Tüm alıştırma soruları

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

In triangle $PQR$, a line segment parallel to $QR$ intersects sides $PQ$ and $PR$ at $S$ and $T$, respectively. If the length of segment $PS$ is $2x - 1$, the length of segment $SQ$ is $3$, the length of segment $PT$ is $x + 2$, and the length of segment $TR$ is $2$, what is the length of segment $PS$?

To create a specific type of mortar, a builder mixes cement and sand in a ratio of $2$ to $5$ by weight. To create a concrete mixture, the builder mixes this mortar with gravel in a ratio of $7$ to $3$ by weight. If the builder prepares $300$ kilograms of the concrete mixture, how many kilograms of sand are needed?

Many marine organisms contribute to the ocean's carbon cycle, but few have as complex an influence as coccolithophores. These single-celled algae construct intricate calcium carbonate shells, a process that absorbs dissolved carbon from surrounding waters. Under standard assumptions, this shell formation should act as a net carbon sink, helping to mitigate atmospheric carbon dioxide levels. However, the biochemical reaction that produces calcium carbonate also releases dissolved carbon dioxide back into the water column. Consequently, coccolithophores can actually function as net carbon sources under certain ecological conditions, complicating models that predict how marine ecosystems respond to global climate shifts.

Which choice best describes the function of the underlined sentence in the text as a whole?

In agricultural chemistry, researchers analyze how soil moisture affects the decomposition of synthetic urea fertilizers, which release ammonia gas ($NH_3$). In a controlled experiment, chemist Dr. Hiroshi Sato observed that in dry soils with less than $10\%$ moisture content, urea decomposition was extremely slow, resulting in minimal $NH_3$ emissions. However, when soil moisture was increased to $30\%$, $NH_3$ emissions rose sharply. Interestingly, in soils saturated with water ($100\%$ moisture content), $NH_3$ emissions fell back to near-zero levels, even though the urea had completely decomposed. Sato noted that water-saturated soils trap dissolved ammonia as ammonium ions ($NH_4^+$) in solution, preventing it from escaping as gas. Which choice most logically completes the text?

A circular tabletop is designed to have a square glass inlay. The square glass inlay is inscribed in the circle and has an area of $64$ square inches. What is the area, in square inches, of the circular tabletop?

Angle $A$ has a measure of $d$ degrees, and angle $B$ has a measure of $r$ radians. The sum of the degree measure of angle $A$ and the degree equivalent of the measure of angle $B$ is $180$. If the measure of angle $A$, in degrees, is $3$ times the degree equivalent of the measure of angle $B$, and $r = k\pi$, what is the value of $k$?

In the $xy$-plane, a circle with its center at the origin $O(0,0)$ is tangent to a line at the point $T(3.6, 4.8)$. The line intersects the $x$-axis at point $P$ and the $y$-axis at point $Q$. What is the perimeter of triangle $OPQ$?

In square $ABCD$, the side length is $12$. Point $E$ lies on side $AB$ such that $AE = 3BE$, and point $F$ lies on side $AD$ such that $AF = FD$. What is the area of triangle $CEF$?

In the $xy$-plane, an angle $\theta$ is in standard position. The terminal ray of the angle is rotated counterclockwise by $\frac{7\pi}{6}$ radians, and then rotated clockwise by $45^\circ$. If the terminal ray of the resulting angle lies on the positive $y$-axis, which of the following could be the value of $\theta$, in degrees?

A student is preparing a presentation on early hominin evidence and has taken the following notes:

* In 1978, paleontologist Mary Leakey discovered fossilized footprints in Laetoli, Tanzania.
* The footprints are preserved in volcanic ash dated to approximately 3.7 million years ago.
* The footprints provide the earliest concrete evidence of upright bipedalism in early hominins.
* The trail consists of footprints made by three individuals of the species *Australopithecus afarensis*.

The student wants to write a sentence that emphasizes both the researcher who discovered the footprints and the approximate age of the footprints. Complete the sentence by filling in the blanks.

The architectural innovations introduced by the ancient Maya during the Classic period, particularly the vaulted arch, ______ a sophisticated understanding of structural engineering. While modern scholars initially believed these structures were purely ornamental, recent archaeological excavations of residential sites reveal that these advanced building techniques were widely utilized to improve the durability of everyday dwellings.

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

While preparing a presentation on the Rosetta Stone, a student took the following notes:
* The Rosetta Stone is an ancient Egyptian stele discovered in $1799$.
* It is made of granodiorite, a coarse-grained igneous rock.
* The stone features a decree issued at Memphis in $196\text{ BCE}$ on behalf of King Ptolemy V.
* The decree is written in three distinct scripts: Ancient Egyptian hieroglyphs, Demotic script, and Ancient Greek.
* French scholar Jean-François Champollion deciphered the hieroglyphs in $1822$.

The student wants to emphasize the physical material of the Rosetta Stone and the scripts used in its inscriptions. Which choice most effectively uses information from the given notes to accomplish this goal?

In the $xy$-plane, the line $y = mx$, where $m$ is a positive constant, is tangent to the circle defined by the equation $x^2 + y^2 - 6x - 8y + 16 = 0$. What is the value of $m$?

In historical musicology, the dissemination of hand-copied musical manuscripts across various European courts during the early Baroque period ______ how stylistic trends spread independently of published compositions. While printed books were highly expensive and difficult to distribute, these informal manuscript copies allowed composers to share their work rapidly with peers across borders. Consequently, regional variations developed much faster than previously assumed.

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

In the $xy$-plane, the graph of the equation $3x^2 + 3y^2 - 24x + 18y - c = 0$, where $c$ is a constant, represents a circle. If the area of the circle is $100\pi$, what is the value of $c$?

In a circle with center $O$, the central angle $AOB$ has a measure of $\frac{4\pi}{5}$ radians. Angle $BOC$ is adjacent to angle $AOB$ such that they form a straight line. What is the degree measure of angle $BOC$?