All practice questions

The polynomial $x^2 + 8x + 15$ can be factored into the form $(x + a)(x + b)$, where $a$ and $b$ are integers such that $a < b$. What is the value of $2a + b$?

If $3x(x - 3) = 2(x + 3) - 12$, what is the product of the two solutions to this equation?

A charity walkathon organizer pledges to donate a base amount of $\$150$ plus $\$2.50$ for every kilometer completed by each participant. On the day of the event, a participant completes a distance that is $3$ kilometers less than twice their training average distance. If the organizer's donation for this participant is $\$212.50$, what is the participant's training average distance, in kilometers?

Line $p$ is defined by the equation $2x + 5y = 10$. Line $q$ is perpendicular to line $p$ and has a $y$-intercept of $-3$. If line $q$ passes through the point $(a, 7)$, what is the value of $a$?

The elevation $y$ (in meters) of a roller coaster track is modeled by the equation $y = (x - 3)^2 - 4$, where $x$ represents the horizontal distance (in meters) from the start of the ride. A straight support beam is designed such that the height of the track is 7 meters less than 7 times the horizontal distance. The support beam connects to the roller coaster track at two points, $(x_1, y_1)$ and $(x_2, y_2)$. What is the value of $x_1 y_1 + x_2 y_2$?

This passage is adapted from an article about the discovery and ecological significance of Movile Cave in Romania.

In $1986$, while surveying the barren, limestone-rich plains of Dobrogea near the Black Sea, Romanian geologist Cristian Lascu was tasked with testing the ground for a new power plant. Instead of finding solid bedrock, Lascu stumbled upon a geological marvel that had remained sealed from the outside world for over five million years: Movile Cave. This subterranean cavity, situated just twenty meters below the surface, would soon rewrite our understanding of how life can thrive on Earth.

Unlike typical caves, which are carved by surface rainwater filtering through limestone, Movile Cave was formed by sulfidic thermal waters rising from deep within the Earth's crust. This unique genesis created a highly hostile environment. The air inside the cave is thick with hydrogen sulfide and carbon dioxide, while oxygen levels are less than half of what is found in the outside atmosphere. To the human lungs, the air is highly toxic and suffocating, and the extreme humidity makes physical exertion incredibly difficult. For decades, scientists believed that sunlight and photosynthesis were the absolute foundation of all food chains on Earth. Movile Cave, however, shattered this biological dogma.

The base of the food web in Movile Cave is not sunlight, but rather a thick, floating mat of autotrophic bacteria that covers the water in its lower chambers. These bacteria engage in chemosynthesis, a process where they oxidize the abundant hydrogen sulfide to produce organic matter, releasing sulfuric acid as a byproduct. This metabolic process provides the primary energy source that sustains a complex and diverse community of organisms, all of which have adapted to live in total, permanent darkness. The water in the cave is warm and mineral-rich, providing a stable, tepid bath that fosters the rapid growth of these bacterial colonies.

Since the cave's discovery, biologists have identified forty-eight distinct animal species residing within its chambers. Remarkably, thirty-three of these species are completely endemic to Movile Cave, meaning they are found nowhere else on the planet. Among these unique inhabitants are blind water scorpions, wingless beetles, and specialized leeches that feed on the local microflora. Over millions of years of isolated evolution, these organisms have discarded unnecessary traits like pigment and eyesight, instead developing elongated antennae, hypersensitive chemical receptors, and advanced sensory organs to navigate their pitch-black world. Many of these creatures are translucent, showing their internal organs through their thin outer membranes.

The discovery of Movile Cave has had profound implications beyond terrestrial biology. Astrobiologists, who study the potential for life on other planets, frequently point to Movile Cave as an analogue for environments on Mars or Jupiter’s moon, Europa. If life can thrive in the dark, toxic depths of Romania without any reliance on solar energy, it is highly plausible that similar chemotrophic organisms could exist beneath the icy surfaces of distant celestial bodies. This has shifted the focus of search-for-life missions toward underground aquifers and subterranean caves rather than surface searches.

To preserve this delicate and invaluable ecosystem, the Romanian government has tightly restricted access to the cave. Only a handful of researchers, equipped with clean suits and specialized breathing apparatuses, are permitted to enter each year. Today, Movile Cave remains a testament to the resilience of life, proving that even in the most hostile and isolated environments, evolution finds a way to flourish. By studying this closed ecosystem, scientists hope to unlock the secrets of Earth's earliest life forms, which likely existed in similar, oxygen-depleted, chemical-rich environments billions of years ago.

According to the passage, of the forty-eight animal species identified in Movile Cave, how many are found nowhere else on Earth?

Researchers studied the effect of dissolved organic carbon (DOC) on the photochemical degradation rate of Pollutant X in water samples exposed to constant UV light. Two initial hypotheses were proposed:

* Hypothesis 1: DOC acts as a photosensitizer, so increasing DOC concentration will continuously increase the degradation rate of Pollutant X.
* Hypothesis 2: DOC acts as a light attenuator (blocking UV light), so increasing DOC concentration will continuously decrease the degradation rate of Pollutant X.

The degradation rate of Pollutant X was measured at various DOC concentrations, and the results are recorded in the table below:

Based on these results, which of the following modified hypotheses best describes the relationship between DOC concentration and the degradation rate of Pollutant X?

Read the passage below:

In 1610, Galileo Galilei directed his newly constructed telescope toward the night sky, revealing Jupiter’s four largest moons. By recording their positions night after night in his leather-bound journal, these celestial bodies were proved to be orbiting a body other than Earth. This groundbreaking discovery directly challenged the geocentric model that had dominated European astronomy for centuries.

Which choice most effectively corrects the underlined portion to ensure the sentence is clear, logical, and grammatically correct?

Read the following passage carefully:

In July 1894, botanist Dr. Samuel Oakes published his seminal research paper on subterranean fern germination. Although the publication marked the formal conclusion of his project, the pivotal breakthrough had occurred four years prior, in 1890, when he discovered a blooming *Cryptocoryne* specimen inside a humid Yorkshire cave. Before leaving the cave to return to his laboratory, Dr. Oakes preserved several root specimens in alcohol. Two years after collecting these initial samples, in the autumn of 1892, he successfully synthesized an artificial growth medium that replicated the cave's mineral composition. Having established this medium, Dr. Oakes spent the following winter of 1893 conducting controlled temperature trials in a greenhouse, which yielded the empirical data compiled for his eventual publication.

Based on the passage, arrange the following events in the correct chronological order from earliest to latest.

A library display shelf has space for $5$ distinct books aligned in a row. The librarian has $3$ different science fiction books and $2$ different biography books. How many different row arrangements of these $5$ books are possible if the $2$ biography books must not be placed next to each other?

If $x = -2$, $y = 5$, and $z = -\frac{1}{2}$, what is the value of the algebraic expression $\frac{x^3 y + z^{-2}}{y - 2x}$?

The Antikythera Puzzle

[1]
In 1901, divers salvaging an ancient Roman shipwreck near the Greek island of Antikythera recovered a corroded, lumpish bronze artifact. For decades, the calcified object remained largely ignored in the storerooms of the National Archaeological Museum in Athens, appearing to be little more than a formless mass of metal and stone. It was not until the mid-twentieth century, when researchers began using X-ray imaging, that the artifact's true nature began to emerge. Beneath its encrusted exterior lay a complex system of dozens of interlocking bronze gears, precisely cut and arranged with mathematical precision.

[2]
This device, now known as the Antikythera Mechanism, is widely considered the world's oldest known analog computer. Dating back to the second century BCE, it was designed to calculate and display astronomical positions, the phases of the moon, and the timing of eclipses with astonishing accuracy. It even tracked the four-year cycle of the ancient Olympic Games. The sophistication of its gear trains, particularly its use of a differential gear to calculate the synodic month, represents a level of mechanical engineering that would not reappear in European clockmaking for another fifteen hundred years.

[3]
The discovery fundamentally challenged long-held historical assumptions about the technological capabilities of ancient Greek civilization. Before the mechanism was fully analyzed, historians generally believed that complex geared technology arose only in the late Middle Ages. The Antikythera Mechanism proved that ancient mechanics had already mastered advanced gearing principles, showing that technological progress is not always a linear, continuous march forward, but can instead be lost to time and subsequently reinvented.

Suppose the writer's goal had been to write an essay that outlines the timeline of the Antikythera Mechanism's investigation and explains how its mechanical complexity challenged prevailing historical narratives regarding ancient engineering. The writer successfully achieved this goal.

The passage below contains a blank indicating a missing punctuation mark. Fill in the blank with the punctuation mark that correctly completes the sentence, maintaining consistency with the punctuation already used to set off the nonessential clause.

A team of plasma physicists conducted an experiment to investigate electrical breakdown phenomena in synthetic atmospheric gas mixtures. In a sealed dielectric chamber, researchers systematically varied the volume ratio of nitrogen ($N_2$) to oxygen ($O_2$) across ten trials while recording the threshold breakdown voltage ($V_{bd}$) required to initiate a spark discharge. Throughout all experimental trials, the spacing between the two planar copper electrodes was fixed at $5.0\text{ mm}$, the total cell pressure was held constant at $101.3\text{ kPa}$, and the ambient temperature was maintained at $298\text{ K}$.

Match each experimental component from this investigation to its correct variable classification.

Four different saline solutions contain varying concentrations of dissolved salt: Solution P contains $\frac{7}{20}$, Solution Q contains $36.5\%$, Solution R contains $\frac{3}{8}$, and Solution S contains $0.38$ salt concentration. Place the four solutions in order from least concentration to greatest concentration.

Read the following passage about art history. Which word most precisely and clearly completes the sentence while maintaining a formal, academic tone?

Arrange the following mathematical expressions in order from least to greatest value.

Which of the following is equivalent to the expression $\frac{(5^3)^4}{5^2}$?

A recipe for fruit punch calls for orange juice and cranberry juice to be mixed in a ratio of $3:4$. If a chef uses $12$ cups of orange juice, how many cups of cranberry juice should be used?

The following paragraphs are from an essay about Douglas Engelbart's 1968 computer demonstration. For the sake of the logic and coherence of the essay, in what order should the paragraphs be arranged?