All practice questions

A surveyor is measuring a triangular plot of land, $ABC$. The distance from point $A$ to point $C$ is $12$ meters, and the distance from point $B$ to point $C$ is $6\sqrt{2}$ meters. If the measure of angle $BAC$ is $30^\circ$, what is the measure, in degrees, of the acute angle $ABC$?

In the semi-arid region of the Karst Plateau, geologists have long monitored the sudden collapse of surface soil, a phenomenon locally known as "swallow holes." While subterranean water flow is the primary long-term driver of this geological erosion, the immediate trigger for these sudden cave-ins is often human activity. Specifically, during the agricultural expansion of the 1970s, farmers installed extensive underground drainage pipes to redirect seasonal runoff away from their crops. This concentrated redirection of water significantly increased the local water pressure against the delicate limestone ceiling of the underground caverns. Consequently, the structural integrity of the limestone weakened rapidly, culminating in the widespread collapses observed today. Although some local residents attributed the cave-ins to recent seismic tremors, geological sensors recorded no significant tectonic activity during the major collapse events. Instead, the data confirmed that the redirecting of surface runoff was the direct catalyst for the structural failures.

According to the passage, the immediate trigger for the sudden collapse of surface soil in the Karst Plateau was which of the following events?

A circle with center $(5, -6)$ is tangent to the $x$-axis in the standard $(x, y)$ coordinate plane. Which of the following is the equation of this circle?

This passage is adapted from an article detailing the history of meteorology in the late nineteenth and early twentieth centuries.

For much of the nineteenth century, meteorologists operated under a foundational assumption: the temperature of the Earth's atmosphere decreased at a constant, uniform rate with increasing altitude. This belief, derived from observations made during manned balloon ascents and mathematical modeling of thermodynamics, held that the air would grow progressively colder until it reached the absolute cold of outer space. However, because human balloonists could rarely survive voyages above twenty-five thousand feet due to hypoxia and extreme cold, empirical data from the highest reaches of the atmosphere remained tantalizingly out of reach.

The impasse was broken by the French meteorologist Léon Teisserenc de Bort. Recognizing the limitations of manned flights, Teisserenc de Bort turned to unmanned, hydrogen-filled paper and varnished-silk balloons known as ballons-sondes, or sounding balloons. These balloons carried lightweight, self-registering instruments—barographs to measure pressure and thermographs to record temperature—suspended in wicker baskets. To protect the temperature sensors from the heating effects of direct solar radiation, which had plagued earlier high-altitude measurements, Teisserenc de Bort designed a double-walled cylindrical brass shield that ventilated the thermometer using the balloon's ascent itself.

Beginning in 1896 from his private observatory in Trappes, near Paris, Teisserenc de Bort launched hundreds of these balloons. The methodology was meticulous: as the balloon rose, the decrease in atmospheric pressure caused the gas inside to expand until the envelope burst. A small parachute then deployed, carrying the instrument package safely back to Earth. Crucially, each basket bore a tag offering a monetary reward to any farmer or villager who recovered the apparatus and returned it to Trappes, ensuring a remarkably high rate of recovery.

As the data accumulated, Teisserenc de Bort noticed a recurring, baffling anomaly. At a certain altitude—roughly eleven kilometers (approximately thirty-six thousand feet) over middle latitudes—the steady decline in temperature abruptly ceased. Instead of continuing to drop, the temperature stabilized, hovering around minus fifty-five degrees Celsius, and occasionally even warmed slightly. Suspecting that the heating of the thermometer by solar radiation was still distorting the data, Teisserenc de Bort initially kept his findings quiet. He conducted subsequent launches exclusively at night to eliminate solar interference entirely. To his astonishment, the nocturnal data confirmed the daytime findings: the isothermal zone was a physical reality, not an instrumental artifact.

In April 1902, Teisserenc de Bort presented his monumental findings to the Paris Academy of Sciences. He proposed that the atmosphere was divided into two distinct regions. The lower layer, which he named the 'troposphere' (from the Greek tropos, meaning 'turning' or 'mixing'), was characterized by convective currents, shifting winds, and a constant decrease in temperature with height—the region where weather occurred. The upper layer, which he named the 'stratosphere' (from the Latin stratum, meaning 'layer'), was a region of relative calm where the temperature remained nearly constant and air moved in horizontal layers without vertical mixing.

Simultaneously and independently, German meteorologist Richard Assmann published similar results obtained using rubber balloons, which could reach higher altitudes than silk ones before bursting. While Assmann's use of rubber envelopes represented a significant technical advance, Teisserenc de Bort is widely credited with the discovery due to the sheer volume of his trials—over two hundred and thirty successful balloon flights by 1902—and his brilliant conceptualization of the two atmospheric zones. His work transformed meteorology from a localized, surface-bound study of weather patterns into a three-dimensional science of the global atmosphere.

Based on the passage, Teisserenc de Bort's decision to launch sounding balloons at night was prompted by a desire to resolve which of the following issues?

An angle $\theta$ is positioned in the second quadrant. If the trigonometric expression $\csc\theta - \cot\theta$ is equal to $4$, what is the value of $17\cos\theta$?

A sector of a circle has a central angle of $\frac{3\pi}{4}$ radians and an area of $3\pi$ square inches. What is the perimeter, in inches, of the sector?

A water tank is being drained of its contents at a constant rate. The depth of the water in the tank is $8.2$ feet after $2$ hours of draining, and it is $5.8$ feet after $6$ hours of draining. What is the slope of the line that represents the depth of the water, in feet, as a function of the draining time, in hours?

In the standard $(x, y)$ coordinate plane, a triangle has vertices at $P(-3, 1)$, $Q(5, 1)$, and $R(1, 7)$. A line segment connects the midpoint of side $PR$ to the midpoint of side $QR$. What is the length of this line segment?

In right triangle $ABC$, the right angle is at vertex $C$. The length of leg $AC$ is $12$ inches. If $\sin(B) = \frac{3}{5}$, what is the length, in inches, of the hypotenuse $AB$?

The following passage is adapted from an essay on the history of atmospheric physics.

By the turn of the twentieth century, physicists were puzzled by a persistent phenomenon: electroscopes, which measure electric charge, slowly discharged over time even when kept in lead-shielded containers. This spontaneous discharge indicated that the air inside was somehow being ionized, or made conductive, by an external source of radiation. The prevailing scientific consensus attributed this ionization to gamma radiation emitting from radioactive materials, such as uranium and radium, embedded in the Earth’s crust. It was assumed that this terrestrial radiation would weaken rapidly as one moved away from the ground, eventually dissipating to zero at high altitudes.

In $1910$, the German physicist Theodor Wulf, a Jesuit priest and pioneer in electroscopy, sought to measure this expected decay. He designed a portable, highly sensitive electroscope and transported it to the top of the Eiffel Tower, approximately $300\text{ meters}$ above the ground. According to Wulf’s calculations, if the radiation originated solely from the Earth, the ionization rate at that height should have dropped to a small fraction of the ground-level value. Instead, his measurements showed that the radiation level at the tower’s summit remained at nearly $64\%$ of the ground value—far higher than could be explained by terrestrial emission alone, given that gamma rays should be heavily absorbed by the intervening air. While Wulf’s results suggested the existence of an additional source of radiation, his findings were largely discounted by contemporary physicists, who blamed instrumental defects or local atmospheric factors for the discrepancy.

Determined to resolve the mystery, the Austrian physicist Victor Hess designed a series of daring experiments. Hess reasoned that the only way to obtain definitive data was to take measurements at altitudes far exceeding the height of any man-made structure. He commissioned the construction of electroscopes capable of withstanding extreme temperature and pressure fluctuations and began a series of balloon ascents starting in $1911$.

Hess’s experimental breakthrough occurred on August 7, $1912$, during his seventh balloon flight. Accompanied by a pilot and a meteorological observer, Hess boarded the balloon *Böhmen* in Aussig, Austria (now in the Czech Republic). He carried three independent ionization chambers, which were hermetically sealed to prevent changes in air pressure from affecting the internal gas. The balloon drifted northwards, eventually reaching a peak altitude of approximately $5{}300\text{ meters}$.

As the balloon ascended, Hess meticulously recorded the ionization rates from his instruments. During the initial phase of the flight, up to an altitude of about $1{}000\text{ meters}$, the radiation levels decreased slightly, aligning with the theory of terrestrial decay. However, as the balloon climbed past $1{}000\text{ meters}$, Hess observed a striking reversal: the ionization rate began to climb steadily. By the time the *Böhmen* reached $5{}300\text{ meters}$, the rate of ionization was about four times greater than it had been at sea level. Because the air at such altitudes was too thin to contain significant quantities of radioactive crustal dust, Hess concluded that a highly penetrating radiation must be entering the atmosphere from above.

Crucially, Hess needed to determine whether this radiation was solar in origin. To test this, he had previously conducted a flight during a near-total solar eclipse on April 17, $1912$. Despite the Sun being almost completely obscured by the Moon, Hess recorded no drop in ionization levels. This critical observation led him to conclude that the source of the radiation was not the Sun, but rather deep space. Hess published his findings later that year, proposing the existence of an undocumented, extra-terrestrial source of radiation. Although his theory was initially met with skepticism, it was eventually confirmed by subsequent experiments, and in $1936$, Hess was awarded the Nobel Prize in Physics for his discovery of what Robert Millikan would later term "cosmic rays."

According to the passage, what did Theodor Wulf observe regarding the radiation level at the top of the Eiffel Tower during his $1910$ experiments?

Beaver dams are remarkable engineering feats that profoundly alter freshwater ecosystems. When beavers construct dams using tree branches, stones, and mud, they block the natural flow of streams. Consequently, the water pools upstream, creating expansive wetlands and ponds. This sudden accumulation of standing water slows down the stream's current, allowing suspended sediment to settle to the bottom rather than being carried downstream. Over time, the trapped sediment builds up rich, nutrient-dense soil layers that encourage the growth of diverse aquatic vegetation. Furthermore, the slow-moving water in these ponds seeps into the surrounding ground, which directly raises the local water table and keeps nearby soil moist even during dry summer months. Although some landowners complain that beaver activity floods low-lying roads, ecologists emphasize that these wetlands act as natural sponges, reducing the severity of downstream flooding during heavy storms.

According to the passage, the settling of suspended sediment to the bottom of beaver ponds is directly caused by which of the following?

The following paragraph is from an essay about nineteenth-century urban planning:

During the late nineteenth century, the growth of municipal sewage systems in European cities did more than just improve public health; it radically transformed urban geography. Prior to these subterranean networks, waste disposal was localized, often relying on cesspools or night-soil collectors whose routes dictated the physical layout of neighborhoods and restricted expansion. The installation of unified underground drainage pipes allowed cities to grow outward without the constant constraint of proximity to waste disposal sites. Consequently, previously uninhabitable lowlands were reclaimed for housing, and the spatial division between affluent and working-class districts began to shift as municipal infrastructure standardized living conditions across municipal boundaries.

Which of the following best states the main idea of this paragraph?

Louis Daguerre’s invention of the daguerreotype in 1839 revolutionized visual representation, though it was initially met with skepticism by traditional painters who feared the mechanical process would render their craft obsolete. Unlike modern photography, which relies on flexible film or digital sensors to capture images, the daguerreotype utilized a highly polished copper plate coated with silver halide. This plate was sensitized with iodine vapor, exposed in a camera obscura, and then developed using mercury fumes. Because each plate was exposed directly in the camera without a negative, the resulting image was a unique, non-reproducible photograph characterized by its astonishing level of detail and metallic sheen. Consequently, early adopters of the technology had to accept that their portraits could not be copied, making each piece a singular heirloom.

According to the passage, early daguerreotype portraits were unique, non-reproducible items because of which of the following factors?

Although standard histories of astronomy often credit Nicolaus Copernicus with single-handedly initiating the heliocentric revolution, a closer examination of his landmark 1543 treatise, De revolutionibus orbium coelestium, reveals a far more conservative thinker bound to ancient traditions. Copernicus indeed placed the Sun at the center of the universe, but he did so not to chart a path toward modern physics, but rather to salvage the classical Greek ideal of uniform circular motion. Frustrated by the Ptolemaic system's reliance on the equant—an imaginary point from which a planet's speed appeared uniform, even though its actual speed along its orbit varied—Copernicus sought a geometrically 'pure' alternative. To maintain perfect circular orbits without the equant, he was forced to introduce a complex web of epicycles and eccentrics, ultimately creating a system nearly as cumbersome as the one he sought to replace. Furthermore, Copernicus retained the Aristotelian belief in solid crystalline spheres carrying the planets, and he remained convinced that the universe was finite and bounded by a sphere of fixed stars. It was not Copernicus, but later figures like Johannes Kepler��who abandoned circular orbits for ellipses—and Galileo Galilei—who dismantled the crystalline spheres—who truly forged the conceptual tools of modern astronomy. Copernicus’s work was less a modern breakthrough than a final, sophisticated attempt to perfect classical geometry.

Based on the passage, which of the following statements best expresses the main idea of the author's analysis of Copernicus?

The following paragraph is excerpted from an essay about historical printing techniques:

[1] Although the invention of the printing press is widely credited to Johannes Gutenberg in fifteenth-century Europe, movable type technology actually originated in East Asia centuries earlier. Bi Sheng developed ceramic movable type in China around 1040, and by the thirteenth century, Korean artisans had advanced this process by using metal type to print documents. While Gutenberg's press revolutionized European literacy, it built upon technological foundations laid down by Asian inventors who had already recognized the efficiency of reusable, individual character pieces.

Based on Paragraph 1 (lines 1–7), which of the following statements best expresses the paragraph's main idea?

For decades, biologists believed that plant communication was restricted to physical contact through root networks. However, recent studies on wild sagebrush have overturned this assumption, demonstrating that these plants actively release volatile organic compounds (VOCs) into the air when damaged by herbivores. When neighboring sagebrush plants detect these airborne chemical signals, they immediately ramp up their own defensive mechanisms, such as producing toxic chemicals that make their leaves unpalatable to insects. Interestingly, this chemical warning system is highly localized; plants only respond to signals from neighbors within a three-foot radius. This proximity requirement suggests that rather than engaging in altruistic warning behavior for the entire community, individual plants are primarily broadcasting signals to their own genetically identical branches, and nearby neighbors simply happen to intercept the message.

According to the passage, the localized nature of the sagebrush warning system suggests that individual plants:

Though early twentieth-century astronomers hypothesized that the interstellar medium was entirely devoid of matter, subsequent spectroscopic analyses revealed a sparse distribution of gas and dust. Specifically, the detection of interstellar absorption lines demonstrated that cold sodium and calcium atoms were absorbing starlight, thereby confirming the existence of diffuse matter between stellar bodies. While these gaseous elements are widely dispersed, their presence is not uniform; rather, they accumulate in discrete clouds that attenuate light from background stars. This attenuation, long mistaken for an absence of stars in certain regions of the sky, actually indicates areas of higher density where interstellar dust scatters shorter wavelengths of light, leaving only the longer, redder wavelengths to pass through to Earth.

Based on the passage, the attenuation of starlight in certain regions of the sky is indicative of which of the following?

To perform a simple paper chromatography experiment to separate the pigments in black ink, scientists follow a precise sequence of steps. First, a thin line is drawn with a pencil about two centimeters from the bottom edge of a strip of chromatography paper. Next, a small dot of the black ink to be analyzed is placed directly on the center of this pencil line and allowed to dry completely. After the ink dries, the paper strip is suspended vertically inside a beaker containing a small amount of solvent, such as rubbing alcohol. It is crucial that the solvent level is low enough so that it does not submerge the ink dot itself. As the solvent rises up the paper through capillary action, it carries the ink pigments with it at different rates. Once the solvent front reaches near the top of the paper, the strip is removed from the beaker and laid flat to dry before the separated color bands are measured and recorded.

According to the procedure described in the passage, what must happen to the black ink dot before the chromatography paper strip is suspended inside the beaker?

To prepare a biological specimen for transmission electron microscopy (TEM), a precise sequence of preparation steps must be followed to preserve the sample's ultrastructure. First, the tissue is subjected to primary fixation, typically using a buffered glutaraldehyde solution, which cross-links proteins and stabilizes the cellular architecture. Subsequently, the sample is washed in a buffer to remove excess fixative before undergoing secondary fixation with osmium tetroxide. This crucial secondary step imparts electron density to cellular lipids, rendering them visible under the microscope's electron beam. Following this, the specimen must be dehydrated. Because the vacuum chamber of a TEM requires a completely dry environment, water in the tissue is gradually replaced by passing the specimen through a graded series of ethanol solutions, ending with absolute ethanol. Once dehydration is complete, the specimen is infiltrated with a liquid epoxy resin. The resin gradually displaces the ethanol over several hours. Finally, the infiltrated specimen is placed in a capsule and heated in an oven to polymerize the resin, creating a hard plastic block that can be sliced into ultra-thin sections using an ultramicrotome. Slicing the specimen prior to complete polymerization would result in the tearing and destruction of the soft tissue.

According to the passage, if a researcher fails to complete the dehydration step before introducing the liquid epoxy resin, which of the following is the most direct consequence during the imaging process?

To reconstruct past climates using tree rings, researchers follow a meticulous multi-step preparation and analysis protocol. First, core samples are extracted from living trees using an increment borer. Before mounting the fragile cores, they must be slowly air-dried in protective paper straws; attempting to mount wet cores leads to structural warping. Once dry, the cores are glued into wooden support channels, ensuring the tracheid cells run vertically to optimize the surface for sanding. The mounted cores are then sanded using progressively finer sandpaper grits—typically starting at 120-grit to level the surface, transitioning to 220-grit, and finishing with 400-grit to polish the cell walls. Prior to cellular measurement under a microscope, any residual sawdust must be cleared from the ring boundaries using compressed air. Measuring uncleaned rings leads to significant errors, as dust particles can mimic cell walls. Subsequently, researchers utilize skeleton plotting to cross-date the samples against a master chronology, aligning narrow rings that represent dry years. Only after cross-dating is complete can researchers confidently input the raw ring-width measurements into standardization software to remove age-related growth trends.

According to the passage, at what point in the preparation and analysis process must researchers clear residual sawdust from the ring boundaries using compressed air?