Tüm alıştırma soruları

Soru 261Soru →

A triangle has a base of length $14$ centimeters and an area of $42$ square centimeters. What is the height, in centimeters, of the triangle perpendicular to this base?

Cevabı ve açıklamayı göster

Cevap: 6

Cevap

The height of the triangle perpendicular to the base is 6 centimeters.

The area of a triangle is given by

\text{Area} = \frac{1}{2} \times \text{base} \times \text{height}

. Substituting

42

for the area and

14

for the base yields

42 = \frac{1}{2} \times 14 \times h

, which simplifies to

42 = 7h

. Dividing

42

by

7

gives

h = 6

.

Adım Adım Çözüm

1

Write down the triangle area formula.

\text{Area} = \frac{1}{2} \times \text{base} \times \text{height}

The area and base are known values, and the goal is to solve for the perpendicular height.

2

Substitute the given values into the formula.

42 = \frac{1}{2} \times 14 \times \text{height}

Plugging in

\text{Area} = 42

and

\text{base} = 14

creates a single-variable algebraic equation.

3

Solve for the height.

42 = 7 \times \text{height} \implies \text{height} = 6

Simplifying

\frac{1}{2} \times 14

to

7

and dividing both sides by

7

yields the height.

Anahtar Kavram

The area of a triangle is calculated using the formula Area = (1/2) * base * height.

Soru 262Soru →

If $x$ is an integer that satisfies both $|5 - 2x| \le 9$ and $|x + 1| > 3$ , what is the least possible value of $x$ ?

Cevabı ve açıklamayı göster

Cevap: 3

Cevap

The least possible value of

x

is 3.

Solving

|5 - 2x| \le 9

gives

-2 \le x \le 7

. Solving

|x + 1| > 3

gives

x > 2

or

x < -4

. Taking the intersection of both regions yields

2 < x \le 7

. The integer values satisfying this combined inequality are

3, 4, 5, 6,

and

7

. The smallest among these integer values is

3

.

Adım Adım Çözüm

1

Solve the first absolute value inequality

|5 - 2x| \le 9

-2 \le x \le 7

Remove the absolute value bars to set up the compound inequality

-9 \le 5 - 2x \le 9

. Subtracting

5

from all parts gives

-14 \le -2x \le 4

. Dividing all parts by

-2

and reversing the inequality signs yields

7 \ge x \ge -2

, or

-2 \le x \le 7

.

2

Solve the second absolute value inequality

|x + 1| > 3

x > 2

or

x < -4

Remove the absolute value bars to create two separate cases:

x + 1 > 3 \implies x > 2

, or

x + 1 < -3 \implies x < -4

.

3

Determine the set of values that satisfy both inequalities simultaneously

2 < x \le 7

The intersection of the interval

[-2, 7]

and

(-\infty, -4) \cup (2, \infty)

is

(2, 7]

, because

x < -4

does not overlap with

[-2, 7]

.

4

Find the smallest integer within the interval

(2, 7]

3

The integers included in the interval

(2, 7]

are

3, 4, 5, 6,

and

7

. Note that

2

is excluded due to the strict inequality

x > 2

. Therefore, the least possible integer value is

3

.

Anahtar Kavram

Linear Inequalities and Absolute Value

Soru 263Soru →

Lines $l_1$ and $l_2$ intersect at point $P$ to form an acute angle measuring $44^\circ$ . Line $b_1$ is the angle bisector of this acute angle. A third line, $l_3$ , is drawn perpendicular to $b_1$ and intersects line $l_1$ at point $Q$ (where $Q \neq P$ ). What is the measure, in degrees, of the acute angle formed by the intersection of line $l_2$ and line $l_3$ ?

22^\circ

46^\circ

68^\circ

78^\circ

112^\circ

Cevabı ve açıklamayı göster

Cevap: $68^\circ$

Cevap

The measure of the acute angle formed by the intersection of line

l_2

and line

l_3

is

68^\circ

.

The correct answer is

68^\circ

. Line

b_1

divides the

44^\circ

angle between

l_1

and

l_2

into two

22^\circ

angles. Because line

l_3

is perpendicular to

b_1

, it forms a right triangle with

l_1

and

b_1

, making the angle between

l_1

and

l_3

equal to

90^\circ - 22^\circ = 68^\circ

. Considering the large triangle formed by lines

l_1

,

l_2

, and

l_3

, two of its angles are

44^\circ

and

68^\circ

. Thus, the third interior angle at the intersection of

l_2

and

l_3

is

180^\circ - (44^\circ + 68^\circ) = 68^\circ

, which is acute.

Adım Adım Çözüm

1

Determine the angle formed by the angle bisector

b_1

with line

l_1

and line

l_2

.

Since line

b_1

bisects the

44^\circ

acute angle between

l_1

and

l_2

, the angle between

l_1

and

b_1

is

\frac{44^\circ}{2} = 22^\circ

, and the angle between

l_2

and

b_1

is also

22^\circ

.

An angle bisector divides an angle into two equal congruent parts.

2

Find the measure of the interior angle between line

l_1

and line

l_3

.

Line

l_3

is perpendicular to

b_1

, forming a right triangle with

l_1

and

b_1

. The interior angle between

l_1

and

l_3

is

180^\circ - 90^\circ - 22^\circ = 68^\circ

.

The sum of interior angles in any triangle is

180^\circ

.

3

Calculate the interior angle at the intersection of line

l_2

and line

l_3

in the main triangle formed by

l_1

,

l_2

, and

l_3

.

The interior angle at the intersection of

l_2

and

l_3

is

180^\circ - 44^\circ - 68^\circ = 68^\circ

.

The interior angles of the triangle formed by lines

l_1

,

l_2

, and

l_3

must sum to

180^\circ

.

4

Verify that the calculated angle is acute.

Since

68^\circ < 90^\circ

, the acute angle formed by lines

l_2

and

l_3

is

68^\circ

.

An angle measuring strictly less than

90^\circ

is defined as an acute angle.

Anahtar Kavram

Angle bisector properties, perpendicular lines, and interior angle sum theorem for triangles.

Soru 264Soru →

Consider the linear equation in one variable

x

:

\frac{2x - 3}{3} - \frac{x - 5}{4} = \frac{x + 9}{6}

If

x

is the solution to this equation, what is the value of

4x - 1

?

5

7

11

19

59

Cevabı ve açıklamayı göster

Cevap: 19

Cevap

19

To solve the linear equation, first multiply all terms by the least common denominator, 12, yielding 4(2x - 3) - 3(x - 5) = 2(x + 9). Expanding both sides yields 8x - 12 - 3x + 15 = 2x + 18. Combining like terms on the left side yields 5x + 3 = 2x + 18. Subtracting 2x and 3 from both sides gives 3x = 15, so x = 5. Substituting x = 5 into the target expression 4x - 1 gives 4(5) - 1 = 19.

Adım Adım Çözüm

1

Clear the fractional denominators by multiplying every term by the least common multiple (LCM) of 3, 4, and 6, which is 12.

12 \cdot \left(\frac{2x - 3}{3}\right) - 12 \cdot \left(\frac{x - 5}{4}\right) = 12 \cdot \left(\frac{x + 9}{6}\right) \implies 4(2x - 3) - 3(x - 5) = 2(x + 9)

Multiplying every term by 12 eliminates all denominators without changing the equation's solution set.

2

Distribute the factors across the binomial terms inside parentheses.

8x - 12 - 3x + 15 = 2x + 18

Distributing -3 across (x - 5) yields -3x + 15 because the product of two negative numbers is positive.

3

Combine like terms on the left side of the equation.

5x + 3 = 2x + 18

Combining 8x - 3x gives 5x, and -12 + 15 gives +3.

4

Isolate x by subtracting 2x and 3 from both sides of the equation.

3x = 15 \implies x = 5

Subtracting 2x from both sides yields 3x + 3 = 18, then subtracting 3 yields 3x = 15, giving x = 5.

5

Substitute x = 5 into the expression 4x - 1 to find the final value.

4(5) - 1 = 20 - 1 = 19

The question asks for the evaluation of 4x - 1 rather than the value of x alone.

Anahtar Kavram

Solving linear equations in one variable involving fractional expressions and evaluating algebraic expressions.

Tahmini Süre:2m 0s

Soru 265Soru →

If $x$ is a real number that satisfies both $|3x - 12| \le 9$ and $|2 - x| \ge 4$ , the maximum possible value of the expression $5 - 2x$ is $M$ and the minimum possible value is $m$ . What is the value of $M - m$ ?

Cevabı ve açıklamayı göster

Cevap: 2

Cevap

The value of

M - m

is 2.

Solving

|3x - 12| \le 9

yields

1 \le x \le 7

. Solving

|2 - x| \ge 4

yields

x \le -2

or

x \ge 6

. The set of

x

-values satisfying both conditions is the intersection

[6, 7]

. Since

5 - 2x

is a linear expression with a negative coefficient, its maximum value

M

occurs at the smallest value of

x

(

x = 6

), giving

M = 5 - 2(6) = -7

. Its minimum value

m

occurs at the largest value of

x

(

x = 7

), giving

m = 5 - 2(7) = -9

. Thus,

M - m = -7 - (-9) = 2

.

Adım Adım Çözüm

1

Solve the inequality

|3x - 12| \le 9

1 \le x \le 7

Expanding absolute value yields

-9 \le 3x - 12 \le 9

. Adding 12 gives

3 \le 3x \le 21

, then dividing by 3 yields

1 \le x \le 7

.

2

Solve the inequality

|2 - x| \ge 4

x \le -2

or

x \ge 6

Absolute value inequality

|u| \ge a

splits into

u \ge a

or

u \le -a

. Here

2 - x \ge 4 \implies x \le -2

, and

2 - x \le -4 \implies x \ge 6

.

3

Determine the overlapping domain for

x

6 \le x \le 7

Combining

1 \le x \le 7

with

x \le -2

or

x \ge 6

leaves only the interval

6 \le x \le 7

.

4

Evaluate maximum

M

and minimum

m

of

5 - 2x

on

6 \le x \le 7

M = -7

and

m = -9

Since

-2x

decreases as

x

increases, the maximum occurs at

x = 6

(

M = 5 - 12 = -7

) and the minimum occurs at

x = 7

(

m = 5 - 14 = -9

).

5

Calculate the difference

M - m

2

Subtracting

m

from

M

gives

-7 - (-9) = 2

.

Anahtar Kavram

Linear Inequalities and Absolute Value

Tahmini Süre:2m 0s

Soru 266Soru →

The table below shows the distribution of scores achieved by 20 students on a final exam:

Score	Frequency
60	3
70	5
80	6
90	4
100	2

What is the interquartile range (IQR) of the scores?

Cevabı ve açıklamayı göster

Cevap: 20

Cevap

The interquartile range (IQR) of the scores is 20.

For a set of 20 ordered scores, the first quartile

Q_1

is the median of the first 10 scores (average of the 5th and 6th values), and the third quartile

Q_3

is the median of the last 10 scores (average of the 15th and 16th values). Using the cumulative frequencies, the 5th and 6th scores are both 70 (

Q_1 = 70

), and the 15th and 16th scores are both 90 (

Q_3 = 90

). Thus, the interquartile range is

IQR = Q_3 - Q_1 = 90 - 70 = 20

.

Adım Adım Çözüm

1

Determine the cumulative frequency to locate quartile positions.

Score 60 occupies positions 1 to 3; Score 70 occupies positions 4 to 8; Score 80 occupies positions 9 to 14; Score 90 occupies positions 15 to 18; Score 100 occupies positions 19 to 20.

Tracking data positions in a frequency distribution allows efficient determination of medians and quartiles without expanding the raw list.

2

Calculate the first quartile (

Q_1

).

Q_1 = \frac{70 + 70}{2} = 70

.

With 20 total values, the lower half consists of the first 10 values (positions 1 through 10). The median of these 10 values is the average of the 5th and 6th values, which are both 70.

3

Calculate the third quartile (

Q_3

).

Q_3 = \frac{90 + 90}{2} = 90

.

The upper half consists of the last 10 values (positions 11 through 20). The median of these 10 values is the average of the 15th and 16th values, which are both 90.

4

Subtract

Q_1

from

Q_3

to find the interquartile range.

IQR = Q_3 - Q_1 = 90 - 70 = 20

.

The interquartile range represents the spread of the middle 50% of the dataset.

Anahtar Kavram

Interquartile Range (IQR) from a Frequency Distribution

Tahmini Süre:1m 30s

Soru 267Soru →

Two water pumps, Pump A and Pump B, were used to drain a reservoir containing 12,000 gallons of water. Pump A operates at a constant rate that is 50 gallons per hour greater than the rate of Pump B. Pump A worked alone for 4 hours, after which both pumps worked together for another 8 hours to completely empty the reservoir. What is the pumping rate of Pump B, in gallons per hour?

Cevabı ve açıklamayı göster

Cevap: 570

Cevap

570

Let

r

represent the rate of Pump B in gallons per hour. Pump A's rate is

(r + 50)

gallons per hour. In the first 4 hours, Pump A drains

4(r + 50) = 4r + 200

gallons. In the next 8 hours, both pumps operate together at a combined rate of

(r + 50) + r = 2r + 50

gallons per hour, draining

8(2r + 50) = 16r + 400

gallons. Adding both quantities gives total volume drained:

(4r + 200) + (16r + 400) = 12,000

. Simplifying gives

20r + 600 = 12,000

, so

20r = 11,400

, which yields

r = 570

gallons per hour.

Adım Adım Çözüm

1

Define variables for the rate of each pump

Rate of Pump B =

r

gal/hr; Rate of Pump A =

r + 50

gal/hr

Establishing a single unknown variable allows setting up a one-variable linear equation.

2

Write expressions for water drained during each time period

Period 1 (Pump A alone for 4 hrs):

4(r + 50) = 4r + 200

; Period 2 (Both pumps for 8 hrs):

8(2r + 50) = 16r + 400

Work done equals rate multiplied by time for each phase of operation.

3

Sum the work done in both periods to equal total volume and solve for

r

20r + 600 = 12,000 \Rightarrow 20r = 11,400 \Rightarrow r = 570

Solving the linear equation yields the exact rate of Pump B.

Anahtar Kavram

Linear Equations in One Variable

Tahmini Süre:2m 0s

Soru 268Soru →

In a circle, an arc of length $4\pi$ corresponds to a central angle of $40^\circ$ . If the area of the sector formed by this central angle is $k\pi$ , what is the value of $k$ ?

Cevabı ve açıklamayı göster

Cevap: 36

Cevap

The value of

k

is 36.

Using the arc length equation

4\pi = \frac{40}{360} \cdot 2\pi r

, we solve for the radius

r = 18

. Substituting

r = 18

into the sector area formula

A = \frac{40}{360} \cdot \pi (18)^2

results in

A = 36\pi

. Thus,

k = 36

.

Adım Adım Çözüm

1

Calculate the radius of the circle using the arc length formula

Radius

r = 18

Arc length is related to central angle and radius by

L = \frac{\theta}{360^\circ} \cdot 2\pi r

. Substituting

L = 4\pi

and

\theta = 40^\circ

gives

4\pi = \frac{1}{9} \cdot 2\pi r \implies r = 18

.

2

Calculate the area of the sector using the radius and central angle

Sector Area

A = 36\pi

Sector area is calculated using

A = \frac{\theta}{360^\circ} \cdot \pi r^2

. Substituting

\theta = 40^\circ

and

r = 18

gives

A = \frac{1}{9} \cdot \pi (18^2) = 36\pi

.

3

Extract the coefficient

k

from

k\pi

k = 36

Comparing

36\pi

to

k\pi

directly yields

k = 36

.

Anahtar Kavram

Relationship between central angle, arc length, radius, and sector area

Soru 269Soru →

Read the passage below:

For decades, archaeologists analyzing ancient Mesoamerican agriculture relied primarily on macrobotanical remains, such as charred seeds and wood fragments, to reconstruct prehistoric crop cultivation. However, this approach systematically underestimated the significance of root crops like cassava and sweet potato, which rarely preserve in acidic tropical soils. To address this evidentiary bias, recent investigations have increasingly deployed phytolith analysis—the microscopic examination of silica bodies deposited within plant tissues. Critics initially contended that phytolith morphotypes are insufficiently distinct to differentiate closely related domesticated species from their wild progenitors. Yet, advanced three-dimensional morphometric modeling has recently demonstrated that phytoliths from domesticated maize can indeed be statistically distinguished from wild teosinte. Consequently, microbotanical analysis has shifted from an auxiliary diagnostic tool to a cornerstone of paleoethnobotanical methodology.

Which of the following best describes the rhetorical function of the sentence 'Critics initially contended that phytolith morphotypes are insufficiently distinct to differentiate closely related domesticated species from their wild progenitors' in the context of the passage as a whole?

It introduces an objection to a newly introduced methodology that the author subsequently shows has been addressed by subsequent technological advances.

It provides empirical evidence that undermines the author's main assertion regarding Mesoamerican agricultural history.

It summarizes the central argument underpinning traditional macrobotanical research frameworks.

It qualifies the author's endorsement of three-dimensional morphometric modeling by highlighting its methodological limitations.

It proves conclusively that microbotanical remains are superior to macrobotanical remains in acidic soil environments.

Cevabı ve açıklamayı göster

Cevap: It introduces an objection to a newly introduced methodology that the author subsequently shows has been addressed by subsequent technological advances.

Cevap

The sentence functions to introduce a potential objection or limitation concerning phytolith analysis, which the author immediately counters in the subsequent sentence by citing advances in three-dimensional morphometric modeling.

The sentence introduces a challenge raised by critics regarding the diagnostic utility of phytolith analysis. The author follows this sentence with the contrast word 'Yet' to show how 3D morphometric modeling resolved this exact limitation, making the option stating that it introduces an objection subsequently shown to be addressed by technological advances the correct description of its function.

Adım Adım Çözüm

1

Analyze the structural position of the target sentence

The target sentence comes immediately after the passage introduces phytolith analysis as a new technique to solve macrobotanical bias.

Understanding what precedes the sentence establishes what method is being discussed.

2

Examine the sentence content and transition words that follow

The sentence notes that critics argued phytolith morphotypes were not distinct enough. The next sentence begins with the contrast word 'Yet' and explains how 3D morphometric modeling resolved this issue.

Tracking pivot words reveals whether the objection is accepted or refuted by the author.

3

Determine the overarching rhetorical role

The sentence sets up a obstacle/criticism to a new technique that is subsequently overcome by technological innovation.

Connecting the sentence to both preceding and succeeding sentences clarifies its function within the paragraph's argument.

Anahtar Kavram

Rhetorical Structure and Sentence Function

Soru 270Soru →

Fill in each blank in the passage below with the word that best completes the sentence logically and coherently.

Aşağıdaki boşlukları doldurun

Far from demonstrating that the system's thermodynamic entropy must strictly increase under all conditions, recent experimental findings suggest that localized microscopic fluctuations can temporarily macroscopic second-law constraints; nevertheless, far from signaling a fundamental of established physical laws, these transient anomalies actually the overarching statistical framework that governs non-equilibrium systems.

Cevabı ve açıklamayı göster

Cevap

Blank 1: 'circumvent' (or 'bypass' / 'evade'); Blank 2: 'repudiation' (or 'refutation' / 'invalidation'); Blank 3: 'reinforce' (or 'buttress' / 'corroborate').

The sentence relies on tracking inter-blank semantic dependencies across two structural pivots: 'Far from' at the beginning and 'nevertheless, far from' in the second clause. For Blank 1, the opening clause contrasts strict monotonic entropy increase with microscopic behavior, requiring a word indicating that fluctuations temporarily get around constraints (such as 'circumvent' or 'bypass'). For Blank 2, 'far from signaling a fundamental [blank]' sets up a concession; the existence of localized anomalies does not imply a complete destruction or rejection of physical laws, making 'repudiation' or 'invalidation' appropriate. Finally, for Blank 3, because non-equilibrium statistical mechanics explicitly accounts for micro-scale fluctuations, these observations actively support and undergird the statistical model, making 'reinforce' or 'buttress' the logically coherent completion.

Adım Adım Çözüm

1

Analyze structural clues and initial contrast for Blank 1

The opening pivot 'Far from demonstrating that... entropy must strictly increase' indicates that microscopic fluctuations behave unexpectedly relative to second-law rules by temporarily getting around or bypassing them.

The contrast setup requires a verb signifying an evasion of strict macroscopic constraints.

2

Track dependency across the contrast pivot for Blank 2

The phrase 'nevertheless, far from signaling a fundamental...' introduces a second concession. Blank 2 requires a noun denoting rejection or invalidation ('repudiation'), which the structural phrase 'far from signaling' then negates.

The sentence clarifies that temporary microscopic anomalies do not constitute a complete rejection or overturning of foundational physical principles.

3

Resolve Blank 3 based on non-equilibrium statistical thermodynamics context

Because statistical mechanics predicts transient microscopic variance, observing these anomalies actually validates and strengthens ('reinforces') the broader statistical theory.

Since the anomalies do not undermine the law, their occurrence harmonizes with and supports the statistical framework.

Anahtar Kavram

Multi-Blank Dependency Tracking

Soru 271Soru →

Fill in each blank in the passage below with the word that best completes the sentence logically and coherently. Which terms correctly complete blank 1, blank 2, and blank 3?

Aşağıdaki boşlukları doldurun

Rather than providing evidence for the long-standing hypothesis of structural neural decay, the newly discovered mechanism of synaptic reconfiguration demonstrates that cellular degeneration can be counteracted; far from certifying that cognitive deterioration is , the study suggests that age-related impairment remains surprisingly when subjected to targeted metabolic interventions.

Cevabı ve açıklamayı göster

Cevap

Blank 1 requires 'conclusive' (or 'definitive'), Blank 2 requires 'inevitable' (or 'inexorable'), and Blank 3 requires 'malleable' (or 'modifiable').

The sentence relies on two main contrast structures ('Rather than...' and 'far from...'). The first blank requires a word denoting decisive proof because the study challenges an existing hypothesis. The second blank requires a word asserting unalterable certainty ('inevitable') which the 'far from certifying' pivot rejects. The third blank completes the logical chain by establishing that cognitive impairment is adaptable ('malleable') given targeted intervention.

Adım Adım Çözüm

1

Analyze the opening contrast signal 'Rather than providing...' in relation to the second clause.

The phrase 'Rather than' indicates a contrast between what the evidence was expected to show versus what synaptic reconfiguration actually demonstrated. Since the discovery shows degeneration can be counteracted, the evidence for structural decay was not final or decisive. Thus, Blank 1 must mean 'conclusive' or 'definitive'.

Tracking structural contrast clues between clauses.

2

Analyze the structural pivot 'far from certifying that...' introducing Blank 2.

The phrase 'far from certifying' signals a negation of an absolute claim about cognitive decline. Because the study shows degeneration can be counteracted, it refutes the notion that decline cannot be stopped. Therefore, Blank 2 requires a term meaning 'inevitable' or 'unavoidable'.

Identifying inter-blank structural dependency and polarity.

3

Analyze the dependency between Blank 2 and Blank 3 across the contrast phrase 'remains surprisingly... depending on targeted interventions'.

If decline is not unavoidable (Blank 2), then its trajectory depends on external interventions, making it flexible or subject to change. Thus, Blank 3 requires a word meaning 'malleable', 'modifiable', or 'contingent'.

Validating semantic coherence across all three interdependent slots.

Anahtar Kavram

Multi-Blank Dependency Tracking

Soru 272Soru →

In analyzing the urban planning decisions of mid-twentieth-century municipal boards, Dr. Aris Thorne challenges the widespread assumption that the aggressive expansion of highway infrastructure was driven solely by economic pragmatism. While proponents initially framed these infrastructure projects as chivalrous endeavors designed to rescue decaying inner-city cores from economic stagnation, archival records reveal a far more contentious reality. The designation of these projects as chivalrous served primarily as a rhetorical veneer, cloaking paternalistic displacement and systematic disenfranchisement in the language of noble benevolence. Consequently, Thorne asserts that the term carried a distinctly patronizing connotation, masking coercive civic policies under the guise of disinterested altruism.

In the context of the passage as a whole, the word "chivalrous" carries which of the following contextual connotations?

An ostensibly noble benevolence that conceals paternalistic and coercive motives

An overtly hostile and malicious campaign intended to dismantle urban communities

A traditional code of heroic virtue and selfless bravery in public service

A purely pragmatic and cost-effective strategy for municipal renovation

An objectively verified record of civic philanthropy supported by historical consensus

Cevabı ve açıklamayı göster

Cevap: An ostensibly noble benevolence that conceals paternalistic and coercive motives

Cevap

The word 'chivalrous' carries the connotation of an ostensibly noble benevolence that conceals paternalistic and coercive motives.

The passage directly explains that applying the word 'chivalrous' to urban planning projects functioned as a 'rhetorical veneer' that cloaked paternalistic displacement and masked coercive civic policies under the appearance of altruism. Therefore, in context, the word connotes a superficial display of noble benevolence that hides patronizing and coercive motives.

Adım Adım Çözüm

1

Identify the target word and surrounding structural context

The target word 'chivalrous' appears in a concession-contrast structure introduced by 'While proponents initially framed...' followed by 'archival records reveal a far more contentious reality.'

Understanding the structural contrast helps determine whether the author intends the word literally or critically.

2

Analyze how the passage defines and modifies the connotation of the target word

The text explicitly defines the usage of 'chivalrous' as a 'rhetorical veneer' that cloaks 'paternalistic displacement' and masks 'coercive civic policies under the guise of disinterested altruism.'

The author directly clarifies the polarity: the appearance of selfless nobility is deceptive and masks patronizing control.

3

Evaluate the options against the contextual evidence

The option describing an ostensibly noble benevolence that conceals paternalistic and coercive motives accurately mirrors the passage's explanation of a deceptive rhetorical veneer.

The correct answer must capture both the outward appearance of nobility and the underlying negative intention established by the passage.

Anahtar Kavram

Identifying Contextual Tone, Polarity, and Connotation

Soru 273Soru →

Archaeologists examining slag remnants from an early Iron Age smelting site hypothesized that local artisans added roasted acacia pods to laterite iron ore as a flux to reduce furnace smelting temperatures. Chemical analysis of the slag revealed unusually high concentrations of calcium oxide and phosphorus pentoxide, elements minimal in raw laterite ore. Because adding organic flux allows laterite ore to melt at significantly lower temperatures achievable by basic charcoal furnaces, the researchers concluded that the presence of these compounds proves the deliberate inclusion of acacia pods in the smelting mixture.

Which of the following statements, if true, strengthen the archaeologists' argument? Select all that apply.

Geçerli olan tümünü seçin

Subspecies of acacia trees native to the region yield seed pods whose ash contains calcium oxide and phosphorus pentoxide in proportions matching those detected in the smelting slag.

Charcoal residues from fuel wood excavated alongside the furnace lacked the elevated calcium and phosphorus levels found in the slag.

Surrounding geological surveys identified nearby limestone deposits rich in calcium oxide that ancient metalworkers frequently utilized in ceramic kiln production.

Artifacts recovered from the site indicate that iron tools manufactured at lower furnace temperatures exhibited greater structural durability than tools forged at higher temperatures.

Iron Age communities in adjacent regions possessed trade networks capable of supplying large quantities of acacia seed pods during peak smelting seasons.

Cevabı ve açıklamayı göster

Cevap: Subspecies of acacia trees native to the region yield seed pods whose ash contains calcium oxide and phosphorus pentoxide in proportions matching those detected in the smelting slag.; Charcoal residues from fuel wood excavated alongside the furnace lacked the elevated calcium and phosphorus levels found in the slag.

Cevap

The argument is strengthened by the statements establishing that acacia pod ash matches the chemical proportions in the slag and that fuel wood charcoal was not the source of the calcium and phosphorus.

The conclusion asserts that acacia seed pods were deliberately added to laterite ore based on chemical evidence in the slag. To strengthen this claim, additional evidence must tie these specific chemical markers directly to acacia pods or rule out alternative origins. The statement matching the calcium and phosphorus ratios of native acacia ash directly to the slag confirms the chemical identity of the additive. Simultaneously, the statement verifying that furnace fuel wood charcoal lacked these elements eliminates timber fuel as a competing source of the compounds, thereby ruling out a key alternative explanation.

Adım Adım Çözüm

1

Analyze the premise and conclusion of the argument.

Premise: Slag contains elevated calcium oxide and phosphorus pentoxide, which lower melting temperatures. Conclusion: Ancient artisans intentionally added acacia pods as flux.

Identifying the gap between detecting specific chemical compounds and attributing them exclusively to acacia pods.

2

Evaluate options for support mechanisms such as matching chemical signatures or ruling out alternative causes.

Demonstrating an exact proportion match with acacia ash directly supports acacia as the source, while showing fuel wood lacked these elements rules out fuel timber as an alternative source.

Both findings increase the probability that acacia pods specifically were added to the furnace.

3

Eliminate options that weaken the argument or address irrelevant background details.

Limestone deposits provide a competing explanation (weakening), tool durability addresses outcome rather than cause, and trade networks do not prove actual usage.

Distractors fail to reinforce the chemical connection between acacia pods and the slag.

Anahtar Kavram

Strengthening causal claims by confirming unique chemical signatures and eliminating alternative potential causes.

Soru 274Soru →

In a circle centered at point $O$ , sector $OAB$ has a central angle of $60^\circ$ and a radius of $12$ . A smaller circle is inscribed inside sector $OAB$ such that it is tangent to radius $OA$ , radius $OB$ , and arc $AB$ . If the area of the region inside sector $OAB$ that lies outside the inscribed circle is expressed in the form $k\pi$ , what is the value of $k$ ?

Cevabı ve açıklamayı göster

Cevap: 8

Cevap

The correct value of

k

is 8.

By using the geometry of the

30^\circ

-

60^\circ

-

90^\circ

right triangle formed by the angle bisector and the radius of tangency, the radius of the inscribed circle is found to be

r = 4

. Subtracting its area (

16\pi

) from the sector's area (

24\pi

) gives

8\pi

, so

k = 8

.

Adım Adım Çözüm

1

Determine the relationship between the radius of the larger circle

R

and the radius of the inscribed circle

r

.

OP = 2r

and

R = 3r

.

The center

P

of the inscribed circle lies on the angle bisector of

\angle AOB = 60^\circ

, creating a

30^\circ

angle with radius

OA

. The perpendicular distance from

P

to radius

OA

is

r

, so

\sin(30^\circ) = \frac{r}{OP} = \frac{1}{2}

, giving

OP = 2r

. Since the inscribed circle touches arc

AB

,

OP + r = R

, so

3r = R

.

2

Calculate the radius

r

of the inscribed circle.

r = 4

.

Given

R = 12

, solving

3r = 12

yields

r = 4

.

3

Compute the area of sector

OAB

.

\text{Area}_{\text{sector}} = 24\pi

.

The formula for the area of a sector is

\frac{\theta}{360^\circ} \pi R^2

. Here,

\frac{60^\circ}{360^\circ} \pi (12^2) = \frac{1}{6} \times 144\pi = 24\pi

.

4

Compute the area of the inscribed circle.

\text{Area}_{\text{circle}} = 16\pi

.

The area of a circle with radius

r = 4

is

\pi r^2 = \pi (4^2) = 16\pi

.

5

Subtract the area of the inscribed circle from the area of sector

OAB

to find

k

.

k = 8

.

\text{Area}_{\text{region}} = 24\pi - 16\pi = 8\pi

, which means

k = 8

.

Anahtar Kavram

Inscribed shapes within sectors, arc length, and sector area relations

Soru 275Soru →

For all $x > 1$ , which of the following expressions is equivalent to $\frac{\sqrt{x^3 \cdot \sqrt[3]{x^2}}}{(x^{-1/3})^2}$ ?

x^{5/2}

x^{7/6}

x^{5/3}

x^{13/3}

x^{35/12}

Cevabı ve açıklamayı göster

Cevap: $x^{5/2}$

Cevap

x^{5/2}

Converting all radical expressions into rational exponent form simplifies the numerator to

(x^{11/3})^{1/2} = x^{11/6}

and the denominator to

x^{-2/3}

. Applying the quotient rule

x^{11/6} / x^{-2/3} = x^{11/6 - (-2/3)}

yields

x^{15/6} = x^{5/2}

.

Adım Adım Çözüm

1

Convert radicals in the numerator to fractional exponents and combine terms inside the square root

x^3 \cdot \sqrt[3]{x^2} = x^3 \cdot x^{2/3} = x^{3 + 2/3} = x^{11/3}

When multiplying exponential expressions with the same base, add the exponents.

2

Apply the outer square root to the simplified expression in the numerator

\sqrt{x^{11/3}} = (x^{11/3})^{1/2} = x^{11/6}

Taking the square root is equivalent to raising an expression to the power of

1/2

.

3

Simplify the denominator using the power of a power rule

(x^{-1/3})^2 = x^{(-1/3) \cdot 2} = x^{-2/3}

When raising a power to another power, multiply the exponents.

4

Divide the simplified numerator by the simplified denominator

\frac{x^{11/6}}{x^{-2/3}} = x^{11/6 - (-2/3)} = x^{11/6 + 4/6} = x^{15/6} = x^{5/2}

When dividing exponential expressions with the same base, subtract the denominator exponent from the numerator exponent.

Anahtar Kavram

Simplifying nested algebraic radicals using rational exponent laws

Tahmini Süre:1m 30s

Soru 276Soru →

Two commercial printing presses, Press A and Press B, operate at constant rates to print a total order of $N$ pages. Press A prints at a constant rate of $x$ pages per minute, while Press B prints at a constant rate that is $25$ pages per minute faster than Press A. Press A begins printing alone. After $20$ minutes, Press B is turned on, and both presses work simultaneously for an additional $30$ minutes. At that point, Press A stops, and Press B works alone for $10$ final minutes to complete the order. If Press B printed exactly $\frac{6}{11}$ of the total number of pages in the order, what is the value of $x$ ?

Cevabı ve açıklamayı göster

Cevap: 50

Cevap

The value of

x

is 50.

To find

x

, calculate the total time each press operated. Press A ran for 20 minutes alone plus 30 minutes with Press B, giving 50 minutes total at

x

pages per minute (

50x

pages). Press B ran for 30 minutes with Press A plus 10 minutes alone, giving 40 minutes total at

(x + 25)

pages per minute (

40x + 1000

pages). The total order size is

N = 90x + 1000

. Setting Press B's output equal to

\frac{6}{11}N

yields

40x + 1000 = \frac{6}{11}(90x + 1000)

. Multiplying both sides by 11 gives

440x + 11000 = 540x + 6000

, which simplifies to

100x = 5000

, giving

x = 50

.

Adım Adım Çözüm

1

Determine total operating times and express pages printed by each press in terms of

x

Press A printed

50x

pages; Press B printed

40(x + 25) = 40x + 1000

pages.

Press A operated for 20 minutes alone plus 30 minutes together (50 minutes total). Press B operated for 30 minutes together plus 10 minutes alone (40 minutes total).

2

Write the expression for total pages

N

N = 50x + (40x + 1000) = 90x + 1000

The total pages in the order is the sum of the pages printed by Press A and Press B.

3

Formulate the linear equation in one variable using the given ratio

40x + 1000 = \frac{6}{11}(90x + 1000)

Press B printed exactly

\frac{6}{11}

of the total pages

N

.

4

Solve the linear equation for

x

x = 50

Multiplying both sides by 11 clears the fraction to give

440x + 11000 = 540x + 6000

, which simplifies to

100x = 5000

.

Anahtar Kavram

Formulating and solving linear equations in one variable from multi-step rate and work scenarios.

Soru 277Soru →

If $x$ is an integer that satisfies the inequality $|6 - 3x| - 4 \le 5$ , which of the following could be the value of $x$ ? Select all such values.

Geçerli olan tümünü seçin

-3

-1

2

4

7

Cevabı ve açıklamayı göster

Cevap: $-1$ ; $2$ ; $4$

Cevap

The correct values of

x

are

-1

,

2

, and

4

.

Isolating the absolute value yields

|6 - 3x| \le 9

, which expands to

-9 \le 6 - 3x \le 9

. Subtracting

6

gives

-15 \le -3x \le 3

. Dividing by

-3

and reversing the inequality signs results in

-1 \le x \le 5

. Among the given choices,

-1

,

2

, and

4

lie within this solution interval

[-1, 5]

.

Adım Adım Çözüm

1

Isolate the absolute value expression

|6 - 3x| \le 9

Add

4

to both sides of the inequality to isolate the absolute value term.

2

Rewrite as a compound inequality

-9 \le 6 - 3x \le 9

The property

|u| \le a

(where

a \ge 0

) expands to

-a \le u \le a

.

3

Subtract 6 from all parts

-15 \le -3x \le 3

Isolate the variable term

-3x

by subtracting

6

across the compound inequality.

4

Divide by -3 and reverse inequality signs

5 \ge x \ge -1

, which is equivalent to

-1 \le x \le 5

Dividing an inequality by a negative quantity requires flipping the inequality direction.

Anahtar Kavram

Solving absolute value inequalities of the form

|ax + b| \le c

by expanding into a compound inequality

-c \le ax + b \le c

and correctly reversing inequality signs when multiplying or dividing by negative numbers.

Soru 278Soru →

Two sides of a triangle have lengths of $5$ centimeters and $9$ centimeters. Which of the following could be the length, in centimeters, of the third side? Select all such lengths.

Geçerli olan tümünü seçin

4

6

10

14

16

Cevabı ve açıklamayı göster

Cevap: $6$ ; $10$

Cevap

The possible lengths for the third side are

6

centimeters and

10

centimeters.

According to the Triangle Inequality Theorem, the length of the third side

x

must be strictly greater than the difference of the two given sides (

9 - 5 = 4

) and strictly less than their sum (

9 + 5 = 14

). This establishes the valid range as

4 < x < 14

. The options

6

centimeters and

10

centimeters are the only values provided that fall strictly within this interval.

Adım Adım Çözüm

1

Apply the Triangle Inequality Theorem to determine the allowable range for the third side.

The length of the third side,

x

, must satisfy

|9 - 5| < x < 9 + 5

, which simplifies to

4 < x < 14

.

The sum of the lengths of any two sides of a non-degenerate triangle must be strictly greater than the length of the remaining side.

2

Check each candidate option against the inequality

4 < x < 14

.

The values

6

and

10

satisfy

4 < x < 14

, whereas

4

,

14

, and

16

do not.

Only numbers strictly within the open interval

(4, 14)

can form a valid triangle with side lengths

5

and

9

.

Anahtar Kavram

Triangle Inequality Theorem

Soru 279Soru →

For all positive real numbers $x$ and $y$ , which of the following expressions are equivalent to $\left(\frac{x^{-2} y^{3/2}}{\sqrt{x^4 y^{-1}}}\right)^{-1/2}$ ? Select all that apply.

Geçerli olan tümünü seçin

x^2 y^{-1}

\sqrt{\frac{x^4}{y^2}}

\frac{y}{x^2}

\frac{x^3 y}{\sqrt{x^2 y^4}}

\frac{x^2}{\sqrt{y}}

Cevabı ve açıklamayı göster

Cevap: $x^2 y^{-1}$ ; $\sqrt{\frac{x^4}{y^2}}$ ; $\frac{x^3 y}{\sqrt{x^2 y^4}}$

Cevap

The expressions equivalent to the given expression are

x^2 y^{-1}

,

\sqrt{\frac{x^4}{y^2}}

, and

\frac{x^3 y}{\sqrt{x^2 y^4}}

.

Simplifying the original expression step-by-step yields

\frac{x^2}{y}

, which is equal to

x^2 y^{-1}

. Taking the square root of

\frac{x^4}{y^2}

gives

\frac{x^2}{y}

, and simplifying

\frac{x^3 y}{\sqrt{x^2 y^4}} = \frac{x^3 y}{x y^2}

also gives

\frac{x^2}{y}

. Thus, these three choices are equivalent to the target expression.

Adım Adım Çözüm

1

Simplify the denominator inside the parentheses

\sqrt{x^4 y^{-1}} = (x^4 y^{-1})^{1/2} = x^2 y^{-1/2}

Apply the power of a product rule and principal square root properties for positive variables.

2

Simplify the fraction inside the parentheses

\frac{x^{-2} y^{3/2}}{x^2 y^{-1/2}} = x^{-2 - 2} y^{3/2 - (-1/2)} = x^{-4} y^2

Subtract exponents of like bases when dividing.

3

Apply the outer exponent

-1/2

(x^{-4} y^2)^{-1/2} = x^{(-4)(-1/2)} y^{(2)(-1/2)} = x^2 y^{-1} = \frac{x^2}{y}

Multiply exponents when raising a power to a power.

4

Evaluate each option against

\frac{x^2}{y}

x^2 y^{-1}

,

\sqrt{\frac{x^4}{y^2}}

, and

\frac{x^3 y}{\sqrt{x^2 y^4}}

all simplify to

\frac{x^2}{y}

.

Check algebraic equivalence for each provided choice.

Anahtar Kavram

Algebraic Exponents and Radicals

Soru 280Soru →

In the geometric plane, line $l_1$ is parallel to line $l_2$ . Point $A$ lies on line $l_1$ and point $B$ lies on line $l_2$ . Point $C$ is located between lines $l_1$ and $l_2$ such that $C$ lies to the right of both $A$ and $B$ .

The acute angle between segment $AC$ and the ray extending to the right from $A$ along line $l_1$ measures $x^\circ$ .

The acute angle between segment $BC$ and the ray extending to the right from $B$ along line $l_2$ measures $y^\circ$ .

The interior angle $\angle ACB = 118^\circ$ , and $y = 2x - 14$ .

Line $k$ passes through point $C$ and is perpendicular to segment $AC$ . Line $k$ intersects line $l_2$ at point $E$ , where point $E$ lies to the right of point $B$ .

What is the measure, in degrees, of the acute angle $\angle CEB$ formed by line $k$ and line $l_2$ ?

16^\circ

44^\circ

46^\circ

62^\circ

74^\circ

Cevabı ve açıklamayı göster

Cevap: $46^\circ$

Cevap

The measure of the acute angle

\angle CEB

is

46^\circ

.

By the parallel lines angle property, drawing a parallel line through vertex

C

shows that

\angle ACB = x + y = 118^\circ

. Substituting

y = 2x - 14

yields

3x - 14 = 118

, giving

x = 44^\circ

. Since line

l_1

is parallel to line

l_2

, segment

AC

intersects line

l_2

at an acute angle of

44^\circ

. Line

k

is constructed perpendicular to segment

AC

, so the acute angle formed by line

k

and line

l_2

is complementary to

44^\circ

, which gives

90^\circ - 44^\circ = 46^\circ

.

Adım Adım Çözüm

1

Set up the parallel lines zig-zag relationship to find

x

and

y

.

\angle ACB = x + y = 118^\circ

By drawing an auxiliary line through

C

parallel to

l_1

and

l_2

, the interior angle

\angle ACB

facing left equals the sum of the alternate interior angles

x

and

y

.

2

Substitute the given algebraic relation

y = 2x - 14

into the sum equation.

x + (2x - 14) = 118 \implies 3x - 14 = 118 \implies 3x = 132 \implies x = 44^\circ

Solving the linear system gives the exact value of angle

x

.

3

Determine the angle that line

AC

makes with line

l_2

.

Line

AC

intersects line

l_2

at an acute angle of

44^\circ

.

Since

l_1 \parallel l_2

, alternate interior angles formed by transversal line

AC

are equal (

x = 44^\circ

).

4

Calculate the acute angle

\angle CEB

between line

k

and line

l_2

.

\angle CEB = 90^\circ - 44^\circ = 46^\circ

Line

k

is perpendicular to segment

AC

, so the angle it forms with line

l_2

is the complementary angle to the angle line

AC

forms with line

l_2

.

Anahtar Kavram

Parallel Line Angle Relationships and Perpendicular Line Complements

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