousness.  That's why the figures in the room seem to disappear along with the light.  

What is an example of an argument that is valid but not deductively valid?

A deductive argument is one that provides a guarantee of the conclusion reached through that argument being true as long as the argument's premises themselves are true. 
So, let's look at an example:

If the Titanic hit an iceberg at 11:40 PM on April 14, 1912, then Donald Trump is a cockroach. 
The Titanic hit an iceberg at 11:40 PM on April 14, 1912.
Therefore, Donald Trump is a cockroach. 

In this example, the structure of the argument itself is valid. It follows the pattern:

If P, then Q.
P.
So Q.

Technically, this argument is valid because according to the logical setup of the argument, it is impossible for its premises to be true while its conclusion is false. That being said, this does not make it deductively sound (which I think is what you were really trying to ask in your question!), as the premises here are false. The Titanic did hit an iceberg at that time and date, but that has no connection to whether or not Donald Trump is a cockroach. Donald Trump is actually not a cockroach, but rather the Republican's candidate for President--a human being. Thus, you have a valid but unsound argument. 
 

College Algebra, Chapter 2, 2.3, Section 2.3, Problem 30

Graph the hyperbola $y^2 - 9x^2 = 1$ by solving for $y$ and graphing the two equations corresponding to the positive and negative squareroots.

$
\begin{equation}
\begin{aligned}
y ^2 - 9x^2 &= 1 && \text{Model}\\
\\
y^2 &= 1 + 9x^2 && \text{Add } 9x^2 \\
\\
y &= \pm \sqrt{1 + 9x^2} && \text{take the square root}
\end{aligned}
\end{equation}
$

Thus, the hyperbola is described by the graphs of two equations
$y = \sqrt{1+9x^2}$ and $y = -\sqrt{1+9x^2}$

The first equation represents the positive half of the hyperbola because $y \geq 0$ while the second represents the negative half. If we graph the first equation in the viewing recatngle $[-2,2]$ by $[-3,3]$ then we get...



The graph of the second equation is



Graphing these half portions together on the same viewing screen, we get the full hyperbola

How did loyalists react to the Navigation Acts?

The Navigation Acts refers to many different laws regarding trade with foreign merchants.  These laws were made by the British to prevent trade by nearby countries and to promote exclusivity.  The Navigation Acts were passed in the middle of the 17th century, and they were not reversed until the 19th century.  Those living in the Thirteen British Colonies were prohibited from trading with Dutch, French, or Spanish merchants according to the laws.
In 1764, the Sugar Act was passed.  It was passed to enforce a tax on molasses.  It also required that some goods could only be sold to Britain.  The economy was weak in the Thirteen Colonies at the time, and the Sugar Act caused feelings of unrest.  Many colonists disagreed with the law.  Colonists who would later become Patriots were vocal protestors of the Sugar Act.  
The Sugar Act most directly impacted merchants and those who were traders on ships.  Many loyalists who worked in these industries disagreed with the Sugar Act.  Though they were loyal to the King, they did not agree with the law.  The loyalists, however, were less vocal protestors.