Tüm alıştırma soruları

Soru 1101Soru →

In many forest ecosystems, the complex symbiotic network of subterranean mycorrhizal fungi and tree roots facilitates the rapid exchange of carbon and nutrients across vast distances. However, a single thread of these fungal filaments, which can extend through meters of soil to connect distant saplings to the main grid, often remains isolated from ______ surrounding web.

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

their

its

it's

they

Cevabı ve açıklamayı göster

Cevap: its

Cevap

its

The correct answer is the singular possessive determiner 'its'. The pronoun must agree in number with its antecedent, the singular noun 'thread' (from the phrase 'a single thread of these fungal filaments'). Additionally, because the pronoun modifies the noun phrase 'surrounding web', the possessive case 'its' is required.

Adım Adım Çözüm

1

Identify the antecedent of the pronoun needed in the blank.

The antecedent is the singular noun 'thread' from the subject noun phrase 'a single thread of these fungal filaments'.

The sentence states that a single thread remains isolated from the surrounding web, so the pronoun must refer back to 'thread'.

2

Determine the required grammatical case for the pronoun.

A possessive determiner is required to modify the noun phrase 'surrounding web'.

The blank directly precedes and modifies the noun phrase 'surrounding web', indicating ownership or association.

3

Select the pronoun that matches both the singular number of the antecedent and the possessive case requirement.

The singular possessive determiner 'its' is the correct choice.

It correctly agrees with the singular, gender-neutral antecedent 'thread' and functions as a possessive determiner.

Anahtar Kavram

Pronoun-Antecedent Agreement and Case

Soru 1102Soru →

A line in the coordinate plane is defined by the function $f(x) = px + q$ , where $p$ and $q$ are constants. This line contains the point $(6, 2)$ and intersects the vertical axis at a point with a negative $y$ -coordinate. If the region bounded by the line and the two coordinate axes is a right triangle with an area of $25$ , what is the value of $p$ ?

\frac{1}{18}

18

2

50

Cevabı ve açıklamayı göster

Cevap: 2

Cevap

2

The correct answer is the option representing the value

2

. Substituting the point

(6, 2)

into the linear function gives

q = 2 - 6p

. The area of the right triangle formed by the intercepts and the origin is given by

\frac{1}{2} \times |\text{base}| \times |\text{height}| = \frac{q^2}{2p}

. Setting this equal to

25

leads to the quadratic equation

18p^2 - 37p + 2 = 0

. Factoring this equation yields

p = 2

or

p = \frac{1}{18}

. Checking the

y

-intercept condition (

q < 0

) shows that only

p = 2

yields a negative

y

-intercept (

q = -10

), whereas

p = \frac{1}{18}

yields a positive

y

-intercept (

q = \frac{5}{3}

).

Adım Adım Çözüm

1

Use the point

(6, 2)

to express

q

in terms of

p

.

q = 2 - 6p

Since the line contains

(6, 2)

, substituting

x = 6

and

f(x) = 2

into the function

f(x) = px + q

must satisfy the equation.

2

Determine the intercepts of the line on the coordinate axes.

The vertical intercept is

(0, q)

and the horizontal intercept is

\left(-\frac{q}{p}, 0\right)

.

The vertical intercept is found by setting

x = 0

, and the horizontal intercept is found by setting

f(x) = 0

.

3

Express the area of the bounded right triangle and set it equal to

25

.

\frac{q^2}{2p} = 25 \Rightarrow q^2 = 50p

The area of a right triangle with vertices at the origin and the intercepts is

\frac{1}{2} \times |\text{base}| \times |\text{height}| = \frac{1}{2} \left(-\frac{q}{p}\right)(-q) = \frac{q^2}{2p}

.

4

Substitute

q = 2 - 6p

into

q^2 = 50p

and solve the quadratic equation.

18p^2 - 37p + 2 = 0 \Rightarrow (18p - 1)(p - 2) = 0 \Rightarrow p = 2

or

p = \frac{1}{18}

.

Substituting the expression for

q

gives a quadratic equation in terms of

p

that can be factored to find its roots.

5

Verify which root satisfies the condition that the vertical intersection point has a negative

y

-coordinate.

For

p = 2

,

q = -10

(valid). For

p = \frac{1}{18}

,

q = \frac{5}{3}

(invalid). Thus,

p = 2

.

The problem states that the vertical axis intersection (

y

-intercept) must have a negative

y

-coordinate (

q < 0

).

Anahtar Kavram

Using coordinate points and geometric properties of intercepts to determine parameters of linear functions

Soru 1103Soru →

In the $xy$ -plane, the graph of a linear function $f$ has a negative slope and a positive $y$ -intercept. The graph of $f$ intersects the $x$ -axis at point $A$ and the $y$ -axis at point $B$ . A second line, $L$ , is perpendicular to the graph of $f$ and passes through the origin. If line $L$ intersects the graph of $f$ at point $P$ such that the ratio of the area of triangle $OAP$ to the area of triangle $OBP$ is $4:9$ , what is the slope of the graph of $f$ ?

-\frac{3}{2}

-\frac{2}{3}

-\frac{9}{4}

-\frac{4}{9}

Cevabı ve açıklamayı göster

Cevap: $-\frac{3}{2}$

Cevap

The slope of the graph of

f

is

-\frac{3}{2}

.

The correct answer is

-\frac{3}{2}

. By setting up the equations of the lines

f(x) = mx + b

and

y = -\frac{1}{m}x

, we can express the coordinates of the intercepts

A

and

B

, as well as the intersection point

P

, in terms of

m

and

b

. Calculating the areas of triangle

OAP

and triangle

OBP

gives

\text{Area}(OAP) = \frac{1}{2} \left(-\frac{b}{m}\right) y_0

and

\text{Area}(OBP) = \frac{1}{2} b x_0

. Since

P

lies on the perpendicular line,

y_0 = -\frac{1}{m}x_0

. Substituting this into the area ratio yields

\frac{\text{Area}(OAP)}{\text{Area}(OBP)} = \frac{1}{m^2}

. Equating this to the given ratio

\frac{4}{9}

results in

m^2 = \frac{9}{4}

. Given that the slope is negative,

m

must be

-\frac{3}{2}

.

Adım Adım Çözüm

1

Define the linear function

f(x)

and the line

L

perpendicular to it.

Let the function be

f(x) = mx + b

, where

m < 0

and

b > 0

. The line

L

perpendicular to the graph of

f

that passes through the origin has the equation

y = -\frac{1}{m}x

.

This establishes the equations of both lines in terms of the slope

m

and

y

-intercept

b

.

2

Determine the coordinates of the intercepts

A

and

B

, and the intersection point

P(x_0, y_0)

.

The

x

-intercept is

A\left(-\frac{b}{m}, 0\right)

and the

y

-intercept is

B(0, b)

. The intersection point

P(x_0, y_0)

is the solution to

mx + b = -\frac{1}{m}x

, which gives

x_0 = -\frac{mb}{m^2+1}

and

y_0 = \frac{b}{m^2+1}

.

Finding these points allows us to express the dimensions of triangles

OAP

and

OBP

.

3

Calculate the areas of triangles

OAP

and

OBP

and find their ratio in terms of

m

.

The base of triangle

OAP

along the

x

-axis is

OA = -\frac{b}{m}

and its height is

y_0

. Thus,

\text{Area}(OAP) = \frac{1}{2} \left(-\frac{b}{m}\right) y_0

. The base of triangle

OBP

along the

y

-axis is

OB = b

and its height is

x_0

. Thus,

\text{Area}(OBP) = \frac{1}{2} b x_0

. The ratio of their areas is

\frac{\text{Area}(OAP)}{\text{Area}(OBP)} = \frac{-\frac{b}{m} y_0}{b x_0} = -\frac{y_0}{m x_0}

. Since

P

lies on

L

, we have

\frac{y_0}{x_0} = -\frac{1}{m}

. Substituting this gives the ratio

\frac{\text{Area}(OAP)}{\text{Area}(OBP)} = -\frac{1}{m}\left(-\frac{1}{m}\right) = \frac{1}{m^2}

.

This simplifies the geometric relationship to a direct relation between the area ratio and the slope of the function.

4

Solve for the slope

m

using the given ratio of

4:9

.

Setting

\frac{1}{m^2} = \frac{4}{9}

yields

m^2 = \frac{9}{4}

. Since the problem states the slope is negative, we take the negative square root to get

m = -\frac{3}{2}

.

This identifies the correct slope value matching the given constraints.

Anahtar Kavram

Using coordinate geometry and system of linear equations to determine slopes and intersections, and relating those to geometric areas on the coordinate plane.

Soru 1104Soru →

Read the passage below. Based on the rules of Standard English conventions, which punctuation mark must fill the blank to correctly set off the nonessential element?

Aşağıdaki boşlukları doldurun

In 1735, the self-taught English clockmaker John Harrison presented his first marine chronometer—a revolutionary timekeeping device designed to accurately calculate longitude at seato the British Board of Longitude. Harrison’s ingenious invention solved a monumental navigation problem that had baffled scientists for centuries, enabling sailors to safely navigate the open ocean by determining their precise east-west position.

Cevabı ve açıklamayı göster

Cevap

An em dash (—)

The correct punctuation mark is an em dash because the nonessential appositive phrase 'a revolutionary timekeeping device designed to accurately calculate longitude at sea' is opened with an em dash. Under Standard English conventions, nonessential elements must be set off with matching pairs of punctuation (comma-comma, dash-dash, or parenthesis-parenthesis). Thus, an em dash must be used to close the phrase to maintain punctuation symmetry.

Adım Adım Çözüm

1

Identify the nonessential element in the sentence.

The phrase 'a revolutionary timekeeping device designed to accurately calculate longitude at sea' is determined to be an appositive phrase that provides nonessential descriptive details about the 'marine chronometer'.

Identifying the nonessential element is necessary to determine where punctuation boundaries must be established.

2

Examine the punctuation mark used to open the nonessential element.

An em dash ('—') is used immediately after 'chronometer' to introduce the appositive phrase.

Standard English conventions require that nonessential elements be set off from the rest of the sentence using a matching pair of punctuation marks.

3

Determine the correct punctuation mark to close the nonessential element.

An em dash ('—') must be placed in the blank before 'to the British Board' to match the opening em dash.

Using any other punctuation mark would violate the rule of punctuation symmetry.

Anahtar Kavram

Punctuation of Nonessential Elements

Soru 1105Soru →

In her analysis of Homer's *Odyssey*, classicist Emily Wilson highlights how the narrative frequently shifts between the perspectives of elite heroes and marginalized figures. Wilson argues that while the former group's members dominate the poem's formal speeches, the other ______ quiet observations provide a crucial counter-narrative, offering a critique of the rigid social hierarchy that governs ancient Ithaca.

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

characters

character's

characters'

character

Cevabı ve açıklamayı göster

Cevap: characters'

Cevap

characters'

The correct answer is the plural possessive noun 'characters''. In the sentence, the word in the blank modifies the noun 'observations' to indicate who is observing, requiring a possessive form. The context refers back to the plural 'marginalized figures,' indicating that a plural possessive noun is needed. For plural nouns ending in 's,' the possessive is formed by adding an apostrophe after the 's.'

Adım Adım Çözüm

1

Identify the noun that the blank word modifies and determine if possession is required.

The word in the blank modifies the noun 'observations'. Because the observations belong to the other characters in the narrative, a possessive form is necessary.

Possessive nouns are used to indicate ownership or association with a following noun.

2

Determine whether the possessive noun should be singular or plural based on the context.

The blank refers back to 'marginalized figures', which is a plural noun phrase.

The grammatical number of the possessive noun must agree with its plural antecedent.

3

Combine the plural and possessive requirements to select the correct spelling and punctuation.

The plural of 'character' is 'characters', and its possessive form is created by adding an apostrophe after the 's' ('characters'').

Plural nouns ending in 's' form the possessive by adding only an apostrophe.

Anahtar Kavram

Plural and Possessive Nouns

Soru 1106Soru →

A project manager uses the linear equation $ax + by = c$ to model the allocation of resources, where $x$ represents the number of hours spent on research, $y$ represents the number of hours spent on development, and $a, b,$ and $c$ are positive constants. Initially, the project manager allocates $40$ hours for research and $30$ hours for development. To optimize the project, the manager decides to increase the research hours by $25\%$ and decrease the development hours by $20\%$ such that the total resource value, $c$ , remains unchanged. Which of the following equations correctly expresses $a$ in terms of $b$ ?

a = 1.67b

a = 0.6b

a = 0.64b

a = 1.6b

Cevabı ve açıklamayı göster

Cevap: $a = 0.6b$

Cevap

The equation

a = 0.6b

correctly expresses

a

in terms of

b

.

The correct equation is

a = 0.6b

. The initial resource allocation is represented by

40a + 30b = c

. Increasing

40

by

25\%

gives

50

, and decreasing

30

by

20\%

gives

24

, which yields the new equation

50a + 24b = c

. Equating these two expressions since

c

is constant gives

40a + 30b = 50a + 24b

. Simplifying this equation results in

6b = 10a

, which gives

a = 0.6b

.

Adım Adım Çözüm

1

Write the linear equation representing the initial allocation of resources.

40a + 30b = c

The initial values are

40

hours for research (

x

) and

30

hours for development (

y

).

2

Calculate the new values of

x

and

y

after the percentage changes are applied.

x = 40 \times 1.25 = 50

and

y = 30 \times 0.80 = 24

Research hours increase by

25\%

and development hours decrease by

20\%

.

3

Write the new linear equation with the updated resource allocations.

50a + 24b = c

The total resource value

c

remains unchanged.

4

Equate the two expressions for

c

and solve for

a

in terms of

b

.

40a + 30b = 50a + 24b \implies 6b = 10a \implies a = 0.6b

Since both equations equal the same constant

c

, we set them equal to each other and isolate the variable

a

.

Anahtar Kavram

Expressing and manipulating linear relationships in two variables with constraints.

Alternatif Yöntem

Instead of keeping

c

general, we can choose a convenient value for

c

, such as

c = 120

. This gives two linear equations in terms of

a

and

b

:

40a + 30b = 120

and

50a + 24b = 120

. Solving this system for the ratio of

a

to

b

yields

a = 1.33

and

b = 2.22

, which confirms that

a/b = 0.6

, or

a = 0.6b

.

Tahmini Süre:3m 0s

Soru 1107Soru →

In his analysis of ancient Mediterranean trade networks, historian Marcus Aureli examines the economic motivations of early merchants. He argues that the rapid expansion of maritime routes during the late Roman Republic was driven not only by a growing domestic demand for exotic agricultural goods but also ______ stronger diplomatic ties with eastern provinces.

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

by a strategic effort to establish

a strategic effort to establish

to make a strategic effort to establish

through a strategic effort to establish

Cevabı ve açıklamayı göster

Cevap: by a strategic effort to establish

Cevap

by a strategic effort to establish

The correct answer maintains parallel structure by using a prepositional phrase starting with the preposition 'by' ('by a strategic effort to establish'). This matches the structure of the first element in the correlative conjunction 'not only... but also,' which is 'by a growing domestic demand for exotic agricultural goods.'

Adım Adım Çözüm

1

Identify the grammatical construction and the parallel elements in the sentence.

The sentence uses the correlative conjunction structure 'not only... but also.'

This correlative conjunction requires that the elements following 'not only' and 'but also' be grammatically parallel.

2

Determine the grammatical form of the first element.

The phrase following 'not only' is 'by a growing domestic demand for exotic agricultural goods,' which is a prepositional phrase starting with the preposition 'by.'

This establishes that the element following 'but also' must also be a prepositional phrase starting with 'by' to maintain parallel structure.

3

Evaluate the choices to find the one that matches this grammatical structure.

The phrase 'by a strategic effort to establish' begins with the preposition 'by' followed by a noun phrase, matching the structure of the first element.

Only this option correctly maintains the prepositional phrase structure required for parallelism in this sentence.

Anahtar Kavram

Parallel Structure with Correlative Conjunctions

Tahmini Süre:1m 0s

Soru 1108Soru →

A student tracks the remaining battery percentage, $y$ , of a tablet after $x$ hours of use. The table below shows the battery percentage at three different times:

Time ( $x$ , hours)	Battery percentage ( $y$ )
0	90
3	75
6	60

Which of the following equations represents the relationship between $x$ and $y$ ?

y = -15x + 90

y = -\frac{1}{5}x + 90

y = -5x + 90

y = 90x - 5

Cevabı ve açıklamayı göster

Cevap: $y = -5x + 90$

Cevap

y = -5x + 90

The correct equation is

y = -5x + 90

. According to the table, when time

x = 0

, the battery percentage

y = 90

, which indicates that the y-intercept of the line is

90

. The slope can be determined using any two points from the table, such as

(0, 90)

and

(3, 75)

:

\text{slope} = \frac{75 - 90}{3 - 0} = \frac{-15}{3} = -5

. Writing this in slope-intercept form gives

y = -5x + 90

.

Adım Adım Çözüm

1

Identify the y-intercept from the table.

The y-intercept

b

is

90

.

When the time

x = 0

hours, the battery percentage

y = 90

. This corresponds to the y-intercept of the linear equation.

2

Calculate the slope (rate of change) using two points from the table.

The slope

m

is

-5

.

Using the points

(0, 90)

and

(3, 75)

, the slope is calculated as the change in

y

divided by the change in

x

:

m = \frac{75 - 90}{3 - 0} = \frac{-15}{3} = -5

.

3

Substitute the slope and y-intercept into the slope-intercept form equation.

y = -5x + 90

Plugging

m = -5

and

b = 90

into

y = mx + b

yields the final equation.

Anahtar Kavram

Determining a linear equation in two variables from a table of values.

Tahmini Süre:1m 0s

Soru 1109Soru →

In 1977, oceanographers exploring the Galapagos Rift made a startling discovery: deep-sea hydrothermal vents teeming with previously unknown life. Before this historic expedition, scientists _______ that life could not exist in the deep ocean without sunlight to power photosynthesis.

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

believe

have believed

had believed

would believe

Cevabı ve açıklamayı göster

Cevap: had believed

Cevap

had believed

The correct option is 'had believed'. The passage describes a series of events in the past: the oceanographers' expedition and discovery occurred in 1977 ('made'). The sentence with the blank describes a belief held by scientists prior to that past expedition ('Before this historic expedition'). To express an action or state that was completed before another past event, standard English conventions require the past perfect tense, which is formed using 'had' followed by the past participle ('had believed').

Adım Adım Çözüm

1

Identify the timeline of events in the passage.

The passage establishes that oceanographers made a discovery in 1977 ('made'), which is in the simple past tense.

Establishing the temporal anchor helps determine the relationship of other verbs in the sentence.

2

Analyze the relationship between the blank and the past events.

The blank describes a belief ('scientists _______ that life could not exist') that occurred and was completed prior to the 1977 expedition ('Before this historic expedition').

Understanding the sequence of events is crucial for choosing the correct verb aspect.

3

Select the appropriate verb tense and aspect.

To describe a state or action that occurred before another action in the past, the past perfect tense ('had' + past participle) is required. Therefore, 'had believed' is the correct form.

Conforming to standard English conventions requires the past perfect aspect for past-of-the-past relationships.

Anahtar Kavram

Verb Tense, Aspect, and Mood

Tahmini Süre:1m 0s

Soru 1110Soru →

If the equation $\frac{3}{4}(8x - 12) - \frac{5}{6}(6x - 18) = -\frac{1}{2}(4x - 8)$ is true, what is the value of $x$ ?

\frac{28}{3}

-\frac{10}{3}

-\frac{2}{3}

\frac{2}{3}

Cevabı ve açıklamayı göster

Cevap: $-\frac{2}{3}$

Cevap

-\frac{2}{3}

The correct answer is found by distributing the coefficients across each set of parentheses, combining like terms on the left side to get

x + 6 = -2x + 4

, adding

2x

to both sides to get

3x + 6 = 4

, subtracting

6

from both sides to get

3x = -2

, and finally dividing by

3

to obtain

-\frac{2}{3}

.

Adım Adım Çözüm

1

Distribute the coefficients to the terms inside the parentheses on both sides of the equation.

6x - 9 - 5x + 15 = -2x + 4

To eliminate the parentheses and prepare the equation for combining like terms.

2

Combine the constant and variable terms on the left side of the equation.

x + 6 = -2x + 4

To simplify the left side of the equation.

3

Add

2x

to both sides of the equation.

3x + 6 = 4

To group all terms containing the variable

x

on the left side.

4

Subtract

6

from both sides, then divide by

3

to solve for

x

.

3x = -2

, which gives

x = -\frac{2}{3}

To isolate

x

and find its value.

Anahtar Kavram

Linear Equations in One Variable

Soru 1111Soru →

For the linear function $f$ , the value of $f(0)$ is $3$ and the value of $f(5)$ is $18$ . What is the slope of the graph of $y = f(x)$ in the $xy$ -plane?

Cevabı ve açıklamayı göster

Cevap: 3

Cevap

The slope of the graph of

y = f(x)

is

3

.

The slope of a linear function can be determined using any two points

(x_1, y_1)

and

(x_2, y_2)

on its graph with the formula

m = \frac{y_2 - y_1}{x_2 - x_1}

. The given function values

f(0) = 3

and

f(5) = 18

correspond to the coordinate points

(0, 3)

and

(5, 18)

respectively. Substituting these coordinates into the formula gives

m = \frac{18 - 3}{5 - 0} = \frac{15}{5} = 3

.

Adım Adım Çözüm

1

Identify the coordinates of two points on the line using the given function values.

The function values

f(0) = 3

and

f(5) = 18

correspond to the points

(0, 3)

and

(5, 18)

on the graph of the function.

In function notation,

f(x) = y

represents a point

(x, y)

on the graph of the function.

2

Calculate the slope using the slope formula.

The slope

m

is calculated as

\frac{18 - 3}{5 - 0} = 3

.

The slope of a line is defined as the change in

y

divided by the change in

x

between any two points on the line.

Anahtar Kavram

Calculating the slope of a linear function from given function values.

Soru 1112Soru →

While botanical expeditions in Borneo have revealed that each of the newly cataloged species of carnivorous pitcher plants displays highly specialized physical adaptations to capture prey, ______ long-term survival in the nutrient-poor soil of the tropical rainforest remains dependent on a delicate symbiotic relationship with mutualistic ants.

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

its

their

they're

its'

Cevabı ve açıklamayı göster

Cevap: its

Cevap

The singular possessive pronoun 'its' correctly refers to the singular antecedent 'each'.

The singular possessive pronoun 'its' correctly agrees with the singular antecedent 'each'. In standard English, indefinite pronouns such as 'each' are grammatically singular, and their grammatical number is not affected by intervening plural prepositional phrases like 'of the newly cataloged species of carnivorous pitcher plants'.

Adım Adım Çözüm

1

Locate the antecedent of the pronoun needed for the blank.

The antecedent is the singular indefinite pronoun 'each' at the start of the clause.

Pronouns must match the grammatical number of their antecedents.

2

Analyze the role of intervening phrases to avoid agreement errors.

The plural nouns 'species' and 'plants' are objects of prepositional phrases and do not change the singular nature of the subject 'each'.

Intervening prepositional phrases should not affect the pronoun-antecedent agreement.

3

Identify the required case and form for the blank.

The blank modifies the noun 'survival', indicating a singular possessive pronoun is required.

The singular possessive pronoun 'its' is the only form that is grammatically correct and matches the singular antecedent.

Anahtar Kavram

Pronoun-Antecedent Agreement and Case

Soru 1113Soru →

If $3(2.5x - 4) - 1.5(x - 2) = 0.8(5x + 10)$ , what is the value of $x$ ?

Cevabı ve açıklamayı göster

Cevap: 8.5

Cevap

8.5

By distributing and simplifying terms, the linear equation simplifies to

2x = 17

, which yields

x = 8.5

.

Adım Adım Çözüm

1

Distribute the constants on both sides of the equation.

7.5x - 12 - 1.5x + 3 = 4x + 8

To simplify the expressions.

2

Combine like terms on the left side.

6x - 9 = 4x + 8

To simplify the left side of the equation.

3

Subtract

4x

from both sides of the equation.

2x - 9 = 8

To group variable terms on one side.

4

Add 9 to both sides of the equation.

2x = 17

To isolate the term with the variable.

5

Divide both sides by 2.

x = 8.5

To find the final value of the variable.

Anahtar Kavram

Linear Equations in One Variable

Alternatif Yöntem

Multiply the entire equation by 10 to clear all decimal values:

30(2.5x - 4) - 15(x - 2) = 8(5x + 10)

. Expanding this yields

75x - 120 - 15x + 30 = 40x + 80

, which simplifies to

60x - 90 = 40x + 80

. Subtracting

40x

and adding 90 to both sides gives

20x = 170

, resulting in

x = 8.5

.

Tahmini Süre:1m 30s

Soru 1114Soru →

In many songbird species, juvenile males acquire their unique songs by imitating adult tutors during a critical, time-limited developmental window. Young zebra finches, for example, must be exposed to tutor songs within their first eighty days of _______ they will fail to develop the complex, species-typical vocalizations required for successful courtship.

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

life, otherwise,

life otherwise

life; otherwise,

life, although

Cevabı ve açıklamayı göster

Cevap: life; otherwise,

Cevap

The correct choice is the option containing 'life; otherwise,' because it uses a semicolon to connect two independent clauses, followed by the conjunctive adverb 'otherwise' and a comma.

The correct choice is the option containing 'life; otherwise,' because it properly links the two independent clauses. A semicolon is used to separate the independent clauses, and the conjunctive adverb 'otherwise' is followed by a comma to establish the consequential relationship between the clauses.

Adım Adım Çözüm

1

Analyze the structure of the text surrounding the blank to identify the boundary between the clauses.

The text contains two independent clauses: 'Young zebra finches, for example, must be exposed to tutor songs within their first eighty days of life' and 'they will fail to develop the complex, species-typical vocalizations required for successful courtship.'

Identifying that both clauses can stand alone as complete sentences helps determine the correct punctuation or coordinating conjunction needed to link them.

2

Determine the logical relationship between the two independent clauses.

The second clause describes the negative consequence of failing to meet the condition stated in the first clause. The transition needs to mean 'if not' or 'or else'.

This establishes that a transitional word expressing consequence or negative condition, such as 'otherwise', is semantically necessary.

3

Select the option that conforms to standard English conventions for joining independent clauses with a conjunctive adverb.

The option using a semicolon followed by 'otherwise' and a comma correctly avoids a comma splice or run-on sentence.

Two independent clauses joined by a conjunctive adverb must be separated by a semicolon (or a period) before the adverb, typically followed by a comma.

Anahtar Kavram

Joining independent clauses using a semicolon and a conjunctive adverb.

Soru 1115Soru →

A linear function $f$ is defined by $f(x) = mx + b$ , where $m$ and $b$ are constants. The graph of $f$ in the $xy$ -plane passes through the point $(2, 5)$ . A second linear function $g$ is defined by $g(x) = f(x - 3) + 4$ . The graph of $g$ has an $x$ -intercept of $(k, 0)$ and a $y$ -intercept of $(0, -k)$ , where $k$ is a nonzero constant. What is the value of $f(8)$ ?

Cevabı ve açıklamayı göster

Cevap: 11

Cevap

11

The correct answer is 11. By using the given point on the graph of

f(x)

, we express its y-intercept

b

in terms of its slope

m

as

b = 5 - 2m

. Substituting this expression into the translation equation

g(x) = f(x - 3) + 4

yields

g(x) = mx - 5m + 9

. Setting the y-intercept of

g(x)

, which is

-5m + 9

, equal to

-k

gives

k = 5m - 9

. Evaluating

g(x)

at its x-intercept

x = k

gives

mk - 5m + 9 = 0

. Substituting the relation

-5m + 9 = -k

into this equation yields

mk - k = 0

, or

k(m - 1) = 0

. Since

k

is nonzero, we find

m = 1

, which in turn gives

b = 3

. Thus, the linear function is

f(x) = x + 3

, and

f(8) = 11

.

Adım Adım Çözüm

1

Express the y-intercept constant

b

in terms of the slope

m

.

b = 5 - 2m

Since the graph of

f(x) = mx + b

passes through

(2, 5)

, substituting

x = 2

and

f(x) = 5

gives

5 = 2m + b

.

2

Write the expression for

g(x)

in terms of

m

.

g(x) = mx - 5m + 9

By definition,

g(x) = f(x - 3) + 4 = m(x - 3) + b + 4

. Substituting

b = 5 - 2m

simplifies the expression to

g(x) = mx - 5m + 9

.

3

Relate the y-intercept of

g(x)

to the parameter

k

.

k = 5m - 9

The y-intercept of

g(x)

is

g(0) = -5m + 9

. Since the y-intercept is given as

(0, -k)

, we set

-k = -5m + 9

, which gives

k = 5m - 9

.

4

Set up an equation using the x-intercept of

g(x)

.

g(k) = mk - 5m + 9 = 0

Since the x-intercept of

g(x)

is

(k, 0)

, substituting

x = k

into the expression for

g(x)

must yield

0

.

5

Solve for the slope

m

.

m = 1

Substituting

-5m + 9 = -k

into the equation

mk - 5m + 9 = 0

gives

mk - k = 0

, which factors as

k(m - 1) = 0

. Since

k

is a nonzero constant, we divide by

k

to get

m - 1 = 0

, so

m = 1

.

6

Find the constant

b

and write the final formula for

f(x)

.

f(x) = x + 3

Substituting

m = 1

back into

b = 5 - 2m

yields

b = 3

. Therefore,

f(x) = x + 3

.

7

Evaluate

f(8)

.

11

Substituting

x = 8

into the function

f(x) = x + 3

gives

f(8) = 8 + 3 = 11

.

Anahtar Kavram

Linear function transformations and intercept properties

Soru 1116Soru →

Complete the passage with the form of the verb 'publish' that correctly completes the sentence to conform to the conventions of Standard English.

Aşağıdaki boşlukları doldurun

While studying Maya hieroglyphs in the 1950s, art historian Tatiana Proskouriakoff noticed patterns of dates that corresponded to the lifespans of historic rulers. Before Proskouriakoff her breakthrough, most scholars in the field of Mesoamerican archaeology had assumed that the glyphs represented only abstract astronomical or religious concepts, rather than records of historical events.

Cevabı ve açıklamayı göster

Cevap

published

The correct answer is 'published' because the sentence describes a sequence of two past events. The action of scholars assuming the glyphs were abstract occurred before Proskouriakoff's breakthrough. Because the earlier action is marked by the past perfect verb 'had assumed', the subsequent past action (the breakthrough) must be expressed in the simple past tense ('published') to establish the correct chronological relationship.

Adım Adım Çözüm

1

Identify the relationship between the two events in the sentence: the scholars' assumption and Proskouriakoff's breakthrough.

The sentence outlines a clear chronological order between two past events.

Establishing the sequence of events determines the correct verb tense relationship.

2

Analyze the verb tense of the earlier event in the timeline.

The earlier action of the scholars' assumption is written in the past perfect tense ('had assumed').

The past perfect tense is used for an action completed before another past action.

3

Select the correct tense for the subsequent event (the breakthrough).

The subsequent past action must be written in the simple past tense ('published').

The simple past tense is required to show that the breakthrough occurred after the period when the scholars assumed otherwise.

Anahtar Kavram

Verb Tense, Aspect, and Mood

Soru 1117Soru →

The equation $3x + 4y = 36$ represents the relationship between the number of small boxes, $x$ , and large boxes, $y$ , that can fit in a delivery van. If the van is loaded with exactly 6 large boxes, what is the number of small boxes that can also fit in the van?

Cevabı ve açıklamayı göster

Cevap: 4

Cevap

The number of small boxes that can also fit in the van is 4.

Substituting

y = 6

into the equation

3x + 4y = 36

gives

3x + 24 = 36

. Subtracting 24 from both sides results in

3x = 12

. Dividing both sides by 3 yields

x = 4

. Thus, the number of small boxes that can fit is 4.

Adım Adım Çözüm

1

Substitute 6 for

y

in the equation

3x + 4y = 36

.

3x + 4(6) = 36

The variable

y

represents the number of large boxes, which is given as 6.

2

Simplify the multiplication in the equation.

3x + 24 = 36

To simplify the terms before solving for

x

.

3

Subtract 24 from both sides of the equation.

3x = 12

To isolate the variable term

3x

on one side of the equation.

4

Divide both sides of the equation by 3.

x = 4

To solve for

x

, the number of small boxes.

Anahtar Kavram

Evaluating and solving a linear equation in two variables given the value of one variable.

Tahmini Süre:45s

Soru 1118Soru →

The graph of the linear function $f$ in the $xy$ -plane is defined by $f(x) = kx + 4$ , where $k$ is a positive constant. The graph of the linear function $g$ is obtained by translating the graph of $f$ left by $3$ units and down by $2$ units. If the $x$ -intercept of the graph of $g$ is $5$ times the $x$ -intercept of the graph of $f$ , what is the value of $k$ ?

Cevabı ve açıklamayı göster

Cevap: 6

Cevap

6

The correct answer is 6. The linear function

g(x)

represents the translated graph, which is expressed as

g(x) = f(x+3) - 2

. Substituting

f(x) = kx + 4

yields

g(x) = k(x+3) + 4 - 2 = kx + 3k + 2

. The

x

-intercept of a graph is the point where the function value is

0

. Solving

f(x) = 0

gives the

x

-intercept of the graph of

f

as

x = -\frac{4}{k}

. Solving

g(x) = 0

gives the

x

-intercept of the graph of

g

as

x = -\frac{3k + 2}{k}

. We are given that the

x

-intercept of the graph of

g

is

5

times the

x

-intercept of the graph of

f

, which translates to the equation

-\frac{3k+2}{k} = 5 \left(-\frac{4}{k}\right)

. Multiplying both sides by

-k

(since

k > 0

,

k \neq 0

) yields

3k + 2 = 20

. Subtracting

2

from both sides gives

3k = 18

, and dividing by

3

results in

k = 6

.

Adım Adım Çözüm

1

Determine the equation of the translated function

g(x)

g(x) = kx + 3k + 2

Shifting the graph of

f(x)

left by

3

units and down by

2

units translates to

g(x) = f(x+3) - 2

.

2

Find the

x

-intercept of the graph of

f

x = -\frac{4}{k}

The

x

-intercept is the input value for which the output of the function is zero, so we solve

f(x) = 0

.

3

Find the

x

-intercept of the graph of

g

x = -\frac{3k + 2}{k}

We set the equation of

g(x)

to zero and solve for

x

in terms of

k

.

4

Set up and solve the equation relating the two

x

-intercepts

k = 6

Using the relation

x_g = 5 \cdot x_f

, we obtain

-\frac{3k + 2}{k} = 5 \left(-\frac{4}{k}\right)

, which simplifies to

3k + 2 = 20

, leading to

k = 6

.

Anahtar Kavram

Function translations and finding intercepts of linear equations

Soru 1119Soru →

The sentence below is missing a punctuation mark. What punctuation mark should be placed in the blank to correctly join the two independent clauses?

Aşağıdaki boşlukları doldurun

The novel was written in the nineteenth century nevertheless, its themes of social justice remain highly relevant to readers today.

Cevabı ve açıklamayı göster

Cevap

A semicolon (;) is the correct punctuation mark to join the two independent clauses before the conjunctive adverb.

A semicolon is the correct punctuation mark because it is used to connect two independent clauses, especially when the second clause begins with a conjunctive adverb like 'nevertheless'.

Adım Adım Çözüm

1

Identify the clauses on either side of the blank.

The clause 'The novel was written in the nineteenth century' and the clause 'nevertheless, its themes of social justice remain highly relevant to readers today' are both independent clauses.

Determining if the clauses are independent is necessary to select the appropriate punctuation.

2

Identify the transition word starting the second clause.

The second clause starts with the transition word (conjunctive adverb) 'nevertheless'.

Transition words dictate specific punctuation requirements when joining independent clauses.

3

Apply the correct punctuation rule for joining independent clauses with a transition word.

A semicolon (;) must be placed before 'nevertheless' to correctly link the two clauses.

A semicolon is required to separate two independent clauses when the second begins with a conjunctive adverb.

Anahtar Kavram

Semicolons with conjunctive adverbs

Soru 1120Soru →

In the $xy$ -plane, a line with a positive slope $m$ and a $y$ -intercept of $(0, b)$ passes through the point $(b, 12)$ , where $b > 0$ . If the area of the triangle bounded by the line, the $x$ -axis, and the $y$ -axis is $4$ , what is the value of $m$ ?

Cevabı ve açıklamayı göster

Cevap: 2

Cevap

2

The line equation is

y = mx + b

. Substituting the point

(b, 12)

gives

12 = b(m + 1)

. The

x

-intercept of the line is at

(-\frac{b}{m}, 0)

, which gives a base length of

\frac{b}{m}

for the right triangle, while the height is

b

. The area of the triangle is

\frac{b^2}{2m} = 4 \implies b^2 = 8m

. Substituting

b = \frac{12}{m + 1}

into this equation yields

\frac{144}{(m + 1)^2} = 8m \implies m(m + 1)^2 = 18

. Solving this for positive values of

m

gives the unique solution

m = 2

.

Adım Adım Çözüm

1

Express the line equation and substitute the given point

(b, 12)

.

b(m + 1) = 12

Since the line has a

y

-intercept of

(0, b)

and a slope of

m

, its equation is

y = mx + b

. Substituting the point

(b, 12)

yields

12 = m(b) + b = b(m + 1)

.

2

Determine the lengths of the base and height of the triangle formed by the line and the coordinate axes.

Base

= \frac{b}{m}

, Height

= b

The

y

-intercept is at

(0, b)

and the

x

-intercept is at

(-\frac{b}{m}, 0)

. Since

b > 0

and

m > 0

, the height is

b

and the base is the absolute value of the

x

-intercept, which is

\frac{b}{m}

.

3

Write the area equation of the triangle and simplify it.

b^2 = 8m

The area of the right triangle is

\frac{1}{2} \times \text{base} \times \text{height} = \frac{1}{2} \left(\frac{b}{m}\right)b = \frac{b^2}{2m}

. Setting this equal to the given area of

4

gives

\frac{b^2}{2m} = 4 \implies b^2 = 8m

.

4

Solve the system of equations for

m

.

m = 2

Substituting

b = \frac{12}{m + 1}

into

b^2 = 8m

yields

\left(\frac{12}{m + 1}\right)^2 = 8m \implies \frac{144}{(m + 1)^2} = 8m \implies m(m + 1)^2 = 18

. Since

m(m + 1)^2

is strictly increasing for positive

m

, testing small integers shows

2(3)^2 = 18

, so

m = 2

is the unique positive real solution.

Anahtar Kavram

Formulating linear equations in slope-intercept form and solving non-linear systems of equations derived from geometric constraints.

Alternatif Yöntem

Instead of algebraically expanding the cubic equation, one can test small positive integers for

m

in the equation

m(m + 1)^2 = 18

. Testing

m = 1

yields

1(2)^2 = 4

(too small), and testing

m = 2

yields

2(3)^2 = 18

(correct). Because

m(m + 1)^2

is strictly increasing for

m > 0

,

m = 2

is the only positive real root.

Tahmini Süre:3m 0s

Cevap

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Anahtar Kavram

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Alternatif Yöntem

Cevap

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Anahtar Kavram

Sık Yapılan Hatalar

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Anahtar Kavram

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Anahtar Kavram

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Cevap

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Anahtar Kavram

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Anahtar Kavram

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Anahtar Kavram

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Anahtar Kavram

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Alternatif Yöntem

Cevap

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Anahtar Kavram

Sık Yapılan Hatalar

Cevap

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Anahtar Kavram

Sık Yapılan Hatalar

Cevap

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Anahtar Kavram

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Anahtar Kavram

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Anahtar Kavram

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Anahtar Kavram

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