Tüm alıştırma soruları

In triangle $PQR$, point $S$ lies on side $PQ$ and point $T$ lies on side $PR$ such that segment $ST$ is parallel to segment $QR$. The length of segment $PS$ is $x$, the length of segment $SQ$ is $6$, the length of segment $ST$ is $x + 2$, and the length of segment $QR$ is $2x + 7$. What is the length of segment $QR$?

Materials scientists studying spider silk have long marvelled at its combination of high tensile strength and extensibility, which allows webs to absorb the impact of flying prey without breaking. ______ researchers have found that dragline silk, the main support thread of a web, can stretch up to forty percent of its original length and support weights comparable to high-grade steel before fracturing.

Which choice completes the text with the most logical transition?

For an acute angle $\theta$, the expression $\frac{\sin^4(\theta) - \cos^4(\theta)}{\sin(\theta) - \cos(\theta)}$ is equal to $\frac{7}{5}$. What is the value of $25\sin(\theta)\cos(\theta)$?

A circle has a central angle of $60^\circ$ that subtends an arc of length $4\pi$. What is the circumference of the circle?

An agricultural department monitored the population of honeybee colonies in two adjacent conservation areas starting in 2018 ($t = 0$). In Area 1, the number of colonies increases by a constant 230 colonies each year. In Area 2, the number of colonies increases by a constant 10% each year. At $t = 0$, both areas had 2,000 colonies. If the model for Area 1 predicts $L$ colonies at $t = 3$ and the model for Area 2 predicts $E$ colonies at $t = 3$, what is the value of $L - E$?

A transportation planner wants to estimate the proportion of commuters in a city who regularly use public transit to travel to work. The city has $60,000$ commuters. The planner conducts a survey by randomly selecting $300$ commuters from a database of registered transit pass holders in the city and finds that $240$ of them regularly use public transit to travel to work. Which of the following is the most appropriate conclusion based on the design of the survey?

Plants often defend against herbivores by emitting volatile organic compounds (VOCs) that recruit predatory insects. Producing VOCs requires significant nitrogen, a nutrient essential for plant growth. In a study of *Solanum lycopersicum* (tomato plants), researchers observed that plants grown in nitrogen-deficient soil reduced their VOC emissions by over 60% when attacked by caterpillars, compared to plants in nitrogen-rich soil. However, the nitrogen-deficient plants showed a corresponding increase in the production of glandular trichomes (hair-like structures on leaves), which physically impede herbivores and are composed primarily of carbon. Because carbon is abundant even in nutrient-poor soils, researchers concluded that under nitrogen limitations, *Solanum lycopersicum* shifts its defense strategy. Which choice most logically completes the text?

Iron is an essential micronutrient for human health, but excess levels can lead to tissue damage. In 2001, researchers discovered that the liver secretes a peptide hormone called hepcidin, which serves as the primary regulator of systemic iron balance. Hepcidin acts by binding to ferroportin, the only known cellular iron exporter found on the surface of macrophages and enterocytes. Upon binding, hepcidin induces the internalization and degradation of ferroportin. This action directly halts the release of iron into the blood plasma, effectively lowering circulating iron levels. Consequently, when hepcidin levels are abnormally low, ferroportin remains active, leading to excessive iron absorption and accumulation in vital organs.

According to the text, how does hepcidin affect circulating iron levels?

In a right triangle, the two acute angles are $\theta$ and $\phi$. If $\sin(\theta) = \frac{7}{25}$ and $\cos(\phi) = \frac{k}{50}$, what is the value of $k$?

For a cubic polynomial function $p$, the graph of $y = p(x)$ in the $xy$-plane has $x$-intercepts at $(1, 0)$ and $(4, 0)$, and $x + 3$ is a factor of $p(x)$. If $p(0) = -24$, what is the value of $p(2)$?

Museum conservators face unique challenges when protecting organic artifacts. In museum curation, the long-term preservation of historic oil paintings on delicate wood panels, which are highly sensitive to sudden fluctuations in relative humidity, ________ a combination of stable microclimates and precise environmental monitoring to prevent the wood from warping. Conservators must therefore rely on advanced climate control systems to maintain stable conditions.

Which choice completes the text so that it conforms to the conventions of Standard English?

Architects and master builders in medieval Europe developed key Gothic architectural innovations not just to achieve aesthetic grandeur, but to solve pressing structural engineering challenges. ______ they introduced the ribbed vault and the pointed arch to distribute the immense weight of stone ceilings more efficiently than the heavy, semi-circular Romanesque vaults could.

Which choice completes the passage with the most logical transition?

The function $h(t) = -16t^2 + v_0 t + h_0$ models the height $h(t)$, in feet, of a model rocket $t$ seconds after launch, where $v_0$ and $h_0$ are constants. The rocket reaches its maximum height of $100$ feet above the ground $2$ seconds after it is launched. Which of the following equations defines $h(t)$?

In a circle with center $O$, central angle $AOB$ has a measure of $\frac{5\pi}{6}$ radians. If the radius of the circle is $12$, the length of arc $AB$ is $k\pi$. What is the value of $k$?

In behavioral economics, the "endowment effect" describes the tendency of individuals to value an item they own more highly than an equivalent item they do not own. To investigate its underlying mechanism, researcher Emily Vance conducted an experiment where one group was given a mug (owners) and another was given money (buyers). Vance hypothesized that if the effect is driven primarily by loss aversion—the pain of losing an item exceeding the pleasure of gaining it—then owners would value the mug much higher than buyers. However, a third group, allowed only to choose between receiving the mug or the money, valued the mug just as highly as the owners did. Which choice most logically completes the text?

Many plant species possess sophisticated defense mechanisms that allow them to warn neighboring foliage of impending herbivore attacks. ______ when an acacia tree is browsed by an antelope, it releases volatile ethylene gas into the air, signaling nearby acacias to immediately ramp up their production of toxic tannins.

Which choice completes the text with the most logical transition?

Which word completes the text with the most logical and precise word?

In the $xy$-plane, the terminal ray of an angle $\theta$ in standard position intersects the unit circle at a point $P$. If the line passing through $P$ and the point $(0, 1)$ has a slope of $-\frac{1}{2}$, what is the value of $\sin(\theta)$?

For decades, historians and archaeologists debated the origins of the complex agricultural terraces in the Andes, with many early researchers asserting that these systems were imported by foreign cultures. However, recent sediment analyses and carbon-dating of the soil have _______ this hypothesis, demonstrating that indigenous communities developed these sophisticated farming techniques independently over centuries.

Which choice completes the text with the most logical and precise word or phrase?

A research group conducted a survey using a random sample of $250$ residents of a town with a total population of $18,000$ residents. Of the residents surveyed, $64\%$ reported that they recycle regularly. The survey has a margin of error of $3\%$. Based on the survey results, what is the difference between the maximum and minimum estimated number of residents in the town who recycle regularly?