The following text is from Theodore Dreiser’s 1900 novel Sister Carrie. Hurstwood, who tricked Carrie into leaving Chicago with him, has just asked her to marry him.
Carrie had never been ill-disposed toward him. Only a moment before she had been listening with some Underlined: complacency End underlined., remembering her old affection for him. He was so handsome, so daring!
Now, however, this feeling had changed to one of opposition, which rose feebly.
As used in the text, what does the word “complacency” most nearly mean?
Correct choice: gratification
Carrie was “remembering her old affection” for Hurstwood and thinking him “so handsome, so daring!” She had been listening with pleasure—an older sense of “complacency” meaning quiet satisfaction or gratification.
A: The modern word “complacent” suggests unconcern, but Carrie was not unconcerned: she was recalling her affection and admiring Hurstwood, which is pleasure, not indifference.
C: Her opposition “rose feebly” afterward, but before that she was listening with pleasure; nothing suggests she had passively given up.
D: Carrie “had never been ill-disposed toward him,” and she was remembering her affection, so she was not listening with suspicion.
Watch for: In older texts, “complacency” often means simple pleasure or satisfaction, without the modern sense of self-satisfied unconcern.
Reading and Writing · Module 1 · Question 2 of 27
A 1993 study reported that college students scored slightly higher on a spatial-reasoning task after listening to ten minutes of a Mozart sonata than after sitting in silence, an improvement that lasted only about fifteen minutes. Within a few years, the finding had been recast in popular accounts as evidence that playing classical music to infants makes them smarter. Psychologists who reviewed the research judged this popular version ______: it offered an appealingly simple lesson only by ignoring how small, brief, and specific the original effect had been.
Which choice completes the text with the most logical and precise word or phrase?
Correct choice: facile
The colon explains the judgment: the popular version was appealingly simple, but only because it ignored the original effect’s limits. “Facile” means easily achieved or stated but superficial, which captures both the appeal and the criticism.
B: “Straightforward” captures the simplicity but carries no criticism; the text says the simplicity was achieved “only by ignoring” the evidence, a fault the word does not express.
C: The popular version was indeed derived from the study, but “derivative” means unoriginal or imitative, and the colon explains a different fault: false simplicity, not lack of originality.
D: “Succinct” praises brevity; the psychologists were faulting the popular version for oversimplifying, not complimenting it for being brief.
Watch for: When a colon explains a judgment, the word must match every part of the explanation—here both “appealingly simple” and “only by ignoring.”
Reading and Writing · Module 1 · Question 3 of 27
The following text is from Charles Dickens’s 1857 novel Little Dorrit. The Circumlocution Office is a fictional government department.
This glorious establishment had been early in the field, when the one sublime principle involving the difficult art of governing a country, was first distinctly revealed to statesmen. It had been foremost to study that bright revelation and to carry its shining influence through the whole of the official proceedings. Underlined: Whatever was required to be done, the Circumlocution Office was beforehand with all the public departments in the art of perceiving—HOW NOT TO DO IT. End underlined.
Through this delicate perception, through the tact with which it invariably seized it, and through the genius with which it always acted on it, the Circumlocution Office had risen to overtop all the public departments; and the public condition had risen to be—what it was.
Which choice best describes the function of the underlined sentence in the text as a whole?
Correct choice: It reveals that the accomplishment the narrator has been lavishly praising is in fact a talent for avoiding action.
The first two sentences praise the office’s “sublime principle” and “bright revelation” without saying what it is. The underlined sentence names it—“HOW NOT TO DO IT”—exposing the praise as ironic: the office’s celebrated skill is avoiding action. The closing “the public condition had risen to be—what it was” continues the irony.
A: This takes the praise literally. The principle is “HOW NOT TO DO IT,” and the final clause (“the public condition had risen to be—what it was”) signals that the public was not well served.
B: The later praise of the office’s “tact” and “genius” continues the same irony rather than withdrawing a criticism; it applies the mock praise to the office’s skill at not acting.
D: The text says the office was “beforehand with all the public departments,” but it never mentions resentment; the other departments are simply outdone in the same pursuit.
Watch for: In satire, a sentence that finally names what has been praised often reveals that the praise was ironic all along.
Reading and Writing · Module 1 · Question 4 of 27
In the 1790s the naturalist Lazzaro Spallanzani found that bats whose eyes had been covered could still fly through a darkened room without striking threads strung across it. Soon afterward, Louis Jurine reported that bats whose ears had been tightly plugged collided with objects. Underlined: Spallanzani then fitted bats with small tubes inserted into their ears; when the tubes were left open, the bats flew normally, but when the tubes were closed, the bats blundered into obstacles. End underlined. Because neither man could hear any sound from the flying bats, the idea that bats find their way by hearing was widely doubted until the late 1930s, when researchers recorded the animals’ high-pitched calls, which lie beyond the range of human hearing.
Which choice best describes the function of the underlined sentence in the text as a whole?
Correct choice: It rules out the possibility that Jurine’s bats collided with objects merely because something was placed in their ears.
Jurine’s plugged bats collided with objects, but plugging an ear might disturb a bat in ways unrelated to hearing. Spallanzani’s tubes were in the bats’ ears in both conditions; only whether sound could pass through changed. Because the bats flew normally with open tubes, the collisions depended on blocking sound, not on the presence of something in the ear.
B: Bats with closed tubes “blundered into obstacles,” which agrees with Jurine’s report; the experiment confirms and strengthens his finding rather than challenging it.
C: The covered-eye experiment succeeded in showing that bats do not need sight; the underlined sentence does not revisit it or explain any failure.
D: The tube experiment shows that navigation depends on hearing, but nothing in it shows that the bats produce sounds or listen for echoes; that was established only when researchers recorded the calls in the 1930s.
Watch for: When an experiment keeps an intervention in place but changes one feature of it, ask which alternative explanation it is designed to exclude.
Reading and Writing · Module 1 · Question 5 of 27
In 1991 psychologists Diane Halpern and Stanley Coren reported that among recently deceased people whose handedness their relatives could recall, left-handers had died about nine years younger, on average, than right-handers, and they suggested that left-handedness shortens life. Underlined: People born in the early twentieth century, however, were far less likely to be identified as left-handed than people born later, since many children of that era were made to write with their right hands. End underlined. A later study that followed a large group of older adults for several years found no difference in survival between left- and right-handers.
Which choice best describes the function of the underlined sentence in the text as a whole?
Correct choice: It points out a difference between generations that could produce Halpern and Coren’s result even if handedness does not affect how long people live.
If older generations include few left-handers, then left-handers among recent deaths will be mostly younger people, and their average age at death will be lower even if handedness has no effect on life span. The sentence supplies this alternative explanation, and the later study’s finding of no difference fits it.
B: The sentence concerns how common left-handedness was in different generations, not whether relatives remembered it correctly.
C: The sentence does imply that less discouragement made left-handedness more common, but its role in the text is to explain Halpern and Coren’s result without any effect on life span, not to explain trends in handedness for their own sake.
D: The sentence says that forced switching made older people less likely to be (recorded as) left-handed; it says nothing about the health or life span of the children who were switched.
Watch for: When a group’s makeup differs by birth year, its average age at death can differ for reasons that have nothing to do with the trait being studied.
Reading and Writing · Module 1 · Question 6 of 27
Text 1
Astronomer Ivo Pell attributes a star’s periodic dimming to a planet passing in front of it. The star’s color remains nearly constant during each dimming event. Pell notes that a cool spot rotating across the star’s visible surface would make the observed light redder, so the color measurements favor his interpretation over the cool-spot explanation.
Text 2
Astronomer Mina Orr agrees that the color measurements weigh against a cool spot. She notes, however, that two stars of nearly the same color could eclipse one another without appreciably changing the combined light’s color. Measurements of the light sources’ motions would be needed to distinguish that possibility from Pell’s planet interpretation.
How would Orr (Text 2) most likely characterize the reasoning in Text 1?
Correct choice: It uses evidence that distinguishes the planet interpretation from one alternative but does not yet distinguish it from another viable alternative.
Orr accepts Pell’s comparison with the cool-spot explanation: the nearly unchanged color is unfavorable to that alternative. Her qualification is that this success does not establish the planet interpretation uniquely. An eclipsing pair of similar-colored stars could also produce dimming without much color change. Pell’s evidence therefore separates two explanations but leaves a third unresolved. Orr’s proposed motion measurements would address the remaining comparison. She is not rejecting the color evidence itself or reporting results from measurements that have not yet been made.
A: This reverses the evidence: color weighs against a cool spot but leaves the similar-colored stellar pair unresolved.
B: No motion measurements have been reported, and Orr does not claim they support a cool spot. She recommends obtaining them.
C: Orr explicitly agrees that the color measurements weigh against a cool spot. Her objection concerns a different alternative.
Watch for: Eliminating one alternative does not establish a favored explanation when another alternative predicts the same evidence.
Reading and Writing · Module 1 · Question 7 of 27
Text 1
Temple inscriptions from the reign of the Egyptian pharaoh Ramesses III describe invaders, whom historians call the Sea Peoples, overrunning lands from Anatolia to Cyprus before an attack on Egypt around 1177 BCE. Many historians concluded that these raiders alone had destroyed the great cities of the eastern Mediterranean at the end of the Bronze Age.
Text 2
Excavations show that the region’s cities were destroyed over several decades rather than in a single wave, and some destruction layers show damage consistent with earthquakes rather than attack. Historian Eric Cline argues that earthquakes, drought and famine, and disrupted trade, together with the raids of the Sea Peoples, weakened the region’s interconnected kingdoms until they collapsed one after another.
Based on the texts, how would Cline (Text 2) most likely respond to the conclusion of the historians described in Text 1?
Correct choice: He would accept that the Sea Peoples contributed to the destruction but reject the idea that their raids alone brought the kingdoms down.
The historians in Text 1 held the Sea Peoples alone responsible for the destruction. Cline lists their raids among several causes—earthquakes, drought and famine, and disrupted trade—that together weakened the kingdoms. He would grant the raids a part while rejecting them as the sole cause.
A: Cline includes “the raids of the Sea Peoples” among the causes of collapse, so he would not deny them any part.
B: Cline accepts that the raids played a part, but he would not agree that they were responsible for the destruction, since he attributes it to several causes acting together.
D: Nothing in Text 2 questions when the attacks happened; Cline disputes only whether the raids alone explain the destruction.
Watch for: When Text 2 lists Text 1’s cause among several, its response is usually “part of the story,” neither full agreement nor rejection.
Reading and Writing · Module 1 · Question 8 of 27
Text 1
In a 1985 study, psychologists Thomas Gilovich, Robert Vallone, and Amos Tversky had college basketball players each take 100 shots from spots where the player made about half of their attempts. For each player, the researchers compared the percentage of shots made immediately after three consecutive makes with the player’s overall percentage. Finding no meaningful difference, they concluded that the widespread belief in a “hot hand”—that players become more likely to score after a string of successes—is an illusion.
Text 2
Economists Joshua Miller and Adam Sanjurjo have identified a subtle bias in such comparisons. Consider a player whose chance of making each shot never changes and who takes 100 shots, as in the 1985 study. If one calculates the percentage of shots made immediately after three consecutive makes, that percentage will, on average, be noticeably lower than the player’s overall percentage.
Based on the texts, how would Miller and Sanjurjo (Text 2) most likely respond to the conclusion in Text 1?
Correct choice: They would argue that finding no difference suggests that the players shot better after three makes than players with unchanging chances would be expected to.
Text 2 shows that for a player whose chances never change, the percentage made after three straight makes is expected to fall below the overall percentage. Text 1’s players, however, made shots after three makes at about their overall rate—better than that expected shortfall. So the data Text 1 took as showing no hot hand actually point toward one.
B: Text 2 describes a bias in short series; nothing suggests that more shots would push the players’ post-streak percentages below their overall percentages.
C: This reverses the bias: the calculation Text 2 describes makes post-streak shooting look worse, not better, than a constant chance would produce, so correcting for it strengthens the case for a hot hand.
D: The spots were chosen so that each player made about half of their attempts; Text 2 raises a problem with the calculation, not with the difficulty of the shots.
Watch for: When Text 2 changes what a result should be compared with, recompute what Text 1’s result means against the new benchmark.
Reading and Writing · Module 1 · Question 9 of 27
The following text is from Sherwood Anderson’s 1919 book Winesburg, Ohio. The narrator describes the central thought of an unpublished book written by an old man.
The thought was involved but a simple statement of it would be something like this:
That in the beginning when the world was young there were a great many thoughts but no such thing as a truth. Man made the truths himself and each truth was a composite of a great many vague thoughts. All about in the world were the truths and they were all beautiful. [...] And then the people came along. Each as he appeared snatched up one of the truths and some who were quite strong snatched up a dozen of them.
It was the truths that made the people grotesques. [...] It was his notion that the moment one of the people took one of the truths to himself, called it his truth, and tried to live his life by it, he became a grotesque and the truth he embraced became a falsehood.
Which choice best states the main idea of the text?
Correct choice: According to the old man’s theory, truths that are beautiful in themselves become falsehoods, and distort people, once someone claims one as his own and lives by it.
The truths are “all beautiful,” yet “it was the truths that made the people grotesques”: the moment a person takes a truth “to himself, called it his truth, and tried to live his life by it,” the person becomes a grotesque and the truth becomes a falsehood. The harm lies in appropriating a truth, not in the truth itself.
A: The text reverses this causation: “It was the truths that made the people grotesques.” People become grotesques by taking up a truth, not because they are grotesque already.
B: The truths “were all beautiful,” and a truth “became a falsehood” only when someone embraced it as his own; it was not false from the beginning.
C: The text notes that some strong people “snatched up a dozen” truths but never says that this spared them; the theory says anyone who takes a truth as his own becomes a grotesque.
Watch for: When a text states a paradox (beautiful truths make people grotesque), the main idea must explain how both halves can hold.
Reading and Writing · Module 1 · Question 10 of 27
In his 1877 essay “An Apology for Idlers,” Robert Louis Stevenson inverts a common judgment of character, treating relentless industry as a sign of inner poverty and a gift for leisure as the mark of a rich, many-sided nature.
Which quotation from “An Apology for Idlers” most effectively illustrates the claim?
Correct choice: “Extreme busyness, whether at school or college, kirk or market, is a symptom of deficient vitality; and a faculty for idleness implies a catholic appetite and a strong sense of personal identity.”
The claim has two parts: relentless industry signals inner poverty, and a gift for leisure marks a rich, many-sided nature. This quotation makes both points: busyness is “a symptom of deficient vitality,” and a faculty for idleness implies “a catholic appetite”—wide-ranging interests—and “a strong sense of personal identity.”
A: This recommends idleness for the young but says nothing about what busyness or idleness reveals about a person’s vitality or interests.
B: This portrays busy people as incurious, which fits the idea of inner poverty, but it says nothing about what a gift for leisure reveals, the claim’s second part.
D: This criticizes constant work for its cost—the neglect of other things—and questions the importance of business, rather than presenting it as a sign of deficient vitality, and it says nothing about idlers’ interests.
Watch for: With a two-part claim, eliminate quotations that support only one part, however apt that part is.
Reading and Writing · Module 1 · Question 11 of 27
Before the 1860s, defenders of spontaneous generation—the idea that living things can arise from nonliving matter—argued that boiled broth stayed free of microbes in a sealed flask only because sealing kept out fresh air, which they believed life requires. Louis Pasteur therefore boiled broth in flasks whose necks he had drawn out into long, downward-bending curves, leaving the ends open. Air could reach the broth, but dust settled in the curves, and the broth remained free of microbes for months. Some critics then proposed that the boiling itself had altered the broth so that it could no longer support microbial life of any kind.
Which finding, if true, would most directly weaken the critics’ proposal?
Correct choice: When a curved-neck flask was tipped so that the boiled broth washed into the curve of the neck and back, microbes soon appeared in the broth.
The critics claim the boiled broth can no longer support microbial life. Tipping the flask exposes the same boiled broth to the dust trapped in the neck, and microbes then appear—so the broth could still support microbes, and what it had lacked was contact with dust-borne microbes.
A: This answers the earlier objection about fresh air, but it fits the critics’ new proposal perfectly well: broth that could no longer support life would also stay free of microbes.
B: If longer boiling altered the broth more thoroughly, broth boiled longer would stay sterile longer; this finding is what the critics’ proposal predicts.
D: Unboiled broth has not been altered by boiling, so microbes appearing in it says nothing about whether boiled broth can still support life.
Watch for: When critics shift to a new objection, the finding that weakens it must address that objection, not the one already answered.
Reading and Writing · Module 1 · Question 12 of 27
In a 1974 study, psychologists Elizabeth Loftus and John Palmer showed 150 participants a short film of a car accident. Afterward, 50 participants were asked, “About how fast were the cars going when they smashed into each other?” Another 50 were asked the same question with “hit” in place of “smashed,” and the remaining 50 were not asked about speed. A week later, all participants were asked whether they had seen any broken glass in the film; there had been none. A student claims that the verb in the speed question changed what participants remembered about the accident itself, not merely the speeds they estimated.
Responses in Loftus and Palmer’s Second Experiment
Question asked after the film
Mean speed estimate (miles per hour)
Participants who reported seeing broken glass (of 50)
Speed question with “smashed”
10.46
16
Speed question with “hit”
8.00
7
No question about speed
—
6
Which choice most effectively uses data from the table to support the student’s claim?
Correct choice: Participants asked with “smashed” were more than twice as likely to report seeing broken glass as participants in either of the other two groups were.
The claim concerns memory of the accident, so the relevant measure is the report of broken glass, which the film did not show. Sixteen of 50 “smashed” participants reported it, compared with 7 of 50 “hit” participants and 6 of 50 participants not asked about speed—more than twice either other group.
A: The figures are accurate (10.46 is about 31 percent above 8.00), but a higher speed estimate could reflect only how participants answered the speed question, which is the alternative the claim sets aside.
B: This has the right shape but misstates the table: the “hit” group reported broken glass 7 times, only slightly more often than the 6 reports from participants not asked about speed.
C: This is accurate, but it shows only that “hit” did not change reports of broken glass; it says nothing about the effect of “smashed,” which is what the claim concerns.
Watch for: When a claim contrasts two outcomes (“not merely …”), choose data on the outcome the claim is really about.
Reading and Writing · Module 1 · Question 13 of 27
The number of farms in the United States reached its peak in the 1930s and then fell steeply as farms grew larger and more mechanized. A student claims that the number of farms per person in the country fell in the first half of the twentieth century—even during the years when the number of farms was still rising.
Number of Farms and Population of the United States
Year
Number of farms (millions)
Population (millions)
1900
5.7
76.2
1935
6.8
127.3
1950
5.4
151.3
2000
2.2
281.4
Which choice most effectively uses data from the table to support the student’s claim?
Correct choice: The number of farms rose from 5.7 million in 1900 to 6.8 million in 1935, but in the same years the population grew from 76.2 million to 127.3 million, a far larger proportional increase.
The claim concerns the years when farms were still increasing, 1900–1935. Farms rose by about 19 percent (5.7 to 6.8 million) while the population rose by about 67 percent (76.2 to 127.3 million), so farms per person fell even as the number of farms grew.
A: This is accurate, but it concerns 1950–2000, when the number of farms was falling; it says nothing about the years when farms were still increasing, which is the part of the claim that needs support.
B: This is accurate, but it describes only the count of farms; without population figures it cannot show that farms per person fell.
C: This has the right shape but overstates the table: population grew by about 67 percent and farms by about 19 percent, so population grew about 3.5 times as fast, not more than five times.
Watch for: A claim about a rate (“per person”) needs both the count and the base; check each figure a sweeping choice relies on.
Reading and Writing · Module 1 · Question 14 of 27
In 1823 the astronomer Heinrich Olbers described a puzzle. Suppose the universe were infinitely large, had existed unchanged forever, and were filled throughout with stars much like the Sun. Then every line of sight from Earth would eventually end at the surface of a star, and the whole night sky would glow as brightly as the surface of the Sun. Dust between the stars cannot remove the difficulty: over endless time, dust absorbing starlight would heat up until it glowed as brightly as the stars themselves. Because the night sky is dark, ______
Which choice most logically completes the text?
Correct choice: at least one of the three suppositions about the universe’s size, age, and contents must be false.
The three suppositions together imply a sky as bright as the Sun’s surface, and dust cannot rescue them. Since the sky is dark, the suppositions cannot all be true—but the argument does not show which one fails.
A: A finite universe would be one way out, but the argument shows only that the three suppositions cannot all hold; nothing in it singles out size as the one that fails.
B: A universe of finite age is the explanation astronomers now favor, but the text’s reasoning does not single out age; it shows only that at least one supposition must fail.
D: The text explains that dust cannot remove the difficulty, because over endless time it would heat up and glow as brightly as the stars.
Watch for: When a set of assumptions leads to a false result, logic shows that at least one assumption fails—not which one.
Reading and Writing · Module 1 · Question 15 of 27
In sealed-bid auctions for leases to drill for oil in the Gulf of Mexico during the 1950s and 1960s, no company knew exactly how much oil a lease held. Each company estimated a lease’s value from its own surveys and bid accordingly, and the lease went to the highest bidder. In 1971 petroleum engineers Ed Capen, Robert Clapp, and William Campbell reported that winning companies had often earned disappointing returns on their leases, even though, the engineers argued, the companies’ estimates had not been too high on average. The engineers explained this apparent paradox by pointing out that ______
Which choice most logically completes the text?
Correct choice: the winning bid usually came from the company whose estimate was highest, and the highest of many estimates tends to exceed a lease’s true value.
Even if estimates are right on average, they scatter above and below a lease’s true value. The auction awards each lease to the highest bid, which usually reflects the highest estimate—one of the ones that overshoot. Winners therefore tend to have overestimated, and disappointing returns follow.
B: The text says nothing about which companies had the most accurate surveys or how cautiously they bid, and this would not explain why winners’ estimates tended to be too high.
C: Competition can raise prices, but each company bid according to its own estimate; if estimates were accurate on average, competition alone would not make winners’ estimates too high, which is the puzzle the engineers explained.
D: Nothing in the text compares drilling costs, and the paradox concerns estimates that were accurate on average yet led to disappointment, which drilling costs would not explain.
Watch for: Selecting the highest of many uncertain estimates picks out the ones most likely to be too high, even when the estimates are unbiased.
Reading and Writing · Module 1 · Question 16 of 27
In 1774 Antoine Lavoisier heated a piece of tin in a sealed glass vessel until part of the tin had turned into a powdery substance called a calx. As others had found, the calx weighed more than the metal it came from, yet the sealed vessel and its contents weighed the same after heating as before. When Lavoisier broke the seal, air rushed into the vessel, which then weighed more than it had. Many chemists of the time held that a metal turns into a calx by giving off a substance called phlogiston. Lavoisier’s results suggest that ______
Which choice most logically completes the text?
Correct choice: the tin gained weight by combining with part of the air sealed inside the vessel, and outside air rushed in to replace the air that had been used up.
The calx is heavier than the tin, yet the sealed vessel’s total weight did not change, so the gain was matched by a loss from something else inside. Air rushing in when the seal was broken shows that some of the air inside had been used up. The tin combined with part of the air, and fresh air replaced it.
B: Absorbing a substance from the glass would keep the sealed weight constant, but it would not explain why air rushed in when the seal was broken, which shows that air inside the vessel had been used up.
C: Phlogiston leaving the tin would not use up any of the air in the vessel, so this cannot explain why air rushed in when the seal was broken.
D: The text says part of the tin had turned into calx while the vessel was sealed and heated, before the seal was broken, so the calx did not form only after air rushed in.
Watch for: In a sealed system, an unchanged total weight means any gain in one part is a loss in another; find what was lost.
Reading and Writing · Module 1 · Question 17 of 27
Genetics, the study of how traits are passed from parents to offspring, ______ its name from a term that the British biologist William Bateson proposed in 1905 in a letter to a fellow scientist.
Which choice completes the text so that it conforms to the conventions of Standard English?
Correct choice: takes
The subject is “Genetics,” the name of a field of study. Such names ending in -ics are singular, so the singular verb “takes” is required; the appositive between commas does not change the subject.
A: “Take” is plural; “genetics” ends in -s but names a single field of study and takes a singular verb.
C: “Are taking” is plural and does not agree with the singular subject “Genetics.”
D: “Have taken” is plural and does not agree with the singular subject “Genetics.”
Watch for: Names of fields ending in -ics (genetics, physics, economics) are singular when they name the field.
Reading and Writing · Module 1 · Question 18 of 27
Benjamin Franklin filled the yearly editions of Poor Richard’s Almanack with short sayings that praised thrift, industry, and plain dealing. “Well done,” ______ “is better than well said.”
Which choice completes the text so that it conforms to the conventions of Standard English?
Correct choice: he wrote in 1737,
The attribution “he wrote in 1737” interrupts a single quoted sentence (“Well done … is better than well said”). An interrupting tag is set off by commas on both sides, so a comma must follow it before the quotation resumes.
A: A semicolon would treat “is better than well said” as a separate independent clause, but it has no subject; it completes the quoted sentence that the tag interrupts.
B: A period would end the sentence and leave “is better than well said” as a fragment beginning with a lowercase verb.
C: Without a comma, the tag runs directly into the resumed quotation; an interrupting tag needs punctuation on both sides.
Watch for: When a speaker tag interrupts one quoted sentence, commas go on both sides of the tag.
Reading and Writing · Module 1 · Question 19 of 27
Alice Ball, a chemist ______ developed an injectable treatment for leprosy at the College of Hawaii, died in 1916 at age twenty-four; the college’s president later published her method without crediting her.
Which choice completes the text so that it conforms to the conventions of Standard English?
Correct choice: who
The blank begins a relative clause describing “a chemist,” a person, and the clause needs a subject for “developed.” The relative pronoun “who” fills both roles.
A: “Which” refers to things or animals, not people; a chemist requires “who.”
C: “She” would create a second independent clause (“a chemist she developed …”) inside the appositive, leaving the sentence ungrammatical.
D: “And she” joins a clause to the noun “a chemist” as if the noun were a clause, producing “Alice Ball, a chemist and she developed …,” which is ungrammatical.
Watch for: A clause that describes a person begins with “who,” not “which.”
Reading and Writing · Module 1 · Question 20 of 27
A 2007 study of English verbs over the past twelve centuries found that frequency protects irregular forms such as “sang” and “went” from becoming regular. The more often an irregular verb is ______ less likely it is to have acquired a regular “-ed” past tense, with rarely used verbs regularizing fastest.
Which choice completes the text so that it conforms to the conventions of Standard English?
Correct choice: used, the
The sentence uses the paired comparative construction “the more …, the less …,” whose two halves are joined by a comma: “The more often an irregular verb is used, the less likely it is.”
A: A semicolon requires an independent clause on each side, but “The more often an irregular verb is used” cannot stand alone; it depends on the second half of the comparison.
C: A period would leave “The more often an irregular verb is used” as a fragment.
D: A colon must follow an independent clause, and “The more often an irregular verb is used” is not one.
Watch for: In “the more …, the more …” sentences, the two halves depend on each other and are joined by a comma.
Reading and Writing · Module 1 · Question 21 of 27
When the Swedish warship Vasa was raised from Stockholm harbor in 1961, it ______ on the harbor floor for 333 years, preserved in part because the harbor’s brackish water was inhospitable to the shipworms that devour submerged wood elsewhere.
Which choice completes the text so that it conforms to the conventions of Standard English?
Correct choice: had lain
The ship rested on the harbor floor, so the verb is “lie” (to recline), whose past participle is “lain.” The past perfect “had lain” places the resting before the 1961 raising.
A: “Laid” is the past participle of “lay” (to put something down), which takes an object; the ship did not lay anything on the harbor floor.
B: “Lay” is the simple past of “lie,” not its past participle, so it cannot follow “had.”
C: “Lied” is the past participle of “lie” meaning to tell a falsehood, which does not fit a sunken ship.
Watch for: “Lie, lay, lain” (recline) and “lay, laid, laid” (put) overlap at “lay”; after “had,” reclining is “lain.”
Reading and Writing · Module 1 · Question 22 of 27
Two days after Samuel Langley’s full-size Aerodrome plunged into the Potomac River in October 1903, an editorial in the New York Times ______ practical flying machine might take “from one million to ten million years” to develop. Ten weeks later, the Wright brothers made the first powered flights at Kitty Hawk.
Which choice completes the text so that it conforms to the conventions of Standard English?
Correct choice: predicted that a
The clause “that a practical flying machine might take …” is the object of the verb “predicted.” No punctuation separates a verb from its object, so “predicted that a” is correct.
A: A colon must follow a complete independent clause, but “an editorial … predicted” is incomplete without the object that the colon would cut off.
C: A comma would separate the verb “predicted” from its object, the clause “that a practical flying machine might take …”; no comma belongs between them.
D: A semicolon separates independent clauses, but “that a practical flying machine might take …” is a dependent clause serving as the object of “predicted.”
Watch for: A “that” clause after a verb such as “predicted,” “argued,” or “found” is the verb’s object and takes no punctuation before it.
Reading and Writing · Module 1 · Question 23 of 27
Sketches and notes that Leonardo da Vinci made around 1493 show a strikingly modern understanding of friction. ______ they state that the force needed to slide an object across a surface is proportional to the object’s weight and does not depend on how much of the object touches the surface—rules that the French physicist Guillaume Amontons would publish two centuries later.
Which choice completes the text with the most logical transition?
Correct choice: In particular,
The first sentence makes a general claim (Leonardo understood friction in a modern way); the second spells out exactly what that understanding was. “In particular” introduces the details of a general statement.
A: “Nevertheless” would signal that the rules hold despite Leonardo’s modern understanding, but the rules are the content of that understanding.
B: “Similarly” would present the rules as a parallel to something already described, but they are details of the claim just made, not a comparable case.
D: “Consequently” would make the rules a result of Leonardo’s understanding, but they are what the understanding consisted of.
Watch for: When a sentence gives the particulars of the claim before it, “In particular” fits; “Consequently” would wrongly make the particulars an effect.
Reading and Writing · Module 1 · Question 24 of 27
In 1869, while surveying the site of the planned Brooklyn Bridge, its designer, engineer John Roebling, suffered an injury that proved fatal. His son Washington took charge and began construction in 1870, but in 1872 he was disabled by decompression sickness from working in the bridge’s underwater foundations. ______ he directed the project from an apartment overlooking the river, relying on his wife, Emily Warren Roebling, to carry his instructions to the engineers until the bridge opened in 1883.
Which choice completes the text with the most logical transition?
Correct choice: Thereafter,
The final sentence describes what Washington Roebling did from the time of his 1872 disability until the bridge opened in 1883. “Thereafter” signals that these events followed the ones just described.
A: “Previously” would place his directing the work from an apartment before his 1872 disability, but that arrangement was a response to it.
B: “Meanwhile” signals events happening at the same time as those just described, but his remote direction began after he was disabled and lasted for years.
C: “Conversely” signals an opposite, but directing the work from an apartment is a continuation of his role after his injury, not a reversal of anything described.
Watch for: For time transitions, place the new sentence on the timeline: before, during, or after the events just described.
Reading and Writing · Module 1 · Question 25 of 27
While researching a topic, a student has taken the following notes:
In 1942 actress Hedy Lamarr and composer George Antheil received a U.S. patent for a “secret communication system.”
The system was designed to keep enemy forces from jamming the radio signals used to guide torpedoes.
It worked by rapidly switching the signal among 88 frequencies in a pattern known only to the sender and the receiver.
Antheil drew on his experience synchronizing player pianos, which have 88 keys.
The U.S. Navy did not adopt the system during World War II, reportedly considering it too cumbersome.
The technique it described, now called frequency hopping, later became a basis for some wireless technologies.
The student wants to explain how the invention was meant to protect a signal from jamming. Which choice most effectively uses relevant information from the notes to accomplish this goal?
Correct choice: Lamarr and Antheil’s system was meant to defeat jamming by switching a torpedo’s guidance signal rapidly among 88 frequencies in a pattern known only to the sender and the receiver.
The goal is to explain how the system protected a signal, which requires its mechanism: rapid switching among 88 frequencies in a pattern known only to sender and receiver. Only this choice describes that mechanism.
A: This states what the system was for—preventing jamming—but not how it accomplished that, which is what the goal asks.
C: This explains where the idea of 88 frequencies came from, not how the system protected a signal from jamming.
D: This describes the system’s reception and legacy rather than how it worked.
Watch for: “How” asks for a mechanism; a choice that states only the purpose (“meant to keep … from jamming”) does not answer it.
Reading and Writing · Module 1 · Question 26 of 27
While researching a topic, a student has taken the following notes:
Navigators in the Marshall Islands traditionally made stick charts from the ribs of coconut palm fronds, lashed together, with small shells tied on.
The curved and straight ribs represented patterns of ocean swells, including the way swells bend around islands; the shells marked islands.
A mattang chart illustrated general principles of how swells behave rather than depicting any real area.
A rebbelib chart depicted a large real region, such as an entire chain of atolls.
Navigators studied the charts on land and memorized them; the charts were not carried on voyages.
The student wants to explain what the charts represented and how navigators used them. Which choice most effectively uses relevant information from the notes to accomplish this goal?
Correct choice: The ribs and shells of Marshall Islands stick charts represented ocean swells and islands, and navigators studied and memorized the charts on land instead of carrying them on voyages.
The goal has two parts: what the charts represented (swells and islands) and how navigators used them (studied and memorized on land, not carried at sea). Only this choice gives both.
A: This contrasts two types of chart but says nothing about how navigators used them.
B: This describes the materials and how the charts were used but not what they represented.
D: This explains what the ribs and shells represented but not how navigators used the charts.
Watch for: Check a detailed choice against each part of the goal; thoroughness on one part does not make up for missing the other.
Reading and Writing · Module 1 · Question 27 of 27
While researching a topic, a student has taken the following notes:
In 1956 physicists Tsung-Dao Lee and Chen Ning Yang suggested that a basic symmetry of nature called parity might not hold in certain radioactive decays.
Parity symmetry implies that a process and its mirror image should occur equally often.
Chien-Shiung Wu tested the suggestion by cooling radioactive cobalt-60 to near absolute zero and lining up its nuclei with a magnetic field.
She found that electrons from the decaying nuclei were emitted more often in one direction than in the opposite direction.
The result showed that parity symmetry is violated in such decays.
Lee and Yang received the 1957 Nobel Prize in Physics for their suggestion.
The student wants to show readers who have never heard of parity how Wu’s result revealed that the symmetry does not always hold. Which choice most effectively uses relevant information from the notes to accomplish this goal?
Correct choice: Parity symmetry implies that a process and its mirror image occur equally often, yet Wu found that electrons from lined-up cobalt-60 nuclei were emitted more often in one direction than in the opposite one.
An audience unfamiliar with parity needs to know what the symmetry requires—mirror-image processes occurring equally often—and then what Wu observed that breaks it: more electrons in one direction than the opposite. Only this choice supplies both, so the reader can see why the result violates the symmetry.
A: This names parity and says the result violated it, but it never explains what parity means or what Wu observed, so an unfamiliar reader cannot see how the result showed a violation.
C: This focuses on Lee and Yang’s suggestion and prize and does not explain either parity or Wu’s observation.
D: This reports Wu’s observation and its conclusion but never explains what parity symmetry requires, so an unfamiliar reader cannot see why unequal emission violates it.
Watch for: “For an audience unfamiliar with X” means the choice must define X, not just name it.
Reading and Writing · Module 2 · Question 1 of 27
The following text is from John Galsworthy’s 1906 novel The Man of Property. Soames Forsyte is suing an architect over the cost of a house. Chankery, a young lawyer on the architect’s side, discusses the case at a dinner; later, Soames’s own lawyer, Boulter, gives Soames his view.
He had, said Chankery, a case coming on with a “very Underlined: nice End underlined. point.” He then explained, preserving every professional discretion, the riddle in Soames’s case. Everyone, he said, to whom he had spoken, thought it a nice point. [...] Boulter, who had the matter in hand, Jobling having died some years before, told him that in his opinion it was rather a nice point; he would like counsel’s opinion on it.
As used in the text, what does the word “nice” most nearly mean?
Correct choice: delicate
Chankery calls the point a “riddle,” lawyers on both sides call it “nice,” and Boulter wants a specialist’s opinion before judging it. A “nice point” here is a delicate one, requiring fine judgment.
A: “Pleasant” is the modern sense of “nice,” but a legal “riddle” on which lawyers want expert opinion is not described as agreeable.
B: Chankery might seem to mean that the point favors his client, but Soames’s own lawyer also calls it a nice point; it cannot favor both sides, so the word describes the question’s difficulty.
D: A point that everyone discusses and that calls for counsel’s opinion is not treated as minor; the lawyers treat it as hard to decide.
Watch for: Older texts often use familiar words in older senses; “nice” once meant precise or finely balanced.
Reading and Writing · Module 2 · Question 2 of 27
When Tim Berners-Lee circulated his 1989 proposal for what would become the World Wide Web, his supervisor wrote “Vague but exciting” on the cover. The proposal presented the web’s central idea—documents linked to one another that anyone on a computer network could reach—in ______ form: it stated the goal clearly, but the software, the rules for transmitting documents, and the format in which to write them were all worked out over the following two years.
Which choice completes the text with the most logical and precise word or phrase?
Correct choice: inchoate
The colon explains the blank: the goal was clear, but the means of achieving it had not yet been developed. “Inchoate” means just begun and not yet fully formed, which fits an idea stated clearly but still undeveloped.
A: “Incoherent” means confused or lacking logical connection, but the proposal “stated the goal clearly”; the supervisor’s “vague” refers to what was missing, not to muddled thinking.
C: “Impractical” would mean the idea could not be carried out, but the text says the missing pieces were simply worked out over the following two years; the proposal was undeveloped, not unworkable.
D: “Definitive” means complete and final, the opposite of a proposal whose essential parts were still to be worked out.
Watch for: Similar-sounding words can differ sharply: “inchoate” (undeveloped) is not “incoherent” (confused).
Reading and Writing · Module 2 · Question 3 of 27
When fragments of a skull and a jaw were unearthed at Piltdown, England, in 1912, many British scientists accepted them as the remains of an early human ancestor. The find paired a large, humanlike braincase with an apelike jaw—the very combination these scientists expected, since they believed that an enlarged brain had led the way in human evolution. As fossils found in Africa and Asia over the following decades revealed ancestors with small brains but upright posture and humanlike teeth, Piltdown fit the record less and less. In 1953 chemical tests exposed it as a forgery that combined a relatively recent human skull with the jaw of an ape.
Which choice best states the main purpose of the text?
Correct choice: To suggest that the Piltdown forgery was accepted partly because it matched what many scientists expected to find
The text stresses that the find was “the very combination these scientists expected,” then shows it fitting the record “less and less” until it was exposed. The account is built to suggest that expectation, not only evidence, explains why the forgery was accepted.
A: The text says only that Piltdown fit the later fossil record less and less; it never says that the African and Asian fossils were dismissed, so this relies on outside knowledge.
C: The tests are mentioned only in the final sentence, and the text does not describe how they worked; they are the end of the story, not its point.
D: The text reports what the forgery combined but says nothing about how the forger obtained the skull or the jaw.
Watch for: A main purpose can be implied by what a text emphasizes; here the phrase “the very combination these scientists expected” carries it.
Reading and Writing · Module 2 · Question 4 of 27
The following text is Siegfried Sassoon’s poem “The General,” from his 1918 collection Counter-Attack and Other Poems. Sassoon served as a British officer in the First World War; Arras is a town in northern France where a major battle was fought in 1917.
“Good-morning; good-morning!” the General said When we met him last week on our way to the line. Now the soldiers he smiled at are most of ’em dead, And we’re cursing his staff for incompetent swine. “He’s a cheery old card,” grunted Harry to Jack As they slogged up to Arras with rifle and pack.
Underlined: But he did for them both by his plan of attack. End underlined.
Which choice best describes the function of the underlined portion in the text as a whole?
Correct choice: It reveals that the genial general caused the deaths of the men who liked him, turning his cheerful greeting into bitter irony.
“Did for them” means killed or ruined them: the general’s “plan of attack” killed Harry and Jack, after Harry called him “a cheery old card.” Set apart after the stanza, the line turns the friendly greeting of the first line into bitter irony.
A: This misreads “did for them” as “did something for them.” The poem says the soldiers the general smiled at “are most of ’em dead,” and the attack is his, so the line reports their deaths, not their rescue.
B: The line says nothing about Harry and Jack’s changing opinion; they last appear with Harry calling the general “a cheery old card,” and the line reports what his plan did to them.
C: The speaker curses the general’s staff earlier, but the last line pins the deaths on “his plan of attack,” the general’s own, rather than shifting blame to the staff.
Watch for: A final line set apart from the rest of a poem often reverses its tone; read idioms (“did for”) in light of the facts already given.
Reading and Writing · Module 2 · Question 5 of 27
Sickle-cell disease, which develops in people who inherit two copies of a particular gene variant, was usually fatal in childhood before modern treatment. Natural selection should make a variant that kills those who inherit two copies before they can reproduce increasingly rare. Underlined: Yet in parts of Africa, more than one person in five carries a single copy of the sickle-cell variant. End underlined. In the 1950s physician Anthony Allison found that people with one copy were less likely than others to suffer severe malaria and that the variant was most common in regions where malaria had long been widespread.
Which choice best describes the function of the underlined sentence in the text as a whole?
Correct choice: It presents an apparent exception to natural selection that the rest of the text shows selection itself explains.
The first two sentences lead us to expect the variant to be rare; the underlined sentence reports that it is common—an apparent exception to natural selection. Allison’s findings then show that carrying one copy protects against severe malaria where malaria is widespread, so selection itself keeps the variant common.
A: Read alone, the underlined sentence seems to challenge natural selection, but Allison’s findings show that selection favors people with one copy where malaria is common, so the sentence does not support this view.
C: The sentence concerns how common the variant is, not how deadly the disease was; the text says the disease was usually fatal in childhood.
D: The text never says why Allison undertook his research; the underlined sentence poses the puzzle his findings resolve.
Watch for: A sentence beginning “Yet” may present a puzzle rather than the author’s position; read on to see how the text resolves it.
Reading and Writing · Module 2 · Question 6 of 27
Text 1
In his 1933 book The Triumph of Mediocrity in Business, the statistician Horace Secrist reported that stores with the highest profits in one year tended, on average, to earn lower profits in later years, while stores with the lowest profits tended to improve. Secrist concluded that competition steadily pushes businesses toward average performance, so that firms grow more alike over time.
Text 2
Reviewing Secrist’s book, the statistician Harold Hotelling observed that the same pattern appears when firms are grouped by their profits in a later year and traced backward: firms with the highest later profits had, on average, earned less in the earlier year, and firms with the lowest later profits had earned more.
Based on the texts, how would Hotelling (Text 2) most likely respond to Secrist’s conclusion in Text 1?
Correct choice: He would argue that the pattern Secrist found cannot show that firms were growing more alike, since the same reasoning applied backward would show them growing apart.
If Secrist’s pattern proved convergence, the identical pattern found when firms are traced backward would prove divergence over the same years—an absurd result. The pattern is simply what happens when firms are grouped by an extreme year, so it cannot show that firms grew more alike.
A: This takes the backward pattern as real evidence of divergence. Hotelling’s point is that the pattern appears in both directions, so neither direction shows a real change in how alike firms are.
B: Hotelling’s observation gives no reason to accept that firms were converging; it shows that the pattern Secrist relied on would appear either way.
D: Neither text suggests that high profits usually rise further; Text 1 reports that stores with the highest profits in one year tended, on average, to earn lower profits in later years, the reverse of this claim.
Watch for: If a pattern would appear whether you look forward or backward in time, it cannot by itself show change in one direction.
Reading and Writing · Module 2 · Question 7 of 27
Text 1
In the 1890s the physician Christiaan Eijkman noticed that chickens at his laboratory in Java developed a paralysis resembling the human disease beriberi when they were fed polished white rice, and that they recovered when fed rice with its outer layers intact. Eijkman concluded that polished rice contains a substance that damages nerves and that the outer layers of the grain contain something that counteracts it.
Text 2
Continuing Eijkman’s work, Gerrit Grijns fed chickens diets that contained no rice at all, such as meat that had been heated for a long time. These chickens also developed the paralysis, and adding the outer layers of rice grains—or certain beans—to their diets prevented it.
Based on the texts, how would Grijns (Text 2) most likely respond to Eijkman’s conclusion in Text 1?
Correct choice: He would argue that the paralysis results from something missing from certain diets rather than from a harmful substance in polished rice.
Grijns’s chickens developed the paralysis on diets containing no rice, so a harmful substance in polished rice cannot be the cause. The disease was prevented by adding rice’s outer layers or beans—different foods that evidently supply something the deficient diets lacked. His results point to a missing nutrient, not a toxin and its antidote.
B: This saves Eijkman’s toxin idea by positing a second toxin, but Text 2 gives no reason to think heated meat contains one; the simpler reading of diets without rice causing the disease, and different foods preventing it, is that something is missing.
C: The outer layers helped in both studies, but Grijns’s chickens got the disease without eating rice, which undercuts the part of Eijkman’s conclusion that blames a substance in polished rice.
D: Neither text mentions the amount of food; Eijkman’s chickens recovered when the kind of rice changed.
Watch for: When the same illness appears on very different diets and is prevented by very different foods, look for a missing ingredient rather than a poison.
Reading and Writing · Module 2 · Question 8 of 27
Text 1
During the 2009 flu season, researchers Nicholas Christakis and James Fowler monitored Harvard College students in two groups: students chosen at random and friends whom the randomly chosen students named. On average, the named friends came down with flu about two weeks earlier than the randomly chosen students did.
Text 2
In 1991 the sociologist Scott Feld showed that in networks of friendships, people’s friends have, on average, more friends than they themselves do. The reason is that a person with many friends appears on many people’s lists of friends, while a person with few friends appears on few lists.
Based on the texts, how would Feld (Text 2) most likely respond to the result reported in Text 1?
Correct choice: He would regard it as expected, since students named as friends tend to have more friends, and so more chances of exposure, than students chosen at random.
Feld’s point implies that people named as friends are, on average, better connected than people chosen at random. Better-connected people meet more people and so are exposed to a spreading illness sooner, which would explain why the named friends caught flu earlier.
B: Text 1 does not say that the randomly chosen students were sick when they named their friends, and they caught flu later than the friends on average.
C: The result concerns how early people caught flu, not whether they caught it at all, and Feld’s point is about who gets named as a friend.
D: This reverses Feld’s point: people named as friends tend to have more friends than the people who named them.
Watch for: Asking people to name friends is itself a way of sampling; ask how that sampling changes who ends up in the group.
Reading and Writing · Module 2 · Question 9 of 27
The following text is from Samuel Butler’s 1903 novel The Way of All Flesh. Years earlier, Christina Allaby and her sisters had played a game of cards to decide which of them would try to marry Theobald Pontifex, a young clergyman; Christina won and later married him.
When the two Miss Allabys came, as they sometimes did, to stay with Christina, they said the life led by their sister and brother-in-law was an idyll. Happy indeed was Christina in her choice, for that she had had a choice was a fiction which soon took root among them—and happy Theobald in his Christina. Somehow or other Christina was always a little shy of cards when her sisters were staying with her, though at other times she enjoyed a game of cribbage or a rubber of whist heartily enough, but her sisters knew they would never be asked to Battersby again if they were to refer to that little matter, and on the whole it was worth their while to be asked to Battersby.
Which choice best states the main idea of the text?
Correct choice: The narrator suggests that the family’s account of a happy, freely chosen marriage is a convenient fiction that the sisters keep up partly for their own benefit.
The sisters call the marriage “an idyll,” and the idea that Christina made a free “choice” is “a fiction which soon took root among them.” Christina avoids cards around her sisters, and they never mention “that little matter” because it was “worth their while to be asked to Battersby.” The account of a happy, chosen marriage is a polite fiction that serves everyone, including the sisters.
A: Christina is “a little shy of cards” around her sisters—a sign of embarrassment about how the match was made—but nothing indicates that she regrets the marriage.
B: This takes the visits at face value and misses the irony: the sisters stay silent about the card game to keep their invitations, and Christina avoids cards when they visit rather than sharing the game with them.
D: The sisters keep quiet about the card game because they want to keep being invited; nothing suggests envy or a wish to expose the story.
Watch for: When a narrator calls a shared story “a fiction,” ask who benefits from keeping it up.
Reading and Writing · Module 2 · Question 10 of 27
In a 2009 experiment, Christopher Bird and Nathan Emery gave captive rooks a tall, narrow tube of water with a worm floating on the surface, too low for the birds to reach. The rooks dropped stones from a nearby pile into the tube until the water rose high enough for them to seize the worm. The researchers hypothesized that the birds understood that adding stones raises the water level, rather than simply dropping objects into the tube until food happened to come within reach.
Which finding, if true, would most directly support the researchers’ hypothesis?
Correct choice: When offered both large and small stones, the rooks chose the large ones, which raise the water level faster.
Dropping objects at random until the worm happened to rise would not favor any particular stone. Choosing the stones that raise the water fastest fits only an understanding that stones displace water, so this finding separates the researchers’ suggestion from the alternative.
A: Starting sooner shows that the birds learned the routine, but a bird that had merely learned to drop objects until food appeared would also start sooner.
B: A bird dropping objects until food came within reach would also stop once it could reach the worm, so stopping does not distinguish the two explanations.
C: A general habit of dropping objects would, if anything, support the alternative the researchers rejected rather than their suggestion.
Watch for: To support one explanation over another, choose a finding that the rival explanation cannot account for.
Reading and Writing · Module 2 · Question 11 of 27
During a severe drought in 1977 on the Galápagos island of Daphne Major, most of the island’s medium ground finches died, and the small, soft seeds they usually eat became scarce, leaving mainly large, hard seeds. Researchers who had been capturing, measuring, and banding the island’s finches for years found that the average beak depth of the population was greater after the drought than before it. They concluded that the increase occurred because birds with deeper beaks, which can crack large, hard seeds, were more likely to survive the drought.
Which finding, if true, would most directly support the researchers’ conclusion?
Correct choice: Banded finches that survived the drought had had deeper beaks, when measured before the drought, than banded finches that died.
The conclusion is that deeper-beaked birds were more likely to survive. The direct test compares, before the drought, the beaks of birds that went on to survive with those of birds that died; survivors having deeper beaks beforehand shows that beak depth predicted survival.
A: Deeper beaks in the next generation suggest the change was inherited, but they do not show why the population changed; the offspring data are consistent with the conclusion without testing it.
B: If individual birds’ beaks grew deeper, the population average could rise without any difference in survival, which would undercut the conclusion rather than support it.
D: The decline of a different, insect-eating species shows that the drought was severe but says nothing about whether beak depth affected the ground finches’ survival.
Watch for: A claim about which individuals survived needs data on individuals measured before the event, not just averages before and after.
Reading and Writing · Module 2 · Question 12 of 27
In a 1959 experiment, psychologists Leon Festinger and James Carlsmith had participants spend an hour on dull, repetitive tasks. Afterward, some participants were paid either $1 or $20 to tell the next participant, who was actually working with the researchers, that the tasks had been enjoyable; participants in a control group were not asked to say anything. Later, all participants rated the tasks on scales from -5 to 5. A student claims that the size of the payment, and not merely the act of describing the dull tasks as enjoyable, affected how participants came to view the tasks.
Participants’ Average Ratings in Festinger and Carlsmith’s Experiment
Which choice most effectively uses data from the graph to support the student’s claim?
Correct choice: Participants paid $1 rated how enjoyable the tasks were at 1.35, compared with -0.05 for participants paid $20.
To show an effect of payment size apart from the act of describing the tasks, compare the two groups that both described the tasks but were paid different amounts. Those paid $1 rated the tasks’ enjoyableness at 1.35, well above the $20 group’s -0.05.
A: The figures are accurate, but the control group neither described the tasks nor was paid, so this comparison cannot separate the effect of payment size from the effect of describing the tasks.
C: These figures are accurate, but they compare the $20 group with the control group, which neither described the tasks nor was paid, so they cannot isolate the effect of payment size.
D: This has the right shape, but the differences are 1.40 points (1.35 versus -0.05) and 1.45 points (1.20 versus -0.25), both less than 1.5.
Watch for: To isolate one factor, compare groups that differ only in that factor—here, two groups that both described the tasks.
Reading and Writing · Module 2 · Question 13 of 27
The mold Neurospora crassa makes the nutrient arginine through a chain of chemical steps that also produces two related substances, ornithine and citrulline. In 1944 geneticists Adrian Srb and Norman Horowitz studied mutant strains that could not grow without added arginine, each blocked at a single step in the chain. Arginine is made at the last step, and a strain grows when given a substance if, and only if, that substance is made at the strain’s blocked step or a later one. The researchers concluded that the mold makes ornithine before citrulline, citing the finding that ______
Growth of Mutant Neurospora Strains Given Different Substances
Strain group
Given ornithine
Given citrulline
Given arginine
Group 1
grew
grew
grew
Group 2
did not grow
grew
grew
Group 3
did not grow
did not grow
grew
Which choice most effectively uses data from the table to complete the text?
Correct choice: strains in group 2 grew when given citrulline but did not grow when given ornithine.
Group 2 strains grew on citrulline but not on ornithine. By the stated rule, citrulline must be made at or after their blocked step and ornithine before it, so ornithine comes before citrulline in the chain.
A: This is accurate, but a strain blocked before both substances would grow on both whichever of them came first, so it cannot establish their order.
B: This would support the conclusion, but it misstates the table: group 3 strains grew only when given arginine, not when given citrulline.
C: This is accurate and shows that arginine is made at or after the step blocked in group 3, but it says nothing about whether ornithine comes before citrulline.
Watch for: To establish the order of two steps, find the group that responds to one substance but not the other.
Reading and Writing · Module 2 · Question 14 of 27
In 1943 Salvador Luria and Max Delbrück asked whether bacteria that survive attack by a virus become resistant in response to the attack or already carry resistance-giving mutations that arose at random beforehand. They grew many separate cultures, each started from a few cells and grown to the same size under identical conditions, exposed each to the same dose of the virus, and compared the numbers of resistant bacteria in these cultures with the numbers in several samples drawn from one large culture. The samples varied little, but most separate cultures contained few or no resistant bacteria and a few contained hundreds. This result suggests that ______
Which choice most logically completes the text?
Correct choice: resistance arose before exposure, with cultures in which a mutation happened to occur early coming to contain many resistant descendants.
If resistance appears only upon exposure, every culture faces the same chance at the same moment, so the counts should be similar, like samples from one culture. The huge differences instead fit mutations that arose at random while the cultures were growing: a mutation that happened early would leave many resistant descendants, and one that happened late or not at all would leave few.
B: Random mutations that arose only after exposure would give every culture the same chance at the same moment, producing modest variation like that among the samples from one culture; only mutations arising during growth explain the few very large counts.
C: If resistance produced at exposure then multiplied, it would multiply in the samples from the single large culture as well, yet those samples varied little; only differences in the cultures’ histories before exposure explain the spread.
D: The samples from one culture serve as the comparison showing how little counts vary by chance; nothing suggests they underestimate anything.
Watch for: Ask when each proposed cause would act: an event during growth can snowball, while an event at exposure cannot.
Reading and Writing · Module 2 · Question 15 of 27
In the 1780s Luigi Galvani found that the leg muscles of a dead frog twitched when a nerve and a muscle were connected by an arc made of two different metals, such as copper and iron. Galvani concluded that animal tissue produces its own electricity, which the metal arc merely released. Alessandro Volta noticed that the twitching was strongest when the two metals were most unlike and weak when they were alike. In 1800 he built a stack of alternating zinc and copper disks separated by brine-soaked cloth that produced a steady electric current with no animal tissue at all. Volta’s findings suggest that ______
Which choice most logically completes the text?
Correct choice: the contact between two different metals may itself have produced the current that made the muscles twitch.
Volta showed that two unlike metals separated by a moist conductor generate current with no tissue present, and that the twitching grew with the difference between the metals. So the metals in Galvani’s arc may have supplied the current themselves—an alternative to Galvani’s explanation, though not proof that tissue produces no electricity.
A: Showing that metals can produce current does not show that tissue produces none; Volta’s findings offer an alternative source, not a disproof of animal electricity.
B: Nothing limits twitching to zinc and copper; Galvani used copper and iron, and the text says the effect depended on how unlike the metals were.
C: Galvani’s experiments used a metal arc touching nerve and muscle, with no brine-soaked cloth; the cloth appears only in Volta’s stack.
Watch for: Evidence for a rival cause undermines the first explanation’s necessity, not necessarily its truth; beware choices that claim a disproof.
Reading and Writing · Module 2 · Question 16 of 27
Seedlings of canary grass bend toward light coming from one side, and the bending occurs in a region well below the tip. In experiments reported in 1880, Charles Darwin and his son Francis covered the tips of some seedlings with small opaque caps; these seedlings did not bend. Seedlings whose tips were covered instead with transparent glass tubes bent normally, as did seedlings whose tips were left exposed while the lower region, where the bending occurs, was shielded from light. Together, these results suggest that ______
Which choice most logically completes the text?
Correct choice: the tip detects the direction of the light and causes the lower region, even when that region is shaded, to bend.
Blocking light from the tip stopped bending, but a transparent covering did not, so the tip must receive light. Shading the lower region did not stop bending, so that region need not receive light. The tip senses the light, and some influence from it makes the shaded region below bend.
A: Seedlings whose lower region was shielded from light still bent, so light need not reach that region.
B: Transparent tubes covered the tips without stopping bending, and shields on the lower region did not stop it either, so covering as such is not the problem.
C: If the lower region detected the light’s direction, shielding it would have prevented bending, but seedlings with shielded lower regions bent normally.
Watch for: With several conditions, test each choice against every result; one contradiction eliminates it.
Reading and Writing · Module 2 · Question 17 of 27
Running water, wind, and the muscles of people and ______ were the chief sources of mechanical power in Europe until steam engines spread through mines and factories in the eighteenth century.
Which choice completes the text so that it conforms to the conventions of Standard English?
Correct choice: animals—these
The sentence opens with a list and then resumes with “these,” which sums up the list as the subject of “were.” A dash marks this break between the introductory list and the clause that sums it up.
A: With no punctuation, the list runs straight into “these were …,” so the sentence seems to have two subjects side by side.
C: Used this way, a semicolon needs an independent clause on each side, but the list before it is only a series of nouns with no verb.
D: “And these were …” would leave the opening list without a verb of its own, so the sentence would begin with a fragment joined to a clause.
Watch for: When a sentence opens with a list and then sums it up with “these” or “all,” a dash marks the turn.
Reading and Writing · Module 2 · Question 18 of 27
In 1947 Thor Heyerdahl and five crewmates sailed the raft Kon-Tiki for 101 days across the Pacific Ocean before running aground on a reef in the Tuamotu ______ had built the raft in Peru from nine balsa logs lashed together with hemp rope.)
Which choice completes the text so that it conforms to the conventions of Standard English?
Correct choice: Islands. (They
The parenthetical ends with its own period inside the closing parenthesis, so it is a separate sentence. The main sentence must therefore end with a period before the opening parenthesis, and the parenthetical sentence begins with a capital letter: “Islands. (They had built …)”.
B: Without a period after “Islands,” the parenthetical becomes part of the main sentence, which then ends with a period inside the parentheses; a parenthetical within a sentence does not take its own period.
C: A comma would join the parenthetical to the main sentence, but the parenthetical is a complete sentence ending with its own period.
D: A semicolon before a parenthesis that closes with its own period leaves the main sentence without an end mark and treats the parenthetical as part of it.
Watch for: A complete sentence in parentheses that stands on its own follows the previous sentence’s period and carries its own period inside the closing parenthesis.
Reading and Writing · Module 2 · Question 19 of 27
Twenty-six miles and 385 yards ______ the official length of the marathon since 1921, when international officials adopted the distance that had been run at the 1908 Olympic Games in London.
Which choice completes the text so that it conforms to the conventions of Standard English?
Correct choice: has been
“Twenty-six miles and 385 yards” names a single distance, the length of the race, so it takes a singular verb. “Since 1921” calls for the present perfect: “has been.”
A: “Have been” treats the measurement as plural items, but it names one distance; an amount considered as a single quantity takes a singular verb.
B: “Were” is plural and past; the distance is still the official length, so the present perfect singular “has been” is needed.
C: “Are” is plural and present tense; the measurement is a single quantity, and “since 1921” calls for the present perfect.
Watch for: A plural-looking amount that names one quantity (a distance, a sum, a period) takes a singular verb.
Reading and Writing · Module 2 · Question 20 of 27
______ most, of Shakespeare’s plays borrow their plots from earlier works, such as Raphael Holinshed’s Chronicles and Thomas North’s translation of Plutarch’s Lives, which Shakespeare reshaped freely.
Which choice completes the text so that it conforms to the conventions of Standard English?
Correct choice: Many, if not
“If not most” is an interrupter set off from the rest of the sentence. Since it closes with a comma after “most,” it must open with a comma after “Many”: “Many, if not most, of Shakespeare’s plays.”
B: Without a comma after “Many,” the interrupter is closed but never opened, and the sentence reads as if “Many if not most” were a single unit followed by a stray comma.
C: An interrupter opened with a dash must close with a dash, but this one closes with a comma after “most.”
D: A semicolon separates independent clauses; “Many” is not a clause, and the semicolon would split the subject from the rest of the sentence.
Watch for: Paired punctuation must match: an interrupter closed with a comma must also open with one.
Reading and Writing · Module 2 · Question 21 of 27
The Lindisfarne Gospels, a richly decorated manuscript of the four Gospels, is thought ______ written and illuminated around 715 by a single artist, Eadfrith, who was then bishop of the island monastery of Lindisfarne.
Which choice completes the text so that it conforms to the conventions of Standard English?
Correct choice: to have been
The thinking happens now, but the writing happened around 715, earlier. An action earlier than the main verb (“is thought”) calls for the perfect infinitive, “to have been.”
A: “To be written” would place the writing at the same time as the present “is thought,” but the manuscript was made around 715.
B: “Is thought having been” is ungrammatical; “is thought” must be followed by an infinitive.
C: “Is thought being written” is ungrammatical and would describe the writing as still in progress.
Watch for: After “is thought,” “is said,” or “is believed,” use “to have been” for something that happened earlier.
Reading and Writing · Module 2 · Question 22 of 27
In Mary Shelley’s 1818 novel Frankenstein, the young scientist Victor Frankenstein assembles a creature from parts of corpses and brings it to life, but horrified by what he has made, he ______ it on the very night it first opens its eyes.
Which choice completes the text so that it conforms to the conventions of Standard English?
Correct choice: abandons
The sentence summarizes the novel’s events in the present tense (“assembles,” “brings,” “has made,” “opens”). The verb in the blank must continue that present tense: “abandons.”
A: “Abandoned” shifts to the past tense in the middle of a summary told in the present (“assembles,” “brings,” “opens”).
B: “Had abandoned” would place the abandonment before the other past events, but the summary is in the present and the abandonment follows the creature’s awakening.
D: “Will abandon” would place the abandonment in the future relative to the present-tense summary, although it happens on the same night the creature opens its eyes.
Watch for: Match the tense the sentence has already established; summaries of a novel’s events are usually told in the present.
Reading and Writing · Module 2 · Question 23 of 27
Critics of the Crystal Palace, the iron-and-glass hall built in London for the Great Exhibition of 1851, complained that it could hardly be called architecture. ______ the building had been assembled in a few months from thousands of identical factory-made parts, and its designer, Joseph Paxton, was a gardener who had built greenhouses rather than a trained architect. But its methods would shape the architecture of railway stations and exhibition halls for decades.
Which choice completes the text with the most logical transition?
Correct choice: Granted,
The blank introduces facts that lend some support to the critics—prefabricated parts, a designer without architectural training—before “But” turns to the building’s influence. “Granted” concedes a point to the other side before the writer’s own point.
A: “Nonetheless” signals that what follows holds despite what came before, but the facts about prefabrication and Paxton’s background support the critics rather than countering them.
B: “Therefore” would present the building’s construction and designer as consequences of the critics’ complaint, reversing the actual relationship: those facts help explain the complaint.
D: “Instead” signals a replacement for something rejected, but the sentence does not replace the critics’ view; it concedes part of it.
Watch for: A sentence that supports the opposing view just before a “But” is usually introduced by a concession word such as “Granted.”
Reading and Writing · Module 2 · Question 24 of 27
Earth’s magnetic field extends far into space, where it deflects most of the charged particles streaming outward from the Sun in the solar wind. ______ those particles would strike the upper atmosphere directly and, over hundreds of millions of years, could strip away much of it—as appears to have happened on Mars, which lost its global magnetic field billions of years ago.
Which choice completes the text with the most logical transition?
Correct choice: Otherwise,
The final sentence describes what would happen if the magnetic field did not deflect the solar wind. “Otherwise” introduces the consequence of the opposite condition.
B: “Similarly” introduces a parallel case, but the final sentence describes what the absence of the magnetic field’s protection would cause, not a comparable example.
C: “In fact” strengthens or intensifies the preceding point, but the stripping of Earth’s atmosphere is not happening; the sentence describes a hypothetical outcome, not a further fact.
D: “Nevertheless” signals something true despite what came before, but the atmosphere is not being stripped despite the protection; it would be stripped only without it.
Watch for: A sentence in the conditional (“would”) that describes the opposite outcome often calls for “Otherwise.”
Reading and Writing · Module 2 · Question 25 of 27
While researching a topic, a student has taken the following notes:
Jacob Lawrence completed The Migration Series, sixty small paintings, in 1941.
The panels depict the movement of African Americans from the rural South to northern cities in the early twentieth century.
Each panel is accompanied by a caption that Lawrence wrote.
In 1942 the Museum of Modern Art in New York and the Phillips Collection in Washington, DC, agreed to buy the series together.
The Museum of Modern Art took the even-numbered panels, and the Phillips Collection took the odd-numbered panels.
The two museums have occasionally reunited the panels for joint exhibitions.
The student wants to explain how the series came to be divided and how the division was made. Which choice most effectively uses relevant information from the notes to accomplish this goal?
Correct choice: In 1942 the Museum of Modern Art and the Phillips Collection jointly bought Lawrence’s The Migration Series, with the Museum of Modern Art taking the even-numbered panels and the Phillips the odd-numbered.
The goal has two parts: how the series came to be divided (the two museums bought it together in 1942) and how the division was made (even-numbered panels to one museum, odd-numbered to the other). Only this choice gives both.
A: This says that two museums own the series but explains neither how that came about nor how the panels were split.
C: This describes the series’ subject and format but does not mention its division at all.
D: This explains how the panels are split but not how the series came to be divided—the joint purchase in 1942.
Watch for: “How it came to be” asks for the cause or event; “how it was made” asks for the method—check that the choice gives both.
Reading and Writing · Module 2 · Question 26 of 27
While researching a topic, a student has taken the following notes:
The Challenger Deep, in the Mariana Trench in the western Pacific Ocean, is the deepest known point in Earth’s oceans.
In 1960 Jacques Piccard and Don Walsh reached the bottom of the Challenger Deep in the bathyscaphe Trieste.
Piccard and Walsh spent about twenty minutes on the bottom and took no photographs, because silt stirred up by their landing clouded the water.
No one else reached the bottom of the Challenger Deep until 2012.
In 2012 filmmaker James Cameron made the descent alone in a vessel called Deepsea Challenger.
Cameron spent about three hours on the bottom, collecting samples and recording video.
The student wants to contrast what the two expeditions were able to record while on the bottom of the Challenger Deep. Which choice most effectively uses relevant information from the notes to accomplish this goal?
Correct choice: Whereas Piccard and Walsh could take no photographs on the bottom because silt stirred up by their landing clouded the water, Cameron recorded video while on the bottom in 2012.
The goal asks for a contrast in what each expedition could record. This choice sets Piccard and Walsh’s lack of photographs against Cameron’s video.
A: This contrasts the two descents, but on time spent at the bottom rather than on what the expeditions recorded.
B: This describes only Cameron’s expedition, so it draws no contrast.
D: This describes only the first expedition, so it draws no contrast.
Watch for: When a goal names the aspect to contrast, a contrast on a different aspect does not meet it.
Reading and Writing · Module 2 · Question 27 of 27
While researching a topic, a student has taken the following notes:
In the 1970s astronomer Vera Rubin measured how fast stars and gas orbit the centers of spiral galaxies.
In the solar system, planets farther from the Sun orbit more slowly than planets closer to it, because most of the system’s mass is concentrated in the Sun.
Because most of a spiral galaxy’s visible mass is concentrated near its center, astronomers expected stars far from the center to orbit more slowly than stars closer in.
Rubin found instead that stars far from the center orbit about as fast as stars closer in.
Her results suggested that galaxies contain large amounts of unseen mass, now called dark matter, extending far beyond their visible edges.
The student wants to explain why Rubin’s finding was surprising. Which choice most effectively uses relevant information from the notes to accomplish this goal?
Correct choice: Because most of a spiral galaxy’s visible mass lies near its center, astronomers expected distant stars to orbit more slowly than stars closer in, but Rubin found that they orbit about as fast.
A finding is surprising only against an expectation. This choice gives the expectation and its reason (visible mass concentrated near the center, so distant stars should orbit more slowly) and then what Rubin found (they orbit about as fast).
B: This reports what Rubin’s results suggested but not what astronomers expected, so it does not show why the finding was surprising.
C: This states what astronomers expected and why, but it never says what Rubin found, so it cannot show that her finding was surprising.
D: This states the finding and its implication but omits the expectation it overturned, so the surprise is left unexplained.
Watch for: To show that something was surprising, a choice must state both the expectation and the result that contradicted it.
Math · Module 1 · Question 1 of 22
The table shows the predicted length
of a coiled ribbon after
identical turns are uncoiled. The relationship is linear. What is the
predicted coiled length after 22 turns are uncoiled?
Uncoiled turns and remaining coiled length
Turns uncoiled n
Coiled length L (cm)
4
103
10
85
Answer: A.
From n = 4 to n = 10, the coiled length decreases by 18 centimeters,
so the slope is -3 centimeters per turn. The rule is
.
At n = 22,
.
Check: Starting at the second table entry, 12 more turns reduce 85 by
,
leaving 49.
B: This is the total decrease over 22 turns: . Subtract it from the initial length to find the length still coiled: .
C: This is the length after 10 turns.
D: This is the initial coiled length.
Watch for: The input counts additional turns; the output is the amount still
coiled, so the slope is negative.
Math · Module 1 · Question 2 of 22
The figure shows a trapezoid whose parallel sides measure and and whose height is . What is the area of the trapezoid?
The area of a trapezoid is . Forgetting the factor of gives . Using only the shorter parallel side gives . Using only the longer one gives .
Math · Module 1 · Question 3 of 22
The graph shows the mass of grain, in kilograms, remaining in a hopper minutes after discharge begins. According to the graph, after how many minutes does the hopper contain kilograms of grain?
Answer: C.
The graph is a line through and , so the hopper starts with kilograms and loses kilograms each minute, giving . Setting gives , so minutes.
Check: In minutes the hopper discharges kilograms, and kilograms are left.
A: is the discharge rate in kilograms per minute, not a number of minutes.
B: is how long it takes to discharge kilograms. The question asks when kilograms are still in the hopper.
D: minutes is when the hopper is empty, which the graph shows as its horizontal intercept.
Watch for: Decide whether a stated amount is what remains or what has already left before writing the equation.
Math · Module 1 · Question 4 of 22
A stamp-foil service bills a fixed preparation charge plus a constant
amount per centimeter of foil. An 80-centimeter order is billed at 94
dollars before any reimbursement. A 140-centimeter order is billed at
127 dollars after a 9-dollar reimbursement. What is the preparation
charge, in dollars?
Answer: B.
The unreimbursed charge for 140 centimeters is 136 dollars. Thus 60
extra centimeters add
dollars, so the rate is 0.70 dollar per centimeter. The fixed charge is
dollars.
Check: The larger bill is
,
as given.
A: This leaves the reimbursement subtracted from the fixed charge.
C: This comes from an incorrect rate.
D: This is the variable cost of the 80-centimeter order.
Watch for: Undo a reimbursement or discount before comparing totals used to
determine a unit rate.
Math · Module 1 · Question 5 of 22
To estimate the mean daily water use of the metered households in an irrigation district, an analyst selected a random sample of households and recorded each household’s use for one day. The sample mean was gallons, with an associated margin of error of gallons at confidence. Which change to the study would most likely have produced a smaller margin of error?
Answer: D.
With the sampling method and the confidence level held fixed, a margin of error shrinks as the sample size grows, because a larger random sample pins down the population mean more precisely. Only choice D increases the sample size while keeping the selection random.
Check: A margin of error is inversely proportional to the square root of the sample size, so quadrupling a random sample from to households would cut the -gallon margin roughly in half, to about gallons, while shrinking it to households would widen the margin to about gallons.
A: Raising the confidence level requires a wider interval in order to sustain that higher level of confidence, so the margin of error grows.
B: A smaller random sample carries less information about the population, so it would widen the margin of error rather than narrow it.
C: Choosing the households with the largest meters abandons random selection and biases the sample toward heavy users. A biased sample does not estimate the district’s mean more precisely.
Watch for: Only more data from the same random process narrows a margin of error; higher confidence widens it and a targeted sample biases it.
Math · Module 1 · Question 6 of 22
Which value is closest to the positive zero of
?
Answer: D.
Graph the function and locate its positive x-intercept. The quadratic
formula also gives
.
Check: The product of the roots is negative, so exactly one root is
positive.
A: This approximates the magnitude of the negative root.
B: This is the axis-of-symmetry input.
C: This is the sum of the roots.
Watch for: Select the x-intercept, not the vertex, and keep the positive
root.
Math · Module 1 · Question 7 of 22
Positive numbers and satisfy and . What is the value of ?
Answer: A.
Let and , so the system becomes and . Halving the second equation gives , and subtracting that from the first gives , so and then . Therefore and , so .
Check: Substituting back into the originals, and , exactly as stated.
B: is , the ratio of the two numbers rather than their product.
C: is .
D: 6 is , the product of the substituted variables and , with no conversion back to and .
Watch for: After solving for a substituted variable, undo the substitution before answering, because the substituted value is usually sitting right there among the choices.
Math · Module 1 · Question 8 of 22
The dot plots show the number of tubers harvested from each plant in a trial plot of seed variety R and in a trial plot of seed variety S. Which statement about the two varieties do the dot plots support?
Answer: C.
Count the dots before comparing anything: variety R has plants and variety S has . For variety R the th and th values are both , so its median is tubers; for variety S the th and th values are both , so its median is tubers. Variety R has the greater median.
Check: Only of variety R’s dots lie below and only lie above it, so has to be the middle value; for variety S, of its dots sit at or below and sit at or above, which pins its middle value at .
A: Range runs from the lowest dot to the highest: variety R spans to , a range of , while variety S spans to , a range of . Variety S is the wider of the two, so this reverses the comparison; the lone dot at is the easiest one to miss.
B: Four variety S plants reached tubers against three variety R plants, so the raw count does favor variety S. But the plots hold different numbers of plants: out of is about , below variety R’s out of , or . This choice compares counts where the wording calls for proportions.
D: Variety S’s dots at and above number , fewer than half of its plants. Reading the tall stacks at and as though they sat at produces this choice.
Watch for: Two dot plots need not hold the same number of dots, so count them first, because every comparison that depends on position or on a share of the whole turns on that count.
Math · Module 1 · Question 9 of 22
The function is defined by . The graph of in the -plane is reflected across the -axis. Which equation defines the reflected graph?
A reflection across the -axis sends each point to , so the reflected graph is .
Substituting for :
Squaring gives , and gives , so the reflected graph is
Math · Module 1 · Question 10 of 22
The number satisfies . How many different integer values can the expression have?
Answer: B.
Adding and dividing by gives . Then
The number need not be an integer, so can be any number between and . The integers strictly between and are through , which is values.
Check: Each of these values is reached: when , and when , both between and .
A: counts the integer values of (,, and ); does not have to be an integer.
C: includes one endpoint, counting through or through , but both inequalities are strict.
D: counts through , treating both strict inequalities as inclusive.
Watch for: The question counts integer values of , not integer values of ; transform the inequality first, then count.
Math · Module 1 · Question 11 of 22
A deck sealant covers square yards per gallon and is sold only in -gallon pails. A rectangular deck measures feet by feet, and it will receive two coats of sealant. What is the least number of pails needed? ( yard feet)
Answer: A.
The deck’s area is square feet. One square yard is square feet, so the area is square yards. Two coats cover square yards, which requires gallons, or pails.
Check: The deck is yards by yards. Each coat uses gallons, so two coats use gallons, exactly pails.
B: is the number of gallons needed for one coat, not the number of pails for two coats.
C: is the number of gallons, not the number of -gallon pails. (Converting square feet to square yards by dividing by also leads to pails.)
D: pails treats square yards per gallon as square feet per gallon: each coat then needs gallons, so two coats need gallons, or pails.
Watch for: Convert area units by the square of the length factor, and answer in the unit asked for (pails, not gallons).
Math · Module 1 · Question 12 of 22
In the given system of equations, and are constants.
The solution to this system is . What is the solution to the following system?
Answer: B.
Substituting into the first system gives and , so and .
The second system is then and . Subtracting times the second equation from the first gives , so and . The solution is .
(Swapping and turns into , so swapping the right sides swaps the solution.)
Check: and .
A: negates the original solution, as if changing the constants reversed the signs.
C: scales the original solution by , which fits the first new equation but not the second.
D: is the solution to the original system; the constants on the right have changed.
Watch for: Find and first, or notice that swapping and exchanges the two left sides.
Math · Module 1 · Question 13 of 22
In the xy-plane, line passes through the point and is perpendicular to the line . What is the y-coordinate of the y-intercept of line ?
Rewriting the given line as shows its slope is , so a perpendicular line has slope . Then gives . Using the original slope gives , and using its opposite gives . Taking the reciprocal without changing the sign, as , gives .
Math · Module 1 · Question 14 of 22
The function is defined by , where is a constant greater than . The difference between the greatest zero of and the least zero of is . What is the value of ?
Enter an integer, decimal, or fraction. Up to 5 characters, or 6 including a leading minus. No units or mixed numbers.
Answer: 5
Group the terms: , so
The zeros are , and . Because , the greatest zero is and the least is , so and .
Check: With , has zeros , and , and .
Watch for: The most common wrong value is , from taking the least zero to be and missing the negative zero from .
Math · Module 1 · Question 15 of 22
A linear function satisfies for all values of . What is the value of ?
Answer: C. Let . The left side becomes . Matching coefficients gives and , so . This intercept is f(0).
Watch for: Each function evaluation contributes its own copy of the intercept.
Math · Module 1 · Question 16 of 22
In the -plane, the graph of , where is a constant, does not intersect the graph of . How many integer values of are possible?
Enter an integer, decimal, or fraction. Up to 5 characters, or 6 including a leading minus. No units or mixed numbers.
Answer: 7
The graphs intersect where , or
They do not intersect when this equation has no real solutions, which happens when its discriminant is negative:
So , which means . The integers in this interval are ,,,,,, and , so there are possible values.
Check: At , the equation is , so the line touches the parabola at ; values must be excluded. At , the line lies below the parabola, which is never lower than .
Watch for: The most common wrong value is , which includes and ; at those values the line is tangent to the parabola, so the graphs do intersect.
Math · Module 1 · Question 17 of 22
For positive constants and , the function is defined by . If , what is the value of ?
Enter an integer, decimal, or fraction. Up to 5 characters, or 6 including a leading minus. No units or mixed numbers.
Answer: 20
Multiply the two outputs:
and . So . Since and are positive, is positive, and .
The individual values of and are not needed; is the average of and , so is the positive square root of the product.
Check: With and , and , whose product is , and .
Watch for: The most common wrong value is , half of the product; because the inputs and average to , the product is the square of .
Math · Module 1 · Question 18 of 22
At a car dealership, of the vehicles in stock are trucks, and the rest are cars. Of the trucks in stock, are electric, and half of the electric vehicles in stock are trucks. If a car in stock is selected at random, what is the probability that the selected car is electric?
Answer: A.
Suppose there are vehicles in stock. Then are trucks and are cars.
There are electric trucks. Because half of the electric vehicles are trucks, there are electric vehicles, and of them are cars.
The probability that a randomly selected car is electric is .
Check: Overall, of the trucks and of the cars are electric. Weighted by trucks and cars: , or of vehicles, half of them trucks.
B: is the probability that a randomly selected vehicle of either kind is electric, .
C: assumes that cars are electric at the same rate as trucks.
D: is the probability that a randomly selected electric vehicle is a car, the reverse conditional probability.
Watch for: Trucks and cars have the same number of electric vehicles but different totals, so divide the electric cars by the number of cars.
Math · Module 1 · Question 19 of 22
In the figure, triangle has . Point lies on side so that . What is the measure, in degrees, of ?
Answer: A.
Let . Because , triangle is isosceles, so . Angle is an exterior angle of triangle , so . Because , triangle is isosceles, so . Because ,. The angles of triangle sum to :
so .
Check: The angles are ,, and . Then and , so triangle has angles ,, and , which matches .
B: This takes to equal instead of , so all three angles of triangle come out equal.
C: This is the measure of each base angle, or .
D: This is the measure of , which is supplementary to .
Watch for: An exterior angle of a triangle equals the sum of the two remote interior angles; use it to carry from one isosceles triangle to the next.
Math · Module 1 · Question 20 of 22
For constants and with , the system of equations and has infinitely many solutions. What is the value of ?
Answer: C.
Rewriting the first equation as , the system has infinitely many solutions only when the second equation is a constant multiple of the first, so . The first two ratios give , so or , and forces . The multiplier is then , so .
Check: With and the equations are and . Dividing the first by 3 and the second by 2 turns both into , so every point on that line solves the system, including and .
A: is the value of that goes with the root , which the condition rules out.
B: 3 is the value of , not of .
D: 22.5 uses , inverting the multiplier that carries the first equation to the second.
Watch for: Infinitely many solutions means one equation is a multiple of the other, so every corresponding pair of coefficients shares the same ratio, constant terms included.
Math · Module 1 · Question 21 of 22
What is the sum of all real solutions to the equation ?
Enter an integer, decimal, or fraction. Up to 5 characters, or 6 including a leading minus. No units or mixed numbers.
Answer: -2
Isolate one radical: . Squaring both sides gives
which simplifies to . Let , so and . Then or , which gives or . Both satisfy the original equation, so the sum is .
Check: At :. At :.
Watch for: The most common wrong value is , which comes from dividing by and losing the solution .
Math · Module 1 · Question 22 of 22
In the -plane, a rectangle has two of its vertices on the -axis and its other two vertices on the graph of . The rectangle lies above the -axis. What is the greatest possible perimeter of such a rectangle?
Enter an integer, decimal, or fraction. Up to 5 characters, or 6 including a leading minus. No units or mixed numbers.
Answer: 26
The top side of the rectangle is horizontal, so its two vertices on the parabola have the same -value and are and for some with . The rectangle’s width is and its height is , so its perimeter is
This quadratic in has a negative leading coefficient, so its greatest value is at . The greatest perimeter is .
Check: At the rectangle is units wide and units tall, with perimeter . Nearby rectangles have smaller perimeters: at and at the perimeter is .
Watch for: The width of the rectangle is , not ; using as the width gives , whose greatest value is . Also maximize the whole perimeter, not half of it, which gives .
Math · Module 2 · Question 1 of 22
The equation models the amount of money , in dollars, remaining on a transit card weeks after the card was loaded. Which of the following is the best interpretation of the value of in this context?
Answer: C.
No money remains when :
so . This value is the number of weeks after the card was loaded when the amount remaining reaches .
Check:, and .
A: The amount spent each week is , the rate of decrease.
B: The amount loaded onto the card is , the value of when .
D: The amount remaining after weeks is , which is negative; it is not .
Watch for: Dividing the starting amount by the weekly decrease gives the number of weeks until the amount reaches .
Math · Module 2 · Question 2 of 22
For and , which of the following is equivalent to ?
Answer: A.
Combine the fractions in the numerator:
Then divide by . Because ,
Check: At , the original expression is , and .
B: cancels with as if they were equal; they are opposites, so the result is negative.
C: is only the simplified numerator; it has not been divided by .
D: rewrites as , which is not a valid way to subtract fractions.
Watch for: The factors and are opposites, so they cancel to , not .
Math · Module 2 · Question 3 of 22
A chemist has milliliters of a solution that is salt by volume. How many milliliters of pure water must be added so that the resulting mixture is salt by volume?
Adding water changes the volume but not the salt, and the salt is milliliters. If is the final volume, then , so and the water added is . Answering with the final volume gives . Applying the drop in concentration to the volume, as , gives , and subtracting that from gives .
Math · Module 2 · Question 4 of 22
A thermostat keeps the temperature of a room, in degrees Fahrenheit, within of its set temperature , so that . When the thermostat is set to temperature , the greatest temperature it allows is degrees Fahrenheit. What is the value of ?
Answer: D.
Because is positive, the inequality is equivalent to
so . The greatest allowed temperature is :
so .
Check: With ,, so the allowed temperatures are from to degrees Fahrenheit, and the greatest is .
A: is the least temperature the thermostat allows when ,, not the set temperature.
B: treats as degrees and subtracts from .
C: subtracts of ; the tolerance is of , not of the greatest temperature.
Watch for: The is a percent of the unknown set temperature , so undo the multiplication by instead of taking of .
Math · Module 2 · Question 5 of 22
For a positive constant , the equation is true for all values of , where and are constants. What is the value of ?
Enter an integer, decimal, or fraction. Up to 5 characters, or 6 including a leading minus. No units or mixed numbers.
Answer: -36
Expand: . Matching the terms gives , so and, since ,. Then and , so .
Check: At : and .
Watch for: The most common wrong values are , from taking both and , and , from taking ; with , both and are negative.
Math · Module 2 · Question 6 of 22
Which interval is the solution set of
?
Answer: C.
Multiply all parts by 4 to obtain
.
Subtract 5:
.
Division by -2 reverses both comparisons, giving
.
Check: The lower endpoint makes the expression 2 and is excluded. The upper
endpoint makes it -3 and is included.
A: This fails to reverse and reorder correctly after division by -2.
B: The endpoint inclusions are reversed.
D: The lower endpoint has the wrong sign.
Watch for: Reverse inequality signs when dividing by a negative number, and
carry each endpoint’s inclusion correctly.
Math · Module 2 · Question 7 of 22
For how many values of the constant do the points ,, and lie on the same line in the -plane?
Answer: D.
The line through and has slope and equation .
The point lies on this line only if , which simplifies to . That is false for every .
Every point lies on the line , which is parallel to and never meets it. So there are such values of .
Check: For example, gives , and the slope from to is , not . For any , the slope from to is , which equals only if .
A: The points do lie on a line of slope , but it is a different, parallel line, , not the line through the two given points.
B: This assumes that a linear condition on always has exactly one solution; here the -terms cancel and leave a false statement.
C: assumes that the collinearity condition gives a quadratic equation in with two solutions; setting the slopes equal and cross-multiplying gives a linear equation here, and it has no solution.
Watch for: Having the same slope is not enough for points to be on the same line; the -intercepts must match too.
Math · Module 2 · Question 8 of 22
A file contains 30 reflectance readings in group A and 50 in group B.
The mean of all 80 original readings is 15. Each group A reading
is replaced by
,
while group B is unchanged. The mean of the 80 updated readings is 21.
What was the original mean of group A?
Enter an integer, decimal, or fraction. Up to 5 characters, or 6 including a leading minus. No units or mixed numbers.
Answer: 13
The total increases from
to
,
a change of 480. Replacing x by 2x + 3 adds x + 3 to each A reading. If
the original A sum is S, the total increase is S + 30(3). Thus S + 90 =
480, so S = 390 and the original A mean is 390/30 = 13.
Check: The original B sum is 810. The updated A sum is
;.
Watch for: Only one group changes; apply the transformation to its sum and
count, not to the entire file’s mean.
Math · Module 2 · Question 9 of 22
Triangle has and . A circle passes through all three vertices of the triangle. What is the radius of the circle?
Answer: B.
Let be the midpoint of . Then and , so .
The center is equally far from and , so it lies on line . Let the radius be , so . Because every angle of the triangle is acute, is inside the triangle, between and , and . In right triangle :
so .
Check:, and .
A: is half of , as if were a diameter; that would require a right angle at .
C: is the altitude , which would be the radius only if the center were at .
D: swaps the roles of and , solving .
Watch for: The center is on the altitude from , but it is not the point where the medians meet; set its distances to and to equal.
Math · Module 2 · Question 10 of 22
The graph of the quadratic function is shown. The function is defined by . What is the sum of all real zeros of ?
Answer: B. 4 The graph shows that the zeros of f are −4 and 9. For , the inner value must equal one of those zeros. It cannot equal −4 over the reals. Setting it equal to 9 gives , or −1 and 5. Their sum is 4.
Watch for: A squared input can reach only nonnegative values, so not every zero of the outer function creates real zeros of the composition.
Math · Module 2 · Question 11 of 22
The shaded region in the graph, including its solid boundary line but not its dashed boundary line, represents all solutions of a system of two linear inequalities. The point lies in the shaded region. Which of the following describes all possible values of ?
Answer: C.
The solid line passes through and , so it is ; the shaded region is on or above it. The dashed line passes through and , so it is ; the shaded region is strictly below it. The system is
and
Substitute and . The first inequality gives , so and . The second gives , so and . Together, .
Check: At the point is , which lies on the solid line, so it is included. At the point is , which lies on the dashed line, so it is excluded.
A: This treats the dashed boundary as part of the region, so it wrongly includes .
B: This reverses which boundary is included: the endpoint comes from the solid line and must be included, while comes from the dashed line and must be excluded.
D: The upper bound is the -coordinate where the two boundary lines meet; it describes the region, not the points of the form .
Watch for: The restriction applies to the point , not to every point with ; substitute both coordinates before solving.
Math · Module 2 · Question 12 of 22
The intensity of light from a small lamp is inversely proportional to the square of the distance from the lamp. When a light sensor is moved meters farther from the lamp, the intensity it measures decreases by . From the sensor’s original position, how many meters toward the lamp must it be moved so that the intensity it measures is greater than the intensity at its original position?
Answer: A.
Let be the original distance. Intensity is proportional to , so moving to multiplies the intensity by . A decrease leaves :
so and . At a new distance , the intensity is multiplied by , and a increase means a factor of :
so and . The sensor must move meters toward the lamp.
Check: From meters to meters, the intensity is multiplied by . From meters to meters, it is multiplied by .
B: This leaves out the square in the second step: gives , a move of meters.
C: This is the sensor’s new distance from the lamp, not the distance it moves.
D: This is the sensor’s original distance from the lamp.
Watch for: Convert each percent change to a factor ( and ), take square roots to get distance ratios, and answer with the distance moved.
Math · Module 2 · Question 13 of 22
In the figure, two circles each have a radius of , and each circle passes through the center of the other. The shaded region lies inside both circles. What is its area?
Answer: D.
Let the centers be and , and let the circles meet at and . Each of ,, and is a radius of length , so triangle is equilateral, and so is triangle . Then .
The part of the shaded region on the side of is a segment of the circle centered at : a sector minus triangle . Triangle is half of rhombus , which is made of the equilateral triangles and , each with area . So triangle has area , and the segment has area
By symmetry, the other part has the same area, so the shaded region has area .
Check: Numerically, . This is more than the area of rhombus ,, which lies inside the region, and less than the area of either circle, .
A: subtracts the area of the whole rhombus from each sector, removing twice as much as it should.
B: is only one of the two equal segments that make up the region.
C: is a single sector, which is neither half of the region nor the whole region.
Watch for: Each half of the region is a sector minus a triangle, and the two halves are equal.
Math · Module 2 · Question 14 of 22
For each value of the constant , the graph of in the -plane is a line. All of these lines pass through the same point . What is the -coordinate of ?
Answer: B.
Group the terms that contain :
A point lies on every line in the family exactly when this is true for every , so both groups must be : and . Adding, , so and . So .
Check: For the line is , or , and for it is , or . These two lines meet at , and for any ,.
A: comes from dropping the right side , which gives and and the point .
C: is the -coordinate of .
D: is the sum of the coordinates of ,.
Watch for: Separate the terms with from the terms without ; each group must equal zero for the point to be on every line.
Math · Module 2 · Question 15 of 22
A data set consists of different numbers, and its mean is . When the numbers are listed in increasing order, the th number is and the th number is . Each number in the data set that is greater than the median is increased by , and the other numbers are not changed. What is the mean of the new data set minus the median of the new data set?
Enter an integer, decimal, or fraction. Up to 5 characters, or 6 including a leading minus. No units or mixed numbers.
Answer: 2
The original median is the average of the th and th numbers, . The numbers greater than are the th through th, which is numbers. Each increases by , so the sum increases by and the mean increases by , to .
The first numbers are unchanged and are at most ; the other are now at least . So the new median is , and the difference is .
Check: The original sum is ; the new sum is , and .
Watch for: The most common wrong value is , which assumes the median stays ; with an even number of values, the median depends on the th number, which changes.
Math · Module 2 · Question 16 of 22
In the -plane, the graph of is a circle, and the graph of , where is a constant, is a parabola. The circle and the parabola intersect at exactly three points. What is the area of the triangle whose vertices are these three points?
Enter an integer, decimal, or fraction. Up to 5 characters, or 6 including a leading minus. No units or mixed numbers.
Answer: 125
Completing the square, the circle is , with center and radius .
Both graphs are symmetric about the -axis, so intersection points off the -axis come in pairs and . Three intersection points require one on the -axis. The parabola meets the -axis only at its vertex , and the circle meets it at and , so or .
Substituting into the circle’s equation gives .
For : or , and only gives a point, so there is point.
For : or . These give and, from , and : points. So .
The triangle has base , from to , and height , so its area is .
Check: The point is on both graphs: , and .
Watch for: The most common wrong value is , which leaves out the factor . Check both candidates for : gives only one intersection point.
Math · Module 2 · Question 17 of 22
In the figure, triangle has a right angle at , and is perpendicular to . If and , what is the length of ?
Answer: A.
In right triangle , the side opposite is and the adjacent side is , so
and . Angles and are both complementary to , so . In right triangle ,, so . Then
so and .
Check:, and . Also , since , as the similar triangles require.
B: This splits in the ratio , the tangent ratio itself, instead of , the square of that ratio: .
C: This is the length of the altitude .
D: This is the length of , the segment adjacent to .
Watch for: The altitude splits the hypotenuse in the ratio of the squares of the legs’ ratio; name each angle’s opposite and adjacent sides inside the small triangle that contains it.
Math · Module 2 · Question 18 of 22
The function is defined by , where ,, and are constants. In the -plane, the parabola has its vertex in quadrant II and crosses the -axis below the -axis. Which of the following must be true?
Answer: D.
The graph crosses the -axis at , so .
The vertex is in quadrant II, so its -coordinate is positive. The point is lower than the vertex, so the vertex is the highest point of the graph: the parabola opens downward and .
The vertex’s -coordinate, , is negative. Because , multiplying by reverses the inequality: , so .
So ,, and are all negative, and .
Check: For example, has vertex in quadrant II and -intercept ; here ,,, and . This graph also crosses the -axis twice.
A: If , the vertex would be the lowest point, but the -intercept is below the vertex, so the parabola opens downward.
B: With , a vertex to the left of the -axis requires ; reverses this sign.
C: A parabola that opens downward with its vertex above the -axis crosses the -axis twice, so has two real solutions.
Watch for: Find the sign of first from the vertex and the -intercept; then the sign of gives the sign of .
Math · Module 2 · Question 19 of 22
In the figure, right triangle has hypotenuse ,, and . Square has vertices and on , vertex on , and vertex on . What is the side length of the square?
Answer: A.
The hypotenuse is , and the altitude from to has length .
Let the side length of the square be . Side is parallel to , so triangle is similar to triangle . Its base is , and its height from is . Corresponding lengths are proportional:
so .
Check: Place ,, and . At height , side is at and side is at ; the distance between them is about .
B: assumes that the square takes up one third of the hypotenuse, which is true only when the two legs are equal.
C: is the side of a square placed in the corner at with two sides along the legs, .
D: is the length of the altitude from to , not the side of the square.
Watch for: The small triangle above the square has height , measured along the altitude to the hypotenuse.
Math · Module 2 · Question 20 of 22
The polynomial function is defined by , where is a constant. When is divided by , the remainder is equal to . What is the sum of the real zeros of ?
Enter an integer, decimal, or fraction. Up to 5 characters, or 6 including a leading minus. No units or mixed numbers.
Answer: -1
By the remainder theorem, the remainder when is divided by is . Also, . So , and :
Testing integer factors of :, so is a factor, and
The quadratic has discriminant , which is negative, so it has no real zeros. The only real zero of is , so the sum of all real zeros is .
Check: Multiplying back gives , the original polynomial. Also, is always positive, so the graph of crosses the -axis only at .
Watch for: The most common wrong value is , which assumes that has three real zeros; the factor has none. Another is , the value of .
Math · Module 2 · Question 21 of 22
A right circular cone has a base radius of centimeters and a height of centimeters. A right circular cylinder is inside the cone: the bottom base of the cylinder lies on the base of the cone, and the circle at the top of the cylinder lies on the lateral surface of the cone. The lateral surface area of the cylinder is square centimeters, and the height of the cylinder is greater than its diameter. What is the total surface area of the cylinder, including both bases, in square centimeters?
Answer: B.
Take a vertical cross section through the cone’s axis. By similar triangles, at a height above the base the cone’s radius is , so the cylinder’s radius and height satisfy . The lateral area gives , so and or . If , then , which is less than the diameter; so and . The total surface area is the lateral area plus two bases: .
Check: With and , the lateral area is and each base has area .
A: adds only one base to the lateral area.
C: adds one base of the other solution, , whose height is less than its diameter.
D: is the total surface area for and , which the condition on height rules out.
Watch for: The lateral-area condition has two solutions; the condition that the height exceeds the diameter selects one.
Math · Module 2 · Question 22 of 22
The quadratic function has a minimum value of , which occurs at . In the -plane, the points and lie on the graph of . What is the greatest possible value of ?
Enter an integer, decimal, or fraction. Up to 5 characters, or 6 including a leading minus. No units or mixed numbers.
Answer: 4
In vertex form, with . The two points give
, so
, so
Dividing the first equation by the second gives , so or .
The first gives , so . The second gives , so . Both give a positive ( and ), so both are possible, and the greatest possible value of is .
Check: For ,: and . For ,: and .
Watch for: The most common wrong value is , which keeps only the positive square root; taking both signs gives a second parabola.
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Reference sheet
AVAILABLE DURING BOTH MODULES
Formula reference
The relationships below match the standard SAT-provided Math reference. This is a newly typeset reference, not a reproduction of the official sheet. [4]
Area and circumference
Circle A = πr2,C = 2πr
Rectangle A = lw
Triangle A = bh
Right triangles
Pythagorean theorem a2 + b2= c2, where c is the hypotenuse.
45°–45°–90° Side lengths s, s, s, in increasing order.
A full circle has 360° or 2π radians. The interior angles of a triangle sum to 180°.
Symbols: r is radius; l is length; w is width; b is base length; h is perpendicular height; A is area; C is circumference; V is volume. Source: official reference linked in the directions, [4].
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