Tüm alıştırma soruları

An architectural firm offers custom townhouse design packages created by making sequential choices across three stages: Exterior Facade Material (4 choices), Floor Plan Layout (5 choices), and Interior Lighting Scheme (6 choices).

However, two design constraints apply:
1. If the first exterior facade material option is chosen, the fifth floor plan layout option cannot be selected.
2. The sixth interior lighting scheme option can only be selected if the fifth floor plan layout option is chosen.

How many different valid townhouse design packages can a client create?

A software consultant charges an enterprise client a fixed setup fee of $200$ dollars plus an hourly rate of $r$ dollars for 1-on-1 technical training. For team training workshops, the consultant charges a fixed setup fee of $400$ dollars plus an hourly rate that is $50\%$ higher than the 1-on-1 rate. If the client was billed a total of $2,400$ dollars for $12$ hours of 1-on-1 training and $8$ hours of team training workshops, what is the value of $r$?

If $2^{a+b} = 12$ and $2^{a-b} = 3$, what is the value of $2^{2a} - 2^{2b}$?

A dataset $X$ consists of five positive integers with a mean of 12, a median of 12, a unique mode of 12, and a range of 8. If $\sigma^2$ represents the variance (the square of the standard deviation) of dataset $X$, what is the maximum possible value of $\sigma^2$?

Linguists examining language divergence have long relied on glottochronology—a quantitative method that assumes a constant, uniform rate of basic vocabulary replacement over time—to construct ancestral phylogenetic trees. Modeled after radiocarbon dating, classic glottochronology posits that core lexical items decay at a predictable rate across all human languages, allowing researchers to estimate the precise date when daughter languages split from a common ancestor.

However, recent computational analyses of large-scale cross-linguistic databases have exposed fundamental limitations in this constant-rate assumption. Lexical retention rates vary dramatically depending on demographic factors, such as population size and social stratification, as well as external contact dynamics like trade and conquest. For instance, small isolated speech communities often undergo rapid lexical turnover due to founder effects or taboo-driven word substitution, whereas dense trade networks can enforce vocabulary stabilization across unrelated language families. By treating vocabulary decay as a universal constant, traditional glottochronology systematically miscalculates divergence timelines, frequently underestimating the antiquity of isolated languages while overestimating that of highly networked groups.

Rather than discarding quantitative phylogenetics entirely, contemporary linguists advocate integrating relaxed clock models borrowed from evolutionary biology. These probabilistic models allow mutation rates to fluctuate across tree branches according to socio-demographic covariates. By replacing the rigid assumption of uniform decay with flexible, parameter-driven rate variations, relaxed clock methodologies reconcile computational precision with the empirical complexities of historical language shift.

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

For decades, theoretical models in evolutionary biology posited that reciprocal altruism—actions that temporarily reduce an organism's fitness to enhance another's with the expectation of future reciprocation—could evolve among non-kin only in stable populations possessing high cognitive capacity for individual recognition. However, recent empirical studies of microbial communities reveal widespread cross-feeding cooperation among genetically distinct bacterial strains that lack memory or cognitive processing. To reconcile this theoretical paradox, contemporary researchers have shifted focus from individual cognitive tracking to spatial self-organization within physical microenvironments. By constructing mathematical simulations of biofilm architecture, evolutionary biologists demonstrated that physical viscosity and localized resource consumption naturally segregate cooperative strains into dense spatial clusters. Within these micro-spatial clusters, the secretor of a costly metabolic byproduct reaps a disproportionate share of the local benefit before the metabolite diffuses to non-cooperative "cheater" strains. Thus, spatial structuring acts as a passive mechanism that produces the same evolutionary stability as cognitive reciprocity without requiring individual recognition. Consequently, the paradigm governing non-kin cooperation is expanding: spatial dynamics are no longer viewed merely as secondary background conditions, but as primary mechanisms that lower the cognitive threshold required for altruistic traits to persist.

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

Read the following short passage and match each structural pivot word or transition phrase (on the left) to its precise rhetorical function within the passage's argument trajectory (on the right):

"Proponents of early industrial mechanization argued that automated spinning frame technology would immediately reduce labor costs across textile manufacturing. However, contemporary records indicate that initial capital outlay for machinery maintenance offset short-term wage savings. Furthermore, while factory owners anticipated rapid adoption, skilled artisans organized widespread boycotts that delayed implementation for nearly a decade."

A committee of $10$ members consists of $5$ senior directors and $5$ junior associates. If $3$ committee members are selected at random one after another without replacement, what is the probability that exactly $2$ of the selected members are senior directors?

A specialty audio equipment distributor purchased a batch of $40$ high-fidelity headphones for a total cost of $\$10,000$. The distributor marked up the wholesale cost price of each unit by $60\%$ to establish its list price. During a seasonal promotion, $30$ of the headphones were sold at a $20\%$ discount off the list price. The remaining $10$ headphones were sold during an end-of-season clearance at a $40\%$ discount off the list price. What was the distributor's net profit percentage on the total purchase of all $40$ headphones?

If $x$ is a positive real number such that $\sqrt{x} - \frac{2}{\sqrt{x}} = 3$, what is the value of $x + \frac{4}{x}$?

In January, a logistics company's total monthly operational expenditure for its electric delivery fleet was $\$C$. In February, the number of vehicles in the fleet increased by $25\%$, while the monthly operational cost per vehicle decreased by $10\%$ due to optimized charging schedules. If the fleet's total operational expenditure in February was $\$33,750$, what was the total operational expenditure, in dollars, in January?

An express cargo train leaves Station A heading toward Station B at a constant speed of $50$ miles per hour. At the exact same time, a local freight train leaves Station B heading toward Station A along a parallel track at a constant speed of $30$ miles per hour. If the total distance between Station A and Station B is $360$ miles, how many miles from Station A will the two trains meet?

For much of the twentieth century, economic historians attributed the rapid post-WWII growth of agricultural productivity in Western Europe almost entirely to state-funded mechanization programs. Under this traditional model, centralized subsidies allowed farmers to adopt heavy machinery and automated harvesters, drastically reducing manual labor requirements and boosting crop yields per acre. Scholars frequently cited official government expenditure logs and parliamentary policy documents as definitive evidence that top-down fiscal intervention was the primary engine driving technological adoption across modernizing rural economies.

However, recent archival analyses of regional farming cooperatives challenge this state-centric narrative. These historical re-evaluations demonstrate that decentralized informal credit networks, organized by local agrarian guilds prior to major state intervention, had already financed widespread equipment modernization. While state subsidies undeniably accelerated the pace of capital accumulation in the late 1950s, nevertheless, they largely reinforced technological trajectories that independent cooperative networks had established decades earlier.

Furthermore, quantitative reassessments of farm productivity metrics reveal that regions relying predominantly on local cooperative capital achieved yield increases comparable to those receiving heavy state intervention. Consequently, attributing the primary catalyst of European agricultural transformation solely to central government policy misrepresents the foundational role of grassroots financial institutions.

In the context of the passage as a whole, the transition introduced by the word "However" at the beginning of the second paragraph serves primarily to:

If $a$, $b$, and $c$ are non-zero real numbers such that $a b^2 c^3 < 0$ and $\frac{a}{b} > 0$, which of the following expressions MUST be negative?

Historically, economic historians attributed the rapid industrial expansion of early nineteenth-century New England primarily to the introduction of power looms and the consolidation of textile manufacturing under a single roof. According to this traditional view, technological centralization drastically reduced transaction costs and boosted labor efficiency, thereby single-handedly creating the competitive edge that propelled the region's commercial dominance.

However, recent archival research into merchant account ledgers paints a far more nuanced picture. While mechanized weaving undeniably enhanced throughput, this technological leap coincided with a fundamental restructuring of regional credit networks and maritime trade routes. In fact, without the simultaneous emergence of flexible merchant banking syndicates willing to absorb seasonal price volatility in raw cotton, large-scale factory operations would have collapsed under the weight of inventory holding costs. Thus, the physical centralization of production was not the primary driver of industrial growth, but rather a secondary adaptation dependent on financial innovation.

Nevertheless, one must caution against overcorrecting by viewing financial capital as the sole catalyst of New England's transformation. Contrary to the assertions of some revisionist scholars who diminish the role of shop-floor technical ingenuity, small-scale machinery modifications devised by local artisans were essential in adapting European machine designs to coarse American fibers. Ultimately, the region’s economic trajectory was shaped neither by isolated technological breakthroughs nor by financial systems alone, but by a complex, interdependent synthesis of institutional finance and decentralized technical adaptation.

Which of the following best describes the structural development of the author's argument across the passage?

A data set consists of $7$ positive integers. The smallest integer in the set is $4$, the range of the set is $19$, the median is $12$, and the set has a unique mode of $8$. What is the maximum possible arithmetic mean of the $7$ integers?

Read the summary descriptions of a three-paragraph passage regarding early twentieth-century urban planning. Match each paragraph designation on the left with its primary structural function on the right.

A boutique clothing retailer purchases a line of jackets for a wholesale price of $x$ dollars. The retailer marks up the wholesale price by $50\%$ to set the original retail price. During a seasonal clearance sale, the original retail price is discounted by $30\%$. If a customer purchases a jacket during the clearance sale for $\$126$, what was the original wholesale price of the jacket, $x$?

A corporate catering service offers two lunch options: a Standard Meal and a Premium Meal. On Monday, an order of $40$ Standard Meals and $20$ Premium Meals cost a total of $\$1,300$. On Tuesday, an order of $20$ Standard Meals and $30$ Premium Meals cost a total of $\$1,250$. What is the total cost of an order consisting of $15$ Standard Meals and $15$ Premium Meals?