All practice questions

Let $N = 2^x \times 3^y \times 7^z$, where $x$, $y$, and $z$ are positive integers. If $N$ is a multiple of $84$ and $N^2$ has exactly $45$ positive integer divisors, what is the value of $x + y + z$?

In a biotechnology laboratory with $200$ research scientists, a survey was conducted regarding the use of three analytical instruments: Mass Spectrometers, NMR Spectrometers, and X-ray Diffractometers. The survey revealed that $95$ scientists use Mass Spectrometers, $80$ use NMR Spectrometers, and $75$ use X-ray Diffractometers. Exactly $20$ scientists use only Mass Spectrometers and NMR Spectrometers, exactly $15$ use only NMR Spectrometers and X-ray Diffractometers, and exactly $10$ use only Mass Spectrometers and X-ray Diffractometers. If $15$ scientists use all three instruments, how many scientists do not use any of these three instruments?

If $x = \sqrt{7 + 4\sqrt{3}}$, what is the value of $x - \frac{1}{x}$?

A bookstore chain recently added coffee cafes inside all of its retail locations to boost overall bookstore sales. During the twelve months following the installation of the cafes, total sales at these bookstores increased by 15 percent. The management concluded that adding the cafes was the primary cause of the increase in sales. Which of the following, if true, most seriously weakens the management's argument?

A set of $7$ positive integers has an arithmetic mean of $16$, a median of $15$, a unique mode of $12$, and a range of $14$. What is the maximum possible value of the largest integer in this set?

A logistics firm generates a 4-character tracking code, $c_1 c_2 c_3 c_4$, for express parcels according to the following rules:
- The first character, $c_1$, must be a non-zero digit ($1$ through $9$).
- The second character, $c_2$, must be a vowel chosen from the set $\{\text{A, E, I, O, U}\}$.
- The third character, $c_3$, must be an even digit ($0, 2, 4, 6, 8$).
- The fourth character, $c_4$, must be a consonant chosen from the set $\{\text{B, F, G, K, M, P}\}$.

If no digit may be repeated within the code, how many different tracking codes can be created?

A high-speed passenger train travels from Station X to Station Y at a constant speed of $60$ miles per hour. On the return trip along the exact same track, due to maintenance, the train travels at a constant speed of $40$ miles per hour. What is the average speed of the train, in miles per hour, for the entire round trip?

A artisanal coffee shop creates a custom blend using three grades of coffee beans: Grade P costing $10 per pound, Grade Q costing$ 15 per pound, and Grade R costing $25 per pound. The weight ratio of Grade P to Grade Q in the mixture is$ 3 : 2 $. If the weighted average cost of the total mixture is$ 18 per pound, what fraction of the total mixture, by weight, consists of Grade R?

Passage:

For much of the twentieth century, monetary historians viewed the establishment of the Bank of Amsterdam in 1609 primarily as a passive response to currency chaos—an institution created solely to stabilize debased coinage by offering accounts denominated in standardized bank money. Under this conventional framework, the bank's historical significance lay exclusively in its pioneer role as a clearinghouse that reduced transaction costs for international merchants.

However, recent archival re-examinations by institutional economists challenge this strictly transactional narrative. Scholars argue that while coin stabilization was indeed the bank's initial mandate, its operational evolution transformed it into a precursor of macroprudential central banking. By systematically managing money supply through bullion operations and maintaining liquidity reserves far exceeding contemporary mercantile norms, the bank actively steered interest rate stability across Northern Europe for over a century.

This conceptual shift does not merely refine historical chronology; it fundamentally recharacterizes early modern financial institutions. Rather than viewing central banking as an invention of late-eighteenth-century industrial states responding to sovereign debt crises, the revisionist synthesis demonstrates that core central-banking functions emerged organically from the private-settlement demands of merchant republics. Thus, the bank's ultimate importance derives not from its static role in coin stabilization, but from its dynamic orchestration of regional credit systems.

Based on the passage above, evaluate the following statement:

The primary purpose of the passage is to re-evaluate a traditional historical perspective regarding an early financial institution by presenting recent revisionist scholarship.

Consider a passage discussing a scientific debate regarding avian evolution. Read the four structural descriptions of the passage's paragraphs below and arrange them in the logical sequence of the author's rhetorical plan from the first paragraph to the last paragraph.

Data set $S$ consists of the four integers $4, 8, 12,$ and $16$. A fifth integer $k$, which is greater than the mean of data set $S$, is added to the set to form data set $T = \{4, 8, 12, 16, k\}$. If the standard deviation of data set $T$ is equal to the standard deviation of data set $S$, what is the value of $k$?

An executive drives from Office A to Office B, a distance of $120$ miles, at a constant speed of $40$ miles per hour. After spending exactly $1$ hour at Office B for a meeting, she returns to Office A along the exact same route at a constant speed of $60$ miles per hour. What is her average speed, in miles per hour, for the entire round trip including the time spent at Office B?

For decades, neoclassical economic theory operated on the assumption that market participants behave as strictly rational actors, driven solely by the goal of utility maximization. According to this traditional paradigm, individuals process all available information without bias, enabling markets to reach equilibrium efficiently. This theoretical framework provided elegant mathematical models for predicting market behavior, establishing a baseline against which economic anomalies were measured.

However, the emergence of behavioral economics in the late twentieth century challenged these foundational assumptions. Drawing on psychological research, behavioral economists demonstrated that human decision-making is systematically subject to cognitive biases and heuristics. For example, concepts such as loss aversion—the tendency to prefer avoiding losses over acquiring equivalent gains—revealed that real-world market actors frequently depart from pure economic rationality. Consequently, behavioral scholars argued that market prices often reflect collective psychological tendencies rather than purely objective fundamental valuations.

Today, contemporary financial literature increasingly attempts to synthesize these two opposing perspectives. Rather than viewing neoclassical theory and behavioral economics as mutually exclusive, modern researchers propose hybrid frameworks. These integrative models acknowledge that while rationality serves as an important long-term anchor for asset pricing, psychological biases induce significant short-term deviations. By delineating the specific conditions under which cognitive distortions override rational calculation, this emerging synthesis offers a more comprehensive understanding of market dynamics.

Which of the following best describes the function of the second paragraph in the passage as a whole?

A chef is arranging 6 distinct dishes—3 appetizers and 3 main courses—in a single straight row on a serving table for a culinary presentation. If all 3 appetizers must be placed adjacent to one another and all 3 main courses must also be placed adjacent to one another, in how many different linear arrangements can the dishes be displayed?

A logistics company models its total daily delivery cost $C$, in dollars, as a linear function of the total distance driven $d$, in miles, given by $C = m \cdot d + b$, where $m$ and $b$ are constants. The total daily cost increases by $\$180$ when the distance driven increases by $120$ miles. If the total daily delivery cost is $\$450$ when $250$ miles are driven, what is the total daily delivery cost when $400$ miles are driven?

For decades, organizational theorists operated on the assumption that strict adherence to the 'unity of command' principle—where each employee reports to a single manager—was indispensable for corporate efficiency. Originating in early twentieth-century administrative doctrine, this hierarchical structure was designed to maintain operational clarity and clear accountability in large-scale industrial enterprises.

However, the rapid diversification of consumer markets in the 1960s exposed severe rigidities in traditional functional hierarchies. In response, enterprise strategists championed the 'matrix structure,' an organizational model that overlays project-based product teams across conventional functional departments. Proponents argued that requiring managers to report dual-linearly—to both a functional director and a project leader—would foster cross-disciplinary innovation and accelerate product development timelines. By dismantling functional silos, matrix management was projected to combine the technical depth of specialized departments with the agility of dedicated project units.

Nevertheless, subsequent empirical studies of corporate governance reveal that the matrix model introduced distinct structural pathologies. Dual reporting lines frequently generated conflicting priorities between functional heads seeking standardized procedures and project directors demanding rapid customized output. Far from streamlining administrative workflows, this systemic friction often paralyzed middle management in protracted consensus-building cycles. Consequently, while the matrix architecture represented a sophisticated attempt to reconcile competing operational demands, its implementation ultimately demonstrated that flexibility achieved at the expense of structural clarity creates severe governance friction.

Which of the following best describes the primary purpose of the passage?

Many evolutionary biologists traditionally held that early mammalian diversification was strictly constrained by the ecological dominance of dinosaurs, occurring in earnest only after the end-Cretaceous mass extinction. According to this traditional view, Mesozoic mammals remained small, nocturnal, and functionally unspecialized throughout the era, filling only marginal ecological niches.

However, recent discoveries of remarkably preserved Mesozoic mammalian fossils have challenged this long-standing narrative. High-resolution biomechanical analyses of these specimens reveal an impressive array of specialized dental and skeletal adaptations—including structural modifications for digging, swimming, and aerial gliding—demonstrating that Mesozoic mammals occupied diverse ecological roles long before the extinction of the dinosaurs.

Consequently, paleontologists are revising their understanding of early mammalian history. Rather than viewing the end-Cretaceous extinction as the initial trigger for mammalian adaptive variation, researchers now regard it as an event that expanded opportunities for lineages that had already established substantial morphological diversity.

Which of the following best describes the function of the second paragraph in the context of the passage as a whole?

Historically, corporate turnaround strategies have prioritized drastic cost-cutting—often termed "retrenchment"—under the premise that eliminating operational inefficiencies is a prerequisite for financial recovery. Proponents of retrenchment contend that immediate reductions in capital expenditure and workforce size stabilize cash flow, thereby creating a secure foundation for subsequent strategic reorientation. However, recent empirical analyses of post-restructuring firm performance challenge this conventional sequencing. These studies demonstrate that while retrenchment may yield short-term solvency, firms that rely primarily on cost suppression frequently suffer severe erosion of dynamic capabilities, ultimately impairing their long-term competitive positioning.

To resolve this operational tension, contemporary organizational theorists propose a "dual-emphasis" paradigm, arguing that retrenchment must be executed concurrently with targeted asset deployment toward innovation-driven capabilities. Rather than viewing efficiency gains and revenue growth as sequential stages, this framework conceptualizes them as interdependent processes. By reallocating assets carved out through efficiency measures into high-potential strategic initiatives, organizations can mitigate the disruptive morale losses and talent drain typically associated with aggressive downsizing. Nonetheless, skeptics argue that simultaneous execution overburdens managerial bandwidth, particularly in turbulent market environments where executive focus is easily fragmented. Ultimately, while the dual-emphasis model offers a compelling theoretical synthesis, its success hinges on whether leadership can maintain rigorous cost controls without stifling the cross-functional agility necessary for value creation.

Which of the following best expresses the primary purpose of the passage?

Read the short passage below and match each excerpt to its structural function within the passage.

Passage:
Evolutionary biologists have long debated whether behavioral plasticity promotes or impedes genetic adaptation. A recent empirical study of urban house finches demonstrates that behavioral shifts in foraging allowed populations to colonize novel habitats prior to any observable genetic changes. This finding supports the broader claim that flexible behavioral responses frequently serve as the primary engine driving evolutionary diversification.