Passage 1

⁽¹⁾ PHI, the Divine Proportion of 1.618, was described by the astronomer Johannes Kepler as one of the “two great treasures of geometry.” (The other is the Pythagorean theorem.)

PHI is the ratio of any two sequential numbers in the Fibonacci

⁽⁵⁾ sequence. If you take the numbers 0 and 1, then create each subse­quent number in the sequence by adding the previous two numbers, you get the Fibonacci sequence. For example, 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144. If you sum the squares of any series of Fibonacci num­bers, they will equal the last Fibonacci number used in the series times

⁽¹⁰⁾ the next Fibonacci number. This property results in the Fibonacci spi­ral seen in everything from seashells to galaxies, and is written math­ematically as: 1² + 1² + 2² + 3² + 5² = 5 × 8.

Plants illustrate the Fibonacci series in the numbers of leaves, the arrangement of leaves around the stem, and in the positioning of

⁽¹⁵⁾ leaves, sections, and seeds. A sunflower seed illustrates this principal

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as the number of clockwise spirals is 55 and the number of counter­clockwise spirals is 89; 89 divided by 55 = 1.618, the Divine Propor­tion. Pinecones and pineapples illustrate similar spirals of successive Fibonacci numbers. ⁽²⁰⁾ PHI is also the ratio of five-sided symmetry. It can be proven by using a basic geometrical figure, the pentagon. This five-sided figure embodies PHI because PHI is the ratio of any diagonal to any side of the pentagon—1.618.

Say you have a regular pentagon ABCDE with equal sides and equal ⁽²⁵⁾ angles. You may draw a diagonal as line AC connecting any two ver-texes of the pentagon. Yo u can then install a total of five such lines, and they are all of equal length. Divide the length of a diagonal AC by the length of a side AB, and you will have an accurate numerical value for PHI—1.618. You can draw a second diagonal line, BC inside the pen-⁽³⁰⁾ tagon so that this new line crosses the first diagonal at point O. What occurs is this: Each diagonal is divided into two parts, and each part is in PHI ratio (1.618) to the other, and to the whole diagonal—the PHI ratio recurs every time any diagonal is divided by another diagonal. When you draw all five pentagon diagonals, they form a five-point ⁽³⁵⁾ star: a pentacle. Inside this star is a smaller, inverted pentagon. Each diagonal is crossed by two other diagonals, and each segment is in PHI ratio to the larger segments and to the whole. Also, the inverted inner pentagon is in PHI ratio to the initial outer pentagon. Thus, PHI is the ratio of five-sided symmetry. ⁽⁴⁰⁾ Inscribe the pentacle star inside a pentagon and you have the pen­tagram, symbol of the ancient Greek School of Mathematics founded by Pythagoras—solid evidence that the ancient Mystery Schools knew about PHI and appreciated the Divine Proportion’s multitude of uses to form our physical and biological worlds.

Passage 2

⁽¹⁾ Langdon turned to face his sea of eager students. “Who can tell me what this number is?”

A long-legged math major in back raised his hand. “That’s the num­ber PHI.” He pronounced it fee.

⁽⁵⁾ “Nice job, Stettner,” Langdon said. “Everyone, meet PHI.” [ . . . ]

“This number PHI,” Langdon continued, “one-point-six-one-eight, is a very important number in art. Who can tell me why?” [ . . . ] “Actually,” Langdon said, [ . . . ] “PHI is generally considered the most beautiful number in the universe.” [ . . . ] As Langdon loaded his slide

⁽¹⁰⁾ projector, he explained that the number PHI was derived from the

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Fibonacci sequence—a progression famous not only because the sum of adjacent terms equaled the next term, but because the quotients of adjacent terms possessed the astonishing property of approaching the number 1.618—PHI! ⁽¹⁵⁾ Despite PHI’s seemingly mystical mathematical origins, Langdon explained, the truly mind-boggling aspect of PHI was its role as a fun­damental building block in nature. Plants, animals, even human beings all possessed dimensional properties that adhered with eerie exactitude to the ratio of PHI to 1. ⁽²⁰⁾ “PHI’s ubiquity in nature clearly exceeds coincidence, and so the ancients assumed the number PHI must have been preordained by the creator of the universe. Early scientists heralded 1.618 as the Divine Proportion.”

[ . . . ] Langdon advanced to the next slide—a close-up of a sun-⁽²⁵⁾ flower’s seed head. “Sunflower seeds grow in opposing spirals. Can you guess the ratio of each rotation’s diameter to the next?

“1.618.”

“Bingo.” Langdon began racing through slides now—spiraled pinecone petals, leaf arrangement on plant stalks, insect segmenta-⁽³⁰⁾ tion—all displaying astonishing obedience to the Divine Proportion.

“This is amazing!” someone cried out.

“Yeah,” someone else said, “but what does it have to do with art?”

[ . . . ] “Nobody understood better than da Vinci the divine struc­ture of the human body. . . . He was the first to show that the human ⁽³⁵⁾ body is literally made of building blocks whose proportional ratios always equal PHI.”

Everyone in class gave him a dubious look.

“Don’t believe me?” . . . Tr y it. Measure the distance from your shoulder to your fingertips, and then divide it by the distance from ⁽⁴⁰⁾ your elbow to your fingertips. PHI again. Another? Hip to floor divided by knee to floor. PHI again. Finger joints. Toes. Spinal divi­sions. PHI, PHI, PHI. My friends, each of you is a walking tribute to the Divine Proportion.” [ . . . .]”In closing,” Langdon said, “we return to symbols.” He drew five intersecting lines that formed a five-pointed ⁽⁴⁵⁾ star. “This symbol is one of the most powerful images you will see this term. Formally known as a pentagram—or pentacle, as the ancients called it—the symbol is considered both divine and magical by many cultures. Can anyone tell me why that may be?”

Stettner, the math major, raised his hand. “Because if you draw a ⁽⁵⁰⁾ pentagram, the lines automatically divide themselves into segments according to the Divine Proportion.”

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Landgon gave the kid a proud nod. “Nice job. Yes, the ratios of line segments in a pentacle all equal PHI, making the symbol the ultimate expression of the Divine Proportion.”

374. The tone of Passage 2 may be described as

a. fascinated discovery.

b. blandly informative.

c. passionate unfolding.

d. droll and jaded.

e. dry and scientific.

375. According to both passages, which of the following are synonyms?

a. pentagon and pentacle

b. pinecones and sunflower seed spirals

c. Divine Proportion and PHI

d. Fibonacci sequence and Divine Proportion

e. Fibonacci sequence and PHI

376. In Passage 2, line 20, ubiquity of PHI most nearly means its

a. rareness in nature.

b. accuracy in nature.

c. commonality in nature.

d. artificiality against nature.

e. purity in an unnatural state.

377. Both passages refer to the “mystical mathematical” side of PHI. Based on the two passages, which statement is NOT another aspect of PHI?

a. PHI is a ratio found in nature.

b. PHI is the area of a regular pentagon.

c. PHI is one of nature’s building blocks.

d. PHI is derived from the Fibonacci sequence.

e. PHI is a math formula.

378. Which of the following techniques is used in Passage 1, lines 13–18 and Passage 2, lines 24–26?

a. explanation of terms

b. comparison of different arguments

c. contrast of opposing views

d. generalized statement

e. illustration by example

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379. All of the following questions can be explicitly answered on the basis of the passage EXCEPT

a. What is the ratio of the length of one’s hip to floor divided by knee to floor?

b. What is the precise mathematical ratio of PHI?

c. What is the ratio of the length of one’s shoulder to fingertips divided by elbow to fingertips?

d. What is the ratio of the length of one’s head to the floor divided by shoulder’s to the floor?

e. What is the ratio of each sunflower seed spiral rotation’s diameter to the next?

380. According to both passages, the terms ancient Mystery Schools (Passage 1, line 43), early scientists (Passage 2, line 22), and ancients (Passage 2, line 46) signify what about the divine proportion?

a. Early scholars felt that the Divine Proportion was a magical number.

b. Early scholars found no scientific basis for the Divine Proportion.

c. Early mystery writers used the Divine Proportion.

d. Early followers of Pythagoras favored the Pythagorean theorem over the divine proportion.

e. Early followers of Kepler used the Divine Proportion in astronomy.

381. Which of the following is NOT true of the pentagon?

a. It is considered both divine and magical by many cultures.

b. It is a geometric figure with five equal sides meeting at five equal angles.

c. It is a geometric figure whereby PHI is the ratio of any diagonal to any side.

d. If you draw an inverted inner pentagon inside a pentagon, it is in PHI ratio to the initial outer pentagon.

e. A polygon having five sides and five interior angles is called a pentagon.

Questions 382–390 are based on the following passage.

The following passage describes the composition and nature of ivory.

⁽¹⁾ Ivory skin, ivory teeth, Ivory Soap, Ivory Snow—we hear “ivory” used all the time to describe something fair, white, and pure. But where does ivory come from, and what exactly is it? Is it natural or man-

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made? Is it a modifier, meaning something pure and white or is it a

⁽⁵⁾ specialized and discrete substance?

Historically, the word ivory has been applied to the tusks of ele­phants. However, the chemical structure of the teeth and tusks of mammals is the same regardless of the species of origin, and the trade in certain teeth and tusks other than elephant is well established and

⁽¹⁰⁾ widespread. Therefore, ivory can correctly be used to describe any mammalian tooth or tusk of commercial interest that is large enough to be carved or scrimshawed. Teeth and tusks have the same origins. Teeth are specialized structures adapted for food mastication. Tusks, which are extremely large teeth projecting beyond the lips, have

⁽¹⁵⁾ evolved from teeth and give certain species an evolutionary advantage that goes beyond chewing and breaking down food in digestible pieces. Furthermore, the tusk can be used to actually secure food through hunting, killing, and then breaking up large chunks of food into manageable bits.

⁽²⁰⁾ The teeth of most mammals consist of a root as well as the tusk proper. Teeth and tusks have the same physical structures: pulp cavity, dentine, cementum, and enamel. The innermost area is the pulp cav­ity. The pulp cavity is an empty space within the tooth that conforms to the shape of the pulp. Odontoblastic cells line the pulp cavity and

⁽²⁵⁾ are responsible for the production of dentine. Dentine, which is the main component of carved ivory objects, forms a layer of consistent thickness around the pulp cavity and comprises the bulk of the tooth and tusk. Dentine is a mineralized connective tissue with an organic matrix of collagenous proteins. The inorganic component of dentine

⁽³⁰⁾ consists of dahllite. Dentine contains a microscopic structure called dentinal tubules which are micro-canals that radiate outward through the dentine from the pulp cavity to the exterior cementum border. These canals have different configurations in different ivories and their diameter ranges between 0.8 and 2.2 microns. Their length is

⁽³⁵⁾ dictated by the radius of the tusk. The three dimensional configura­tion of the dentinal tubules is under genetic control and is therefore a characteristic unique to the order of the mammal.

Exterior to the dentine lies the cementum layer. Cementum forms a layer surrounding the dentine of tooth and tusk roots. Its main func-⁽⁴⁰⁾ tion is to adhere the tooth and tusk root to the mandibular and max­illary jaw bones. Incremental lines are commonly seen in cementum. Enamel, the hardest animal tissue, covers the surface of the tooth or tusk which receives the most wear, such as the tip or crown. Ameloblasts are responsible for the formation of enamel and are lost

⁽⁴⁵⁾ after the enamel process is complete. Enamel exhibits a prismatic struc-

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ture with prisms that run perpendicular to the crown or tip. Enamel

prism patterns can have both taxonomic and evolutionary significance.

Tooth and tusk ivory can be carved into an almost infinite variety of

shapes and objects. A small example of carved ivory objects are small

⁽⁵⁰⁾ statuary, netsukes, jewelry, flatware handles, furniture inlays, and piano

keys. Additionally, wart hog tusks, and teeth from sperm whales, killer

whales, and hippos can also be scrimshawed or superficially carved, thus

retaining their original shapes as morphologically recognizable objects.

The identification of ivory and ivory substitutes is based on the

⁽⁵⁵⁾ physical and chemical class characteristics of these materials. A com­mon approach to identification is to use the macroscopic and micro­scopic physical characteristics of ivory in combination with a simple chemical test using ultraviolet light.

382. In line 5, what does the term discrete most nearly mean?

a. tactful

b. distinct

c. careful

d. prudent

e. judicious

383. Which of the following titles is most appropriate for this passage?

a. Ivory: An Endangered Species

b. Elephants, Ivory, and Widespread Hunting in Africa

c. Ivory: Is It Organic or Inorganic?

d. Uncovering the Aspects of Natural Ivory

e. Scrimshaw: A Study of the Art of Ivory Carving

384. The word scrimshawed in line 12 and line 52 most nearly means

a. floated.

b. waxed.

c. carved.

d. sunk.

e. buoyed.

385. Which of the following choices is NOT part of the physical structure of teeth?

a. pulp cavity

b. dentine

c. cementum

d. tusk

e. enamel

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386. As used in line 13, what is the best synonym for mastication?

a. digestion

b. tasting

c. biting

d. chewing

e. preparation

387. Which sentence best describes dentinal tubules?

a. Dentinal tubules are a layer surrounding the dentine of tooth and tusk roots.

b. Dentinal tubules are micro-canals that radiate outward through the dentine from the pulp cavity to the exterior cementum border.

c. Dentinal tubules are responsible for the formation of enamel and are lost after the enamel process is complete.

d. Dentinal tubules cover the surface of the tooth or tusk which receives the most wear, such as the tip or crown.

e. Dentinal tubules are extremely large teeth projecting beyond the lips that have evolved from teeth and give certain species an evolutionary advantage.

388. According to the passage, all of the following are organic substances EXCEPT

a. cementum.

b. dentine.

c. dahllite.

d. ameloblasts.

e. collagen.

389. According to the passage, how can natural ivory be authenticated?

a. by ultraviolet light

b. by gamma rays

c. by physical observation

d. by osmosis

e. by scrimshaw

390. According to the passage, which statement is NOT true of enamel?

a. It is an organic substance.

b. It is the hardest of animal tissues.

c. It should never be exposed to ultraviolet light.

d. It structure is prismatic.

e. It is formed with the aid of ameloblasts.

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Questions 391–399 are based on the following passage.

This passage is about the process by which scientists prove theories, the scientific method.

(1) The scientific method usually refers to either a series or a collection of processes that are considered characteristic of scientific investiga­tion and of the acquisition of new scientific knowledge. The essential elements of the scientific method are:

(5) Observe: Observe or read about a phenomenon.

Hypothesize: Wonder about your observations, and invent a

hypothesis, or a guess, which could explain the phenomenon or

set of facts that you have observed. Test: Conduct tests to try out your hypothesis. (10) Predict: Use the logical consequences of your hypothesis to pre­dict observations of new phenomena or results of new

measurements. Experiment: Perform experiments to test the accuracy of these

predictions. (15) Conclude: Accept or refute your hypothesis.

Evaluate: Search for other possible explanations of the result until

you can show that your guess was indeed the explanation, with

confidence. Formulate new hypothesis: as required.

(20) This idealized process is often misinterpreted as applying to scien­tists individually rather than to the scientific enterprise as a whole. Sci­ence is a social activity, and one scientist’s theory or proposal cannot become accepted unless it has been published, peer reviewed, criti­cized, and finally accepted by the scientific community.

(25) Observation

The scientific method begins with observation. Observation often demands careful measurement. It also requires the establishment of an operational definition of measurements and other concepts before the experiment begins.

(30) Hypothesis

To explain the observation, scientists use whatever they can (their own creativity, ideas from other fields, or even systematic guessing) to come up with possible explanations for the phenomenon under

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study. Deductive reasoning is the way in which predictions are used ⁽³⁵⁾ to test a hypothesis.