Why does the ghost fall ill?

The answer depends upon your definition of "ill." In the story, the author describes the ghost as having fallen "ill" on two occasions.
The first occasion happens after the ghost tries to put on his customary suit of mail. Apparently, he hasn't worn it in a long time, and he has forgotten how thoroughly heavy and unwieldy it is. The result is that he becomes "completely overpowered by the weight of the huge breastplate and steel casque" and ends up scraping his knees horribly and "bruising the knuckles of his right hand."
The author relates that "For some days after this he (the ghost) was extremely ill, and hardly stirred out of his room at all, except to keep the blood-stain in proper repair." So, the ghost becomes indisposed after he injures himself while trying to put on the suit of mail.
The second occasion happens when the ghost tries to frighten the Otis family with his "Headless Earl" disguise. He hasn't worn the disguise in seventy years and thinks that he can do a tolerable job in frightening the Otis children with it. At the appointed time, the ghost rigs himself up in his outfit (complete with big leather riding-boots and a pistol) and makes his way to the twins' room.
When he gets to the room, he finds the door ajar. As he swings the door open, a jug of cold water comes crashing down on him, "wetting him to the skin." Because he is drenched with the cold water during the cooler hours of the night, the ghost catches cold; this is the reason that he falls ill a second time.

The shock to his nervous system was so great that he fled back to his room as hard as he could go, and the next day he was laid up with a severe cold. The only thing that at all consoled him in the whole affair was the fact that he had not brought his head with him, for, had he done so, the consequences might have been very serious.

Describe Chips's encounter with Katherine.

Chips has always appeared to be a confirmed bachelor. He has reached the age of forty-eight and is still not married, very rare for men of that time. Given his somewhat unflattering opinion of modern women, it seems his bachelorhood will be a permanent condition. All that changes, however, during a summer vacation in the Lake District. One day he notices a young lady standing on a dangerous rock, waving her hand. Chips immediately thinks she is in danger, so, being a gallant old chap, he rushes to her rescue. Unfortunately, poor old Chips sprains his ankle in the attempt. In any case, the young lady wasn't in trouble, after all.
Katherine—for it is she—was simply waving to a friend, but she feels that Chips's injury is kind of her fault. So she helps him get to a safe place where he can rest his ankle. There they get acquainted. Chips is instantly taken by this vision of loveliness. Yet not only is Katherine rather pretty, she's also very much a modern woman, someone highly intelligent and with ideas of her own. Previously, Chips—very much a man of his time—had dismissed such women as "monstrous creatures." But now, in the presence of Katherine, he gains a whole new perspective on things. This is the start of a truly beautiful relationship.

Is there a tension between the values of these warrior peoples and the suggested Christianity of the poet in the poem Beowulf?

There may be a tension between New Testament Christianity as we understand it and the values of the Dark Age Danes portrayed in Beowulf.  But the brand of Christianity apparently held by the poet seems to fit very well with the demands of his warrior culture.
When thinking of Christianity, most modern readers think first of Jesus and His New Testament teaching, especially the Sermon on the Mount. (See Matthew 5:1 – 7:27.)  It is this sermon which contains the most famous sayings of Jesus, and also those which sound the most pacifistic, such as “Blessed are the meek,” “Turn the other cheek,” and “If someone wants to take your tunic, let him have your cloak as well.”  Taken in isolation, these sayings would seem to be urging the exact opposite of the lifestyle led by Beowulf and company, whose core values seem to be bravery, boastfulness, and vengeance.
But actually, this sharp contrast depends on caricatures of both the epic of Beowulf and of Jesus’ teaching.
The warriors in Beowulf are indeed tough, violent, and not friends of mercy.  However, this does not mean they have no moral code.  They value loyalty to their king or leader, loyalty to their family and clan (sometimes expressed in vengeance), generosity (the king gives gifts), and self-sacrifice (Beowulf’s exploits are done not only to prove himself but to protect the weak who are being slaughtered by Grendel and the other monsters).   Though violent, they are not just barbarians living in chaos.
Jesus’ teaching is also much more rich and complex than “let people walk all over you.”  If we look at the totality of his teaching, He preaches a way of life that calls for a great deal of loyalty, generosity, and courage to carry it out.  And though, in the Sermon on the Mount, He was warning us against being contemptuous, selfish, and greedy, He does not ever say that we may never defend ourselves or go to war.  For example, in Luke 3:10 – 14, John the Baptist, who is calling the people to “repent” and prepare for Jesus’ coming, does not tell Roman soldiers or even tax collectors to leave their jobs; He tells them to do their jobs in a fair way.  This is also the tenor of Jesus’ teaching.
As for the poet’s understanding of Christianity, he does not mention Christ's teachings.  He has clearly heard of the God of the Bible.  He is familiar with certain (not all?) Old Testament events; for example, he says that Grendel and other monsters are descended from Cain. 
The poet’s understanding of God actually elevates his warrior values to a beautiful philosophy.  In an extended discussion, the poet points out that God alone knows, and chooses, when it is a man’s time to die.  This should make a man brave in battle, and it should make him die quietly and uncomplaining, knowing that he cannot stop his death.  Though the poet does not use the word, he is articulating the theological doctrine of God’s sovereignty.  This is indeed a doctrine that can give people peace in the face of terrible events and can help to make us calm and brave, if we believe it.   For example, in Revelation 1:17 – 18, Jesus says, “Do not be afraid … I hold the keys of Death and Hades.”  
Both in Christianity and in the values of Beowulf, death is not the worst thing that can happen to a person.
This idea of God being over all also influences how kings are viewed in Beowulf.  Again, we are not dealing with barbarians where the only law is the king’s whim.  Rather, the king is also governed by certain values.  Though there are good and bad kings, and bad kings are not immediately deposed, there is a sense of a value system that is higher than any individual king, and by which kings can be judged.
All in all, the “Christianity” or theology of the Beowulf poet fits in surprisingly well with his warrior culture, and seems to strengthen all that is best in it.

Why do many of the characters in Long Day's Journey into Night feel dead or like ghosts?

Most of the main characters in Long Day's Journey Into Night retain a ghost-like appearance in that they are haunted. The sad, faded world they inhabit is no longer occupied by flesh and blood human beings, but by flickering shadows, half-existing in a drab, melancholic netherworld. Indeed, famed literary critic Harold Bloom once said that the play could be re-titled "The Ghost Sonata" and he was surely right.
Throughout the long, tragic, drawn-out journey to the call of the midnight bell we are introduced to characters no more than ghosts of their former selves. Mary is no longer the bright, refined convent girl of years gone by. Now she is little more than a pathetic, shambling morphine addict, an object of pity and contempt. James's heyday as a great stage actor is long since gone. He appears oblivious to the impending financial hardship about to befall his family, determined to keep his money tied up in property to avoid the worse-than-death fate of the poorhouse. And Jamie's promise as an actor has been shamefully squandered in a life of drunken dissipation.
Amidst all the human wreckage is the figure of Edmund. Potentially, he provides a sense of hope, a chance to transcend the chilled confines of this haunted summer house. His poetic reveries lift him up out of his grim surroundings to a much brighter place, and bring before his glittering imagination "a saint's vision of beatitude." But it's all too brief. Though on the cusp of literary fame, for now Edmund crashes straight back down to the drink-sodden kingdom of the living dead. As he says: "You have to go on living as a ghost." Before long, Edmund too will be a shadow of his former self.
And in the final, heartrending scene, as a wasted Mary pathetically clutches her wedding dress, we see what Edmund means by "a ghost within a ghost." Mary, now irretrievably lost in a drug-induced haze babbles incoherently about her younger days. The unquiet spirit of Mary's charmed youth is still there, somewhere, buried deep beneath the haggard wanness of her ghostly complexion. In the silvery gray apparition of Mary Tyrone are united the ghosts of past and present. But tragically, the past and present combine to point towards a haunted future not just for Mary, but for the whole of the Tyrone family.
 
 

What was the black spot?

The black spot is another of Robert Louis Stevenson's wonderful literary inventions, and it has been an integral part of pirate and criminal folklore ever since. In Treasure Island, although the pirates are a bunch of bloodthirsty criminals, they do still live by a kind of code. This code is not always successful, greed for loot and treasure often gets in the way, but for the most part, they stick to the old pirate ways.
One of the time-honored pirate traditions is the black spot. The idea is that when a captain is about to be deposed, he is given a slip of paper with a black spot on it. This is the pirate equivalent of a judgement in a court of law. When a captain receives this piece of paper, he knows that he has violated the pirate code and must be punished accordingly.
Long John Silver's crew have the audacity to present him with the black spot. But Silver has been at this game too long to be spooked by it. The black spot given to the old sea dog has been painted on a page crudely torn from a Bible. Like most seamen of the time, the pirates in Treasure Island are deeply superstitious, and painting a black spot on a page from the Bible is considered bad luck. As a result, Silver knows that he really has nothing to fear from his men.

What subjects and themes explored in The Phantom of the Opera are relevant in society today?

There are a lot of themes in Phantom of the Opera which resonate in modern society. One which comes to mind is the theme of social ostracism and how it can contribute to warping people's behavior. The Phantom's quest for revenge is, in his mind, justified, because he has been mistreated by society. He thinks that he will not be accepted for who he is, and, therefore, has to wear a mask in order to make society accept him. This is a theme that, in its general sense, is applicable in every society; many people feel they need to "wear a mask" in a metaphorical sense in order to fit in, whether or not this is really true. In more nuanced senses, we could identify the Phantom's behavior with the thread that often connects modern high school shooters and terrorists——people who feel they have been ostracized from society, usually young white men who have found it difficult to make friends and fit in, commit depraved acts in order to draw attention to themselves because they feel that, otherwise, they are ignored. In The Phantom of the Opera, the theme of anger driven by the hero's sense of rejection is certainly relevant to us today and a concern to which we have not yet found a solution.

One of the subjects and themes of The Phantom of the Opera that is still relevant today is the choice that people must make between professional success and personal happiness. Christine agrees to be beholden to Erik, the phantom, so that she can achieve success in the opera, but Erik forces her to be  a virtual prisoner to him and to give up her true love, Raoul. The parallel to today's society is that many people sacrifice personal happiness, such as time spent doing what they love or time spent with their families and loved ones, to achieve success in the professional world. Another theme is the meaning of true love. At first, Erik feels that he wants to entrap Christine and kill off her lover, Raoul, but he comes to realize that true love means letting Christine be free to marry as she wishes and allowing her to live the life she wants. 

Precalculus, Chapter 3, 3.1, Section 3.1, Problem 39

Please look at the attached picture down below for the table of values.
The graph should look like this: