All practice questions

If $9^{x+1} - 9^x = 216$ , what is the value of $4^x$ ?

Show answer & explanation

Answer: 8

Answer

The value of

4^x

is 8.

Factoring

9^x

from

9^{x+1} - 9^x

gives

9^x(9 - 1) = 216

, or

8 \cdot 9^x = 216

. Dividing by 8 yields

9^x = 27

. Expressing both sides with base 3 gives

3^{2x} = 3^3

, so

2x = 3

and

x = \frac{3}{2}

. Substituting this value into

4^x

results in

4^{3/2} = (\sqrt{4})^3 = 2^3 = 8

.

Step-by-Step Solution

1

Factor out common exponent terms from the left side of the equation

9^x(9^1 - 1) = 216 \implies 8 \cdot 9^x = 216

Using the exponent property

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

, rewrite

9^{x+1}

as

9^x \cdot 9^1

to factor out

9^x

.

2

Isolate the exponential term

9^x = 27

Dividing both sides of

8 \cdot 9^x = 216

by 8 yields

27

.

3

Convert both sides to a common prime base of 3

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

Since bases are equal, exponents must be equal.

4

Evaluate the target expression

4^x

4^{3/2} = (4^{1/2})^3 = 2^3 = 8

Substitute

x = \frac{3}{2}

into

4^x

and apply the rule

a^{m/n} = (\sqrt[n]{a})^m

.

Key Concept

Solving exponential equations by factoring and equating powers with a common base.

Question 1022Question →

A data set consists of five positive integers $a, b, c, d,$ and $e$ , written in non-decreasing order ( $a \le b \le c \le d \le e$ ). The set has a unique mode of $8$ , a median of $10$ , and an arithmetic mean of $12$ . If the range of the set is $14$ , what is the value of the largest integer $e$ ?

17

20

22

24

26

Show answer & explanation

Answer: 22

Answer

22

The median of a 5-element ordered set is the middle element, so

c = 10

. Because 8 is the unique mode and is strictly less than 10, the first two numbers must both be 8 (

a = 8, b = 8

). The range is 14, meaning the difference between the maximum element

e

and the minimum element

a

is 14. Therefore,

e = 8 + 14 = 22

.

Step-by-Step Solution

1

Determine the median value

c

c = 10

In an ordered 5-element set, the median is the 3rd element.

2

Determine the values of

a

and

b

using the unique mode

a = 8

and

b = 8

Since 8 is the unique mode and

8 < 10

, 8 must appear at least twice in the first two positions.

3

Calculate the largest integer

e

using the given range

e = a + 14 = 8 + 14 = 22

The range of a set is the difference between the maximum (

e

) and minimum (

a

) values.

4

Verify consistency with the mean

Sum

= 5 \times 12 = 60

;

d = 60 - (8 + 8 + 10 + 22) = 12

, which satisfies

10 \le 12 \le 22

.

Ensures all set conditions (mean of 12, non-decreasing order, unique mode) are satisfied.

Key Concept

Relating Mean, Median, Mode, and Range in Ordered Data Sets

Question 1023Question →

In astrophysics, the precise mechanism governing the formation of massive stars—those with masses exceeding eight solar masses—remained debated for decades. According to the core accretion model, a dense circumstellar disk continuously feeds gas onto a protostellar core until radiation pressure from the growing star halts further inflow. However, early theoretical calculations suggested that intense radiation pressure would blast away surrounding gas and dust before the star could accrete sufficient mass. To resolve this apparent conflict, recent three-dimensional hydrodynamical simulations revealed that accretion disks develop narrow, collimated channels through which radiation escapes, allowing infalling gas to continue accumulating along the equatorial plane. Thus, these simulations demonstrate that core accretion remains a viable framework for massive star formation.

Which of the following best states the primary purpose of the passage?

To explain how recent computational simulations resolve a theoretical obstacle in massive star formation.

To argue that the core accretion model should be completely replaced by a new astrophysical theory.

To detail the specific process by which radiation pressure blasts away surrounding circumstellar gas.

To contrast theoretical calculations with direct empirical telescopic observations of protostars.

To demonstrate that radiation pressure is irrelevant to the growth of protostellar cores.

Show answer & explanation

Answer: To explain how recent computational simulations resolve a theoretical obstacle in massive star formation.

Answer

The primary purpose of the passage is to explain how recent computational simulations resolve a theoretical obstacle in massive star formation.

The passage follows a classic problem-solution structure: it introduces the core accretion model, explains a potential flaw (radiation pressure blowing away gas), and describes how recent 3D hydrodynamical simulations resolved this issue by showing how gas can continue to fall along the equatorial plane. Therefore, the main purpose is to explain how these simulations resolve a theoretical obstacle in stellar formation.

Step-by-Step Solution

1

Identify the main topic and structural flow of the passage.

The passage begins with the core accretion model, introduces a theoretical conflict involving radiation pressure, and ends with new 3D simulations that resolve this conflict.

Tracking structural transitions reveals the author's core thesis and reason for writing.

2

Evaluate the author's ultimate conclusion.

The author concludes that core accretion 'remains a viable framework' because simulations showed how radiation escapes without stopping gas inflow.

The primary purpose must reflect the main resolution reached at the end of the text.

3

Match the overall narrative to the choice that captures scope and tone accurately.

The statement emphasizing how recent simulations resolve the theoretical obstacle accurately captures the full scope of the passage.

Primary purpose answer choices must encompass the entire scope without overgeneralizing or focusing on minor details.

Key Concept

Identifying Primary Purpose in Scientific Passages

Question 1024Question →

A wholesaler purchased $80$ identical smartphones for a total cost of $\$16,000$ . The wholesaler marked up the cost price of each smartphone by $50\%$ to establish the regular retail price. During a promotional sale, $20\%$ of the smartphones were sold at a $20\%$ discount off the regular retail price, and the remaining smartphones were sold at the full regular retail price. What was the total profit earned by the wholesaler on all $80$ smartphones?

\$5,760

\$6,400

\$7,040

\$7,360

\$8,000

Show answer & explanation

Answer: $\$7,040$

Answer

The total profit earned by the wholesaler on all

80

smartphones was

\$7,040

.

The cost per smartphone is

\$200

. Marking up by

50\%

gives a regular retail price of

\$300

.

16

smartphones (

20\%

of

80

) are sold at a

20\%

discount for

\$240

each, generating

\$3,840

in revenue. The remaining

64

smartphones are sold at

\$300

each, generating

\$19,200

in revenue. The total revenue is

\$23,040

, and subtracting the initial total cost of

\$16,000

yields a net profit of

\$7,040

.

Step-by-Step Solution

1

Calculate the cost price per unit and the regular retail price per unit.

Cost per unit =

\frac{\$16,000}{80} = \$200

. Regular retail price =

\$200 \times (1 + 0.50) = \$300

.

Markup is applied to the cost price per unit.

2

Calculate the number of units sold at a discount and at regular price, along with their respective selling prices.

Discounted units =

80 \times 0.20 = 16

units. Regular units =

80 - 16 = 64

units. Discounted price =

\$300 \times (1 - 0.20) = \$240

.

The

20\%

discount is taken off the regular retail price of

\$300

.

3

Calculate total revenue and total profit.

Total revenue =

(16 \times \$240) + (64 \times \$300) = \$3,840 + \$19,200 = \$23,040

. Total profit =

\$23,040 - \$16,000 = \$7,040

.

Profit is total revenue minus total cost.

Key Concept

Successive percentage markup and discount calculations with split inventory

Estimated Time:2m 0s

Question 1025Question →

If

x

is a real number that satisfies the absolute value equation

|2x - 7| = 3x - 11

what is the value of

x^2 - 3x

?

2.16

4

16

28

270

Show answer & explanation

Answer: $4$

Answer

The value of

x^2 - 3x

is

4

.

Solving the absolute value equation

|2x - 7| = 3x - 11

yields two potential roots:

x = 4

and

x = 3.6

. Checking both in the original equation shows that

x = 3.6

makes the right-hand side negative (

-0.2

), making it an extraneous solution. The only valid solution is

x = 4

. Substituting

x = 4

into

x^2 - 3x

yields

4^2 - 3(4) = 16 - 12 = 4

.

Step-by-Step Solution

1

Set up equations based on the definition of absolute value

Case 1:

2x - 7 = 3x - 11

; Case 2:

2x - 7 = -(3x - 11)

An absolute value equation

|A| = B

splits into

A = B

or

A = -B

, with the requirement that

B \ge 0

.

2

Solve Case 1:

2x - 7 = 3x - 11

3x - 2x = 11 - 7 \implies x = 4

Isolate the variable

x

by algebraic rearrangement.

3

Solve Case 2:

2x - 7 = -3x + 11

5x = 18 \implies x = \frac{18}{5} = 3.6

Isolate

x

for the negative case.

4

Check both potential solutions in the original equation

|2x - 7| = 3x - 11

For

x = 4

:

|2(4) - 7| = |1| = 1

and

3(4) - 11 = 1

(Valid).
For

x = 3.6

:

|2(3.6) - 7| = |0.2| = 0.2

but

3(3.6) - 11 = -0.2

(Extraneous).

The right side

3x - 11

must be non-negative. Since

3(3.6) - 11 = -0.2 < 0

,

x = 3.6

is extraneous.

5

Evaluate the target expression

x^2 - 3x

using the valid root

x = 4

4^2 - 3(4) = 16 - 12 = 4

Substitute the single valid root into the requested expression.

Key Concept

Absolute Value Linear Equations and Extraneous Solution Checking

Question 1026Question →

A panel of $150$ sommeliers evaluated three vintages of wine: Cabernet, Pinot Noir, and Syrah. Every sommelier rated at least one vintage as Exceptional. Overall, $79$ sommeliers rated Cabernet as Exceptional, $71$ rated Pinot Noir as Exceptional, and $84$ rated Syrah as Exceptional. Exactly $18$ sommeliers rated all three vintages as Exceptional, and exactly $27$ rated ONLY Cabernet as Exceptional. If the total number of sommeliers who rated BOTH Cabernet and Pinot Noir as Exceptional is twice the number of sommeliers who rated ONLY Pinot Noir and Syrah as Exceptional, how many sommeliers rated ONLY Cabernet and Syrah as Exceptional?

Show answer & explanation

Answer: 24

Answer

24

Applying the 3-set Inclusion-Exclusion Principle determines that

48

sommeliers rated exactly two vintages. Combining the Cabernet set total (

79

) with the given ONLY Cabernet count (

27

) establishes that

cp + cs = 34

. Substituting

cp = 2ps - 18

into these two relationships forms a system of linear equations (

3ps + cs = 66

and

2ps + cs = 52

). Solving this system yields

ps = 14

and

cs = 24

sommeliers who rated ONLY Cabernet and Syrah as Exceptional.

Step-by-Step Solution

1

Calculate the sum of all pairwise set overlaps using the Inclusion-Exclusion Principle.

|C \cap P| + |P \cap S| + |C \cap S| = 102

The total union is equal to the sum of individual set sizes minus the sum of pairwise intersections plus the three-set intersection.

2

Determine the sum of regions representing sommeliers who rated exactly two vintages.

cp + ps + cs = 48

Each pairwise intersection consists of an 'exactly two' region plus the 'all three' region (

18

). Subtracting

3 \times 18 = 54

from

102

leaves

48

.

3

Set up linear equations using the given ratio and Cabernet set total.

3ps + cs = 66

and

2ps + cs = 52

Expressing

cp = 2ps - 18

and substituting it into

cp + ps + cs = 48

gives the first equation; substituting into the Cabernet total

27 + cp + cs + 18 = 79

gives the second equation.

4

Solve the system of equations for the target region

cs

.

ps = 14

and

cs = 24

Subtracting the two equations yields

ps = 14

, and substituting back gives

cs = 24

.

Key Concept

Three-set Venn diagram algebraic modeling and inclusion-exclusion principle

Question 1027Question →

For decades, paleoanthropologists attributed the rapid demise of megafaunal populations in Late Pleistocene North America almost exclusively to the 'overkill hypothesis,' which posits that human arrival and subsequent overhunting directly precipitated systemic extinctions. Adherents of this model underscored the tight chronological alignment between the proliferation of Clovis projectile technology and the abrupt disappearance of large mammals. However, recent high-resolution palynological and isotopic analyses have fundamentally challenged this monocausal paradigm by demonstrating that severe climatic volatility during the Bølling-Allerød to Younger Dryas transition caused widespread vegetation shifts and habitat fragmentation prior to significant human demographic expansion.

In response to these findings, a revisionist school of thought has emerged, contending that climatic shifts were the sole driver of the extinction event and dismissing human agency as negligible. Yet, this counter-theory proves equally reductive, as it fails to explain why earlier, comparably severe glacial transitions did not trigger equivalent ecological collapses. A more robust synthesis suggests that environmental stress severely compromised megafaunal population density and genetic diversity, leaving these already vulnerable species incapable of absorbing even modest, localized hunting pressures. Ultimately, the extinction cascade was neither a simple anthropogenic catastrophe nor a purely climate-driven collapse, but rather the product of a synergistic interaction between ecological instability and targeted human predation.

Which of the following best states the primary purpose of the passage?

To evaluate two opposing explanations for a prehistoric extinction event and present an integrative perspective that synthesizes both factors.

To detail the chronological alignment between Clovis projectile technology and the decline of North American megafauna.

To prove that climatic volatility during the Younger Dryas transition was the exclusive driver of megafaunal extinctions.

To forcefully denounce the overkill hypothesis as entirely unscientific and unsupported by archaeological evidence.

To demonstrate that earlier glacial transitions caused greater habitat fragmentation than the Bølling-Allerød transition.

Show answer & explanation

Answer: To evaluate two opposing explanations for a prehistoric extinction event and present an integrative perspective that synthesizes both factors.

Answer

The primary purpose of the passage is to evaluate two opposing explanations for a prehistoric extinction event and present an integrative perspective that synthesizes both factors.

The passage begins by outlining the long-held overkill hypothesis, introduces climate evidence challenging it, discusses a revisionist climate-only model, refutes both single-cause models, and concludes by presenting a combined synergistic model. The option stating that the primary purpose is to evaluate two opposing explanations and present an integrative synthesis accurately captures this complete rhetorical arc.

Step-by-Step Solution

1

Analyze the structural movement of Paragraph 1

Identified the traditional 'overkill hypothesis' (human cause) followed by a pivot marker ('However') introducing evidence of climate volatility.

Tracking structural transitions reveals how the author introduces competing viewpoints.

2

Analyze the structural movement of Paragraph 2

Identified the climate-only revisionist view, followed by the author's rebuttal ('equally reductive') and final synthesis ('synergistic interaction').

Determining authorial stance relative to cited viewpoints is essential for identifying primary purpose.

3

Synthesize the overarching thesis

The author rejects both monocausal extreme positions in favor of a combined model where climate stress rendered populations vulnerable to human hunting.

The main idea must cover the full scope of the passage without being too narrow or too extreme.

Key Concept

Identifying Primary Purpose in Multi-Viewpoint Passages

Question 1028Question →

In a university computer science department, students belong to one of three groups: Undergraduates, Master's students, or Doctoral candidates. The ratio of the number of Undergraduates to Master's students is $3 : 2$ , and the ratio of the number of Master's students to Doctoral candidates is $4 : 1$ . The average number of weekly research hours is $6$ hours for Undergraduates and $14$ hours for Master's students. If the combined average number of weekly research hours across all three groups is $11$ hours, what is the average number of weekly research hours for a Doctoral candidate?

13

18

20

21

29

Show answer & explanation

Answer: 29

Answer

The average number of weekly research hours for a Doctoral candidate is 29 hours.

To find the average weekly research hours for Doctoral candidates, first combine the group ratios. Given

U : M = 3 : 2

and

M : D = 4 : 1

, scale

U : M

to

6 : 4

so that the

M

term is identical in both ratios. This yields a single combined ratio

U : M : D = 6 : 4 : 1

, giving a total of

6 + 4 + 1 = 11

equal parts.

Next, calculate the total research hours across all parts:

6(6) + 4(14) + 1(x) = 36 + 56 + x = 92 + x

. Setting the overall weighted average to

11

yields

\frac{92 + x}{11} = 11

, which simplifies to

92 + x = 121

, so

x = 29

.

Step-by-Step Solution

1

Determine the combined ratio of students across the three groups.

Undergraduates : Master's : Doctoral =

6 : 4 : 1

.

We are given

U : M = 3 : 2

and

M : D = 4 : 1

. To combine these into a single ratio

U : M : D

, scale

U : M

by multiplying by

2

so that the

M

term matches:

U : M = 6 : 4

. Thus,

U : M : D = 6 : 4 : 1

.

2

Express total students and total research hours in terms of a multiplier

k

.

Total students =

11k

; Total hours =

36k + 56k + kx = 92k + kx

.

Let the number of students be

U = 6k

,

M = 4k

, and

D = 1k

. The total research hours contributed by Undergraduates is

6k \times 6 = 36k

, by Master's students is

4k \times 14 = 56k

, and by Doctoral candidates is

1k \times x = kx

.

3

Set up the weighted average formula and solve for

x

.

x = 29

.

The combined average is given by

\frac{\text{Total Hours}}{\text{Total Students}} = 11

. Substituting our values gives

\frac{92k + kx}{11k} = 11 \implies \frac{92 + x}{11} = 11 \implies 92 + x = 121 \implies x = 29

.

Key Concept

Weighted Average and Combined Sets with Compound Ratios

Estimated Time:2m 0s

Question 1029Question →

An artisan tea shop allows customers to order a custom tea blend prepared through three sequential choices:

1. Base Tea: Choose $1$ of $4$ available tea types (Black, Green, Oolong, or White).
2. Flavor Infusion: Choose $1$ of $5$ available herbal flavors. However, if Black tea is chosen as the base, only $3$ of these $5$ herbal flavors can be selected.
3. Sweetener: Choose $1$ of $3$ natural sweeteners (Honey, Stevia, or Maple Syrup), or choose to have no sweetener.

How many different custom tea blends can a customer create?

54

60

72

80

96

Show answer & explanation

Answer: $72$

Answer

72 different custom tea blends can be created.

To solve this counting problem with conditional restrictions, break the calculation into two mutually exclusive cases based on the base tea chosen:

1. Case 1: Black Tea Base
- Base tea options:

1

(Black)
- Flavor infusion options:

3

(restricted subset)
- Sweetener options:

4

(

3

sweeteners +

1

option for no sweetener)
- Combinations =

1 \times 3 \times 4 = 12

2. Case 2: Non-Black Tea Base (Green, Oolong, or White)
- Base tea options:

3

- Flavor infusion options:

5

(all available)
- Sweetener options:

4

- Combinations =

3 \times 5 \times 4 = 60

Adding the combinations from these two mutually exclusive cases yields

12 + 60 = 72

total custom tea blends.

Step-by-Step Solution

1

Identify total options available for the sweetener stage.

There are

3

sweetener choices plus

1

choice for 'no sweetener', making

3 + 1 = 4

total options for Stage 3.

Choosing 'no sweetener' is a distinct decision outcome that must be counted.

2

Calculate the number of blends when Black tea is chosen as the base.

Number of Black tea blends =

1 \text{ (Black base)} \times 3 \text{ (flavors)} \times 4 \text{ (sweetener choices)} = 12

.

When Black tea is chosen, only

3

flavor infusions are permitted.

3

Calculate the number of blends when any non-Black tea is chosen as the base.

Number of non-Black tea blends =

3 \text{ (Green, Oolong, White bases)} \times 5 \text{ (flavors)} \times 4 \text{ (sweetener choices)} = 60

.

There are

3

non-Black base choices, and each can be paired with any of the

5

flavor infusions.

4

Sum the combinations from both mutually exclusive cases.

Total blends =

12 + 60 = 72

.

Since selecting Black tea and selecting non-Black tea are mutually exclusive cases, their individual counts are added.

Key Concept

Fundamental Counting Principle with Conditional Restrictions

Estimated Time:1m 45s

Question 1030Question →

For over three decades, corporate strategists evaluated global supply chain resilience primarily through the lens of cost minimization and operational speed, widely adopting just-in-time (JIT) inventory management protocols to eliminate buffer storage. While this lean approach significantly enhanced short-term capital efficiency and reduced holding expenditures, recent large-scale market disruptions have exposed its underlying fragility. Skeptics argue that hyper-optimized, zero-buffer supply networks lack the structural elasticity necessary to absorb unexpected geopolitical friction, natural disasters, or sudden demand volatility. Consequently, a growing faction of management theorists advocates a complete reversal toward "just-in-case" (JIC) paradigms, which prioritize extensive safety stocks and localized supplier redundancy to guarantee continuity.

However, discarding lean principles altogether introduces severe financial penalties, including capital immobilization, inventory obsolescence, and ballooning warehousing expenses. A more sophisticated alternative, developed by contemporary operations analysts, proposes a hybrid model known as "dynamic risk-tiered buffering." Rather than deploying blanket inventory reserves across all product lines, this strategy utilizes real-time predictive data to dynamically adjust buffer thresholds strictly for high-criticality, single-source components while preserving lean management for easily substitutable inputs. Therefore, the central challenge facing modern enterprise operations is not choosing between absolute lean efficiency and total redundant capacity, but rather implementing a flexible, risk-differentiated inventory allocation framework.

Which of the following best states the primary purpose of the passage?

To evaluate two opposing supply chain strategies and present a hybrid approach that addresses the limitations of both.

To argue that just-in-time inventory management should be entirely abandoned in favor of localized safety stocks.

To detail the specific holding expenditures and warehousing expenses associated with maintaining excess inventory reserves.

To demonstrate how predictive real-time analytics can eliminate all operational and financial risks in global manufacturing.

To criticize operations analysts for underestimating the vulnerability of hyper-optimized supply networks.

Show answer & explanation

Answer: To evaluate two opposing supply chain strategies and present a hybrid approach that addresses the limitations of both.

Answer

The primary purpose of the passage is to evaluate two opposing supply chain strategies and present a hybrid approach that addresses the limitations of both.

The passage begins by outlining the lean JIT model and the opposing JIC alternative proposed by critics. It then uses a transition ('However') to point out the flaws of total JIC adoption and introduces 'dynamic risk-tiered buffering' as a flexible hybrid framework. Thus, the option stating that the purpose is to evaluate two opposing strategies and present a hybrid approach accurately synthesizes the full scope and intent of the text.

Step-by-Step Solution

1

Analyze the structural thesis of Paragraph 1

Paragraph 1 introduces the traditional Just-in-Time (JIT) strategy, notes its vulnerabilities during market disruptions, and introduces an opposing reaction favoring Just-in-Case (JIC) inventory models.

Understanding the initial context and contrasting viewpoints sets the foundation of the discussion.

2

Identify the structural pivot in Paragraph 2

Paragraph 2 begins with 'However', signaling a pivot: completely abandoning lean principles for JIC has major drawbacks. It then introduces 'dynamic risk-tiered buffering' as a hybrid solution.

The author uses this pivot to move from a binary debate to promoting a balanced, risk-differentiated framework.

3

Synthesize the overall main idea and evaluate the option choices

The main purpose encompasses contrasting JIT and JIC while introducing dynamic risk-tiered buffering to reconcile their respective shortcomings.

The correct primary purpose answer choice must reflect the full scope of the passage rather than focusing on a single viewpoint or detail.

Key Concept

Identifying Primary Purpose in Multi-Viewpoint Passages

Question 1031Question →

Passage:
For over a century, art historians framed the nineteenth-century Gothic Revival in Western Europe primarily as a reactionary aesthetic retreat—a nostalgic rejection of industrialization and a longing for medieval spiritual unity. Standard accounts emphasized how proponents like A.W.N. Pugin viewed medieval architecture not merely as a stylistic choice, but as a moral imperative against the dehumanizing effects of modern mechanization. However, recent scholarly reevaluations argue that this traditional view oversimplifies a complex cultural phenomenon. Far from being a purely regressive movement, Gothic Revival architecture actively engaged with contemporary technological innovations and socioeconomic realities. Revisionist scholars highlight that Gothic Revival architects were among the earliest adopters of mass-produced structural iron and prefabricated decorative tiles, seamlessly integrating modern industrial materials beneath intricate medievalized facades. Furthermore, rather than attempting to dismantle capitalist labor structures, Gothic Revival theorists frequently adapted their rhetoric to justify municipal infrastructure spending, framing Gothic civic buildings as symbols of modern civic pride and institutional efficiency. Thus, while the movement's rhetoric certainly invoked a romanticized past, its actual practice functioned as a sophisticated negotiation with—and synthesis of—nineteenth-century modernity. Consequently, understanding the Gothic Revival requires recognizing that its proponents did not seek to escape the industrial present, but rather sought to sanitize and legitimize it through a culturally resonant historical idiom.

Statement: The primary purpose of the passage is to challenge a traditional historical interpretation of the Gothic Revival by demonstrating that the movement actively integrated and responded to nineteenth-century industrial modernity.

True

False

Show answer & explanation

Answer: True

Answer

True. The primary purpose of the passage is to reevaluate the conventional view of the Gothic Revival, arguing that it engaged with and synthesized industrial modernity rather than simply rejecting it.

The statement correctly identifies the central goal of the passage, which is to challenge the traditional view of the Gothic Revival as a nostalgic escape by demonstrating its active synthesis of industrial modernity.

Step-by-Step Solution

1

Analyze the structural organization and rhetorical flow of the passage.

The passage presents an established historical consensus in the first two sentences, introduces a major contrast pivot ('However, recent scholarly reevaluations...'), and uses the remaining sentences to elaborate on the revised perspective.

Tracking structural contrast markers reveals whether the author intends to support, critique, or contextualize the initial claims.

2

Identify the author's primary thesis and supporting arguments.

The author argues that the Gothic Revival was not merely a nostalgic escape from mechanization, but a sophisticated negotiation with nineteenth-century technology (e.g., structural iron, prefabricated tiles) and modern socioeconomic structures (e.g., municipal spending).

The main idea must synthesize both the central claim and the scope of the evidence presented.

3

Evaluate the statement against the passage's primary purpose.

The statement accurately reflects the passage's core objective: reframing the Gothic Revival from a purely reactionary movement to one that integrated and responded to industrial modernity.

Comparing the statement directly to the passage's overarching thesis confirms that the statement is true.

Key Concept

Identifying Primary Purpose and Main Idea

Question 1032Question →

Set $S$ consists of consecutive integers from $a$ to $b$ , inclusive, such that $a < 0 < b$ . The number of negative integers in set $S$ is equal to three times the number of positive integers in set $S$ . If the sum of all elements in set $S$ is $-68$ , how many integers are in set $S$ ?

15

16

17

21

13

Show answer & explanation

Answer: 17

Answer

17

Set

S

consists of

12

negative integers (

-12

through

-1

),

4

positive integers (

1

through

4

), and the integer

0

. The sum of positive integers is

10

and the sum of negative integers is

-78

, giving a total sum of

-68

. The total number of integers in the set is

12 + 4 + 1 = 17

.

Step-by-Step Solution

1

Define the number of terms and bounds of set

S

in terms of positive integers

b

.

The positive integers are

1, 2, \dots, b

, so there are

b

positive integers. The negative integers are

-1, -2, \dots, a

. Since the number of negative integers is

3

times the number of positive integers, there are

3b

negative integers, which means

a = -3b

. Set

S

also includes

0

.

Expressing all elements in terms of

b

allows setting up a single variable equation.

2

Calculate the sum of all elements in set

S

as a function of

b

.

The sum of the positive integers is

\frac{b(b+1)}{2}

. The sum of the negative integers is

-\frac{3b(3b+1)}{2}

. The total sum is

\frac{b(b+1)}{2} - \frac{3b(3b+1)}{2} = \frac{b^2 + b - 9b^2 - 3b}{2} = \frac{-8b^2 - 2b}{2} = -4b^2 - b

.

The sum of consecutive integers from

1

to

n

is

\frac{n(n+1)}{2}

.

3

Solve for

b

using the given sum of

-68

.

-4b^2 - b = -68 \implies 4b^2 + b - 68 = 0

. Factoring gives

(4b + 17)(b - 4) = 0

. Since

b

must be a positive integer,

b = 4

. Therefore,

a = -3(4) = -12

.

Solving the quadratic equation identifies the exact boundaries of the set.

4

Calculate the total number of integers in set

S

.

The set contains integers from

-12

to

4

, inclusive. The total count of integers is

4 - (-12) + 1 = 17

.

The number of integers in an inclusive range

[a, b]

is given by

b - a + 1

.

Key Concept

Consecutive Integers and Number Sets

Question 1033Question →

A commercial airline conducted a fleet audit of $180$ transoceanic aircraft to evaluate their emergency navigation systems: Satellite Positioning ( $S$ ), Inertial Guidance ( $I$ ), and Terrestrial Radio Navigation ( $T$ ). The audit revealed the following data:
- $95$ aircraft are equipped with Satellite Positioning.
- $80$ aircraft are equipped with Inertial Guidance.
- $75$ aircraft are equipped with Terrestrial Radio Navigation.
- $30$ aircraft are equipped with Satellite Positioning and Inertial Guidance, but not Terrestrial Radio Navigation.
- $20$ aircraft are equipped with Inertial Guidance and Terrestrial Radio Navigation, but not Satellite Positioning.
- $25$ aircraft are equipped with Satellite Positioning and Terrestrial Radio Navigation, but not Inertial Guidance.
- $15$ aircraft are not equipped with any of these three navigation systems.

How many aircraft in the fleet are equipped with exactly one of these three navigation systems?

70

75

85

95

100

Show answer & explanation

Answer: $85$

Answer

The number of aircraft equipped with exactly one navigation system is

85

.

Subtracting the

15

unequipped aircraft from the total fleet of

180

gives

165

aircraft equipped with at least one navigation system. Letting

x

represent the number of aircraft with all three systems, the single-system counts are

(95 - 30 - 25 - x) = 40 - x

for Satellite Positioning only,

(80 - 30 - 20 - x) = 30 - x

for Inertial Guidance only, and

(75 - 25 - 20 - x) = 30 - x

for Terrestrial Radio Navigation only. Summing all seven mutually exclusive regions yields

(40 - x) + (30 - x) + (30 - x) + 30 + 20 + 25 + x = 165

, which simplifies to

175 - 2x = 165

, giving

x = 5

. Substituting

x = 5

into the single-system expressions gives

35 + 25 + 25 = 85

.

Step-by-Step Solution

1

Calculate the total number of aircraft equipped with at least one system.

180 - 15 = 165

aircraft.

Subtracting the

15

aircraft with no systems from the fleet total gives the union of sets

S

,

I

, and

T

.

2

Express the regions of the 3-set Venn diagram in terms of

x

, where

x

is the number of aircraft equipped with all three systems.

Only

S = 40 - x

, Only

I = 30 - x

, Only

T = 30 - x

.

For set

S

:

95 - (30 + 25 + x) = 40 - x

. For set

I

:

80 - (30 + 20 + x) = 30 - x

. For set

T

:

75 - (25 + 20 + x) = 30 - x

.

3

Set up an equation for the total number of aircraft with at least one system to solve for

x

.

(40 - x) + (30 - x) + (30 - x) + 30 + 20 + 25 + x = 165 \implies 175 - 2x = 165 \implies x = 5

.

Summing all

7

non-overlapping regions inside the Venn diagram equals the total union of

165

.

4

Compute the sum of aircraft equipped with exactly one system.

(Only

S

) + (Only

I

) + (Only

T

) =

(40 - 5) + (30 - 5) + (30 - 5) = 35 + 25 + 25 = 85

.

Substitute

x = 5

into each of the single-system region expressions.

Key Concept

Three-Set Inclusion-Exclusion Principle and Venn Diagram Region Decomposition

Estimated Time:2m 0s

Question 1034Question →

A cyclist travels from City A to City B at a constant speed of $20$ miles per hour. On the return trip along the exact same route, the cyclist increases their speed by $25\%$ . If the total time taken for the round trip is $9$ hours, what is the distance, in miles, between City A and City B?

Show answer & explanation

Answer: 100

Answer

The distance between City A and City B is 100 miles.

To find the one-way distance

d

, determine the return speed as

20 \times 1.25 = 25

miles per hour. The outbound time is

\frac{d}{20}

hours and the return time is

\frac{d}{25}

hours. Adding these gives

\frac{d}{20} + \frac{d}{25} = 9

, which simplifies to

\frac{9d}{100} = 9

, so

d = 100

miles.

Step-by-Step Solution

1

Calculate the speed on the return trip.

The return speed is

20 \times 1.25 = 25

miles per hour.

The return speed increases by

25\%

over the outbound speed of

20

miles per hour.

2

Express outbound and inbound travel times using distance

d

.

Outbound time

t_1 = \frac{d}{20}

hours, Inbound time

t_2 = \frac{d}{25}

hours.

Using the rate-time-distance formula

t = \frac{d}{r}

.

3

Set up and solve the total time equation.

\frac{d}{20} + \frac{d}{25} = 9 \implies \frac{9d}{100} = 9 \implies d = 100

miles.

The total duration of the round trip is given as

9

hours.

Key Concept

Rate, Time, and Distance Relationship in Round Trips

Question 1035Question →

Passage:
Traditional antitrust analysis relies heavily on the hypothetical monopolist test to define relevant product markets, assessing whether a firm could profitably sustain a small but significant non-transitory increase in price (SSNIP). However, economists specializing in digital platforms argue that applying standard SSNIP metrics to multi-sided digital ecosystems fundamentally mischaracterizes market dynamics. In two-sided transaction platforms, such as payment networks or ridesharing services, cross-system network externalities interlock the demands of distinct user groups. Consequently, pricing strategies on one side of the market—often subsidized to zero—are intrinsically interdependent with premium pricing on the counter-side.

Critics of traditional market definition protocols contend that evaluating one side of a two-sided platform in isolation inevitably leads to erroneous market power assessments. For instance, treating a zero-price consumer search engine interface as a standalone market obscures the cross-subsidization provided by advertiser fees. Conversely, alternative scholars warn against over-correcting by conflating distinct sides into a single amorphous market, which risks overlooking anti-competitive exclusion targeting a specific user subset.

Ultimately, resolving this debate requires shifting regulatory focus from static price-centric market boundaries to dynamic ecosystem-wide competitive interdependencies. Rather than abandoning established antitrust principles, economic analysts must recalibrate existing diagnostic metrics to capture feedback loops across multi-sided user networks.

Statement: Based on the passage, the primary purpose of the text is to advocate for the complete abandonment of traditional antitrust principles in favor of an entirely new legal framework created exclusively for digital platforms.

True

False

Show answer & explanation

Answer: False

Answer

The statement is False. The primary purpose of the passage is to recommend adapting and recalibrating current antitrust tools for multi-sided markets, explicitly rejecting the complete abandonment of traditional principles.

The statement is false because it exaggerates the author's position into an extreme claim. The passage concludes with a clear recommendation to adapt existing antitrust tools rather than discard them.

Step-by-Step Solution

1

Identify the overall structure and thesis of the passage.

The passage highlights limitations in applying traditional SSNIP tests to multi-sided platforms (Paragraph 1), presents contrasting risks of under-defining or over-amalgamating markets (Paragraph 2), and resolves the debate in the conclusion (Paragraph 3).

Tracking structural shifts across paragraphs isolates the author's ultimate objective.

2

Analyze the author's recommended resolution in the final paragraph.

The author advocates shifting focus to dynamic interdependencies and explicitly states that analysts should recalibrate existing metrics '[r]ather than abandoning established antitrust principles.'

The conclusion contains the definitive statement of primary purpose.

3

Compare the author's thesis with the proposed statement.

The statement claims the passage advocates for 'complete abandonment' and 'an entirely new legal framework,' which directly contradicts the author's explicit stance.

Determining direct alignment or contradiction verifies the truth value of the statement.

Key Concept

Identifying Primary Purpose and Main Idea

Question 1036Question →

A financial advisory board of $6$ members is to be selected from a pool of $6$ senior analysts and $5$ junior analysts. The board must include at least $2$ senior analysts and at least $2$ junior analysts. However, two specific senior analysts, $A$ and $B$ , cannot both serve on the board together. How many different $6$ -member boards can be formed under these conditions?

300

320

325

336

425

Show answer & explanation

Answer: 320

Answer

The total number of valid 6-member boards that can be formed is 320.

To find the number of valid boards, we first calculate all possible 6-member boards satisfying the minimum criteria of having at least 2 senior analysts and at least 2 junior analysts. The possible (senior, junior) distributions are (4,2), (3,3), and (2,4). Calculating each case gives

\binom{6}{4}\binom{5}{2} = 150

,

\binom{6}{3}\binom{5}{3} = 200

, and

\binom{6}{2}\binom{5}{4} = 75

, for a total of

150 + 200 + 75 = 425

boards. Next, we determine how many of these boards contain both senior analysts A and B. Fixing A and B requires choosing 4 more members from the remaining 4 seniors and 5 juniors such that the total junior count is at least 2. The valid remaining senior choices

s'

are 0, 1, or 2, yielding

\binom{4}{0}\binom{5}{4} = 5

,

\binom{4}{1}\binom{5}{3} = 40

, and

\binom{4}{2}\binom{5}{2} = 60

, totaling

5 + 40 + 60 = 105

boards. Subtracting these forbidden boards from 425 yields

425 - 105 = 320

.

Step-by-Step Solution

1

Calculate total valid boards with at least 2 senior analysts and at least 2 junior analysts without restriction on A and B.

Total valid boards without conflict constraint = 425

The valid (Senior, Junior) distributions for a 6-member board are (4,2), (3,3), and (2,4). Their combinations are

\binom{6}{4}\binom{5}{2} = 150

,

\binom{6}{3}\binom{5}{3} = 200

, and

\binom{6}{2}\binom{5}{4} = 75

, giving

150 + 200 + 75 = 425

.

2

Calculate the number of valid boards that contain BOTH senior analysts A and B.

Number of invalid boards containing both A and B = 105

If A and B are both selected, 2 senior spots are fixed. We need 4 remaining members from the remaining 4 seniors and 5 juniors. To satisfy the junior constraint (at least 2 juniors), the additional senior count

s'

can be 0, 1, or 2:

s'=0 \implies \binom{4}{0}\binom{5}{4}=5

,

s'=1 \implies \binom{4}{1}\binom{5}{3}=40

,

s'=2 \implies \binom{4}{2}\binom{5}{2}=60

. Summing gives

5 + 40 + 60 = 105

.

3

Subtract the conflicting boards containing both A and B from the total valid boards.

Valid boards = 425 - 105 = 320

Complementary counting gives the exact number of boards satisfying all conditions.

Key Concept

Combinations with Subgroup Constraints and Complementary Counting

Estimated Time:2m 0s

Question 1037Question →

A data set consists of seven integers written in ascending order: $10, 12, 15, x, 26, 29, 34$ . If the arithmetic mean of the data set is equal to its median, what is the value of $x$ ?

18

20.5

21

23

25.2

Show answer & explanation

Answer: $21$

Answer

The value of

x

is

21

.

In an ordered set of seven numbers, the median is the 4th number, which is

x

. The arithmetic mean of the seven numbers is

\frac{10 + 12 + 15 + x + 26 + 29 + 34}{7} = \frac{126 + x}{7}

. Equating the mean to the median gives

\frac{126 + x}{7} = x

. Solving this linear equation yields

126 + x = 7x \implies 6x = 126 \implies x = 21

. Since

21

lies between

15

and

26

, it maintains the ascending order of the set.

Step-by-Step Solution

1

Identify the median of the ordered set

Median =

x

For an ordered set with an odd number of elements (

n = 7

), the median is the middle element, which is the

\frac{7 + 1}{2} = 4

th element. Since the elements are in ascending order, the 4th element is

x

.

2

Calculate the sum of all elements in terms of

x

Sum =

126 + x

The sum of the known six numbers is

10 + 12 + 15 + 26 + 29 + 34 = 126

. Adding

x

gives a total sum of

126 + x

.

3

Set up the equation equating mean and median

\frac{126 + x}{7} = x

The arithmetic mean is defined as the total sum divided by the total number of elements (

7

). We are given that Mean = Median.

4

Solve for

x

and verify ordering bounds

x = 21

Multiplying both sides by

7

gives

126 + x = 7x

, so

6x = 126

, yielding

x = 21

. Checking the ascending order condition

15 \le x \le 26

,

21

satisfies the condition

15 \le 21 \le 26

.

Key Concept

Mean and Median of an Ordered Set

Estimated Time:1m 30s

Question 1038Question →

If $x$ and $y$ are integers such that $-8 \le x \le 4$ and $-2 \le y \le 5$ , $x y^2 > 0$ , and $x^3 y < 0$ , how many distinct integer values are possible for the product $x y$ ?

Show answer & explanation

Answer: 6

Answer

The total number of distinct integer values for the product

x y

is 6.

Analyzing the given inequalities reveals sign constraints on both variables:

x y^2 > 0

forces

x > 0

(since

y^2 > 0

for non-zero

y

), limiting

x

to integer values

1, 2, 3, 4

. Then

x^3 y < 0

forces

y < 0

(since

x^3 > 0

), limiting

y

to integer values

-2, -1

. Multiplying each value of

x

by each value of

y

gives the products

-1, -2, -3, -4, -6, -8

, which constitutes exactly 6 distinct values.

Step-by-Step Solution

1

Determine the sign and allowable integer values of

x

x \in \{1, 2, 3, 4\}

Since

y^2

is strictly positive for any non-zero real number

y

,

x y^2 > 0

implies

x > 0

and

y \neq 0

.

2

Determine the sign and allowable integer values of

y

y \in \{-2, -1\}

Because

x > 0

,

x^3

is also positive. For

x^3 y < 0

,

y

must be negative.

3

Calculate products for all valid pairs of

(x, y)

and remove duplicates

The distinct product values are

-1, -2, -3, -4, -6, -8

, giving 6 values in total.

Evaluating

x y

for

x \in \{1, 2, 3, 4\}

and

y \in \{-2, -1\}

yields

-1, -2, -3, -4

when

y = -1

and

-2, -4, -6, -8

when

y = -2

.

Key Concept

Even powers of non-zero numbers are always positive, while odd powers preserve the sign of the base number.

Question 1039Question →

A 5-digit passcode is to be created using five distinct digits chosen from the set $\{1, 2, 3, 4, 5, 6, 7\}$ . The passcode must include both the digit $3$ and the digit $5$ , and the digit $3$ must appear somewhere to the left of the digit $5$ in the passcode. How many such 5-digit passcodes can be formed?

Show answer & explanation

Answer: 600

Answer

600

To form a valid passcode, first select 3 digits from the 5 available digits

\{1, 2, 4, 6, 7\}

, which can be done in

\binom{5}{3} = 10

ways. Each chosen set of 5 distinct digits (including 3 and 5) can be arranged in

5! = 120

ways. Because the digits 3 and 5 are distinct, the digit 3 appears before the digit 5 in exactly half of these arrangements (

120 / 2 = 60

). Multiplying the 10 combinations of digits by the 60 valid arrangements gives a total of

600

valid passcodes.

Step-by-Step Solution

1

Determine the number of ways to choose the remaining digits.

\binom{5}{3} = 10

ways to choose 3 additional digits from

\{1, 2, 4, 6, 7\}

.

Since the passcode must contain both 3 and 5, 3 additional distinct digits must be selected from the 5 available remaining digits.

2

Calculate the total permutations of the 5 chosen digits.

5! = 120

total permutations.

Any set of 5 distinct digits can be arranged into a 5-digit sequence in

5!

ways.

3

Apply the positional restriction using symmetry.

\frac{120}{2} = 60

valid arrangements per set of digits.

In exactly half of all permutations containing both 3 and 5, the digit 3 appears before the digit 5.

4

Compute the total number of valid passcodes.

10 \times 60 = 600

passcodes.

Multiply the number of digit selections by the number of valid orderings per selection.

Key Concept

Combining selection (combinations) with symmetry-restricted arrangements (permutations)

Estimated Time:2m 0s

Question 1040Question →

For all real numbers $x > 1$ , the expression $\frac{\sqrt{x+1} + \sqrt{x-1}}{\sqrt{x+1} - \sqrt{x-1}}$ is equal to $x + \sqrt{x^2 - 1}$ .

True

False

Show answer & explanation

Answer: True

Answer

The statement is true because rationalizing the denominator of the algebraic radical expression simplifies directly to

x + \sqrt{x^2 - 1}

.

Rationalizing the radical expression by multiplying the numerator and denominator by the conjugate

(\sqrt{x+1} + \sqrt{x-1})

leads directly to the simplified form

x + \sqrt{x^2 - 1}

.

Step-by-Step Solution

1

Multiply the numerator and denominator by the conjugate of the denominator,

(\sqrt{x+1} + \sqrt{x-1})

.

\frac{(\sqrt{x+1} + \sqrt{x-1})^2}{(\sqrt{x+1} - \sqrt{x-1})(\sqrt{x+1} + \sqrt{x-1})}

Rationalizing the denominator removes radicals from the denominator using the difference of squares identity.

2

Expand the numerator using

(a+b)^2 = a^2 + 2ab + b^2

.

(\sqrt{x+1})^2 + 2\sqrt{x+1}\sqrt{x-1} + (\sqrt{x-1})^2 = (x+1) + 2\sqrt{x^2-1} + (x-1) = 2x + 2\sqrt{x^2-1}

Simplifying sums of squared radicals and combining like terms.

3

Expand the denominator using

(a-b)(a+b) = a^2 - b^2

.

(\sqrt{x+1})^2 - (\sqrt{x-1})^2 = (x+1) - (x-1) = 2

Evaluating the difference of squares of square root terms.

4

Divide the expanded numerator by the simplified denominator.

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

Factoring out 2 from the numerator cancels the denominator of 2.

Key Concept

Rationalizing radical expressions using conjugates and difference of squares.

Answer

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