Team+Adam

Adam 26. This is a question of the Main Point type, and it functions to test the reader’s overall understanding of the passage. Knowing the meaning of the words treatise, aesthetic and theorem will aid in the answering of the question. In order to answer the question the reader must have fully read and understood the passage and its main point. In terms of procedure, the reader must read and and annotate the passage for the main ideas and important context clues of the author’s purpose.

A- While different viewpoints are mentioned they do not make a large appearance in the passage, and certainly are not analyzed enough for this statement to describe the passage the best. B- Computers are mentioned only at the very end of the passage making this statement irrelevant to the passage as a whole. C- This is the correct answer because the passage talks about the various aspects and features of mathematical proofs for its entire duration. D- The passage makes no mention of explaining the steps for a mathematical proof making this choice irrelevant to describing the passage as a whole E - This answer is the distractor option. A treatise is the wrong way to describe this passage because it does not deal with the subject systematically, but rather by jumping to several topics and one example. Also, although the passage mentions the aesthetic component of proofs, it does not completely focus on it and rather focuses on the qualities that contribute to a “beautiful” proof.

27. This question is of the purpose type, and it tests the reader’s ability to comprehend the author’s purpose of including a quote. To answer this question the reader must be able to use context clues to understand what syllogy means, or have a prior knowledge of its meaning. The reader must then apply this knowledge to understand what the question is asking. Procedurally the reader must go back and re-read the lines provided in the question, as well as the other text around it to gain an understanding of why the author chose to include the quote.

A- This answer is correct because it clearly states the device cited in the passage that is used to describe the Bill Clinton reference. B- There is no mention of a logical fallacy in the passage making this answer irrelevant. C- The passage has no political point so this answer is incorrect. This could be a possible distractor as people generally associate Bill Clinton with politics, although the passage’s aim is not to do so. D- The passage is not grimly mocking or cynical so the use of sardonic would not fit into the author’s purpose for using the quote. E- The passage does not seek to differentiate between math and politics so there would be no reason for the author to do so with this quote.

36. This question refers to the development of paragraphs 2 and 3. Its purpose is to test the reader’s ability to understand how the author developed these two paragraphs using a certain strategy. No prior knowledge is required to answer this question. To answer this question procedurally the reader must go back to the two paragraphs and read them specifically focusing on the development of ideas. After doing so the reader must then compare what they noted to the possible answers.

A- Although the passage does define the Four Color Map Theorem, this is not the primary aspect of the paragraphs, therefore making this answer incorrect. B- The author uses these two paragraphs to support their argument with examples, and not to make an argument, which makes this incorrect. C- The author does not attempt persuasion in these 2 paragraphs, making this answer incorrect D- The author does not compare or contrast anything in these two paragraphs making this answer incorrect E- The author both narrates and analyzes the information he has provided in these two paragraphs making this the correct choice.

Devin: 28. This question is of the content type, and its function is to test the reader’s comprehension of a series of quotations from relevant experts in the field. In order to correctly answer the question, the reader needs to know the meaning of the words “inevitability” and “unexpectedness”. The reader must be also be able to realize which characteristics would be the opposite of those words, as the question asks for qualities of a proof that does not follow the provided description. In terms of procedure, the reader must be able to read through a series of separated quotations and piece them together to form an overall meaning.

A- The passage does not mention anything in regard to a proof’s usefulness, so this answer is irrelevant and false. B- The passage states that a proof can be “messy”, which can be interpreted as having a lot of useless steps. Therefore, this answer satisfies the qualities of a proof and is incorrect. C- This answer is false because the passage states that a proof must show its “aesthetic” side and be visually and logically pleasing; a clearly comprehensible proof certainly satisfies this. D- This answer is false because the passage states that one cannot “escape from the conclusion”, so a proof that is resistant to challenge does satisfy this. E- This answer is the correct answer because it displays the opposite of “inevitability” and “unexpectedness”. Something that is completely predictable is very obvious and not unexpected at all.

29. This question refers to contrasts and functions to test a reader’s understanding of the content. In terms of declarative knowledge, the reader must know the definitions of all words used and their contexts, and understand the concept of contrast. Also, the reader must at times be able to decipher colloquial, as in the case of “straight from the Book”. In terms of procedural knowledge, the reader must be able to go through the first paragraph and understand each of the points made in order to understand and find contrast between the words and ideas used in the paragraph. Also, the reader must be able to find the context of words and their connotations, such as in the case of “science” and “art”, in order to further demonstrate an ability to notice contrast.

A- Science and art are described in the passage as different fields and each is described with contrasting examples in the passage. Therefore, this example does show contrast. B- The words “obvious” and “expected” are used with the conjunction “or” (which signifies mutual exclusivity and therefore contrast), and by definition mean opposite things. Thus, this example does show contrast. C- The words “trivial” and “beautiful” are used with the conjunction “or” (which signifies mutual exclusivity and therefore contrast), and by definition mean opposite things. Thus, this example does show contrast. D- The phrases “surprising” and “straight from the Book” are used with the conjunction “or”, which signifies mutual exclusivity and therefore contrast. Also, once the reader considers the fact that “straight from the Book” means very standard and expected, the two phrases mean opposite things. Thus, this example does show contrast. E- This answer is correct because “odd” and “surprising” are connected by the conjunction “and”, which signifies similar meanings. Also, by definition, the words are synonyms and refer to an unexpected or unusual circumstance. Additionally, in context, these words describe a proof that has a degree of uncertainty and since these words both pertain to that, they do not show contrast.

30. This question is of the identification type, in which the reader must be able to understand the strategies and devices used by the author in paragraph two. In terms of declarative knowledge, one needs to understand the phrase “inherent simplicity” and the word “tenacity”. In terms of procedural knowledge, the reader greatly benefits by understanding process of elimination, and needs to be be able to condense paragraphs into its main, overall components.

I- This is false because the paragraph clearly states that the proof was extremely difficult to prove and therefore not inherently simple. II- This is correct because the passage focuses on attempts to prove the Four Color Map Theorem and discusses the Theorem's content and its and lack of a true, concise proof. III - This is correct because the paragraph gives the specific example of E. F. Moore, a mathematician who demonstrates his tenacity by repeatedly attempting to prove the Four Color Map Theorem even after multitudinous failures.

The answer choice must therefore state only II and III are true, so D is the correct answer since the other choices do not name only II and III.

= James Lin =

31) Question 31 is a Word Meaning in Context question. To answer this question, one needs to understand the meaning of the word “contrarian.” The meaning can be deduced from the related “contrary,” which is defined as opposite in nature, direction, or meaning. Referring back to the passage, one needs to distinguish the majority, those who attempted to prove the Theorem, from the minority, the “contrarians” of line 34. The student must know how to analyze the context in which the word appears, and recognize key words; that is, read before the line reference from the beginning of the sentence in line 29 and determine that the “contrarians” were “few” in number. Overall, the reader must be able to find the general connotation of a word and use this connotation and its surrounding contextual clues in order to deduce its meaning in relation to the passage.

A is incorrect; nowhere does the phrase “old-fashioned” appear, or are contrarians suggested to be “old-fashioned;” in fact, the contrarians took the less commonly accepted viewpoint.

B is incorrect; the “contrarians” did not “deny the complexity of the four color map;” rather, they attempted to disprove it.

C is correct; the “contrarians” took an “unorthodox perspective” by attempting to disprove rather than prove the theorem, as was the goal of the majority of contemporaneous mathematicians.

D is incorrect; nowhere does the line reference suggest the contrarians “disliked most mathematical procedures.” After the line reference, the author includes an example of one such contrarian whose diligence affirms an appreciation of mathematical procedures.

E is incorrect; it may be considered a distractor option as while the contrarians may have cooperated with other mathematicians, such an implication was not present in the reading. The contrarians had views contrary to the majority and could be said to have worked against the majority of other mathematicians.

32) Question 32 is a Phrase Meaning question. One needs to infer the figurative meaning of “Mount Everest;” that is, a seemingly insurmountable challenge. One needs to connect the usage of this figurative phrase to the subject of the whole passage, and recognize that the challenge in question was proving the Four Color Map Theorem. Specifically, in the very next sentence the literal meaning of “mathematical Mount Everest” is given: “the proof of the Four Color Map Theorem.” One needs to know how to analyze the subject of the passage to determine the meaning of the metaphor. One needs to be able to afterward connect the metaphor back to the subject.

A is incorrect; the paragraph referenced begins by saying that the quest for a counterexample was in vain. The line reference itself explains the Four Color Map Theorem had been proven.

B is incorrect; the five-color map was a type of counterexample, the search for which was as futile as the search for any other, and incorrect for the same reason as A.

C is correct; the “mathematical Mount Everest” conquered was the Four Color Map Theorem, the topic of the passage whose proof invalidated any search for counterexamples.

D is incorrect; high-speed computers were indeed used to prove the Theorem, but the computers themselves were not the challenge or “mathematical Mount Everest” conquered. They were the means and not the ends.

E is incorrect for the same reason as D; the 1500 fundamental cases were elements of the proof of the Theorem, and not the challenge of proving the Four Color Map Theorem itself.

= Ghanash V. = 33. In context, the phrase “fundamental cases”(line 62) means (A) basic patterns (B) primary numbers (C) specific reasons (D) historical illustrations (E) unchanging proofs

This question is a vocabulary in context question. The answer to this question is A. In order to answer this type of question one must go back to the referred lines in the question and read several sentences before and after what phrase or word one is looking for. In terms of declarative knowledge, the reader must understand the meanings of the answers and the words used in the passage, as well be able to realize that the connotation of the word is provided in the passage through “at most four colors”. A is correct because the paragraph talks about how computers were able to paint 1500 “fundamental cases”, or, as the answer states, “basic patterns”. Answer B is incorrect because the computer creates color maps, which are not primary numbers. Answer C is incorrect because the computer maps illustrations of cases, not reasons for them. Answer D is incorrect because although it has “illustrations” which may relate to the maps, the mention of “historical” makes it incorrect. Answer E is the distractor option because although it relates to the “fundamental cases”, the cases themselves were not proofs and were rather used to formulate a rather long and tedious proof.

Ghanash V 34. At the end of paragraph three, the author includes a quotation from Erdos (lines 69-73) primarily to show that Erdos (A) admires those who developed the theorem (B) knows little about the theorem (C) reserves judgment on the proof (D) dislikes some aspects of the proof (E) begs to differ with the results of the proof

This question is purpose type question. The function of the question is to show how the following quotations provide a purpose regarding Erdos. One needs to read the third paragraph and refer to the quotations in lines (69-73) and infer what it shows. Based of the third paragraph, Erdos says he is not an expert on the four color theorem but assumes it is correct. However, he also states “it’s not beautiful. I’d prefer to see a proof that gives insight into why four colors are sufficient.” These quotes show that he dislikes a certain aspect of the proof and this shows that D is correct. A is incorrect because he does not mention his feelings toward those who created the proof. Answer B is the distractor option, as Erdos acknowledges the fact that he does not have an abundance of information on the proof, but is incorrect because this fact is not the primary function of the quotation. C is incorrect because does judge the proof and says that it lacks beauty. Answer E is incorrect because there is no place in the passage where he states that he necessarily disagrees with the proof’s result. Rather, he would prefers to see a proof that provides insight into why the four colors are sufficient in a concise and clear manner, therefore making D correct.

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= CRAIG THOROGOOD =

35) This question is of the purpose type, asking what purpose the second and third paragraphs serve in relation to the entire piece. In order to successfully answer this question, one would need to understand what is being asked of them (i.e. finding out what those paragraphs do). When examining the answer choices, one would also need to know what a syllogism is. One also needs to know how to look for context clues as to the purpose of each paragraph. Additionally, the reader must be able to read the paragraphs and extract main goals and purposes (through a high-level reading) in order to find overall points and answer the question successfully.

A- This choice is clearly wrong, but may be a distractor. Although paragraphs two and three mention two mathematicians in particular, one of them does not have any “strength” over the other. Although Moore could not find a map that required five colors, neither could Graham.

B- This choice is wrong because the second and third paragraphs explain the Four Color map theorem. It is the first paragraph that deals with syllogisms.

C- This choice is wrong because only one proof is mentioned, and the results of the investigations correlated with what the proof stated

D- This choice is correct, and is the best choice, because these paragraphs explain why the Four Color Map Theorem is not beautiful, because there is no explanation for why it works. These paragraphs explain why this Theorem is not an ideal proof.

E- This choice is clearly wrong. Erdos’s characteristics are not mentioned. He may be portrayed as a bit of a perfectionist through the last few sentences of the paragraph, but that is in no way the main gist of the two paragraphs as a whole.

39) This question is of the rhetorical strategies type. The reader needs to be able to locate (or not locate) all these strategies in the piece, and know what all of them are. The reader must also be able to potentially use process of elimination to narrow down the answer choices, as well as read the passage at a very broad level and pinpoint devices rather than reading extremely closely and analyzing minor parts.

A- This choice is wrong because the author uses many quotations from different mathematicians.

B- The author does not define any words by use of their dictionary definition in the passage, therefore B is the correct answer.

C- The author describes the history of the Four Color Map Theorem, therefore this answer is not correct.

D- The Four Color Map Theorem story is an extended anecdote, therefore this answer is not correct.

E- Erdos compares beautiful proofs to music, therefore this answer is not correct

_ = Robbie Sandhu =

37) This question deals with the main topic or main point of a certain paragraph and it makes the reader reread a certain part of the passage to make sure the reader understands what the paragraph is mainly focused on (a broad view rather than a very specific, detailed one). In addition, this question requires knowledge of many rhetorical terms, for example one of the answers “paradox”, can throw some readers off if they do not know the definition of this rhetorical term. In other words in order to answer this question the reader must know a decent amount of rhetorical terms. The reader must be able to then apply these rhetorical terms and realize how the author develops his idea through the specific device in the last paragraph.

A is correct because the author is first talking about why the words “Beauty” and “insight”, used by Erdos and his colleagues, is difficult to explain but then the author provides an example or comparison of Beethoven’s symphony. In other words the author is further explaining what he first stated by using parallel cases to help explain what was originally being talked about. B is incorrect because within the last paragraph there is no content to suggest that humor is being used, everything is serious. C is incorrect because throughout the last paragraph there are no use of words that would convey a meaning that is the opposite of its literal meaning. D is incorrect but it may be a distractor because within the first sentence, the author talks about how hard it is to explain the words “beauty” and “insight” and at a first glance the reader may completely disagree with that but as the reader reads on he/she may see that some truth is to it. This is where readers get tripped up because there is no truth to what the author is saying, the author is comparing these two words to a completely different topic and saying that if Beethoven’s ninth symphony is hard to explain then these two words are also hard to explain. E is incorrect because there is no content that suggests that an inanimate object or animal that is being given human characteristics.

38) This question is of the tone type because it is asking the reader what he/she thinks the tone of the passage could be described as. In order to answer this question, the reader must have read the whole passage or skimmed the passage to get enough information to understand the topic of the passage and a general tone of the passage. In order to find the tone of the passage, the reader must be able to realize the use of connotative words and phrases as well as be able to find the overall opinion or idea of the passage. In addition the reader must be familiar with the terms provided in the question such as “confrontational” or “objective”. A is incorrect because the author is direct about the information throughout the passage meaning that the author completely understands the topic he/she is writing about is not “puzzled” it.  B is correct because as I said before the author is direct throughout his/her writing, the author does not involve opinions or observations; the author states facts. C is incorrect because the author is not expressing any doubt. D is incorrect because the author is not trying to “fight” with anyone the author is stating facts and explaining them. E is incorrect but it could be a distractor because the author describes an argument as “childish” but throughout his passage he is does not give the feeling of being superior than others, he is explaining facts from a biography of a mathematician.