All practice questions

An industrial designer is drafting a template for a rectangular solar panel. The total area of the panel is represented by the polynomial $4x^3(3x^2 - 2x + 5)$ square centimeters. A rectangular sensor cutout with an area of $(2x^2 - 3)^2$ square centimeters is removed from the panel. Which of the following polynomials represents the remaining area, in square centimeters, of the solar panel in terms of $x$?

A vertex of a polygon in the standard $(x, y)$ coordinate plane undergoes two transformations: it is first reflected across the line $y = x$, and then it is translated $4$ units to the right. If the coordinates of the final image of the vertex are $(1, 5)$, what were the coordinates of the vertex before the transformations?

In a convex polygon, the measures of the interior angles form an arithmetic progression. The smallest interior angle measures $120^\circ$, and the common difference between consecutive interior angles is $5^\circ$. What is the number of sides of this polygon?

A board game spinner has a pointer of length $6\text{ inches}$ that pivots at the center of the circular board. If the sector of the board representing 'Lose a Turn' has a central angle of $\frac{\pi}{3}$ radians, what is the area, in square inches, of this sector?

An observer stands at point $P$ on horizontal ground and measures the angle of elevation to the top of a vertical tower, $T$, as $\theta$. The observer then walks a distance of $d$ meters directly toward the base of the tower to point $Q$. From point $Q$, the angle of elevation to the top of the tower is $2\theta$. If $\cos(\theta) = \frac{4}{5}$, what is the ratio of the height of the tower to the distance $d$?

A surveyor is mapping a triangular park. Starting at point $A$, she walks due east for $80$ meters to point $B$. She then turns $120^\circ$ to her left and walks in a straight line to point $C$, which is located directly north of point $A$. What is the straight-line distance, in meters, from point $B$ to point $C$?

The equation $x^2 + y^2 - 12x + 8y + 3 = 0$ defines a circle in the standard $(x, y)$ coordinate plane. What is the radius of this circle?

The following passage is about the historic discovery of a rare fish species.

Discovery of the Living Fossil

On December 22, 1938, Marjorie Courtenay-Latimer, the young curator of the East London Museum in South Africa, walked down to the harbor docks. She had received a call from Captain Hendrik Goosen of the trawler *Nerine*, who had returned from a fishing trip near the mouth of the Chalumna River and wanted to know if she wished to examine the catch for any specimens of interest. Amidst a pile of sharks and common marine life, Courtenay-Latimer noticed a peculiar protrusion: a five-foot-long, heavy-scaled fish of an intense iridescent blue, with limb-like lobed fins and a strange, puppy-like second tail. Intrigued, she carted the heavy creature back to her museum in a taxi.

Recognizing that the specimen was unlike anything described in South African reference books, Courtenay-Latimer immediately attempted to find a way to preserve it. She contacted the local municipal offices in hopes of using their cold storage facilities, but her request was rejected by officials who deemed the rotting carcass a health hazard. Desperate, she wrote a letter containing a brief description and a hand-drawn sketch of the fish to James Leonard Brierley Smith, a chemistry lecturer at Rhodes University in Grahamstown who was also a passionate self-taught ichthyologist.

Because the South African summer heat was intense and the museum lacked formal preservation equipment, Courtenay-Latimer realized she could not wait indefinitely for a response. By the third day, the fish had begun to decompose rapidly. She made the difficult decision to take the specimen to a local taxidermist, Robert Center, instructing him to skin the fish and mount it. Unfortunately, because the mounting process required removing the soft tissues, Center discarded the internal organs and skeleton, preserving only the skin and a few skeletal elements in the head.

Meanwhile, J.L.B. Smith was away on vacation and did not receive Courtenay-Latimer's letter until January 3, 1939. Upon opening the envelope and viewing the sketch, Smith was struck by a shock of recognition. The drawing bore an uncanny resemblance to fossilized coelacanths, a prehistoric lineage of lobe-finned fish thought to have gone extinct at the end of the Cretaceous period, roughly 66 million years ago. Smith sent an urgent telegram advising Courtenay-Latimer to preserve the remains at all costs, but the message arrived long after the internal structures had already been discarded.

Smith finally arrived at the East London Museum on February 16, 1939. The moment he laid eyes on the mounted specimen, his suspicions were confirmed. Despite the tragic loss of the internal organs, the external features—most notably the heavy cosmoid scales and the jointed, limb-like fins—clearly identified it as a living coelacanth. He named the species *Latimeria chalumnae* in honor of the curator and the river near which it was caught. The discovery sent shockwaves through the global scientific community, challenging long-held assumptions about the fossil record and the evolution of tetrapods.

Realizing that a single taxidermied specimen was insufficient to fully study the fish's biology, Smith embarked on a quest to locate another living coelacanth. He printed thousands of bilingual leaflets offering a reward and distributed them along the southeastern coast of Africa. The search took fourteen years of patient waiting and false alarms. Finally, in December 1952, a second coelacanth was caught by fishers in the Comoros Islands, hundreds of miles north of the original site. This specimen was successfully preserved intact, allowing Smith and other scientists to study the internal anatomy of this living fossil for the first time.

Based on the passage, in what chronological order did the events below occur, from earliest to latest?

A robotic arm starts at the point $(1,0)$ on the unit circle and rotates counterclockwise by $\frac{11\pi}{6}$ radians. It then rotates clockwise by $120^\circ$. At which of the following coordinates on the unit circle does the arm's tip end?

In nearshore marine ecosystems, the sea otter plays a critical role in maintaining the ecological balance of coastal kelp forests. Kelp forests are underwater areas with a high density of kelp canopy, supporting a diverse array of marine life. Sea otters feed primarily on sea urchins, which are herbivorous invertebrates that graze heavily on kelp. Without sea otters to control the urchin population, these invertebrates quickly multiply and consume entire forests of kelp, creating barren seafloors known as 'urchin barrens.' While sea otters also consume crabs, clams, and other invertebrates, their predation on urchins is their most ecologically significant behavior. By keeping the urchin population in check, sea otters indirectly protect the habitat of hundreds of other species, including various fish, snails, and crabs that rely on the kelp for shelter and food. Ultimately, the presence of sea otters is essential to preserving the overall health and biodiversity of the entire nearshore marine community.

Which of the following statements best expresses the main idea of the passage?

Julian adjusted the brass vernier of his sextant, the cold salt spray of the Atlantic misting his spectacles. For three months, he had walked the jagged perimeter of Cape Despair, tasked by the Royal Society with producing the first "definitive" map of the peninsula. His predecessors had relied on hearsay and rough sketches, but Julian brought the modern weight of trigonometry, astronomical observations, and a pocket chronometer that ticked with the relentless accuracy of the age.

To the Society in London, the coast was a mathematical puzzle to be solved, a boundary line that could be pinned to a grid. Yet, with every line Julian inked onto his vellum sheets, he felt a quiet, encroaching frustration. The cape refused to remain static. At high tide, the sandy spit to the south vanished entirely under the frothing brine; at low tide, it revealed a ribbed expanse of mud and kelp that seemed to stretch the very geography he was trying to capture.

"A map is a contract between the traveler and the terrain," his mentor, Master Sterling, had once told him. "It must be reliable. If a rock is there today, it must be there tomorrow."

But Julian was beginning to see the lie in Sterling’s philosophy. The rocks were there, certainly, but their relationship to the shore was a fluid negotiation. One afternoon, he met an elderly fisherman named Tom, who was mending lobster pots outside a ramshackle hut that Julian had marked simply as "Shed: Uninhabited."

"You’re putting the tide pool on your paper, young master?" Tom asked, squinting at Julian’s sketches.

"I am marking the mean high-water mark," Julian replied, his pencil poised.

Tom laughed, a sound like dry gravel shifting. "The high-water mark is wherever the sea decides to sleep on a stormy night, boy. Last winter, she slept in my parlor. This summer, she won't even touch the garden gate. You can't draw a line where the water and the land are always dancing."

Julian dismissed the old man's words at the time, attributing them to the unscientific mind of the local peasantry. He returned to his calculations, verifying the latitude by the transit of Venus, and aligning his baselines with scrupulous care. He wanted his map to be perfect, an absolute representation of reality.

However, as the autumn gales began to howl across the cliffs, the landscape itself staged a rebellion. A massive landslide near the northern bluff sheared off a hundred feet of granite, sending it crashing into the surf below. The path Julian had walked just a week prior, mapping its exact elevation and curvature, no longer existed. In its place was a raw, gaping scar and a new bay that his instruments had never predicted.

Sitting in his drafty inn room that night, Julian looked at his masterfully rendered draft. The lines were crisp, the calculations flawless, yet the map felt increasingly like a tombstone—a monument to a Cape Despair that had existed for only a fleeting moment in late August. He realized that the very act of mapping was not an act of capturing reality, but of freezing it, of stripping away its vitality to make it comprehensible to minds that feared change.

He thought of Tom's "dancing" shoreline. The fisherman did not need a map; he carried the coast in his bones, understanding it as a living process rather than a static grid. Julian looked at the blank space he had left near the western marsh, where the paths were too marshy for his heavy surveying equipment. Instead of attempting to measure it tomorrow, he dipped his pen in the ink and wrote in a fine, cursive hand: *Here the land yields to the tide, shifting with the seasons.*

It was a departure from the strict guidelines of the Royal Society, a small concession to the truth of the place. In that single line, Julian acknowledged that the true nature of the cape lay not in its fixed coordinates, but in its perpetual state of becoming.

Based on the passage, which of the following statements best describes the central theme of Julian's experience?

Passage

The dust in the archives of the historical society had a distinct, sweet scent, like dried apples. For weeks, Marcus had worked in the shadow of the grand mahogany desk where Dr. Helen Vance—his predecessor and former mentor—had compiled her final, unfinished monograph on the local ironworks. Her notes were legendary for their meticulous margins, dense with cross-references, but Marcus found himself drawn instead to the blank spaces. On the back of a receipt from a local apothecary dated November 1892, Vance had sketched a rough map of the valley, marking a site she called 'The Foundry's Shadow.' She had never mentioned this site in her published papers or her lectures. Yet, Marcus noticed that her diary entries from that same month abruptly ceased their usual complaints about funding and university politics, replaced instead by cryptic logs of soil acidity and water temperature. A week later, she had abruptly packed her trunks and requested an indefinite leave of absence, citing 'exhaustion.' Standing by the window, looking out at the overgrown path leading toward the river, Marcus realized that Vance’s departure was not a retreat, but a pursuit.

Based on the passage, which of the following best describes the relationship between Helen Vance’s discovery of the site she labeled 'The Foundry's Shadow' and her subsequent leave of absence?

The invention of the printing press in the fifteenth century was a watershed moment in human history. Prior to its creation, books were painstakingly copied by hand, a process that limited access to knowledge to a small elite of scholars and nobles. Johannes Gutenberg's machine, however, transformed this landscape. By automating the reproduction of text, the press allowed ideas to spread across Europe with unprecedented speed.

This technological leap paved the way for the scientific revolution, as scholars from different regions could now easily share their findings and build upon one another’s work. Rather than hoarding information, societies began to value the open exchange of observations. Consequently, the printing press did not merely change how books were made; it fundamentally altered how humanity accumulated and verified knowledge, showing that access to information could reshape the social order. Without this rapid exchange of text, the discoveries of the sixteenth and seventeenth centuries would have likely remained isolated, local events rather than a global movement.

In the context of the passage, the figurative expression 'paved the way for' (paragraph 2) most nearly means which of the following?

In the early twentieth century, the classical music landscape underwent a radical bifurcation as composers sought alternatives to late-Romantic tonality. Impressionism, spearheaded by Claude Debussy, aimed to evoke fleeting sensory experiences and atmospheres. Debussy accomplished this by using non-functional harmony, whole-tone scales, and shimmering orchestral textures, which softened the boundaries of traditional key centers without fully abandoning them. To the listener, Impressionist music felt suspended and weightless, prioritizing immediate sonic color over linear narrative.

Conversely, Expressionism, led by Arnold Schoenberg, rejected traditional tonality entirely in favor of atonality and, later, the twelve-tone technique. Rather than depicting external impressions of nature, Expressionists turned inward, seeking to represent raw, subconscious psychological states. Schoenberg's works featured highly dissonant, fragmented melodies and abrupt dynamic shifts, intentionally jarring the listener to convey existential anxiety. While Debussy sought to expand and color the existing harmonic palette to soothe the senses, Schoenberg dismantled it systematically to confront the psyche.

According to the passage, which of the following statements best describes how Claude Debussy's compositional approach differs from Arnold Schoenberg's?

For decades, the dominant paradigm in cognitive science treated the human mind as a dry-docked computer, parsing symbolic inputs through rigid algorithmic pathways. When connectionist models emerged in the late twentieth century, proposing that intelligence instead emerges dynamically from vast, interconnected neural networks, traditionalists reacted with defensive skepticism. They dismissed these new architectures as mere 'assemblages'—a term loaded with the implication of arbitrary, unguided construction. To the computational orthodox, connectionism lacked the elegant, top-down blueprints of classical cognitive architectures, offering instead what they perceived as a messy, bottom-up heap of nodes. By branding connectionist networks with this term, traditionalists sought to underscore the absence of structured, rule-based representation, framing the new theory not as a sophisticated alternative, but as a regression into mechanical chaos.

In the context of the passage, the traditionalists' branding of connectionist networks as 'assemblages' primarily serves to do which of the following?

Passage:

For three weeks, I sat in the corner of Julian's workshop, a silent witness to the resurrection of the 17th-century atlas. My task was simple: catalog the fragments of leather and vellum that Julian discarded as he worked to rebind the massive volume. To the untrained eye, Julian's movements might have seemed almost destructive. He sliced through centuries-old threads with a scalpel, peeled back layers of adhesive with a damp spatula, and subjected the fragile pages to baths of distilled water. Yet, watching him, I realized that his apparent ruthlessness was guided by a profound reverence. He knew exactly how much tension a waterlogged page could bear before its fibers parted, and he could distinguish between the grit of historic soot and the destructive bloom of active mold by touch alone. Julian rarely spoke to me, and when he did, it was only to demand a specific archive folder or to complain about the fluctuating humidity of the room. The public would eventually see only the restored masterpiece behind glass at the metropolitan library, praising the genius of the preservation team. But in the quiet hum of the workshop, I saw the true cost of preservation: a meticulous, unsung dialogue between Julian's calloused hands and the crumbling thoughts of a long-dead cartographer.

Based on the passage, which of the following statements best characterizes the narrator's perspective and relationship to Julian's work?

Early conservationists in the late nineteenth century feared that the unregulated expansion of steam-powered timber mills would consume the surrounding forest entirely. However, the unexpected onset of a national economic depression served to check the industry’s rapid growth. With investment capital suddenly scarce, logging companies were forced to halt their plans to lay down new railroad lines into the valley's pristine interior. This sudden pause in industrial development allowed the local vegetation to recover, demonstrating that ecological preservation can sometimes result from market failures rather than deliberate policy.

As it is used in the passage, the word "check" most nearly means which of the following?

The following passage is adapted from an article about the history of paleontology.

Paragraph 1
For nearly a century after the first dinosaur fossils were classified in the nineteenth century, scientists viewed these creatures through a strictly reptilian lens. Dinosaurs were reconstructed as slow, lumbering ectotherms—cold-blooded animals reliant on external heat sources to regulate their body temperature. This consensus was largely based on structural similarities between dinosaur skeletons and those of modern lizards. Because modern lizards have low metabolic rates and exhibit sluggish behavior in cold conditions, early paleontologists assumed that dinosaurs must have operated under similar physiological constraints.

Paragraph 2
This paradigm shifted dramatically in the late 1960s with the discovery of Deinonychus, a small, bird-like predator. Paleontologist John Ostrom pointed out that Deinonychus possessed a highly specialized skeleton designed for speed, agility, and active hunting. Ostrom argued that such high-energy behavior would be impossible for an ectothermic animal, suggesting instead that dinosaurs possessed a warm-blooded, endothermic metabolism. This discovery sparked the "Dinosaur Renaissance," a period of intense scientific debate that fundamentally reevaluated dinosaur biology.

Based on the passage, which of the following best describes how Paragraph 2 functions in relation to Paragraph 1?

Isabella Bird’s travels through the rugged interior of Rocky Mountain territory in 1873 were characterized not by a search for standard tourist comforts, but by an overt, self-proclaimed aversion to domestic tranquility. While her contemporaries in England settled into predictable social routines, Bird sought out the most arduous terrains, writing in her letters that the predictable rhythms of home life made her feel 'physically stifled.' In her journals, she explicitly described herself as 'constitutionally restless,' a trait that drove her to ride horseback through blizzards and camp in open canyons. This internal restlessness was not a symptom of anxiety, but rather a profound craving for physical challenge. Although some biographers have retroactively labeled her as reckless, Bird’s own accounts reveal a woman who planned her routes with careful deliberation, even if those routes led directly into the most uninhabited corners of the American West.

Based on the passage, Isabella Bird explicitly characterized herself as having which of the following traits?

Dr. Sarah Finch carefully extracted the delicate tissue sample from the ghost orchid variant she had just located in the high canopy of the Chocó forest. This moment was the culmination of a quest that had begun five years prior, when she first stumbled across a faded drawing of the flower in the margins of a nineteenth-century journal in Madrid. That discovery in the archives had immediately prompted her to secure funding from the Botanical Society, a grueling process that took nearly two years of rejected proposals. But as she packaged the sample, her mind flashed forward to the upcoming presentation at the Geneva Symposium, where she would finally vindicate her theory of high-altitude orchid evolution. Sitting on the forest floor, she remembered her first expedition to Ecuador three years ago, which had ended in near-disaster when a sudden mudslide swept away all their collection gear, forcing a hasty retreat. Now, looking at the glowing white petals in the twilight, Finch knew the years of setbacks were finally behind her.

Based on the passage, what is the correct chronological order of the following events in Finch's quest, from earliest to latest?