All practice questions

In quadrilateral $ABCD$, $\angle B = 90^\circ$ and $\angle D = 90^\circ$. If $AB = 12$ units, $BC = 16$ units, and $AD = 10$ units, what is the length, in units, of side $CD$?

If $\sin \theta + \cos \theta = \sqrt{1.5}$, what is the value of $\tan \theta + \cot \theta$?

What is the exact value of $\cos\left(2\arctan(-3)\right)$ expressed as a decimal?

A region $ABCD$ is bounded by two concentric circular arcs of radii $R$ and $r$ (where $R > r$) and two radial line segments, all sharing a central angle of $72^\circ$. The area of region $ABCD$ is $60\pi\text{ cm}^2$, and the total perimeter of region $ABCD$ is $(12\pi + 20)\text{ cm}$. What is the value of the outer radius $R$, in centimeters?

Passage A
In the early 1920s, commercial aviation in the United States relied almost entirely on "contact flying"—pilots navigating by identifying rivers, railway lines, and towns from the cockpit. To enable night operations, the U.S. Post Office Department initiated the construction of the transcontinental lighted airway, a network of rotating acetylene beacons spaced along flight paths. These ground-based visual markers transformed airmail transport, proving that scheduled night flying was commercially viable. While primitive by modern standards, this illuminated network established the essential principle of designated air corridors and provided the operational backbone that allowed commercial aviation to expand rapidly before complex electronic systems existed.

Passage B
While visual light beacons offered a temporary solution for fair-weather night flight in the 1920s, they failed fundamentally whenever fog, low clouds, or heavy precipitation obscured the ground. The true breakthrough for reliable, all-weather commercial aviation arrived with the introduction of the Four-Course Radio Range system in the late 1920s. By transmitting directional radio signals that pilots listened to via audio tones in their headsets, radio navigation freed aviators from line-of-sight visual cues. Rather than merely extending daytime visual methods into the night as light beacons did, radio technology revolutionized aeronautics by establishing the paradigm of instrument flight, rendering visual ground infrastructure obsolete for long-distance transit.

Which choice best describes the relationship between the primary function of Passage A and the primary function of Passage B?

Read the following passage carefully:

Before Dr. Maya Lin embarked on her 2018 expedition to the Antarctic Peninsula to sample subglacial meltwater, she spent two years synthesizing satellite telemetry collected during the 2012 Polar Sentinel mission. Her interest in subglacial hydrology had actually been sparked years prior, when she read Dr. Alistair Vance’s 2004 seminal paper on sub-ice lakes while completing her undergraduate thesis in 2007. Upon arriving at McMurdo Station in early 2018, Lin discovered that her primary drill rig had suffered severe mechanical failure during transit—a setback reminiscent of her failed 2015 pilot study in Greenland, where extreme weather destroyed her temperature sensors. Rather than aborting the 2018 mission, Lin pivoted to deploying autonomous thermal probes, a technique she had perfected while consulting for a deep-sea geothermal project in 2016. By late 2019, after analyzing the data recovered by the thermal probes, Lin published her landmark study proving the existence of interconnected subglacial rivers.

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

Read the passage below and evaluate the statement that follows.

In early twentieth-century organology, the study of musical instruments, scholars asserted that the structural evolution of bowed stringed instruments in early modern Europe was driven almost exclusively by deliberate acoustic refinement aimed at maximizing sound projection in large public halls. Proponents of this view frequently pointed to the consistent thickness profiles and specific arching patterns observed in classic Italian violins as evidence of early master luthiers possessing an advanced theoretical understanding of vibrational mechanics.

However, recent historical analysis of sixteenth-century artisan guild records and regional trade ledgers presents a contrasting narrative. These documents indicate that timber selection—specifically the preference for high-altitude alpine spruce—was heavily dictated by strict municipal logging quotas, seasonal transport logistics, and guild-enforced material standards rather than theoretical acoustic modeling. Furthermore, workshop manuals from the period emphasize standardized geometric templates derived from traditional woodworking apprenticeships rather than acoustic experimentation. Consequently, contemporary historians argue that attributing past instrument design innovations primarily to sophisticated acoustic engineering overestimates the empirical intentions of historical artisans.

Statement: The author's argument relies on the unstated assumption that municipal logging quotas and guild material standards were not themselves originally established to optimize the acoustic performance of the instruments.

The following passage is adapted from an article on mid-twentieth-century oceanography and plate tectonics:

In the summer of 1947, oceanographer Maurice Ewing led an expedition aboard the research vessel Atlantis to map the Mid-Atlantic Ridge. At the time, prevailing geological theory held that the ocean floor consisted of a thick layer of continental granite covered by miles of accumulated sediment. Ewing utilized seismic refraction measurements, sending acoustic shockwaves from small explosive detonations down to the sea floor and recording the echo timings with hydrophones.

The results shattered long-held assumptions. Ewing discovered that the oceanic crust was composed entirely of dense basalt, not granite. Furthermore, the sediment layer measured less than 1,000 feet thick, a fraction of the 12,000-foot thickness predicted by conventional models of an ancient, static ocean floor. This thin sediment layer indicated that the Atlantic floor was geologically young, providing crucial empirical evidence that eventually supported the theory of seafloor spreading.

According to the passage, what specific equipment did Ewing use aboard the Atlantis to record the echoes of the acoustic shockwaves?

Passage:
Passive solar heating and active solar heating both capture solar radiation to provide warmth to interior spaces, but they employ distinct physical mechanisms. Passive solar systems rely entirely on the architectural elements of a building—such as south-facing glass windows and high-density masonry floors—to collect and distribute heat naturally without mechanical intervention. In contrast, active solar systems require external mechanical or electrical devices, including liquid collectors, pumps, and circulating fans, to absorb thermal energy and transport it throughout the structure.

Based on the passage, what is one explicit difference between passive solar heating and active solar heating?

If $\alpha = \arccos\left(-\frac{5}{13}\right)$, what is the exact value of $\tan\left(\alpha - \frac{\pi}{4}\right)$?

Passage A
For over a century, epigraphers studying damaged ancient stone inscriptions relied primarily on squeezes—paper-mâché impressions pressed into carved surfaces—and manual transcription. While these traditional techniques preserved physical dimensions, they often failed to capture subtle incision depths worn down by centuries of weathering. Scholars frequently engaged in subjective debates over ambiguous letterforms, as static line drawings could not account for variations in lighting or erosion patterns. Consequently, epigraphic interpretation remained bound to the physical accessibility of the artifact and the individual researcher's visual acuity.

Passage B
Recent advances in Reflectance Transformation Imaging (RTI) and neural network analysis have fundamentally altered how historians analyze degraded inscriptions. By synthesizing multiple digital photographs taken from varying light angles, RTI creates interactive surface models that allow researchers to dynamically manipulate illumination and exaggerate micro-topographical details. Rather than replacing traditional epigraphers, these digital tools supplement manual analysis by isolating consistent geometric traces of carved strokes that the human eye might overlook under ambient light. As a result, readings once deemed speculative can now be evaluated against quantitative surface data.

Which of the following best describes the relationship between the primary function of Passage A and the primary function of Passage B?

Read the four short argument excerpts regarding historic preservation policies below. Match each argument excerpt on the left with the core unstated assumption on which its conclusion fundamentally relies on the right.

The following passage is adapted from an essay on the history of experimental low-temperature physics:

In April 1911, Dutch physicist Heike Kamerlingh Onnes and his research team at Leiden University conducted groundbreaking electrical resistance experiments on metals at cryogenic temperatures. Three years earlier, in 1908, Onnes had become the first scientist to liquefy helium, achieving a temperature of $4.2\text{ K}$ ($-268.95^\circ\text{C}$). This breakthrough allowed his team to systematically investigate how electrical resistance behaved in conductors near absolute zero.

Initial theoretical opinions regarding electrical conductivity at ultra-low temperatures were sharply divided among early twentieth-century physicists. Lord Kelvin had hypothesized that as temperature decreased toward absolute zero, conduction electrons would freeze in place, causing electrical resistance to increase toward infinity. Conversely, Onnes originally conjectured that electrical resistance would steadily diminish and smoothly approach zero as thermal motion ceased.

To test these competing hypotheses, Onnes and his assistant Gilles Holst constructed a closed glass capillary tube containing purified mercury. Mercury was selected because it could be repeatedly distilled to eliminate chemical impurities that might distort resistance measurements. The capillary tube was submerged in a double-walled bath of liquid helium inside a vacuum-insulated cryostat. As the team systematically reduced the vapor pressure over the liquid helium bath to lower the temperature, Holst monitored the potential difference across the mercury capillary using a sensitive Wheatstone bridge.

At a temperature of approximately $4.20\text{ K}$, the electrical resistance of the mercury sample did not drop gradually as Onnes had anticipated. Instead, within a temperature interval of less than $0.01\text{ K}$, the resistance plummeted abruptly from $0.11\text{ ohms}$ to less than one-hundred-thousandth of an ohm ($10^{-5}\text{ ohms}$), falling below the detection threshold of their instruments. Initially suspecting a short circuit in the wiring connections outside the cryostat, Holst and Onnes spent hours inspecting the external terminal leads before confirming that the mercury had entered a novel state of zero resistance, which Onnes subsequently named 'superconductivity' in 1912.

According to the passage, why did Kamerlingh Onnes and his team specifically choose mercury for their low-temperature electrical resistance experiments?

Passage Excerpt:
Glaciers move across landscapes through two primary mechanisms: basal sliding and internal deformation. Basal sliding occurs when liquid water under immense pressure acts as a lubricant beneath the glacier, allowing the entire ice mass to melt slightly at its base and slide over bedrock. In contrast, internal deformation occurs when individual ice crystals within the glacier slip past one another under the weight of overlying ice without requiring subglacial liquid water. While basal sliding dominates in warmer temperate glaciers where basal ice reaches the melting point, internal deformation is the principal movement mechanism in cold polar glaciers, where the ice remains frozen solid to the bedrock below.

Based on the passage, what is one explicitly stated difference between basal sliding and internal deformation?

In right triangle $ABC$, the right angle is located at vertex $B$. Point $D$ lies on leg $BC$ such that $AB = 12$ inches and $BD = 9$ inches. If segment $AD$ is equal in length to segment $DC$, what is the length of leg $BC$, in inches?

In right triangle $PQR$, the right angle is at vertex $Q$. Point $S$ lies on leg $PQ$ such that the length of $PS$ is $7$ units and the length of $SQ$ is $5$ units. If $\tan(\angle PRQ) = \frac{4}{3}$, what is the value of $\cos(\angle SRQ)$?

Which of the following expressions is equivalent to $\frac{\tan \theta + \cot \theta}{\csc \theta}$ for all values of $\theta$ where the expression is defined?

Read the following passage:

In 1903, botanist Clara Hallowell began her research into native fern species in Maine. After publishing her preliminary survey in 1906, she traveled to Europe in 1908 to consult with herbarium curators. Upon returning home in 1910, Hallowell established the state's first protected botanical reserve before releasing her definitive textbook on regional flora in 1914.

Based on the passage, which of the following events occurred immediately after Clara Hallowell returned home in 1910?

Read the following passage and answer the question below.

In the spring of 1904, horologist Julien Mercier published his comprehensive manual on restoring marine chronometers, a work he began writing only after completing the delicate restoration of the HMS Resolution’s clock. Two years prior to that restoration, however, Mercier had discovered the clock’s missing brass escapement wheel in a forgotten crate at the Portsmouth naval dockyards. He did not immediately clean or reassemble the piece; instead, he spent eight months analyzing the alloy’s metallurgical composition to ensure compatibility with modern preservation solvents. Only when those tests concluded in late 1901 did Mercier return to his bench to rebuild the gear train. Interestingly, before he ever set foot in Portsmouth or laid eyes on the escapement wheel, Mercier had served an intensive six-year apprenticeship under master watchmaker Gabriel Laurent in Paris, during which he first conceptualized his non-invasive solvent technique.

According to the passage, which of the following events occurred EARLIEST in Julien Mercier's career?

Passage A
For decades, marine bioacousticians modeled baleen whale vocalizations primarily through a rigid evolutionary lens, framing humpback and blue whale songs as instinctual acoustic signals designed for long-range navigation and mate selection. Early acoustic studies prioritized quantifiable physical properties—such as frequency modulation, amplitude, and propagation distance through oceanic sound channels—treating vocal variations as simple genetic divergence between isolated populations. Under this framework, vocal patterns were viewed as static within a lineage, altering only across evolutionary timescales as genetic mutations dictated physical differences in vocal anatomy.

Passage B
Recent longitudinal tracking of humpback whale populations across the South Pacific challenges the assumption that marine acoustic communication is strictly genetically hardwired. Researchers observed entire migratory pods rapidly adopting new, complex song structures introduced by roving individual whales from neighboring western basins within a single breeding season. This swift horizontal transmission of vocal novelty across geographic boundaries suggests that whale song represents a dynamic cultural phenomenon governed by social learning rather than slow genetic shift. Consequently, bioacoustics must expand beyond mechanistic physical models to incorporate social dynamics and regional dialects.

Which of the following best describes the functional relationship between Passage B and Passage A?