All practice questions

Question 241Question →

A constant value of $7$ is added to each number in a dataset to form a new dataset. Which of the following statistical measures of the new dataset are equal to the corresponding measures of the original dataset? Select all that apply.

Select all that apply

The range

The standard deviation

The interquartile range

The mean

The median

Show answer & explanation

Answer: The range; The standard deviation; The interquartile range

Answer

The range, the standard deviation, and the interquartile range of the new dataset are equal to those of the original dataset.

Adding a constant to every value in a dataset shifts the entire distribution uniformly without altering its overall shape or spread. Consequently, measures of dispersion—including the range, standard deviation, and interquartile range—remain completely unchanged.

Step-by-Step Solution

1

Evaluate the effect of adding a constant on measures of central tendency

If each data point

x_i

is replaced by

x_i + 7

, the mean

\mu

becomes

\mu + 7

and the median

M

becomes

M + 7

.

Measures of central tendency shift directly by the added constant, so they are not equal to the original measures.

2

Evaluate the effect of adding a constant on measures of dispersion

The new range is

(x_{\text{max}} + 7) - (x_{\text{min}} + 7) = x_{\text{max}} - x_{\text{min}}

. The new interquartile range is

(Q_3 + 7) - (Q_1 + 7) = Q_3 - Q_1

. Each deviation from the mean is

(x_i + 7) - (\mu + 7) = x_i - \mu

, so the standard deviation is unchanged.

Measures of dispersion reflect the relative distances between data points or distance from the mean, which are unaffected by shifting the whole distribution.

Key Concept

Effect of adding a constant on measures of central tendency versus measures of dispersion

Question 242Question →

In 1864, Scottish polymath James Croll proposed that glacial epochs were triggered by periodic variations in Earth’s orbital eccentricity, which altered the distribution of solar radiation. Unlike earlier qualitative hypotheses that attributed glaciation solely to shifting ocean currents or topographic shifts, Croll attempted a rigorous quantitative mechanism, calculating how eccentricity-driven winters in aphelion would amplify albedo feedbacks through ice accumulation. However, Croll’s model predicted that glacial periods should alternate asynchronously between the Northern and Southern Hemispheres and that the most recent glacial epoch ended roughly 80,000 years ago. While nineteenth-century geologists initially welcomed Croll’s physical framework for establishing an absolute geological chronology, stratigraphers studying post-glacial erosion rates and Niagara Falls recession soon argued that the last ice sheet retreated far more recently—likely between 10,000 and 15,000 years ago. Furthermore, field evidence from transatlantic fossil floras suggested synchronous interhemispheric cooling rather than hemispheric alternation. Consequently, despite Croll’s pioneering integration of celestial mechanics with terrestrial feedback loops, late-nineteenth-century consensus sidelined his timeline, preferring empirical stratigraphic estimates until Milutin Milankovitch later refined the orbital parameters in the 1920s.

Consider each of the three choices separately and select all that apply.

Which of the following can be inferred from the passage regarding nineteenth-century geologists’ reaction to Croll’s theory?

Select all that apply

Their initial reception of Croll's model was influenced by the prospect of acquiring an absolute chronological standard for geological history.

They ultimately rejected Croll's model because they disputed the general principle that astronomical forces could alter Earth's climate.

Their physical evidence regarding the timing of the most recent glacial retreat differed significantly from the timeframe derived from Croll's calculations.

Show answer & explanation

Answer: Their initial reception of Croll's model was influenced by the prospect of acquiring an absolute chronological standard for geological history.; Their physical evidence regarding the timing of the most recent glacial retreat differed significantly from the timeframe derived from Croll's calculations.

Answer

The statement regarding initial interest in an absolute geological chronology and the statement regarding the discrepancy between empirical evidence and Croll's calculated timeline are both supported by the text.

The passage explicitly states that geologists initially welcomed Croll's framework because it offered an absolute geological chronology, supporting the inference regarding their initial motivation. Additionally, the passage contrasts Croll's calculated 80,000-year mark with stratigraphers' empirical finding of 10,000–15,000 years, directly supporting the inference that physical evidence contradicted his calculated timeframe.

Step-by-Step Solution

1

Analyze the passage statement about the initial reception of Croll's work.

The text explicitly states geologists welcomed his framework 'for establishing an absolute geological chronology.'

This confirms that the possibility of having a quantitative timeline was an appealing factor in their initial reception.

2

Evaluate the reason for the eventual rejection of Croll's model.

Rejection stemmed from conflicting empirical data (retreat 10,000–15,000 years ago vs. Croll's 80,000 years, and synchronous vs. asynchronous cooling).

There is no indication that geologists doubted astronomical influence in principle; rather, Croll's specific timeline and interhemispheric predictions failed empirical tests.

3

Compare stratigraphers' timing evidence with Croll's model output.

Stratigraphers estimated retreat at 10,000 to 15,000 years ago, contrasting sharply with Croll's calculated 80,000 years.

This direct quantitative disagreement validates the inference that empirical evidence contradicted Croll's theoretical timeframe.

Key Concept

Inferences and Implicit Meaning in Reading Comprehension

Estimated Time:2m 30s

Question 243Question →

In mid-nineteenth-century glaciology, Louis Agassiz proposed that extensive ice sheets had once covered northern Europe, attributing the transport of massive erratic boulders to glacial drift rather than catastrophic floods. While earlier naturalists like Leopold von Buch argued that these erratics were dispersed by high-velocity currents of liquid water carrying debris, Agassiz cited the presence of polished bedrock surfaces and fine striations beneath the boulders as direct evidence of ice abrasion. Agassiz asserted that liquid water alone lacked the frictional mass required to scratch dense granite, whereas slow-moving glaciers bearing embedded rock fragments naturally carved these distinct linear grooves into the underlying strata.

According to the passage, Agassiz cited which of the following as evidence that erratic boulders were transported by glaciers rather than liquid water?

The occurrence of fine scratches and polished surfaces on the bedrock beneath the boulders

The presence of high-velocity liquid water currents carrying debris

The total absence of granite fragments within the surrounding glacial drift

The discovery that catastrophic floods occurred prior to the formation of ice sheets

The rapid movement of ice sheets across the northern landscape

Show answer & explanation

Answer: The occurrence of fine scratches and polished surfaces on the bedrock beneath the boulders

Answer

Agassiz cited the occurrence of fine scratches (striations) and polished surfaces on the underlying bedrock as evidence of glacial transport.

The passage explicitly indicates that Agassiz supported his glacial drift hypothesis by pointing to polished bedrock surfaces and fine striations (scratches) beneath the boulders as evidence of ice abrasion.

Step-by-Step Solution

1

Identify the target keyword and concept in the prompt.

The prompt asks specifically for the evidence Louis Agassiz cited to support glacial transport over liquid water transport.

Focusing on the specific claim attributed to Agassiz prevents confusing his position with competing views.

2

Locate the sentence in the passage detailing Agassiz's specific evidence.

The text states: 'Agassiz cited the presence of polished bedrock surfaces and fine striations beneath the boulders as direct evidence of ice abrasion.'

Explicit detail retrieval questions require finding direct evidence stated explicitly in the text.

3

Match the text statement with the paraphrased option.

'Fine striations and polished bedrock surfaces' directly corresponds to 'fine scratches and polished surfaces on the bedrock beneath the boulders.'

The correct choice paraphrases the explicit facts given in the passage.

Key Concept

Explicit Detail Retrieval

Question 244Question →

In GRE Analytical Writing essays, selecting strategically between active and passive voice determines sentence clarity, agency attribution, and persuasive force. Match each problematic sentence construction on the left with the primary rhetorical defect or stylistic flaw caused by its passive voice usage on the right.

Click a left item, then click its matching right item

Show answer & explanation

Answer

Each passive sentence construction corresponds to a specific rhetorical flaw: agentless passive audit statements obscure institutional agency; passive constructions with 'by' phrases create prepositional drag; passive scientific methodology creates anonymity; and passive cause-and-effect statements dilute logical argument momentum.

Each sentence illustrates a classic rhetorical weakness associated with unstrategic passive voice usage in academic writing: agentless passives hide responsibility, passive prepositional phrases create wordiness, passive experimental descriptions obscure methodology, and passive causal chains weaken argument logic.

Step-by-Step Solution

1

Analyze the audit sentence ('financial discrepancies were discovered...') for subject-verb relations and agency.

Identified that no subject is performing the actions 'discovered' or 'made', hiding who is responsible.

Omitting the agent in policy contexts leads to Agency Obscuration.

2

Examine the sentence regarding legislative reform ('It is contended by proponents...').

Noted that the real actors ('proponents', 'decentralized governance') are buried inside 'by' prepositional phrases.

Demoting active agents into prepositional phrases creates unnecessary wordiness and Prepositional Drag.

3

Evaluate the clinical trial statement ('patient responses were recorded...').

Observed multiple consecutive passive verbs describing procedure without attributing the actions to researchers.

Stacking passive verbs in empirical contexts obscures ownership of the methodology.

4

Analyze the causal sentence ('Because insufficient evidence was presented...').

Determined that both the premise and outcome lack active subjects performing the action.

Framing cause and effect passively reduces persuasive impact and creates Causal Dilution.

Key Concept

Strategic Active vs. Passive Voice Usage

Question 245Question →

A regional health authority plans to introduce mobile vaccination units to rural districts in order to increase overall regional vaccination rates. The authority reasons that because low vaccination rates in these districts are primarily caused by the long distance residents must travel to fixed clinics, bringing mobile clinics directly to rural towns will significantly raise regional vaccination coverage.

Which of the following statements represent necessary assumptions upon which the health authority's reasoning depends? Select all that apply.

Select all that apply

Residents in rural districts who refrain from vaccination because of travel distances will choose to utilize the mobile units when they become available locally.

Mobile vaccination units provide a higher quality of overall medical care than fixed rural clinics.

The deployment of mobile vaccination units will not lead to a reduction in the services or capacity of existing fixed clinics.

Rural districts currently exhibit lower overall rates of preventable infectious diseases than urban districts.

Mobile vaccination units are significantly cheaper to operate per patient than building new permanent clinics.

Show answer & explanation

Answer: Residents in rural districts who refrain from vaccination because of travel distances will choose to utilize the mobile units when they become available locally.; The deployment of mobile vaccination units will not lead to a reduction in the services or capacity of existing fixed clinics.

Answer

The argument depends on assuming that distance-constrained rural residents will actually use the mobile units and that deploying these units will not diminish existing clinic services.

The correct statements are necessary for the conclusion to hold. First, removing the barrier of travel distance only succeeds if residents actually take advantage of local mobile units. Second, for overall regional rates to rise, gains from mobile units must not be offset by losses in fixed clinic capacity.

Step-by-Step Solution

1

Identify the conclusion and main premise of the argument.

Conclusion: Mobile vaccination units will significantly raise regional vaccination coverage. Premise: Distance to fixed clinics is the primary cause of low rural vaccination rates.

Understanding the premise-to-conclusion link is necessary to isolate unstated assumptions.

2

Apply the Negation Test to candidate statements.

Negating the statement about resident utilization reveals that if residents do not use mobile units, coverage will not rise. Negating the statement about existing clinic services reveals that if existing services drop, net coverage will not rise.

A necessary assumption, when negated, logically destroys the validity of the conclusion.

Key Concept

Identifying Underlying Assumptions using the Negation Test

Question 246Question →

In a circle centered at point $O$ , sector $AOB$ has a radius of length $r$ , a central angle of $\theta$ degrees, an arc length of $L$ , an area of $A_s$ , and a perimeter of $P$ . If the ratio of the sector area to the arc length is $\frac{A_s}{L} = 4$ , and the ratio of the sector area to the sector perimeter is $\frac{A_s}{P} = \frac{4}{3}$ , which of the following statements must be true? Select all that apply.

Select all that apply

The radius

r

of the circle is

8

.

The area

A_s

of sector

AOB

is

32

.

The central angle

\theta

is greater than

60^\circ

.

The perimeter

P

of sector

AOB

is

24

.

The arc length

L

of sector

AOB

is

8\pi

.

Show answer & explanation

Answer: The radius $r$ of the circle is $8$ .; The area $A_s$ of sector $AOB$ is $32$ .; The perimeter $P$ of sector $AOB$ is $24$ .

Answer

The true statements are that the radius of the circle is 8, the area of the sector is 32, and the perimeter of the sector is 24.

The ratio of sector area to arc length simplifies directly to half the radius,

\frac{r}{2} = 4

, giving a radius of

8

. Expressing the sector perimeter as the sum of the arc length and two radii (

L + 16

) and using

\frac{A_s}{P} = \frac{4L}{L + 16} = \frac{4}{3}

yields

L = 8

. From this, the sector area is

A_s = 32

and the sector perimeter is

P = 24

.

Step-by-Step Solution

1

Relate sector area

A_s

and arc length

L

to find the radius

r

.

A_s = \frac{\theta}{360}\pi r^2

and

L = \frac{\theta}{360}2\pi r

, so

\frac{A_s}{L} = \frac{r}{2} = 4 \implies r = 8

.

Dividing the sector area formula by the arc length formula cancels the central angle fraction and

\pi

.

2

Set up the ratio equation for

\frac{A_s}{P}

using

P = L + 2r

.

Since

r = 8

,

P = L + 16

. Given

A_s = 4L

, we have

\frac{4L}{L + 16} = \frac{4}{3} \implies 3L = L + 16 \implies 2L = 16 \implies L = 8

.

Expressing both sector area and perimeter in terms of arc length allows direct solution for

L

.

3

Calculate sector area

A_s

and perimeter

P

.

A_s = 4(8) = 32

and

P = 8 + 2(8) = 24

.

Substituting

L = 8

and

r = 8

gives the exact values for area and perimeter.

4

Determine the central angle

\theta

.

8 = \frac{\theta}{360} \cdot 2\pi(8) \implies 8 = \frac{16\pi \theta}{360} \implies \theta = \left(\frac{90}{\pi}\right)^\circ \approx 28.65^\circ

.

Plugging

L = 8

and

r = 8

into the arc length formula gives

\theta \approx 28.65^\circ

, which is less than

60^\circ

.

Key Concept

Relating sector area, arc length, and perimeter using proportional ratios and fundamental circle formulas.

Question 247Question →

In the linear equation below,

k

is a constant:

\frac{x - k}{2} + \frac{2x + 1}{5} = \frac{3x - 4}{10}

If

x

must be a positive integer, which of the following values could be

k

? Select all such values.

Select all that apply

-6

6

10

12

15

Show answer & explanation

Answer: $6$ ; $12$

Answer

The values of

k

that satisfy the condition are

6

and

12

.

Clearing denominators gives

6x = 5k - 6

, or

x = \frac{5k - 6}{6}

. For

x

to be a positive integer,

5k - 6

must be positive and divisible by

6

. This requires

k

to be a positive multiple of

6

. The options

6

and

12

satisfy both criteria, yielding

x = 4

and

x = 9

respectively.

Step-by-Step Solution

1

Clear denominators by multiplying the entire equation by the least common denominator (LCD), which is 10.

5(x - k) + 2(2x + 1) = 3x - 4

Eliminating fractions simplifies further algebraic manipulation.

2

Expand terms and combine like terms on the left side of the equation.

5x - 5k + 4x + 2 = 3x - 4 \implies 9x - 5k + 2 = 3x - 4

Distribute constants across parentheses to isolate variable expressions.

3

Isolate

x

in terms of

k

.

6x = 5k - 6 \implies x = \frac{5k - 6}{6}

Express

x

as an explicit linear function of

k

to test divisibility and positivity conditions.

4

Determine constraints on

k

for

x

to be a positive integer.

5k

must be divisible by

6

(so

k

must be a multiple of

6

) and

5k - 6 > 0 \implies k > 1.2

.

Since

5

and

6

share no common factors,

k

must be a positive multiple of

6

.

5

Evaluate the choices against the constraints.

Among the options, only

6

and

12

are positive multiples of

6

, producing

x = 4

and

x = 9

respectively.

Both values produce integer solutions greater than zero.

Key Concept

Solving single-variable linear equations with rational coefficients and applying divisibility and inequality constraints.

Question 248Question →

If $x$ and $y$ are real numbers satisfying the inequalities $|2x - 7| \le 5$ and $|3y + 2| < 8$ , which of the following inequalities must be true? Select all that apply.

Select all that apply

x - y > -1

xy < 12

x + y > 0

|y| < 3

y - x > -3

Show answer & explanation

Answer: $x - y > -1$ ; $xy < 12$

Answer

The inequalities that must be true are

x - y > -1

and

xy < 12

.

Solving

|2x - 7| \le 5

yields

1 \le x \le 6

, and solving

|3y + 2| < 8

yields

-\frac{10}{3} < y < 2

. Combining

x \ge 1

with

-y > -2

gives

x - y > -1

, which is always true. Furthermore, since

x

is positive and bounded above by

6

while

y

is bounded above by

2

, the product

xy

must be strictly less than

12

.

Step-by-Step Solution

1

Solve the absolute value inequality for

x

.

-5 \le 2x - 7 \le 5 \implies 2 \le 2x \le 12 \implies 1 \le x \le 6

.

Unfolding the absolute value

|2x - 7| \le 5

gives a compound linear inequality.

2

Solve the absolute value inequality for

y

.

-8 < 3y + 2 < 8 \implies -10 < 3y < 6 \implies -\frac{10}{3} < y < 2

.

Unfolding the absolute value

|3y + 2| < 8

gives a strict compound linear inequality.

3

Evaluate the inequality

x - y > -1

.

Since

x \ge 1

and

y < 2 \implies -y > -2

, adding the inequalities gives

x + (-y) > 1 + (-2) = -1

.

This establishes that

x - y > -1

is always true.

4

Evaluate the inequality

xy < 12

.

Since

1 \le x \le 6

(all positive) and

y < 2

, if

y > 0

,

xy < 6 \cdot 2 = 12

. If

y \le 0

,

xy \le 0 < 12

.

In all cases within the domain,

xy < 12

holds strictly.

5

Test counterexamples for the remaining statements.

For

x + y > 0

,

x=1, y=-3 \implies x+y=-2 \ngtr 0

. For

|y| < 3

,

y=-3.2 \implies |-3.2|=3.2 \nless 3

. For

y - x > -3

,

x=6, y=0 \implies y-x=-6 \ngtr -3

.

Counterexamples disprove that these remaining statements must be true.

Key Concept

Linear Inequalities and Absolute Value Bounds

Question 249Question →

A dataset $S$ consists of $9$ distinct integers. The median of $S$ is $20$ , and the mean of $S$ is $24$ . The mean of the $4$ smallest integers in $S$ is $10$ . If the largest integer in $S$ is removed to form a new dataset $R$ consisting of $8$ integers, what is the maximum possible median of dataset $R$ ?

18.5

19.0

19.5

20.0

20.5

Show answer & explanation

Answer: $19.5$

Answer

The maximum possible median of dataset

R

is

19.5

.

In the sorted set of

9

distinct integers, the median is the

5

th element, which is

20

. Removing the largest value leaves

8

integers, whose median is the average of the

4

th and

5

th elements. Because all integers are distinct, the

4

th element can be at most

19

. The maximum median is therefore

\frac{19 + 20}{2} = 19.5

.

Step-by-Step Solution

1

Identify the position of the median in the original dataset

S

.

In a dataset of

9

sorted distinct integers

x_1 < x_2 < x_3 < x_4 < x_5 < x_6 < x_7 < x_8 < x_9

, the median is the

5

th element, so

x_5 = 20

.

For an odd number of ordered elements

n=9

, the median is at index

\frac{9+1}{2} = 5

.

2

Determine the formula for the median of the modified dataset

R

.

Removing the largest element

x_9

leaves

8

ordered elements

x_1, x_2, x_3, x_4, x_5, x_6, x_7, x_8

. The median of

R

is

\frac{x_4 + x_5}{2} = \frac{x_4 + 20}{2}

.

For an even number of ordered elements

n=8

, the median is the average of the two middle terms at indices

4

and

5

.

3

Maximize the value of

x_4

.

Since all integers are distinct and

x_5 = 20

,

x_4

must be an integer strictly less than

20

. Thus, the maximum integer value

x_4

can take is

19

.

Maximizing

x_4

maximizes the average

\frac{x_4 + 20}{2}

.

4

Verify that

x_4 = 19

is achievable under the given mean constraint.

The sum of the smallest

4

integers is

4 \times 10 = 40

. With

x_4 = 19

, we require

x_1 + x_2 + x_3 = 21

. Choosing distinct integers such as

x_1 = 1

,

x_2 = 2

, and

x_3 = 18

satisfies

1 + 2 + 18 = 21

and

x_1 < x_2 < x_3 < x_4

.

Confirming feasibility ensures the upper bound

19

is valid.

5

Calculate the maximum median of dataset

R

.

\frac{19 + 20}{2} = 19.5

.

Substitute the maximum value of

x_4 = 19

into the median expression.

Key Concept

Properties of median for odd vs. even datasets and optimization under distinctness constraints

Question 250Question →

A solid right circular cylinder has a base radius of $r$ and a height of $4r$ . A solid sphere has a radius of $R$ . If the total surface area of the sphere is equal to the total surface area of the cylinder, what is the ratio of the volume of the sphere to the volume of the cylinder?

\frac{5\sqrt{10}}{12}

\frac{\sqrt{10}}{2}

\frac{2\sqrt{2}}{3}

\frac{5\sqrt{10}}{6}

\frac{5}{3}

Show answer & explanation

Answer: $\frac{5\sqrt{10}}{12}$

Answer

The ratio of the volume of the sphere to the volume of the cylinder is

\frac{5\sqrt{10}}{12}

.

The total surface area of the cylinder is the sum of its lateral area and two circular bases:

2\pi r(4r) + 2\pi r^2 = 10\pi r^2

. Setting this equal to the sphere's surface area

4\pi R^2

yields

R/r = \sqrt{5/2} = \sqrt{10}/2

. The ratio of the sphere's volume

\frac{4}{3}\pi R^3

to the cylinder's volume

\pi r^2 (4r) = 4\pi r^3

is

\frac{1}{3}(R/r)^3 = \frac{1}{3} \left(\frac{\sqrt{10}}{2}\right)^3 = \frac{5\sqrt{10}}{12}

.

Step-by-Step Solution

1

Calculate the total surface area of the cylinder.

\text{TSA}_{\text{cyl}} = 2\pi r^2 + 2\pi r h = 2\pi r^2 + 2\pi r(4r) = 10\pi r^2

A cylinder's total surface area consists of two circular bases (

2\times\pi r^2

) and the lateral surface area (

2\pi r h

).

2

Equate the total surface area of the sphere to the total surface area of the cylinder to find the ratio of

R

to

r

.

4\pi R^2 = 10\pi r^2 \implies R^2 = \frac{5}{2}r^2 \implies \frac{R}{r} = \sqrt{\frac{5}{2}} = \frac{\sqrt{10}}{2}

The total surface area of a sphere of radius

R

is

4\pi R^2

.

3

Express the volumes of both figures in terms of

r

and

R

.

V_{\text{sph}} = \frac{4}{3}\pi R^3

and

V_{\text{cyl}} = \pi r^2 h = \pi r^2 (4r) = 4\pi r^3

The volume of a sphere is

\frac{4}{3}\pi R^3

and the volume of a cylinder is

\pi r^2 h

.

4

Compute the ratio of the volume of the sphere to the volume of the cylinder.

\frac{V_{\text{sph}}}{V_{\text{cyl}}} = \frac{\frac{4}{3}\pi R^3}{4\pi r^3} = \frac{1}{3}\left(\frac{R}{r}\right)^3 = \frac{1}{3}\left(\frac{\sqrt{10}}{2}\right)^3 = \frac{1}{3} \cdot \frac{10\sqrt{10}}{8} = \frac{5\sqrt{10}}{12}

Substitute

\frac{R}{r} = \frac{\sqrt{10}}{2}

into the ratio expression.

Key Concept

Relating 3D surface area formulas to volume formulas for cylinders and spheres

Estimated Time:2m 0s

Question 251Question →

Which of the following values of $x$ are solutions to the equation $(x - 2)^{\frac{4}{3}} - 5(x - 2)^{\frac{2}{3}} + 4 = 0$ ? Select all that apply.

Select all that apply

-6

1

10

-2

6

Show answer & explanation

Answer: -6; 1; 10

Answer

The correct values of

x

that satisfy the equation are

-6

,

1

, and

10

.

Substituting

u = (x - 2)^{\frac{2}{3}}

yields

u^2 - 5u + 4 = 0

, which factors as

(u - 1)(u - 4) = 0

, giving

u = 1

and

u = 4

. Solving

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

gives

(x - 2)^2 = 1 \implies x - 2 = \pm 1

, yielding

x = 3

and

x = 1

. Solving

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

gives

(x - 2)^2 = 64 \implies x - 2 = \pm 8

, yielding

x = 10

and

x = -6

. Thus, the values

-6

,

1

, and

10

are all valid solutions.

Step-by-Step Solution

1

Perform a substitution to rewrite the equation in quadratic form.

Let

u = (x - 2)^{\frac{2}{3}}

. Then

u^2 = (x - 2)^{\frac{4}{3}}

, giving

u^2 - 5u + 4 = 0

.

Recognizing quadratic structure simplifies equations with rational exponents.

2

Solve the quadratic equation for

u

.

(u - 1)(u - 4) = 0 \implies u = 1

or

u = 4

.

Factoring determines the values of the substituted variable

u

.

3

Solve for

x

when

u = 1

.

(x - 2)^{\frac{2}{3}} = 1 \implies (x - 2)^2 = 1^3 = 1 \implies x - 2 = \pm 1 \implies x = 3

or

x = 1

.

Raising both sides to the power of

\frac{3}{2}

requires taking both positive and negative roots because the numerator of the power is even.

4

Solve for

x

when

u = 4

.

(x - 2)^{\frac{2}{3}} = 4 \implies (x - 2)^2 = 4^3 = 64 \implies x - 2 = \pm 8 \implies x = 10

or

x = -6

.

Squaring and taking square roots yields two solutions,

10

and

-6

.

5

Match calculated solutions with the given choices.

The solutions present among the options are

-6

,

1

, and

10

.

Comparing all valid algebraic solutions to the available choices identifies all correct options.

Key Concept

Solving quadratic-form equations with fractional exponents and accounting for negative base branches when taking even roots.

Question 252Question →

If $a > 0$ , which of the following is equivalent to the expression $\sqrt{\frac{a^8 \cdot a^4}{a^{-2}}}$ ?

a^7

a^5

a^6

a^{14}

a^{10}

Show answer & explanation

Answer: $a^7$

Answer

a^7

Multiplying the terms in the numerator gives

a^{12}

. Dividing by

a^{-2}

gives

a^{12 - (-2)} = a^{14}

. Taking the square root of

a^{14}

gives

(a^{14})^{1/2} = a^7

.

Step-by-Step Solution

1

Simplify the numerator inside the square root using the product rule

a^m \cdot a^n = a^{m+n}

.

a^8 \cdot a^4 = a^{8+4} = a^{12}

Powers with the same base are multiplied by adding their exponents.

2

Divide by the denominator using the quotient rule

\frac{a^m}{a^n} = a^{m-n}

.

\frac{a^{12}}{a^{-2}} = a^{12 - (-2)} = a^{14}

Dividing powers with the same base requires subtracting the lower exponent from the upper exponent.

3

Apply the fractional exponent rule for radicals

\sqrt{x} = x^{\frac{1}{2}}

.

\sqrt{a^{14}} = (a^{14})^{\frac{1}{2}} = a^{14 \cdot \frac{1}{2}} = a^7

Taking the square root of a power is equivalent to multiplying the exponent by

\frac{1}{2}

.

Key Concept

Simplifying expressions using exponent laws and fractional radical powers

Question 253Question →

In a group of 200 student test scores, a score of $S$ is located at the 75th percentile of the distribution. Which of the following statements must be true?

Score

S

is equal to 75.

Approximately 75 students scored at or below

S

.

Approximately 150 students scored at or below

S

.

Score

S

is 75 percent greater than the average score of the distribution.

The standard deviation of the test scores is equal to 75.

Show answer & explanation

Answer: Approximately 150 students scored at or below $S$ .

Answer

Approximately 150 students scored at or below

S

.

The 75th percentile of a distribution indicates that approximately

75\%

of the data points lie at or below that value. For a group of 200 students,

75\%

of 200 is calculated as

0.75 \times 200 = 150

. Therefore, approximately 150 students achieved a score at or below

S

.

Step-by-Step Solution

1

Understand the definition of a percentile rank

The 75th percentile means that approximately

75\%

of all observations in the dataset are less than or equal to that score.

By definition, the

p

-th percentile represents the value below which

p

percent of the observations fall.

2

Calculate 75% of the total number of students

0.75 \times 200 = 150

students.

The dataset contains 200 total student scores, so

75\%

of 200 yields the number of students at or below score

S

.

Key Concept

Percentile Position

Estimated Time:45s

Question 254Question →

Dataset $X$ consists of seven numerical values: $\{12, 16, 20, 24, 28, 32, 36\}$ . Dataset $Y$ is formed by replacing the value $36$ in Dataset $X$ with $56$ , while keeping all other six values the same. Which of the following statements correctly compares the interquartile range (IQR) and standard deviation of Dataset $Y$ to those of Dataset $X$ ?

The IQR remains unchanged, while the standard deviation increases.

Both the IQR and the standard deviation increase.

The IQR increases, while the standard deviation remains unchanged.

Both the IQR and the standard deviation remain unchanged.

The IQR decreases, while the standard deviation increases.

Show answer & explanation

Answer: The IQR remains unchanged, while the standard deviation increases.

Answer

The IQR remains unchanged, while the standard deviation increases.

The statement specifying that the IQR remains unchanged while the standard deviation increases is correct because the quartiles Q1 (16) and Q3 (32) are unaffected by changing the largest value from 36 to 56, maintaining an IQR of 16. In contrast, standard deviation measures spread relative to the mean, so pulling the maximum value further out increases overall deviation.

Step-by-Step Solution

1

Calculate the IQR of Dataset X

For Dataset X = {12, 16, 20, 24, 28, 32, 36}, the median is 24. The lower half is {12, 16, 20} with Q1 = 16. The upper half is {28, 32, 36} with Q3 = 32. Thus, IQR_X = 32 - 16 = 16.

IQR is calculated as the difference between the third quartile (Q3) and the first quartile (Q1).

2

Calculate the IQR of Dataset Y

For Dataset Y = {12, 16, 20, 24, 28, 32, 56}, the median is still 24. The lower half remains {12, 16, 20} (Q1 = 16). The upper half becomes {28, 32, 56}, so Q3 is still 32. Thus, IQR_Y = 32 - 16 = 16.

The third quartile is the middle number of the upper half, which remains 32 regardless of replacing 36 with 56.

3

Compare the standard deviation of Dataset X and Dataset Y

Replacing 36 with a significantly higher value 56 increases the distance of the maximum data point from the mean, increasing overall variance and thus increasing the standard deviation.

Standard deviation measures the average distance of data points from the mean and is highly sensitive to extreme values/outliers.

Key Concept

Sensitivity of Measures of Dispersion to Outliers

Question 255Question →

Two straight lines, $L_1$ and $L_2$ , intersect at point $O$ . One of the angles formed by their intersection measures $(3x - 15)^\circ$ , and the vertically opposite angle measures $(x + 25)^\circ$ . What is the degree measure of an angle adjacent to $(3x - 15)^\circ$ ?

20^\circ

45^\circ

135^\circ

67.5^\circ

112.5^\circ

Show answer & explanation

Answer: $135^\circ$

Answer

135^\circ

Since vertically opposite angles are equal, setting

3x - 15 = x + 25

yields

x = 20

. Substituting

x = 20

into

3x - 15

gives an angle of

45^\circ

. Because adjacent angles on intersecting lines form a straight line (a linear pair), they are supplementary. Therefore, the adjacent angle measures

180^\circ - 45^\circ = 135^\circ

.

Step-by-Step Solution

1

Set the vertically opposite angle expressions equal to each other.

3x - 15 = x + 25

Vertically opposite angles formed by two intersecting lines are equal in measure.

2

Solve for the variable

x

.

2x = 40 \implies x = 20

Subtract

x

and add

15

to both sides of the equation.

3

Calculate the degree measure of the angle

(3x - 15)^\circ

.

3(20) - 15 = 60 - 15 = 45^\circ

Substitute

x = 20

back into the angle expression.

4

Calculate the measure of an adjacent angle.

180^\circ - 45^\circ = 135^\circ

Adjacent angles along a straight line are supplementary and sum to

180^\circ

.

Key Concept

Vertically opposite angles are equal, and adjacent angles forming a linear pair are supplementary.

Question 256Question →

If $x$ is a real number that satisfies the inequality $|2x - 7| < 5$ , which of the following represents all possible values of the expression $1 - 3x$ ?

-17 < 1 - 3x < -2

-2 < 1 - 3x < 17

-17 < 1 - 3x < 1

-17 < 1 - 3x < 17

-5 < 1 - 3x < 5

Show answer & explanation

Answer: $-17 < 1 - 3x < -2$

Answer

-17 < 1 - 3x < -2

Solving

|2x - 7| < 5

gives

-5 < 2x - 7 < 5

. Adding

7

to all three parts yields

2 < 2x < 12

, which simplifies to

1 < x < 6

. Multiplying this inequality by

-3

reverses the direction of the inequalities, resulting in

-18 < -3x < -3

. Adding

1

to each part gives

-17 < 1 - 3x < -2

. Therefore, the range of possible values for the expression is strictly between

-17

and

-2

.

Step-by-Step Solution

1

Express the absolute value inequality as a compound inequality.

-5 < 2x - 7 < 5

An inequality of the form

|u| < k

for

k > 0

is equivalent to

-k < u < k

.

2

Isolate

x

in the compound inequality.

1 < x < 6

Add

7

to all parts to get

2 < 2x < 12

, then divide all parts by

2

to obtain

1 < x < 6

.

3

Multiply the compound inequality by

-3

.

-18 < -3x < -3

Multiplying an inequality by a negative number reverses the direction of the inequality signs:

-3(6) < -3(x) < -3(1)

.

4

Add

1

to all parts of the compound inequality.

-17 < 1 - 3x < -2

Adding a constant to an inequality preserves the inequality direction.

Key Concept

Linear Inequalities and Absolute Value Transformations

Estimated Time:1m 30s

Question 257Question →

A sector of a circle with a radius of $12$ units has an area of $24\pi$ square units. What is the measure, in degrees, of the central angle of the sector?

Show answer & explanation

Answer: 60

Answer

The measure of the central angle of the sector is

60^\circ

.

The area of the full circle is

\pi r^2 = \pi (12)^2 = 144\pi

. The sector area represents a fraction of the total area, specifically

\frac{24\pi}{144\pi} = \frac{1}{6}

. Multiplying this fraction by the full circle's central angle of

360^\circ

gives

\frac{1}{6} \times 360^\circ = 60^\circ

.

Step-by-Step Solution

1

Find the total area of the circle

The total area of the circle is

\pi \times 12^2 = 144\pi

.

The area of a full circle with radius

r

is given by

A = \pi r^2

.

2

Relate the sector area to the total circle area

The fraction of the circle represented by the sector is

\frac{24\pi}{144\pi} = \frac{1}{6}

.

The area of a sector is proportional to the fraction of the total central angle (

360^\circ

) it subtends.

3

Calculate the central angle in degrees

\theta = \frac{1}{6} \times 360^\circ = 60^\circ.

Multiply the fraction of the circle by

360^\circ

to get the central angle measure.

Key Concept

Relationship between central angle measure, total circle area, and sector area

Question 258Question →

If $x$ is a real number that satisfies the inequality $||2x - 5| - 7| \le 4$ , and $y = |3 - x|$ , what is the difference between the maximum possible value and the minimum possible value of $y$ ?

3

4

5

6

7

Show answer & explanation

Answer: 5

Answer

The difference between the maximum possible value and the minimum possible value of

y

is 5.

Solving

||2x - 5| - 7| \le 4

gives

3 \le |2x - 5| \le 11

, which restricts

x

to the disconnected domain

[-3, 1] \cup [4, 8]

. Evaluating

y = |3 - x|

across these intervals gives a maximum value of 6 (at

x = -3

) and a minimum value of 1 (at

x = 4

). The difference between the maximum and minimum values is

6 - 1 = 5

.

Step-by-Step Solution

1

Unpack the outer absolute value inequality

||2x - 5| - 7| \le 4

.

-4 \le |2x - 5| - 7 \le 4 \implies 3 \le |2x - 5| \le 11

.

An absolute value inequality

|u| \le a

(for

a \ge 0

) is equivalent to

-a \le u \le a

.

2

Solve the double inequality

3 \le |2x - 5| \le 11

by splitting it into two conditions.

Condition 1:

|2x - 5| \le 11 \implies -11 \le 2x - 5 \le 11 \implies -3 \le x \le 8

.
Condition 2:

|2x - 5| \ge 3 \implies 2x - 5 \ge 3

or

2x - 5 \le -3 \implies x \ge 4

or

x \le 1

.

The quantity

|2x - 5|

must simultaneously satisfy upper and lower absolute value bounds.

3

Intersect Condition 1 and Condition 2 to determine the complete domain of

x

.

x \in [-3, 1] \cup [4, 8]

.

Values in the open interval

(1, 4)

make

|2x - 5| < 3

and must be excluded from the domain.

4

Evaluate the range of

y = |3 - x|

over the valid domain of

x

.

On

[-3, 1]

,

y = 3 - x

decreases from

3 - (-3) = 6

to

3 - 1 = 2

, giving

y \in [2, 6]

.
On

[4, 8]

,

y = x - 3

increases from

4 - 3 = 1

to

8 - 3 = 5

, giving

y \in [1, 5]

.
The complete range of

y

is

[1, 6]

.

Combining the output ranges of both disjoint intervals yields all possible values for

y

.

5

Calculate the difference between the maximum and minimum values of

y

.

Maximum

y = 6

, Minimum

y = 1

, Difference =

6 - 1 = 5

.

Subtracting the minimum value 1 from the maximum value 6 gives the required difference.

Key Concept

Solving compound nested absolute value inequalities and finding the extreme values of a transformed function over disconnected solution intervals.

Estimated Time:2m 0s

Question 259Question →

If $2^{x + 3} \cdot 4^{x - 1} = 16^x$ , what is the value of $x$ ?

Show answer & explanation

Answer: 1

Answer

The value of

x

is 1.

Rewriting

4^{x-1}

as

2^{2x-2}

and

16^x

as

2^{4x}

transforms the left side into

2^{x+3} \cdot 2^{2x-2} = 2^{3x+1}

. Setting exponents equal gives

3x + 1 = 4x

, which simplifies to

x = 1

.

Step-by-Step Solution

1

Express all terms with a common base of 2

2^{x+3} \cdot (2^2)^{x-1} = (2^4)^x

Converting

4

to

2^2

and

16

to

2^4

allows all terms to share the base 2.

2

Apply the power of a power rule

(a^m)^n = a^{m \cdot n}

and the product rule

a^m \cdot a^n = a^{m+n}

2^{3x+1} = 2^{4x}

Multiplying exponents gives

(2^2)^{x-1} = 2^{2x-2}

and

(2^4)^x = 2^{4x}

. Adding exponents on the left gives

(x+3) + (2x-2) = 3x+1

.

3

Equate the exponents and solve for

x

x = 1

Since

2^A = 2^B

implies

A = B

, setting

3x + 1 = 4x

directly yields

x = 1

.

Key Concept

Solving exponential equations using common bases and exponent properties

Question 260Question →

The high temperatures, in degrees Fahrenheit, recorded in a city over a 7-day period were 64, 58, 75, 67, 61, 83, and 72. What is the interquartile range of these temperatures, in degrees Fahrenheit?

Show answer & explanation

Answer: 14

Answer

The interquartile range of the recorded temperatures is 14.

To calculate the interquartile range, first arrange the dataset in ascending order: 58, 61, 64, 67, 72, 75, 83. The median of the 7 values is 67. The first quartile (

Q_1

) is the median of the lower three values (58, 61, 64), which is 61. The third quartile (

Q_3

) is the median of the upper three values (72, 75, 83), which is 75. Subtracting

Q_1

from

Q_3

yields an interquartile range of

75 - 61 = 14

.

Step-by-Step Solution

1

Order the dataset from least to greatest

58, 61, 64, 67, 72, 75, 83

Finding quartiles requires data to be arranged in ascending order.

2

Find the first quartile (

Q_1

) and third quartile (

Q_3

)

Q_1 = 61

and

Q_3 = 75

The median of the dataset (the 4th value) is 67. The lower half of the data consists of 58, 61, 64 (median 61), and the upper half consists of 72, 75, 83 (median 75).

3

Compute the difference between

Q_3

and

Q_1

75 - 61 = 14

The interquartile range is defined as

IQR = Q_3 - Q_1

.

Key Concept

Interquartile Range (IQR)

Answer

Step-by-Step Solution

Key Concept

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